NASA/TM-1999-208769 C' d "'?

jt

Steady Secondary Flows Generated by Periodic

Compression and Expansion of an Ideal Gas

in a Pulse Tube

Jeffrey M. Lee

September 1999

https://ntrs.nasa.gov/search.jsp?R=20000034896 2018-05-14T14:05:19+00:00Z
The NASA STI Program Office... in Profile

Since its founding, NASA has been dedicated to the

advancement of aeronautics and space science. The

NASA Scientific and Technical Information (STI)

Program Office plays a key part in helping NASA

maintain this important role.

The NASA STI Program Office is operated by

Langley Research Center, the Lead Center for

NASA's scientific and technical information. The

NASA STI Program Office provides access to the

NASA STI Database, the largest collection of

aeronautical and space science STI in the world.

The Program Office is also NASA's institutional

mechanism for disseminating the results of its

research and development activities. These results

are published by NASA in the NASA STI Report

Series, which includes the following report types:

TECHNICAL PUBLICATION. Reports of

completed research or a major significant phase

of research that present the results of NASA

programs and include extensive data or theoretical analysis. Includes compilations of significant

scientific and technical data and information

deemed to be of continuing reference value.

NASA's counterpart of peer-reviewed formal

professional papers but has less stringent

limitations on manuscript length and extent

of graphic presentations.

• TECHNICAL MEMORANDUM. Scientific and

technical findings that are preliminary or of

specialized interest, e.g., quick release reports,

working papers, and bibliographies that contain

minimal annotation. Does not contain extensive

analysis.

• CONTRACTOR REPORT. Scientific and

technical findings by NASA-sponsored

contractors and grantees.

CONFERENCE PUBLICATION. Collected

papers from scientific and technical conferences, symposia, seminars, or other meetings

sponsored or cosponsored by NASA.

SPECIAL PUBLICATION. Scientific, technical,

or historical information from NASA programs,

projects, and missions, often concerned with

subjects having substantial public interest.

TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific and

technical material pertinent to NASA's mission.

Specialized services that complement the STI

Program Office's diverse offerings include creating

custom thesauri, building customized databases,

organizing and publishing research results.., even

providing videos.

For more information about the NASA STI

Program Office, see the following:

• Access the NASA STI Program Home Page at

http.'//www: sti.nasa, gov

• E-mail your question via the Internet to

help @sti.nasa.gov

• Fax your question to the NASA Access Help

Desk at (301) 621-0134

• Telephone the NASA Access Help Desk at

(301) 621-0390

Write to:

NASA Access Help Desk

NASA Center for AeroSpace Information

7121 Standard Drive

Hanover, MD 21076-1320
NASA/TM-1999-208769

Steady Secondary Flows Generated by Periodic

Compression and Expansion of an Ideal Gas

in a Pulse Tube

Jeffrey M. Lee

Ames Research Center, Moffett Field, California

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, California 94035-1000

September 1999
Acknowledgments

There are many persons I would like to acknowledge for their contributions.

Professor Klaus D. Timmerhaus and my advisory committee, Professors William B. Krantz,

Robert Sani, and Howard Snyder of the University of Colorado at Boulder deserve many thanks for their

critical assessments and insightful suggestion. Special thanks to Becky Phillips for pushing my paper

work and registration through in the eleventh hour.

Dr. Ray Radebaugh of the National Institute of Standards and Technology served on the advisory

committee and supplied the use of NIST facilities early-on for the initial flow visualization studies.

Dr. Nicholas Rott contributed by offering his expertise on thermoacoustic systems.

Dr. Peter Kittel of the Cryogenics Group at NASA Ames Research Center served on the advisory

committee and was a strong advocate of this work. Drs. Pat Roach, Ali Kashani, and Ben Helvensteijn

also of the Cryogenics Group deserve special recognition for several stimulating discussions that led to

the hashing-out of many technical details. Dr. Meng Wang guided my understanding of perturbation

expansions. Mic Clark was instrumental in the construction of the experimental system. And thanks to

my Branch Chief, Dr. Craig McCreight who patiently supported my "extended" program.

This work was funded by and conducted at NASA Ames Research Center.

NASA Center for AeroSpace Information

7121 Standard Drive

Hanover, MD 21076-1320

(301 ) 621-0390

Available from:

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22161

(703) 487-4650
CONTENTS

SYMBOLS ...............................................................................................................................

SUMMARY .............................................................................................................................

1. INTRODUCTION ..............................................................................................................

Mechanics of Mean-Steady Secondary Transport in Oscillating Systems ...................

Background ..................................................................................................................

Conventions Used .........................................................................................................

Scope of This Study .....................................................................................................

2. GOVERNING EQUATIONS ............................................................................................

Equations of Change for an Oscillating Fluid ..............................................................

Relevant Transport Quantities ......................................................................................

Conditions for Acoustic and Anelastic Flows ..............................................................

3. TWO-DIMENSIONAL AXISYMMETRIC FORMULATION ........................................

System ..........................................................................................................................

Equations ......................................................................................................................

Complex Embedding of the First-Order Equations ......................................................

4. SOLUTION FOR THE THERMALLY STRONG CASE, To, z = 0 .................................

Leading Order Results ..................................................................................................

Mean-Steady Secondary Flow ......................................................................................

Discussion ....................................................................................................................

5. EXPERIMENTAL MEASUREMENTS OF THE MEAN-STEADY

PARTICLE VELOCITY ....................................................................................................

Experimental System ....................................................................................................

Comparison with Theory ..............................................................................................

Discussion ....................................................................................................................

6. CONCLUDING REMARKS AND FUTURE WORK ......................................................

V

1

3

3

5

12

12

15

15

16

17

21

21

22

30

35

35

46

64

67

67

68

81

83

111
APPENDIX A--SCALING .....................................................................................................

APPENDIX B--PARTICLE VELOCITY ..............................................................................

APPENDIX C--ANELASTIC APPROXIMATION ..............................................................

APPENDIX D--TWO-DIMENSIONAL ANELASTIC EXPANSION

AND ORDERING .........................................................................................

APPENDIX E--REDUCING THE FIRST-ORDER EQUATIONS USING

COMPLEX EMBEDDING ...........................................................................

APPENDIX F--SOLUTION FOR THE THERMALLY STRONG CASE,

Vro =o .........................................................................................................

APPENDIX G--COMPUTATIONAL PROGRAM ...............................................................

APPENDIX H--EXPERIMENTAL DATA ............................................................................

REFERENCES .........................................................................................................................

87

91

93

95

101

105

113

139

151

iv
SYMBOLS

Note:

A

a

Cp

Cv

d

aC

f

Fo

H

h

i

J.

J

k

L

l

M

mn

n

T

Pr

P

Q

q

All quantities are dimensionless unless designated with a superscript asterisk, in which case

they are dimensional.

area normal to flow

speed of sound

specific heat capacity at constant pressure

specific heat capacity at constant volume

displacement of gas at end of tube

general diffusion term

frequency

* /1 * .2

Fourier number for the tube wall, Fo = aw/_O9 I )

enthalpy

specific enthalpy

imaginary number, i =

imaginary component of a complex number

nth order Bessel function

mass flux vector

thermal conductivity of material

length of tube

thickness of tube wall

Mach number, M = U_/a*

dimensionless radial function defined in table 5, equations (h), (i), and (j)

unit normal vector

anelastic pressure

Prandtl number, Pr = v'/a*

pressure

heat

conduction heat flux, q = -kVT

v
*

Re

r

rw

S

S

s

T

Tw

t

UL

Wa

V

W

Z

ideal gas constant

Reynolds number, Re = U_ r_ / v* = eVa L*/r*w

real component of a complex quantity

radial coordinate

inner radius of tube

entropy

surface area enclosing domain

specific entropy

gas temperature

temperature at interface between the gas and tube wall

time

velocity ratio, UL = UL/U_

Velocity components

'U higher-order Eulerian velocity vector

u Eulerian velocity vector

u axial Eulerian velocity

fi axial Eulerian velocity phasor

v radial Eulerian velocity

radial Eulerian velocity phasor

u axial higher order Eulerian velocity

Up Lagrangian particle velocity vector

Up axial Lagrangian particle velocity

Up radial Lagrangian particle velocity

"v radial higher order Eulerian velocity

•2 - *

Valensi number, Va = rw co /v

domain volume

work

axial coordinate

vi
O_

F

Fw

E

0

K

it

#

#2

v

t_

P

I

¢

Cr

ev

Z

Ill

(.o

thermal diffusivity

thermalexpansioncoefncient,/ =-(i/p)(ap/ar)p

gas domain length ratio F =(rw/eL" )

tube wall length ratio F w =(f/eZL ")

heat capacity ratio, 1, = Cp/Cv

inverse Strouhal number (expansion parameter), e = U_/(w'L') =d_/L"

Bessel function ratio (n = Jn/Jo, n = O, 1

temperature of tube wall

bulk modulus, x= p(Op/Op) T

elasticity parameter, it = 7M2/e = T(U_/a" O)(co'L'/a'O)

dynamic viscosity

second viscosity

kinematic viscosity

entropy of the tube wall

density

stress tensor

dissipation function

phase angle

phase angle between UO (velocity at z = 0) and temperature

phase angle between UO (velocity at z = 0) and UL (velocity at z = 1)

arbitrary thermodynamic variable

stream function

angular frequency

Superscripts

- mean-steady, time average over a period

vii
!

0

¢C

complex quantity containing amplitude and phase angle

dimensional quantity (no asterisk represents dimensionless quantity)

amplitude of a real quantity

exact derivative

definite integral

complex conjugate

Subscripts

0 zeroth-order, quantity at z = 0, reference quantity

1 first-order

2 second- order

3 third-order

a aftercooler

c cold end of pulse tube

g gas

h hot end of pulse tube

L quantity at z = 1

osc oscillating quantity

w tube wall

Dimensionless Numbers

E

;t

Fo

M

Pr

Re

UL

Va

inverse Strouhal number (expansion parameter), ¢ : U_/(O9"L*):d 0 /L"

elasticity parameter, 2, = )'M 2/¢ = ?'(U_ / a_)(og"L" / a_ )

• /[ ...2\

Fourier number, Fo = aw/[o9 t )

Mach number, M = U_/a"

Prandtl number, Pr = v'/o:*

Reynolds number, Re = U_ r_, / v* = tVa L"/r_,

velocity ratio, U L = UL/U'o

• 2 */ •

Valensi number, Va = rw o9 / v

viii
STEADY SECONDARY FLOWS GENERATED BY

PERIODIC COMPRESSION AND EXPANSION

OF AN IDEAL GAS IN A PULSE TUBE

Jeffrey M. Lee

Ames Research Center

SUMMARY

This study establishes.a consistent set of differential equations for use in describing the

steady secondary flows generated by periodic compression and expansion of an ideal gas in pulse

tubes. A small-amplitude series expansion solution in the inverse Strouhal number at the anelastic

limit is proposed for the two-dimensional axisymmetric mass, momentum, and energy equations.

The anelastic approach applies when shock and acoustic energies are small compared with the

energy needed to compress and expand the gas, such as for pulse tubes.

Seven independent dimensionless numbers are used to scale the system. The reciprocal

Strouhal number and Valensi number are used to linearize the mass and momentum equations. The

Fourier number is used to characterize heat transfer within the tube wall. The Mach number, the

Prandtl number, the velocity amplitude, and the velocity phase angle at the tube ends complete the

dimensionless scales.

The ordered equations show that the zeroth-, first-, and second-order equations, are coupled

through the zeroth-order temperature. An analytic solution is obtained in the strong temperature

limit where the zeroth-order temperature is constant. The solution shows that periodic heat transfer

between the gas and tube, characterized by the complex Nusselt number, is independent of axial

velocity boundary conditions and Fourier number. Steady velocities increase linearly for small

Valensi number and can be of order 1 for large Valensi number. Decreasing heat transfer between

the gas and the tube decreases steady velocities for orifice pulse tubes. The opposite is true for basic

pulse tubes. A conversion of steady work flow to heat flow occurs whenever temperature, velocity,

or phase-angle gradients are present. Steady enthalpy flow is reduced by heat transfer and is scaled

by the Prandtl times Valensi numbers.

Particle velocities from a smoke-wire experiment were compared with predictions for basic

and orifice pulse tube configurations. The theory predicted the observed mass streaming and flow

reversals between the centerline and diffusion layers. The results indicate that the theory is valid for

pulse tubes and that it can be used to solve for the zeroth-order temperature, to compute enthalpy

flows, and to determine losses associated with steady secondary streaming.

1. INTRODUCTION

A gas subject to periodic compression and expansion generates higher-order steady

secondary flows. The steady flows can manifest themselves as mass, momentum, and energy

streaming. The pulse tube refrigerator, in which the generated steady flow ofenthalpy can lead to

temperature differences of over 200 K, is a practical use of this type of transport.

The behavior results from nonlinear steady secondary transport. What appears to be strictly a

linear periodic system-- oscillating temperatures and oscillating mass flows-- in fact, gives rise to

mean-steady enthalpy flows. These enthalpy flows produce the observed refrigeration effect. The

term "mean-steady flow" refers to the time-averaged secondary flow.l Other types of mean-steady

flows, or streaming, include momentum streaming (such as acoustic streaming) and mass-species

streaming. Although the magnitude of mean-steady flows may be small, the fact that they are steady

and unidirectional can lead to sizable gradients over extended times.

This work examines mean-steady secondary transport for the pulse tube refrigerator. The

phenomenon is examined in the limit of a linear anelastic approximation. Oscillating anelastic 2

flows are characterized by low Mach numbers and oscillating frequencies that are much lower than

the resonance frequency defined by the system geometry. The anelastic approximation filters shocks

from the fluid equations while retaining the effects of density variations resulting from "slow"

compression and expansion of the gas. This applies when the energies of acoustic waves and shock

waves are negligible relative to the energy needed to compress and expand the bulk gas. 3 This work

investigates steady secondary momentum and energy flows of an ideal compressible gas in the limit

of linear anelasticity.

Mechanics of Mean-Steady Secondary Transport in Oscillating Systems

Mean- steady secondary flows have been investigated for a number of systems. For such

flows, the nonlinear advection terms of the appropriate transport equation produce the "driving

force" for the mean-steady flow. Mean-steady flows can produce energy streaming, mass streaming

(refs. 3-5), and even a mechanism for separating different mass species in multicomponent mixtures

(refs. 6, 7). An example of mass streaming for a single component fluid is shown in figure 1.

Figure 1(a) shows the observed mean-steady pathlines extemal to an extended cylinder oscillating

along its diameter in water, and figure 1(b) shows the corresponding calculated pathlines.

In this section, a simple linear mathematical model for describing mean-steady secondary

flows is developed. Consider the unsteady advection-diffusion equation where u is the velocity, Z is

1This is different from 'quasi-steady' flow which refers to periodic oscillations, or 'steady-state' flow which is

absolutely constant in time.

2Anelastic flows are described in more detail by Sherman (ref. 1) and Paolucci (ref. 2).

3The arguments are similar to Boussinesq flow, except that the flow is driven by applied pressure forces instead of

buoyancy forces.
_1--11,

(a) (b)

Figure 1. Example of mean-steady flows in oscillating systems:from Schlichting's oscillating

cylinder (ref 8).

a thermodynamic quantity (such as temperature or pressure), and dis the term containing diffusion

of Z,

8

 + Tzz(ZU)=d (1)

Now consider a time-periodic series expansion solution of u and 27that is expanded in the small

parameter e, valid for e << 1:

u = fio(Z)COSOOt + eUl(Z,t) + O(e 2) (2)

Z = X,o(Z)COS(Ogt + (p)+ eZl(Z,t)+O(_ 2) (3)

d = do(z,t)+ gdl(z,t) + 0(8 2 ) (4)

where _ is the phase angle between u and Z. Substituting the series solution into the differential

equation and equating terms of like order in e results in

O(1): do = _t _0 cos(ogt + q_) (5)

=8

--_ Z l(Z,t ) + -_z [fio COSCOt"Zo COS(COt+ 9)] (6) o(e): dl

where O means "of order." Equation (5) is the linear oscillating equation of O(1). Its solution

allows evaluation of equation (6) which describes a secondary flow of O(e). By time-averaging

equation (6) over a period the relation for the mean-steady flow is obtained:

4
- 03 }d d,= cos

dI -_zz co I 21 1/o2 )

(7)

_ °3(fi0Z0) cos ¢ - fi0Z0 sin q_03--_-_

& &

Equation (7) shows that the secondary flow depends on the gradients of amplitude and phase angle.

Two limiting flows are characterized by being either of the standing wave type, in which

phase is independent of position 03(P/03z= 0, or the progressive wave type, in which amplitude is

independent of position 03(fi0;_o)/03z = 0. An acoustic oscillator is an example of the standing wave

type. An example of the progressive wave type is water of uniform depth oscillating near the shore

of a pond. The oscillating water generates mean-steady momentum forces that form periodic ridges

in the fine sediment.

The above illustrates the mathematical basis for a linearized pulse tube model. The timeaveraged product of velocity (kinematic quantity) and temperature (thermodynamic quantity) results

in a mean-steady unidirectional flow of enthalpy. The mean-steady enthalpy flow will be non-zero

when there are gradients in the phase angle and product amplitudes. This can be accomplished by

heat transfer between the gas and the tube wall such that the thermal and viscous penetration depths

are not equal, or by independently controlling the amplitude and phase angle of velocity at the tube

ends. The first case is the enthalpy flow mechanism in the basic pulse tube, and the second is the

mechanism in the orifice pulse tube.

Background

The following is a chronological summary of pulse tube development. A historical brief on

pulse tube development is given by Longsworth fief. 9), Kittel fief. 10), and Radebaugh (ref. 1 1).

Ames Research Center's pulse tube home page 4 provides a comprehensive list of published pulse

tube research papers through 1994. A variation of the pulse tube is the acoustic refrigerator, a

detailed account of which was prepared by Swift (ref. 12).

Gifford and Longsworth

In 1963 Gifford and Longsworth reported on a new type of regenerative refrigerator (ref. 13). Their

"pulse tube refrigerator," aptly named because of the use of pressure pulses to alternately compress

and expand the gas, was innovative and very promising because there were no cold moving parts

which could limit reliability. Comparable systems at that time were based on Stirling systems which

required both a compressor and a cold expander for operation. Figure 2 shows a schematic of this

early basic pulse tube (BPT). The BPT consists of a hot heat exchanger, an open tube, a cold heat

exchanger, a regenerator, an aftercooler, and a reciprocating compressor. Work is supplied by the

compressor, heat is rejected at Th and Ta, and cooling is produced at Tc.

4http://i rtek.arc, nasa.gov/C ryoGroup/PTDatabase/database, htm

5
Compressor

I

w

Aftercooler HE Cold HE Hot HE

I Regenerator Tube I_

i I i

Qa, Ta Qc, Tc Qh, Th

Figure 2. Basic pulse tube.

Longsworth's experiments demonstrated the performance of a BPT, obtaining temperatures

of 169 K with a single stage and of 123 K with two stages using helium (ref. 14).

Gifford and Longsworth formulated a "step-wise" heat-transfer theory to describe their

results (ref. 15), and recently, de Boer extended the analysis (ref. 16). The basis of his model

consists of four steps: adiabatic compression of the gas and displacement toward the hot end of the

pulse tube; isobaric heat transfer from the hot gas to the cooler tube wall; adiabatic expansion of the

gas and displacement back toward the cold end of the pulse tube; and isobaric heat transfer from the

wall of the pulse tube to the cooler gas. This description enables one to visualize how an "energy

packet" can migrate between the gas and the tube wall in such a fashion as to result in a net transport

of energy from one end of the tube to the other. Heat transfer between the tube wall--which acts as

a second thermodynamic medium--and the gas is required to produce the phase shift necessary for

the flow of enthalpy, as explained above. After a decade of research, however, interest waned

because the BPT was unable to obtain temperatures and efficiencies comparable to those of Stirling

cryocoolers.

Wheatley, Swift, Hoffler

Following the BPT, Wheatley et al. at the Los Alamos National Laboratory began

investigating "naturally irreversible engines," that is, those processes in which a downhill flow of

entropy, say for example, heat conduction, can be used for conversion into mechanical energy, such

as a thermally driven engine or prime mover (ref. 17). This work is still in progress by Swift at

Los Alamos. A complete account of the theory of acoustic engines given by Swift (ref. 12) and a

similitude analysis has been done by Olsen and Swift (ref. 18).

The work on heat engines was applied to refrigeration by reversing the thermodynamic

process. Hoffier examined an acoustic refrigerator in which a pressure driver was placed at one end

of the system shown in figure 3 (ref. 19). Temperatures of 193 K were demonstrated using helium,

with the stack acting as the second thermodynamic medium serving the same purpose as with the

tube of the BPT.

6
Hot HE Cold HE

Compressorl St_;ck I

W

Qh, Th Qc, Tc

Figure 3. Acoustic refrigerator.

Rott

Previous to Wheatley's work, Rott formulated a set of linear acoustic equations which

included transverse diffusion (refs. 20, 21). Mi.iller, studying under Rott, detailed much of the

analyses in reference 22. The equations were examined for a number of conditions, including

stability limits for thermally driven oscillations (ref. 23), tubes with variable cross section (ref. 24),

gas-liquid oscillations (ref. 25), and efficiency analysis of thermoacoustic oscillations operating as

heat engines (ref. 26). Of primary interest are Rott's papers on second-order mean-steady heat flux

(refs. 27-29), which focus on enthalpy and mass streaming in the limit of constant axial temperature.

Rott has termed this limit the "strong" temperature case (Rott, N. 1993: personal communication).

Merkli

Merkli and Thomann (ref. 30) experimentally investigated an acoustic refrigerator and

examined their results using Rott's acoustic equations. They found good correlation between

experiment and prediction, with local heating at velocity nodes and cooling at antinodes. This and

the experimental work of Wheatley laid the theoretical foundation for acoustic refrigerators and

acoustic engines as accepted today.

Mikulin

The fundamental principle behind the acoustic devices and the BPT requires heat transfer

between a solid boundary and the working gas. This necessitates a spacing between solid boundaries

of the order of the thermal and viscous diffusion lengths. However, this limits enthalpy flow,

because temperature and velocity amplitudes and phase-angle are restrained and because the

dynamics are governed by transverse diffusion. However, Mikulin et al. showed that enthalpy

transport need not require diffusion to produce the required phase shift between gas temperature

and velocity, but that the phase and amplitude relation between velocity and temperature can be

separately managed by controlling the boundary conditions at the tube ends (ref. 31). In their

experiments, Mikulin et al. demonstrated the first orifice pulse tube (OPT) refrigerator, obtaining

temperatures of 105 K with air.

A variation of the OPT device is shown in figure 4. The approach Mikulin et al. used was to

place an orifice and reservoir volume at the closed end of a basic pulse tube, thus allowing for a

7
finite gas flow. The result is to greatly increase enthalpy flow at the hot end s and to change the

phase angle between velocity and temperature at the cold end so that pressure and velocity are closer

to being in-phase. This eliminates the need for a second thermodynamic medium and the inherent

restrictions associated with diffusion. It was also accomplished without the need for any type of

mechanical device. The OPT results in lower temperatures, increased cooling, and higher

efficiencies. Since the tube of the ideal OPT does not require heat transfer to transport energy,

the transport process within the tube is ideally reversible.

Aftercooler HE

Compressor I I

W

Cold HE

Regenerator

I

Hot HE

Tube Orifice

I

Qa, Ta Qc, Tc Qh, Th

Reservoir Vol

I

I

Figure 4. Orifice pulse tube.

Radebaugh

After reviewing Mikulin's work, Radebaugh and his coworkers recognized the intrinsic

value of a cooler without any cold moving mechanical parts, and so began work on the OPT as it is

known today. Their initial work demonstrated a single-stage 60 K refrigerator using helium, a

substantially lower temperature than had been achieved by any previous PT device (ref. 32). Other

investigators have since demonstrated temperatures of 4 K (ref. 33) and even lower by using

multiple-stages (ref. 34).

A description of the transport process for OPT devices was first given by Storch et al. based

on a one-dimensional thermodynamic model that assumed adiabatic processes within the tube

(ref. 35). A primary advantage of the one-dimensional (l-D) model is the convenience in using

phasor diagrams to describe the phase relation between oscillating temperature, pressure, and mass

flow. David et al. extended one-dimensional analysis for arbitrary time-dependence of pressure

(ref. 36), and Kittel reviewed and extended the theory for entropy and work flows, temperature

gradients, thermal conduction, and viscosity (ref. 37). For large systems in which transverse

diffusion is confined to thin-boundary layers, this approach works well. However, for smaller

systems in which diffusion is significant, large discrepancies arise (from 2 to 5 times, depending on

tube diameter-to-length ratio (ref. 35)).

Beyond one dimension

The one-dimensional description of the transport process within pulse tubes is unable to

describe transverse transport. To account for transverse effects, lumped-parameter corrections to the

1-D model have been used. The complex Nusselt number developed by K. Lee (ref. 38) and

5Whereas in the BPT, enthalpy flow goes to zero at the closed end of the tube because the velocity goes to zero.
extended by Kornhauser and Smith (refs. 39, 40) was used by Roach and Kashani (ref. 41). The

complex Nusselt number allows for lumped-parameter corrections to the oscillating heat transfer

between the gas and the tube wall by accounting for amplitude and phase-shift owing to diffusion.

A number of investigators have taken the 2-D dissipative acoustic equations of Rott (ref. 20)

and applied them to pulse tubes. Though the dynamics of the pulse tube are not within the constraints of the acoustic approximation, nevertheless, the corrections offered by considering diffusion

do increase the predictability.

Jeong calculated 2-D steady secondary flows between flat plates with velocity oscillations

for a BPT-configured system (ref. 42). His leading-order solution is in the limit of a boundary-layer

approximation (thin diffusion layer). He applies the leading-order solution to the higher-order meansteady problem and extends the solution over the complete transverse domain, with the solution for

the core region corresponding well to the mean-steady parabolic solution of Rott. Previous solutions

of the dissipative acoustic equations for a cylindrical geometry by J. Lee and coworkers have shown

similar results for mean-steady mass flow, and have been extended to enthalpy flow (ref. 43).

Xiao formulated the problem in the acoustic limit. His three papers present a general set of

2-D acoustic equations with transverse effects averaged out (by integrating over the area normal to

axial flow), thereby obtaining equations amenable to solution for "flow quantities" (integrated

quantities as opposed to local vector quantities) (refs. 44-46). Xiao's subsequent solutions are

offered for isothermal zeroth-order temperature and adiabatic conditions at the tube wall. His

analysis has yielded insightful understanding of axial mean-steady work, heat, and energy flows,

and is a step above a 1-D analysis because it includes lower-order transverse diffusion; however, it

cannot be used to determine the vector fields, nor the higher-order mean-steady transverse diffusion

heat transfer.

Bauwens takes an anelastic approach of the leading-order problem for very narrow tubes and

small Mach numbers, and arrives at a solution for the axial temperature profile and mean-steady

enthalpy flow (refs. 47, 48). Velocity conditions at the tube ends are treated as independent, and the

temperature of the tube wall is fixed (high heat capacity). His analysis is applicable for very small

Peclet numbers of the gas (small tube diameters) such as for regenerators.

Only a few experimental investigations have been conducted to validate the predictions

offered by the above solutions. Linear oscillating flow for low speeds was measured by Shiraishi

et al. (ref. 49) and Nakamura et al. (ref. 50). Velocity phase shifts in the diffusion layer have been

observed, corresponding to a Stokes solution of the linearized momentum equation. Hoffman et al.

(ref. 51) compared pulse tube performance with the acoustic solutions of Xiao (refs. 44--46) and

obtained good agreement. Previous work by J. Lee and coworkers had shown the existence of meansteady streaming with characteristic lengths of the tube length, in agreement with higher-order

solutions obtained from linear analysis (ref. 52).

Other phase-shifting mechanisms

The OPT goes beyond the capabilities ofa BPT by utilizing a valve and reservoir volume at

the hot end to increased phase shift and mass flow. The valve/reservoir combination, however, is

still limiting. Two other new pulse tube configurations have the potential to further increase

performance: the double-inlet pulse tube and the inertance pulse tube.
Double-Inlet Pulse Tube

The double-inlet pulse tube, illustrated in figure 5, adds a flow path from the compressor to

the tube hot end. This allows the gas to be compressed from the hot end of the tube, thereby

increasing the phase angle and reducing the mass flow through the regenerator. Reducing the mass

flow through the regenerator reduces enthalpy flow losses. Zhu et al. (ref. 53) first demonstrated the

advantages of the double-inlet pulse tube, and Lewis and Radebaugh (ref. 54) recently obtained a

temperature of 35 K for a 4 - cm 3 compressor using a double-inlet. Seo et al. measured axial and

radial temperatures for the basic, orifice, and double-inlet pulse tubes and examined the phase and

amplitude differences between the three (ref. 55). Shiraishi et al. measured axial mass flow between

the three pulse tubes and showed that the double-inlet decreases flow in the middle of the tube while

maintaining generally the same flow at the ends of the tube (ref. 56). This suggests that a primary

advantage of the double-inlet is to increase phase angles.

Compressor

I

W

Aftercooler HE Cold HE

Regenerator

I

Qa, Ta Qc, Tc

Tube

I

Hot HE

Orifice

I

Qh, T__

Reservoir Vol

I

I

By-pass inlet

Figure 5. Double-inlet pulse tube.

The concept of the double- inlet can be extended to multi- inlets by providing flow

communication between the regenerator and pulse tube at locations with similar temperatures. In

theory, the effect is a multi- stage device with cooling stages at the multi- inlets, but in practice, it is

difficult to obtain the proper flow amplitudes and phase angles.

Inertanee Pulse Tube

Use of an inertance tube to replace the orifice has been reported by Godshalk et al. (ref. 57),

Zhu et al. (ref. 58), Gardner and Swift (ref. 59), and Roach and Kashani (ref. 60). The inertance

pulse tube is shown in figure 6. The idea is to use the inertia of the gas in a tong tube to provide

added phase shift, analogous to the inductance in an electrical circuit. Hence the term inertance, a

combination of inertia and inductance. Matching the gas inertia in the inertance tube to the gas

spring compliance of the compressor, regenerator, and tube combination can result in near resonant

operation with higher performance.

10
Compressor

I

W

Aftercooler HE Cold HE

I Regenerator I

Hot HE

Tube

I

NQa, Ta Qc, Tc Qh, Th

Reservoir

Volume

Figure 6. Inertance pulse tube.

Inertance

tube

A qualitative comparison of different modeling approaches is shown in table 1.

Two-dimensional anelastic modeling is seen to yield a high return on modeling effort. With 2-D

modeling, transverse diffusion effects (which represent viscous effects and heat transfer between the

gas and the tube wall) can be evaluated to obtain transverse temperatures profiles and higher-order

mean-steady flows. The information obtained from a 2-D model is significantly more than from a

1-D model, and anelastic approximations are computationally less intensive than fully compressible

CFD (computational fluid dynamic) codes.

Table 1. Qualitative comparison of different modeling approaches.

Model comparison chart

Primary measures

Refrigeration (enthalpy flow)

Integral

Phasor

-4

Temperature, axial profile

Temperature, transverse profile

Mean-steady secondary flows

Heat transfer between tube and gas

Secondary measures

Temperature-dependent properties

Oscillating temperature at tube ends

Nonlinearities

Buoyancy effects (free convection)

Flow end effects

Qualitative measures

System optimization

Transient simulation

Physics are easily understandable

Differential CFD

analytic Compressible

I-D 2-D 1-D 2-D

Anelastic

3-D 2-D

-4 -4 -4 -4 -4 -4

q -4 -4 -4 q .4

.4 .4 _/ -4

,J q q q

-_ q .4 q

-V q _/ -4 -4 -4

-4 -4 q q -4 -4

q q .4

q q q q

q q -4

-_ V q -4 q q

,J -J -4 q

I1
Conventions Used

The following conventions are used for variables in this study:

u = scalar

U = vector

"c= tensor

u" = dimensional

u = dimensionless

o(e)

Z,t = OX / Ot

lightface italic are scalars

boldface are vectors

lightface non-italic Greek are tensors

with asterisk denotes a dimensional quantity

without asterisk denotes a nondimensional quantity

of order of e

partial differential notation

Complex embedding will be used to eliminate the time variable. For periodic Z,

Z = _ [x, eit] = _I_ [ff( ei(t+O)]

where t = co't', _ is complex, _, is a real number, _ is the phase angle, and 9_

component of the complex quantity.

(8)

extracts the real

Scope of This Study

Previous attempts to model the dynamic workings of pulse tube coolers have typically used

thermodynamic or acoustic analysis. The first, being an integral approach, fails in its attempt to

capture local transport. It is also inherently one-dimensional, and so neglects transverse thermal and

viscous diffusion. The consequence is to overs implify phase angles and overestimate velocity,

temperature, and pressure amplitudes, resulting in overprediction of enthalpy flows, particularly for

small pulse tubes in which fields dominated by diffusion constitute a significant amount of the

mean-steady flow. To account for transverse diffusion, lumped-parameter heat-transfer relations

have been introduced with some success.

The second approach, acoustic analysis, examines the differential acoustic equations first

formulated by Rott..The transverse diffusion terms are retained, thus allowing for transverse 2-D

effects. Typically, however, the solution for the linear system has been simplified by approximating

the diffusion layer as being very thin, thereby obtaining the pressure function at the 1-D inviscid

limit. This is applicable to pulse tubes that have large Valensi numbers. The higher-order meansteady flows have been examined for flow in a BPT. However, for the OPT, the transverse effects

have been integrated-out, thereby making vector fields unobtainable. For these cases, the acoustic

ordering is not completely consistent with demonstrated pulse tubes since pulse tubes operate at

frequencies considerably below resonance. This is not a weakness, however, since the acoustic

12
solutionsaresmoothandwell-behaved,andsocanbeusedin engineeringmodels.Noneof the

previouswork hascomparedmeasuredmean-steadyvelocityfields to predictionsin orderto

validatethesolutions.

This studygoesbeyondpreviousanalysesby calculatingtheanelastic2-Dvectorfieldsfor

enthalpyandvelocity,andtransversetemperature.Includedaretheeffectsof heattransferbetween

thegasandthetubewall of finite thickness.Finally, experimentallymeasuredmean-steadyparticle

velocitiesarecomparedwith thetheory'spredictions.

Section2 focusesonthefluid equationsandtheconditionsin whichtheycanbe

approximatedfor acoustic,anelastic,andlinearanelasticflows.Theenthalpyflow, mean-steady

particlevelocity,andentropyequations,usefulin this study,arepresented.

Section3 reducesthefluid equationsfor linearanelasticflow of agasin atube.Thesystem

takenis two-dimensionalaxisymmetric.Heattransferbetweenthegasandathin-walledtubeis

included.An expansionseriessolutionistaken,andtheleading-orderproblemandthemean-steady

higher-orderequationsarepresented.Thelinearizedequationsarethensimplifiedby usingcomplex

embeddingfor time.

Section4presentsthelinearizedsolutionsforthestrongtemperaturecasein whichthe

zeroth-ordertemperatureis constant.Examinedareoscillatingflows for velocity andtemperature,

oscillatingheattransferbetweenthegasandthetubewall, andoscillatingshear.Correlationsand

useof thecomplexNusseltnumberaregiven.Mean-steadyflowsarecalculatedfor theEulerianand

Lagrangian(particle)velocities,enthalpyflux fields,andtemperature.Theinteractionbetween

enthalpyflow, work flow, andheatconductionis discussed.

In section5 thepredictedmean-steadyparticlevelocitiesarecomparedwith measured

velocities.Themeasuredvelocitiesareobtainedusingsmoke-wireflow visualizationfor air

containedin atransparenttubewith oscillatingcompressorsat eachend.

Studyresultsandsuggestionsfor futurework aresummarizedin section6.

13

2. GOVERNING EQUATIONS

This chapter examines and simplifies the governing fluid equations for slow oscillating

compressible flows. The equations are scaled, and dimensionless numbers are def'med. Next,

secondary transport quantities are derived, and then series solutions are identified for the limiting

cases of acoustic and anelastic flow.

Equations of Change for an Oscillating Fluid

The three governing equations for a compressible viscous fluid are def'med by the laws of

mass, momentum, and energy conservation. Supplementing the fluid equations are four auxiliary

relations for density, enthalpy (energy), thermal conductivity, and viscosity in terms of pressure and

temperature. Linear diffusion transport is assumed: Newtonian flow relating shear and velocity

gradient through viscosity, and Fourier heat transfer relating heat conduction and temperature

gradient through thermal conductivity. Stokes's hypothesis for the bulk viscosity is assumed, and

body forces are neglected. A detailed formulation of the governing fluid equations can be found in

Batchelor (ref. 61). 1

The system and the boundary conditions dictate how the equations are scaled. Details of

scaling are given in appendix A and a dimensional analysis is provided by J. Lee et al. (ref. 62).

Consider an ideal gas contained in a system with characteristic length L'. At each end of the tube the

gas is displaced a distance d_, with frequency f" and characteristic velocity amplitude Ub. The

imposed oscillating displacements also produce oscillations in the velocity vector u', pressure p',

density p', and temperature T'. Pressure, density, and temperature oscillate about mean reference

bulk values p_, p_, and Tb, respectively, which are taken as the scaling constants. The scaling for

u" is U_ and the scaling for time is the inverse angular frequency, co" = 27rf'. Using the above

scaling constants, the conservation equations for mass, momentum and energy are, respectively,

where

O= P,t + EV •(pU)

1 1V.

pu s + epu. Vu = --_ Vp - Va x

M 2

oTt+sou.VT=Y----_l(p,t+tu. Vp) 1 V.q+(r+l)-_-aX:Vu

y Pr Va

"c=p2(V.u)l+ 21.t defu; 3/./2+2/./=0; def u = l [vu + (Vu) tr ]

(9)

(10)

(11)

(12)

tFor additional references, see Sherman (ref. 1), White (ref. 63), or Bird et al. (ref. 64).

15
and#2 is the second viscosity, q = -kVT is the conduction heat flux vector, and x is the viscous

stress tensor. The dimensionless parameters are the inverse Strouhal number, e ; the Much number,

M; the Prandtl number, Pr; the Valensi number, Va; and the elasticity parameter, _, which is a

measure of the resilience of the gas (ref. 2). The parameters are defined as

e=- U'O M 2 = U°2 Pr= v*. Va= --L'2c°" _ =_M2 (13)

co'L" y R'T" 0 a v" t

where v ° is the kinematic viscosity, a" is the thermal diffusivity, 7is the heat capacity ratio

C'p/C'v, a" = _ is the speed of sound for an ideal gas, and R" is the ideal gas constant with

the ideal gas law reducing to

p - p T (14)

Note that only two of the three parameters e, M, and _ are mutually independent. For sinusoidal

motion of velocity at the tube ends, 8 = db/L'. Rewriting A, = ),Mco'L'/a" shows that it is

composed of the Much number and the ratio of the oscillation frequency to the system acoustic

resonance frequency. For the problems we will address, M << 1, co'L'/a" << 1, and y = O(1).

Relevant Transport Quantities

Mean-Steady Enthalpy Flow

The local mean-steady enthalpy flow is evaluated by integrating the enthalpy flux over the

cross-sectional area normal to the flow and time-averaging over a period,

H= I _pn.uTdt dA (15)

A

where the overbar represents mean- steady values, and n is the unit vector pointing in the positive

direction and normal to the cross-sectional area A. Enthalpy flux is scaled as h_ = p'oU'oTbCp and

enthalpy flow as H_ = h_A'.

Mean-Steady Particle Velocity

For absolute steady-state conditions, the stream function represents lines of constant mass

flow (ref. 63) and is identical to the path along which particles travel (particle path). However, for

oscillating flow in which quadratic products of the linear solutions produce higher-order meansteady flows, the mean-steady stream function does not represent the mean-steady particle path. To

find the mean-steady particle path, the mean-steady particle velocity must be determined.

The instantaneous particle velocity, Up(X,t), is the Lagrangian velocity, as opposed to

u(x,t), which is the Eulerian velocity. For small displacement, Up(X,t) can be obtained by a

Taylor expansion of u(x, t) about the initial position xi at t = 0. Details are given by Batchelor in

16
reference61andtheyarereviewedin appendixB. Themean-steadyparticlevelocity for an

oscillatingflow is givenas

Up(X) = u(xi)+ U(X, _') d_'. Vu

X i

(16)

where the first term on the right-hand side is the mean-steady field velocity at the initial position x i

at t = 0, and the second term is the velocity component due to time-averaged particle oscillations

that transverse _(x).

The mean- steady Lagrangian transport of other quantities has a similar form. Let Z be such a

quantity (for example, enthalpy h; entropy s; or the acceleration u,t + u. Vu). Then the mean-steady

value of Z associated with the particle Velocity is

_p{X) = X(xi) + n(x, "t')d't'- _7z

Xi

(17)

Entropy

The entropy equation is given by

M 2

ps,t + epu. Vs =

1

V.q (18)

Pr Va

where • = x:Vu is the dissipation function. The entropy equation is an alternative form of the

energy equation and is useful for optimization studies.

Conditions for Acoustic and Anelastic Flows

For the limiting condition of very small e, equations (9)--(11) can be reduced to a set of linear

equations that are amenable to an expansion series solution. The parameter A serves by helping to

identify the distinguished limit between e and M.

Acoustic Flow

The equations can be reduced to an acoustic form for 2 = O(1) and e << 1. Here the firstorder pressure Pl (x,t), is coupled between the mass conservation and momentum equations. The

acoustic equations are linear (ref. 65) and allow a series solution in e where M = e 0"5 is the

distinguished limit. The expansion for p(x,t) is

p(x,t) = p0(x) + e pl(x,t) + e 2p2(x,t) + O(e 3) (19)

17
Equation(19) showsthatthefirst correctionto thepressure,Pl, is spatially dependent, and that

pressure gradients are of the order of the bulk temporal pressure oscillations. The rest of the

variables are also expanded in e,

p(X,t) = Po(X) + ePl(X,t) + _2/92(X,t ) + O(e 3) (20)

T(x,t) = To(x) + e rz(x,t) + e2T2(x,t) + O(e 3) (21)

u(x,t) = uo(X,t) + eUl(X,t) + O(e 2 ) (22)

U(x,t) = Vo(X,t)+ e Ul(X,t)+ O(e 2) (23)

Anelastic Flow

An anelastic approximation of the fluid equations neglects density changes due to pressure

gradients. Its effect is to "remove acoustic phenomena from theoretical considerations" (ref. 1) and it

can be used to "filter sound" from the fluid equations with the advantage that the equations do not

propagate numerical shocks (ref. 2). The anelastic approximation applies when shock and acoustic

energies are small relative to the energy needed to compress and expand the gas.

The condition _, < e << 1, defines the anelastic approximation. The corresponding

distinguished limit is M < e. The pressure is expanded as

p(x,t) = 2P (t) + _, p2(x,t) + higher order terms (24)

The anelastic equations are similar to the acoustic equations except that now there is no coupling

(through the pressure gradient) between the mass conservation equation and the momentum equation

at leading-order, that is, at leading-order 2P(t) is not present in the momentum equation, but remains

in the mass and energy conservation equations, and in the equation of state. The next pressure term

is O(A); it is the kinematic pressure that drives the flow in the momentum equation. A consequence

of this ordering is that there is an inaccuracy in the mass conservation equation of O(_). Appendix C

details the anelastic scaling.

The added unknown T (t) requires an additional relation. The relation comes from the

integral form of the energy equation. For a calorically perfect gas (ideal gas with constant heat

capacity), the integral equation is

PrZV a n.VTdS-@ P(t)n.u dS

(25)

where n is the outward pointing unit normal vector, S is the surface area enclosing the domain, and

"V is the domain volume. Equation (25) defines T (t) in terms of boundary conditions only, hence,

7:' (t) is the thermodynamic equilibrium pressure and is the integral equivalent of the first law used

in macroscopic thermodynamics.

Now consider a subset of the above case in which _ << e << 1. An appropriate expansion of

pressure is

p(x,t) = PO + e pl(t) + _ P2(x't) + e,_ p3(x,t) + O(e 2,_) (26)

18
Theexpansiongivenby equation(26)isapplicablefor pulsetubeswhereT (t) of equation (24) has

been split into two terms, Po + epl(t). The first term represents O(1) time-dependence of pressure,

such as when the pulse tube is cooling or warming; for quasi-steady conditions it is constant. The

second term represents O(e) time-dependence given by the oscillating pressure in the pulse tube. As

before, the thermodynamic pressure Pl is determined from the energy integral, equation (25). The

expansion in e of the other variables is given by equations (20)--(23).

Linear Anelastic Flow

The added condition eVa << 1 allows linearization of momentum, that is, a linear anelastic

approximation. The product eVa is the Reynolds number. In this study, the linear anelastic problem

is investigated. Though the acoustic problem is not addressed here, it has been addressed elsewhere

(ref. 20); it is, however, summarized, along with the anelastic equations, in table 3 of section 3.

19

3. TWO-DIMENSIONAL AXISYMMETRIC FORMULATION

In this section, the two-dimensional axisymmetric linear anelastic problem is derived for

oscillating flow in a tube. Heat transfer between the gas and a tube wall of thin but f'mite thickness is

considered. Mean-steady relations are derived and summarized for both the anelastic and acoustic

limits. Finally, the equations are simplified for sinusoidal time using complex embedding.

System

Consider an ideal gas contained in a long cylindrical tube of f'mite wall thickness. The

thickness of the tube wall is very thin relative to the tube radius. The system is thermally insulated,

and the gas velocities at the tube ends are of small amplitude and sinusoidal. Figure 7 is a sketch of

the system. A summary of the assumptions and approximations made in the analysis is as follows:

1. Two-dimensional, axisymmetric cylindrical geometry

2. Inert single component ideal gas

3. Constant properties--thermal conductivity, heat capacity, dynamic viscosity

4. Enthalpy a function of temperature only

5. Stokes hypothesis for second viscosity, 3/.t 2 + 2_ = 0

6. rw/L" << 1 implying that the radial momentum equation is negligible, P,r - 0, and that

p= p(z,t)

7. No body forces

8. 1" << rw so that the tube wall domain can be approximated with rectangular Cartesian

coordinates

9. Sinusoidal time-periodic velocity boundary conditions

at r= 0,

(10 eit

ADIABATIC AT WALL

OUTER SURFACE

J TUBE WALL

y=l _ / DOMAIN

y__o r=l .................................... i/ GAS DOMAIN

r=0

Z=0 Z=I atr= 0,

(lLei(t+O_)

Figure 7. Two-dimensional axisymmetric system for oscillating flow.

21
Theproblemis dividedintotwo domains.In thegasdomain,only thegasinsidethetubeis

considered;it extendsfrom r = 0 to r = 1, and from z = 0 to z = 1. In the tube-wall domain, only the

tube wall thickness is considered; it extends from y = 0 to y = 1, and from z = 0 to z = 1. The two

problems are coupled through the boundary conditions between the gas and the tube wall which

require that the temperature and heat flux be continuous across the interface. In the gas domain, the

thermodynamic variables of temperature T and density p are functions of spatial coordinates r and z,

and of time t. For the tube-wall domain, the tube-wall temperature 0 is a function of y, z, and t.

The normalized velocity field for the gas domain is composed of the axial velocity u and the radial

velocity v. The periodic boundary condition at r = 0 and z = 0 is u = 0 o e it and at r = 0 and z = 1 it

is u= UL ei(t+¢v) where i= x/-Z-1. The velocity is scaled by 0_ so that 00 = 1 and _r L = UL/U_).

Finally, at the outer surface of the tube (y = 1) the radial heat flux is zero (adiabatic).

Equations

The dimensionless fluid equations for mass and energy conservation, the equation of motion,

and energy conservation for the tube wall are

p,, +E (pU),z

(( 7 ) )= 1 (rU'r),r (Pu),t + £" P ,r + (puU),z --_P,z "_ "Va r

TP't+E[ (pl)_r)'r + (PU)'z] = )'y- 1EUP, z +PrqCa_, 1 [(rTr)r I r' _'e2FZrzz +(_'-l)-_a 2@

(27)

(28)

(29)

p = pT (30)

O,t=Fo(Oyy +e4F2wOzz ) (31)

F2 =(rw/EL*) 2

= Vhe1.sttworel.tion ,r rw, ofO( ).This where and

W \ l ]

ordering for equations (29) and (31) applies when the axial conduction of energy due to the tube

wall is two orders higher than the corresponding conduction of the gas. The equations do not allow

for azimuthal flow; however, it is noted that one should not necessarily rule out the possibility of

azimuthal flow in pulse tubes.

In addition to the dimensionless numbers previously introduced (e, M, Pr, Va, and ;I,) there

are several other dimensionless quantities that result from the boundary conditions and the system

geometry. These are the velocity amplitude ratio, I.)L; the normalized velocity phase angle, q_u,

which is the phase angle between the velocities U_ and U_ where U_ leads U_, and 0 < CU < 1; the

22
* // .2 *_.

Fourier number for the tube wall, Fo= tZw/ll o9 ), and dimensionless length ratios, rw/L" and

l'/L*. Table 2 lists the complete set of dimensionless parameters.

Table 2. Dimensionless numbers.

E

;t

M

Wa

Pr

Fo

rw/C

r/c

OL

Ou

B

H

Range Name Definition Comments

10 -3 to Expansion -. ....

10_1 parameter UO//(CO E)=d_)/E tubeRati° lengthlength of displacement L', co" = 27rf" d_to

10-8 to Elasticity

10- 5 parameter e a a

10 -5 to Mach

10-3 number Uo/a*

1 to 10 2 Valensi *2 */ *

number rw co / v

0.7 Prandtl

number v*/ a"

0 to 102 Fourier • //,2 ._

number aw/_l co )

10-1 Gas domain

length ratio

10-2 Tube wall

length ratio

0to 1 Velocity -, -,

ratio U L / U_)

-0.5 to Velocity

0 phase angle

Normalized

enthalpy n*/nref

flow

Cv

Ratio of velocity at z = 0 to speed

of sound

Squared ratio of tube inner radius

to viscous diffusion length

Squared ratio of viscous to thermal

diffusion length of gas

Squared ratio of tube-wall thermal

diffusion length to tube-wall

thickness

Ratio of tube radius to tube length

Ratio of tube-wall thickness to tube

length

Ratio of velocity amplitude at z = 1

to amplitude at z = 0

Velocity phase angle where Uo at

z = 0 leads U/. at z = 1

Ratio of mean- steady enthalpy

flow to oscillating enthalpy flow

Href = POlOuOCpJrrw *_* _., _* .2

To illustrate the values of the dimensionless numbers, consider a small-sized pulse tube

operating at 10 Hz, with a stainless steel tube of wall thickness f= 0.01 cm, inner radius

rw 0.35 cm, length L'= 5 cm; thermal diffusivity of the tube wall ct w = 0.045 cm2/sec; helium gas

mean pressure at 106 Pa; kinematic viscosity v ° = 0.124 cm2/sec; Prandtl number Pr = 0.7; speed of

sound a'= 103 rrdsec; and gas displacement d_ = 0.04 cm. The calculated dimensionless numbers

become e = 8x10 -3, Va = 62, eVa = 0.497, M = 2.5x10 -5, _ = 4.8x10 -8, Fo = 7.2, and

23
M2/Va = O(10-11) << E2. The value of eVa shows that these conditions are at the limit of the

linear approximation, and the relation for M2/Va shows that viscous dissipation is not important.

Now consider a pulse tube operating with a more typical e = 0.1. This results in eVa = 6.2,

which is not strictly within the constraints of the linear approximation. However, if the measured

values of mean-steady flow for eVa > 1 are in agreement with calculations using a linear approach,

then an argument can be made for describing mean-steady flows for pulse tubes using a linear

theory. It will be shown that this is the case.

Series Expansion

The anelastic criterion that applies to pulse tubes is

_, << e << 1 (32)

We are interested in the quasi-steady periodic solution for the leading-order problem.

Equations (27)-(31) are linearized using a two-parameter series expansion for pressure in e and 2,

p(z,t) = 1+ e pl(t)+ _. p2(z,t)+ eA p3(z,t)+ O(e2A) (33)

and an expansion in e for the other variables,

p(r,z,t) = Po( Z) + e P l(r,z,t) + e2p2(r,z, t) + O( e3 ) (34)

T(r,z,t) = TO(Z)+ e Tl(r,z,t) + e2T2(r,z,t) + O(e 3) (35)

u(r,z,t)=uo(r,z,t)+ eul(r,z,t)+O(e 2) (36)

v(r,z,t) = 190(r,z,t) + e vl(r,z,t) + O(e 2) (37)

For details as to why the expansion takes the form of integer powers in e and A, refer to appendix D.

Equation (33) reflects that the leading-order equation of state is

l = po = Po(z)To(z) (38)

and the first-order correction for pressure is

pl=Pl(t) (39)

Details for obtaining equation (39) are also given in appendix D.

First-Order Oscillating Equations

The resulting first-order equations for mass and energy conservation, the equation of state,

and the energy equation for the tube wall are, respectively,

(pouor),r

Pl,t + _ (PoUO),z = 0 (40)

r

24
( o or).r _ 1 (rr"r).r pl, + (poUo) z (41)

y r Pr Va r

Pl = PoT1 + PlTo (42)

Ol,t = Fo Ol,yy (43)

The condition eVa << 1 allows linearization of momentum. Ordering the momentum equation shows

that the flow is driven by the p2(z,t) pressure gradient

1 (rUo,r),r

(Pouo),t =-P2,z + (44) Va r

The energy integral over the gas domain defines the relation for pl(t)

p,l(t) = y_ (.___}__.l fn-Tl, rdS-In.pouodS) (45)

V_ PrVa_ S

where the prime denotes an exact derivative. Equation (45) shows that the thermodynamic

oscillating pressure pl(t), results from the forced oscillations at the tube ends and from periodic

radial heat conduction at the tube walls. _ Details of the expansion are given in appendix D.

The above coupled set of linear equations still does not completely define the basic state

problem. This is because the zeroth-order temperature, TO(Z ), is coupled to the mean-steady,

second-order energy equation. To completely define the basic state problem, the mean-steady

equations of next order must be determined.

Mean-Steady Equations

Mean-steady velocity component due to Reynolds stresses, E 1

The mean-steady, second-order momentum equation is given by

1 (rUl,r)r (46)

(Po_oUO)'r _(pou-_),z =--P3,z ÷ Va '

r r

The left-hand side of equation (46) contains the Reynolds stresses. The right-hand side is the

resulting mean-steady velocity _1, and the mean-steady pressure P3- These two are unknown. A

second relation comes from the zero-net-mass-flow constraint, which requires that the integration

of the steady axial mass flux over the area normal to the axial flow be zero,

1

0 = I (pOul + PluO) rdr (47)

0

:Periodic axial heat transfer at the tube ends is not considered here.

25
Theabovetwoequationsareusedto solvefor _-1andP3. The mass conservation equation is then

used to solve for Vl,

(PO_l r + PlVor),r

O= r _'(POUl+PlUO)'z

(48)

Mean-steady temperature distribution and the equation of state

The mean-steady, second-order energy equation is used to fend T2,

1 (rr r)r

Pr Va r + r2To,zz} = Va "(pOUl + pluo)

(49)

where F 2 = O(1) and (V a ") is the axisymmetric divergence operator, that is,

1 --

V a "(POUl + pluo) = (Po-_I + pluO),z + r(rpoul + rpluO),r" Here we see that To(Z ) is coupled to the

second- order energy equation. The equation of state is used to find the density

-P2 = Po _2 + Pl TI + P2 TO (50)

The gas-domain equation is coupled to the tube-wall domain

0 = 02,yy (51)

which implies 02 = 0 after requiring that the temperature boundary condition at y = 1 be adiabatic.

The zeroth-order temperature of the tube wall, Oo,zz, is of O(e 4) so it does not enter into the

problem at O(e2).

The complete leading-order problem requires To(Z ) to be solved simultaneously at zerothorder (eq. (38)), oscillating first-order (eq. (42)), and mean-steady, second-order (eq. (49)). Table 3

summarizes the leading-order and mean-steady equations and boundary conditions.

26
Table 3. Summary of equations: leading order and mean-steady boundary conditions."

Zeroth -order relations Eq.

PO = I Zeroth-order equation of state (a)

po = po(z)ro(z) (b)

Oo(z)= to(z) (c)

First-order oscillating equations

1 (rUo, r),r

(pOuO),t = -P2,z +

Va r

Pl,t + Va" (pouo) = o

( (rTl,r) r

' 1 ' vo.(pouo)

Pl,t = _1 Pr Va r

Pl,t=_ _Va_n'Tl,rdS-_n'pouodS

S S ,

Pl = poTI + PlTo

Ol,t = Fo Ol,yy

Mean-steady equations

Wa r

1

O= _ (pO'_l * pluo)rdr

0

V a •(pOll1 + pluO) = 0

(rr ,r)r

Pr Va r

1--2

-- - Va'(POU, * P_O)- p--_aZO, zz

P2 = PO _2 + PlT1 + P2TO

0 = 02,yy

O(A) pressure gradient drives the O(1)

velocity

Mass conservation

Energy

Volume integrated energy

Equation of state

Energy equation for tube wall

Solve with the axial mass flow

constraint to obtain p3and _'l average

Axial mass flow constraint

Single quadrature to find U1

Second-order mean-steady energy

equation

(d)

(e)

(f)

(g)

(h)

(i)

(J)

(k)

(i)

(m)

(n)

(o)

aThe acoustic problem is obtained by replacing P2 in eq. (d) with Pl (z,t)

-----_--, replacing P3 in eq. (j) with

p2(z,t)

, and eliminating eq. (g).

27
Table 3. Concluded.

Boundary conditions for oscillating problem"

O=uo, r = 1)0 at r=0

O=u 0 at r=l

0= O0 at r=l

uo=Uo= LOoei' ]

uo=UL= _ •(JL eit]

at z=0andr=0

at z=l andr=0

O= T1,r at r=0

T 1 = 01 and

kgTl, r = kwOl,y at y = 0 and r = 1

0 = 01,y at y = 1

Symmetry

No slip

No-penetration condition

Two boundary conditions in z needed for

U0,zz

Symmetry

Temperature and heat flux continuity

Adiabatic at outer wall

(p)

Boundary conditions for mean-steady problem

0 = Ul,r ='_1 at r=0

0=_ 1 at r=l

0=_ 1 at r=l

TO = Tlz=O at Z = 0

TO = TIz=l at z = 1

at r=O

aty=Oandr=l

m

0 = 02,y at y=l

Symmetry

No slip and isothermal

No-penetration condition

Two boundary conditions in z

needed for TO,zz

Symmetry

Results from solution of eq. (o) and

applying boundary condition where it is

adiabatic at outer wall

(q)

28
Comments

The complete problem involves 15 variables (u O, v O, Ul, _1, PO, Pl, -P2 TO, T1, T2, 01,

02, Pl, P2, P3) in the 15 equations listed by equations (b) and (d) - (q) in table 3. There are 12

continuum equations, one integral equation (g), and two boundary constraints (p) and (q). The

boundary constraints are independent equations, since in the differential equation for mass conservation, vo has only a single derivative and need satisfy only a single boundary condition. Note that the

zeroth-, first-, and second-order equations must be solved simultaneously for the leading-order

problem because To is coupled between the zeroth- and first- order equations of state, and the meansteady energy equation.

Thermally strong approximation, TO,z = 0

The problem can be greatly simplified for the case of negligible zeroth-order axial

temperature gradient, To, z = 0. This would apply for a strongly imposed heat sink/source on the

system. This leads to decoupling of the zeroth-order equation of state from the mean-steady energy

equations. The zeroth-order equation of state is exactly known (Po = PO = TO = 1) so that the firstorder oscillating equations become self-contained and can be solved independently. The result is

seven equations (eqs. (d) to (i) and boundary constraint eq. (p)) with seven unknowns (uo, vo, Pl, P2,

Pl, T1, 01). This simplified case applies to the experimental investigations of section 5.

Acoustic limit

As shown in section 2, the acoustic limit requires A, -- e << 1 which results in

p(z,t) = 1 + epl(z,t ) + e2p2(z,t) + O(e3). The equations in table 3 can be rewritten in acoustic form

by replacing p2(z,t) in equation (d) with pl(z,t); replacing p3(z,t) in equation (j) with p2(z,t);

and eliminating equation (g).

Entropy production

The mean-steady generation of entropy can be used to optimize enthalpy flow by

minimizing entropy production. The entropy equation is expanded to second-order and integrated

over the domain. The zeroth- and first-order entropy equations do not contribute to mean-steady

entropy production since they describe only linear oscillating entropy transfer in the form of heat

transfer and entropy advection, and not generation. 2 Entropy generation is quadratic at secondorder.

Integrating the second-order entropy equation over the domain results in the steady

generation of entropy $2,

2Entropy generation is different than accumulation which can come in at first-order if the flow of heat at each end of

the tube is different. This would result in a bulk temperature transient. In this case, the first-order accumulation of

entropy is given by $1 = 1" _ n- T°'z dS which requires the system to be heating o1 cooling.

Pr Va T O

s

29
V

- To 7"0 t, ro ) ] s ro }

- f..(po.ost+po ,+o

S

(52)

Viscous dissipation does not come in at second-order--irreversibilities are due only to heat transfer.

The entropy produced in the tube wall must be added. The entropy equation for the wall at

second-order is

O2=Fo , +F dV w + _n. O2"----LdSw (53)

[3,'., &,,, TO

The total entropy generated by the combined gas and tube wall system is the sum, (5 2 + O2)-In the

limit of an isothermal or an adiabatic wall boundary condition, the wall-domain problem need not be

considered, and the gas-domain equation, 5 2, by itself, is sufficient because in both idealized cases

there are no temperature gradients present in the wall.

Complex Embedding of the First-Order Equations

To obtain a periodic solution for the linear equations of table 3, complex embedding can be

used to eliminate the time-dependence. The problem is then solved in the complex plane with the

real part of the complex solution being of physical significance. The complex solution takes the

form

Z= ¢ll[_,(x)e it] (54)

where Z represents the real part of the complex function ,_(x)e it . In general, 2(x)is spatially

dependent and is itself complex. Appendix E details the procedure for simplifying the first-order

equations; only the important points are summarized here.

The first- order momentum equation given in table 3, equation (d), is reduced to a solution in

z. The solution for uo is explicit in r with the z-dependence contained in P2(z) and Po(Z), both

unknown at this time:

fio=ii,'2(z)r. ,. i z;,f-V-a)] (55)

p--_tl- _ot r,

30
where

_n(r,z;cr)= n=0,1 (56)

with Jn being the Bessel function of the nth-order. In a similar manner, the energy equation can be

solved explicitly in r with unknown dependence in z. The energy equation is rearranged using the

equation of state and the mass conservation equation to arrive at

7"1= _ f_lTo[ 1 - _0( r,z; _V--ff)] + 7"weiCT _0( r,z; _-V-a )

-ToT_{[l-(o(r,z;,_-'_-ff)]-(__l)[_o(r,z;_--a)-_o(r,z;_)] }

(57)

where 7_w is the amplitude and _T is the phase angle of the temperature at the interface between the

gas and the tube wall. The first term on the right-hand side of equation (57) contains the factor

7-1 _ITo" This factor represents the leading term of a series expansion of the thermodynamic

7

relation for isentropic compression of an ideal gas, dT = )"- 1 dp. The first term also includes the

T 7 P

Bessel function expression [1-(o(r,z; _-,/-_-_)], which describes the transverse diffusion

temperature fall-offbetween the center of the gas and the tube wall.

The second term of equation (57) is the effect due to the heat transfer between the gas and

the tube wall. For an isothermal tube wall, Tw = 0. For a non- isothermal wall of finite thickness, T1

is coupled to the temperature 01 of the tube wall. The governing equation for the tube wall is given

in table 3 by equation (i). The solution is

where Z =

O -- weiO [ei Y--e r's n Yll, COS (58)

The third term is due to advection of gas along the axial temperature gradient. The

dependence on TO suggests that the oscillating temperature T1 should increase with local mean

temperature. Experimental measurements of the oscillating temperature in a pulse tube suggest this

is the case (ref. 35, p. 54). For Pr _ 1, L' Hospital's rule shows that the term remains finite. Moreover, for a BPT with Pr ----)1 and Tw = 0, the phase shift between temperature and velocity--

provided by the difference between thermal and viscous diffusion lengths--is everywhere

orthogonal, which results in zero enthalpy flux. However, for an OPT, a phase shift is imposed by

the velocity conditions at the tube ends, hence, enthalpy flux remains finite for Pr = 1.

31
Equation(57) showsthattheunknowntemperature,7_1,isnow describedasa functionof the

axialtemperatureTO(Z), the kinematic pressure /32(z), the bulk oscillating pressure/31, and the

temperature amplitude Tw and phase angle OTbetween the gas and tube wall. Equation (57) is

combined with the zeroth- and first-order equations of state and with the first-order mass conservation equation to obtain a second-order ordinary differential equation for vO which is explicit in r and

unknown in z:

•{( 190 = t m I r,z; 2

+ (ml(r'z;'4_Q--_) l-'r -_P-f-r " [m3(r'z; " _--ff) - m3(r'z; P_-P--_-ff)])T_/3'21L (59)

+ trz l l+m trz; l oeiO }

where

m,lr, (60)

m2(r,z; P_-_--ff)=-[-f_y+_m3(r,z;'_-V--a)] (61)

1

m3(r,z;cr)= cr __o(z) _l(r,z;cr) (62)

Details about the derivation of equation (59) are given in appendix E. Equation (62) shows that

when TO,z = O, PO is constant and that vo becomes independent of z.The boundary condition v0 = 0

at r = 1 is used with equation (69) to obtain a quasi-linear, second-order ordinary differential

equation for/32(z) with unknown constant/31.

0=/3_ +Jml(l'z;4-P-W_) prlm3(X,z;_V-ff ) ....m3(X'z;_PrrP-_aVa)]]'ln '

[ ml(l'z;_7_-ff) Pr-l[ml(l'z;'4_--_) ml(l'z;'fQ--ff) 1_ TO(z)] /3'2

(63)

m2(l'z; P__) /31 + m3(l'z;_-_--_) l"w ei_T

+ ml(1,Z;_--ff)T O ml(1,z;-_-'a ) T2

This is the pressure equation for/32(z). The volume-integrated energy equation is required for the

unknown /31,

i/31=_lPr--_a!n'T1'rdS-_n'Ct°dSs

(64)

32
Table4 summarizesthereducedsetof oscillatingfirst-orderequations.Theseequations,alongwith

themean-steadyequationsandboundaryconditionsof table3, completetheformulationof the

linearanelasticproblem.

Table4. Reducedsetof oscillatingfirst-orderequations.

u° = i P2(Z) [lp-"_t_°(r'z; _--a )]

T1 = :y -__.__l _lTo[ l _ _o(r,z; _) ] + _.weiC)T_o(r,z; p_-p-_-ff)

?"

,o rz; o r.z; )lt

ei lisin y l - Twe_#r

cosX JJ

{( z-¢_)_o^.. f)O = _ ml r,z; P2

+ ( ml(r'z;_--ff) Pr-1Pr [m3(r,z;._/-_-a)_m3(r,z;p_-_---ff)])Td_

+m2(r,z; P._/-P-_-a)[_l+m3(r,z; P_P-_--a)_o eiC)T }

0 = f_+J ml(l'z;P4-_V-ff) Pr [m._(1,z;q _V-ff) m3(l,z; Pr_--{Va)] l '

l ml(l'z;_fQ--a) Pr-1 _ ml(1,z;_f-V-a ) jl[_nr°<z)] _

m2(1,z; P_-_--ff)[_1 m3(I'z; P_/-P-'_-_)Tw eiC)T

+ _lO,Z;Z-_)_o' _ _o_

i[_1= _ n'7"l,rdS- n.uodS --_

S

(a)

(b)

(c)

(d)

(e)

(0

33
Considerationsfor the Leading-Order Temperature, To(Z )

Solving the most general case requires simultaneous solution of the zeroth-, first-, and

mean-steady, second-order equations because of the temperature coupling of TO(Z ). However, there

are three limiting cases that specify To(z), a priori, thus allowing decoupling of the In'st-order and

mean-steady equations. Following is a summary of the general case along with the three limiting

cases:

1. To(Z ) is unknown

This is the most general case in which TO(Z ) is part of the solution to the set of coupled zeroth-,

first-, and second- order equations.

I

2. (InTo) =constant_ To(Z)= To(O) T°(1) "

Here To(z ) is exponential. This requires solving a coupled set of quasi-linear differential

equations.

3. TD = constant =, (In T0)" = m

mz+a

Here To(z ) is linear in z with slope m. This case also requires solving a coupled set of quasilinear differential equations.

4. T_)=(lnTo)'=O

Here To is a constant. This is the thermally strong case, implying that Po = PO = TO = 1, where

thermal sources and sinks are imposed on the system to maintain a constant temperature TO.

Case 4 describes the thermally strong condition and is the simplest since the pressure

equation is reduced to a second-order ordinary differential equation with constant coefficients,

which allows an analytic solution. In addition,/31 is now one of two unknown constants in the

pressure equation and is found by using the second boundary condition on uo; hence, the volume

integral of the energy equation is not required (eq. (f) in table 4). The remaining part of this study

will focus on the thermally strong condition of Case 4.

34
4. SOLUTION FOR THE THERMALLY STRONG CASE, To, z = 0

The equation sets in tables 3 and 4 can be analytically solved for the case of VT 0 = 0. The

condition VT 0 = 0 implies PO = Po = To = 1 and thus allows the quasi- linear differential equation

for pressure (eq. (e) in table 4) to be reduced to one with constant coefficients. This is Case 4

described in the previous section--- a hypothetical condition in which heat sources and sinks

maintain TO constant. Rott (1993: personal communication) termed this case thermally dominant or

"strong." The solution is tabulated in table 5, and solution details are given in appendix F.

Leading-Order Results

In this section, flow fields are calculated for BPT (UL = 0) and OPT ( UL = 1) configurations

as a function of Va, _OU,and Fo for fixed values ofe = 0.002, ),= 5/3, (1"/I212 ~ O(e4),

(rw/L*) 2 ~ O(e21, _l,~ O(10-91, and M 2 ~ O(10-111. For these conditions, M2/Va < O(10-12 t

so viscous dissipation is not important.

Table 5 shows that the pressure, equation (a), and the temperature, equation (c), are constant

in z and depend only on the velocity boundary conditions and the tube-wall temperature. There is no

dependence on pressure gradient; this is a result of the anelastic approach, and reflects thermodynamic equilibrium. The axial velocity, equation (b), also reflects thermodynamic equilibrium in its

linear z-dependence, while the radial velocity, equation (d), is constant in z. The temperature of the

tube-wall domain is given by equation (f), and the effect of heat transfer between the gas and the

tube wall is contained in the terms represented by i'w eiCr . Appendix G lists the code used for

computing the solutions.

Comparison of/31 and 751

The equation for integrated bulk oscillating pressure is given by equation (f) in table 4. After

integration, it is

pr-_a [ X-ULeiOV 751 =-2iy rT"l,rlr= 1 -m/(1;_a) 1 -_0(0_a ) (65)

This is more general than the solution pressure given by equation (a) in table 5. For the present

solution, equation (65) is not required for determining the bulk oscillating pressure/31, because the

bulk oscillating pressure is an unknown constant in the leading-order momentum equation and is

determined by the velocity boundary conditions at the tube ends (details in appendix F). The

pressure /31 that results is the thermodynamic pressure calculated along the centerline of the tube. In

contrast, the integrated pressure 751 of equation (65) is determined by integrating over the volume

^ ^

domain of the gas. This results in a discrepancy between P! and Pl. The difference should be of

order of the thermal diffusion layer thickness, o(' _ /.

35
Table5.Summaryof leading-ordersolutionsfor strong-temperaturecase,TO,z = O.

Pl =-i 1-OLei% ml(1;x/-_)

1- (0(0;-_/-_)m2 (1;-_-_--a)

1-;'o(r;,rv_)!

"

m21'l; roll 1;-_/-_ i J1 - (0(r;_-P-_)l Px/-P-{_ t t 1 - _0(0;x/-_ ) 1-ULeiOU)

" m 3 1;_-r-V--a + _o(r;_t- _'-_ ! ,---!r_-_o(r;_-_t]_,_,eiOT

Y m2[1;4PrVa) t j

^ mlll;x/-V--ff) m2(r;P_--r-v-a} ml(r;_V-a)! (l_OLeiCU)

1)0 = 1_ _0, 0;._V--ff)'im2(1;.,j-P-_ i ml(1;_V-a)'

[m_(r;_7_-_)

^

pl=PI-T1

01= _weiCr -e,ZY _ eiXl isin zy ) -

k cosz .

Eq.

(a)

(b)

(c)

(d)

(e)

(f_

Jn(ra-_-iPo(Z)) Jo(O) = 1

(n(r,z; Cr) = n=0,1

So(o_l s,(o)=o

m,(r,z;_--a)=-.'_ -r _ m3(r,z;_--_) ]

m2(r,z;x/-p-_-a)=-i r 7-1 (r, Z;P-x/-P--r-Q_) --+ m 3

L2Y 7

1

m3(r,z;G)= __l(r,z; Cr)

m,(O,z;4V_)--O

m2(O,z; P4-_)=O

m3(O, z; cr) = 0

(g)

(h)

(i)

0)

36
The bulk pressures /31 and 731 for a BPT (UL = 0) with e = 2 x 10 -3 and 1 = 6.46 x 10 -9

are given in figure 8. The difference between/31 and 731 for small PrVa is significant, being about

40% for PrVa = 0.7. As PrVa increases (7< PrVa < 21), the difference is less than 5%, and for

PrVa = 70 the difference is less than 1%. The difference between/31 and 731 is consistent with a

PrVa scaling.

0.75

0.5

0.25

-0.25

-0.5

-O.75

PrVa = 0.7

/ \

/ k 0

\_.__// -0

-0

k /

/

"x i -0

PrVa = 7

.4

.6 _'_._J

1

0.5

-0.

PrVa = 21

Q .

o0'._ 0'. .6 t

-0.

PrVa =70

Figure 8. Pressure vs. time for f21 (--) and f_l (--) with _, = 6.46 x 10 -9, 7"w = 0, 0 L = 0.

Leading-Order Temperature and Pressure Phasors

Figure 9 shows how heat transfer between the gas and the tube wall affects the temperature,

pressure, and velocity phasors 1 for a BPT and an OPT. Phasors for/31, UO, UL, and Tb for the

conditions Fo ---) 0 and Fo = 100, and Va = 1, 30, and 100 are shown where 7"b is the bulk

oscillating temperature (temperature averaged over the tube cross-sectional area),

1

7"b= 2_rTldr (66)

0

tPhasors are complex vector quantities whose magnitude and angular position, when plotted in the complex plane,

represent, respectively, the amplitude and phase angle of a sinusoid.

37
The BPT is a standing wave device and so the velocity phasor _0 has a constant phase angle

along the tube length; hence _0 lies along the real axis. Because _0 lies only along the real axis, the

temperature phasor Tb is out of phase with the velocity. This is seen in figures 9(a)-9(c) for a BPT.

The result is very small enthalpy fluxes since enthalpy flux depends on the cosine of the phase angle

between temperature and velocity according to the general relation given by equation (7) where Zo

is the temperature phasor and u 0 is the velocity phasor.

Three more observations are illustrated by figure 9. The first is that for the BPT phasor

diagrams, figures 9(a)-9(c), which show that l"b and _o are closer to being in-phase for Fo --_ 0

than for Fo = 100. This is due to the large heat transfer between the gas and the tube wall for

Fo _ 0. The condition Fo _ 0 represents isothermal wall conditions. The condition Fo = 100

represents near adiabatic conditions on the gas. That is, the condition Fo = 100 represents a very

thin-walled tube relative to the thermal penetration in the tube wall. This illustrates how heat

transfer is required for the BPT in order to obtain a favorable phase shift between temperature and

velocity.

The second interesting observation is shown for the OPT in figures 9(c)-9(e). The results are

for 0 L = 1 and CU = -0.1. The velocity phasor at z = 0 is U0, which lies along the real axis with

amplitude 1, and the velocity phasor at z - 1 is UL- The shaded area between Uo and UL represents

all velocity phasors between z = 0 and z -- 1, hence the OPT is more of a progressive wave device

because of the presence of phase-angle gradients.

The OPT shows that 7_b can be adjusted through the velocity boundary conditions so that it

is more nearly in-phase with t_O. This is most easily seen in figures 9(d) and 9(e) for Va - 30 and

Va - 100. Here Tb is more in-phase with _o when compared with the BPT. Also, it is apparent that

for the OPT the adiabatic conditions imposed on the gas by the tube wall result in the temperature

and velocity phasors becoming more in phase. This is most easily seen for Va = 100 and Fo - 100

where Tb is now in the shaded area representing the velocity phasors within the tube. For the

isothermal wall condition of Fo _ 0, Tb is pushed out of phase (out of the shaded region) with

velocity. This results in less enthalpy flow. Heat transfer, then, is not advantageous for an OPT.

Third, when there is significant heat transfer between the gas and tube wall, the pressure and

temperature phasors move out of phase relative to each other for both BPT and OPT. This is seen

by comparing/31 and 7"b for Fo --_ 0, with/31 and Tb for Fo = 100. For Fo = 100, /31 and _'b are

nearly in phase (Va = 30 and Va = 100), whereas for Fo _ 0, they are not in phase. Calculations

show that the phase shift can be about 20%. This is important because in simple I-D models it is

often assumed that there are adiabatic conditions on the gas and so there is a presumption that

temperature is always in phase with pressure. A discrepancy between experimental measurements

and model predictions may then arise. Most pulse tubes operate at Fo = O(1), which is closer to

isothermal wall conditions.

38
;)

<::;_

II

N

o

<_

o

o

i

0

<_>

o

H

<_

<-,. -, _ ,<> <: _,

o, _ <._ _, ,7,

o

i

II

7\

T

T

<_ \

,'4

0 0 (::' 0

|

<[...?

o

/

o

o

o

8

¢

]

_g

TT,

? ? ? ?

_-%_ _ [,,,,- -_II _= -_._=

<J

I

II

II

%

H

E_

=<

_<_

e,D :"-
Transverse Diffusion of Oscillating Velocity and Temperature

Velocity

Figure 10 shows uo(r) at z = 0 for Va = land UL = 0 at t= 0, 0.1, 0.2, 0.3, and 0.4. For

small Va, the profile is parabolic as in steady-state flow. Increasing Va to Va = 10 shows that the

velocity near the wall begins to lead the velocity at the centerline. Further increasing Va to 30 and to

100 shows that the velocity very near the wall substantially leads that of the centerline, and that the

velocity in the vicinity of the centerline begins to flatten and take on an inviscid profile, with the

velocity amplitude near the wall leading and overshooting the velocity at the centerline. Figure 11

shows vo(r). For small Va, v0 is in phase (with itself) at all r- locations. Increasing Va, however,

results in v0 near the wall leading _0 at the centerline region.

1,

0.5

1

0.5

uo

Va =1

t=O 1

0.5

t-0.2 , _ r

0'.2 0.4 _1

t=0.3 _ -o

t =0.4

Va=30

uo t=O 1

t=O.l __ 0.5

0.2 0.4 0. _---_i

t=_ -0.5

t= 0.4 -I

-i

Va =10

uo

t=O

o'.2 o.4' 0.6 _2_"1

t=0.3

t=0.4

Va =100

uo

t=O

t=O.1

t = 0.2

0.2 0.4 0.6 O.

t = 0.3

Figure 10. Oscillating axial velocity, uo (r).

40
Va =1 Va =10

o.11% _ I uO t=O

°°5r 0.o

_o.o5t ..... /

-o.o75_ _ -°.

_0.1t t=0.4 1 f

t

Va = 30 Va =100

IuO /..._-_'_.--0 o.i 0.075

0.05 0.025

-0.025

-0.05

-0.05

-0.i[ t _ -0.075_0.i

u o

t=O,l

Figure 11. Oscillating radial velocity, vO (r).

Temperature

Figure 12 shows the Tl(r) profile for PrVa = 0.7, 7, 21, and 70 with Fo _ 0 and Fo = 100.

For the isothermal wall condition of Fo ---) 0, the temperature profiles are similar to the velocity

profiles of figure 10, just scaled by the Prandtl number. For the near-adiabatic case ofFo = 100, in

all cases of PrVa, the temperature at the wall is seen to float. This is because the wall temperature

responds to the imposed oscillating gas temperature--the wall temperature is not fixed as it is for an

isothermal wall. The ability of the wall temperature to float for the adiabatic wall allows for the

temperature at the centerline region to be larger when compared with that of the isothermal wall.

This is apparent for PrVa = 0.7 and PrVa - 7. As Va increases, the centerline temperature for

Fo ---) 0 begins to approach that ofFo = 100 in both amplitude and phase. This is because diffusion

is now confined near the wall in the boundary layer. Figure 12 shows that the effect of the floating

wall temperature on gas temperature is primarily confined to the diffusion layer, and that an

adiabatic wall condition will have a greater relative effect on gas temperature amplitude for small

PrVa than for large PrVa.

41
0.06

0.04

0.02

0.4

0.2

-0.

-0.

T1

PrVa = 0.7

*--_Aa_q.2

0.2

.4 o. I

. ?"

0.4 0,6 0 _"_'_l_t L._V2- _.5

0.2

-0.I

.---.-"_'_.6 -0.2

PrV a =21

r1

t = 0.2 __r o. 4

0.2

0.2

• I=0.5 -0.2

_ -0,4

t= 0.6

PrV a =7

rl

.z.._.t_--0.2

0.2 0.4 o_0.5

0.6

r/

PrV a =70

t=0.2

t =0.4

0.2

t=0.5

t =0.6

Figure 12. Oscillating temperature, Tl(r ). Temperature profiles for Fo --->0 are pinned at r = 1,

and proflles for Fo = 100 (identified with '* ')float at r = 1.

Oscillating Heat Transfer and Oscillating Shear

Complex Nusselt number for oscillating heat transfer

The thermal diffusion and temperature phase shifts are now examined within the context of

the complex Nusselt number proposed by K. Lee (ref. 38). His work focused on BPT-type systems.

We now extend our analysis for OPT systems. The complex Nusselt number iqu is defined as the

heat flux at the wall divided by the difference in temperature between the tube wall and the bulk gas,

^ "

Iqu = qwe't (67)

(Zw-_'b)e it

= -Tl, r r=l (68)

1

Zl r=l - 2f rT"1 dr

0

where Ow = -]'l, rJr=l, 7"w = 7"1r=l' and i?b is defined by equation (66).

42
1

Figure 13 shows phasors for l_lu, Tb, AT= 7"w - 21 rT1 dr and Ow as functions of

0

PrVa = 0.7, 21, and 70, and Fo --_ 0 and Fo = 100 for the BPT and OPT, respectively. For

PrVa = 0.7 and Fo -_ 0, the above phasors are of very small amplitude, and are generally in phase

(or 180 ° out of phase) with D 0 (the phasor of D o is along the positive real axis and is of unit

amplitude). Steady-state heat transfer coefficients can be used under these conditions. Continuing

with Fo --_ 0, as Va increases to PrVa = 21 and beyond, phase angles also increase with Ow lagging

D O. At Va = 70, _'u = -0.397, and for Va > 70, O_ u -- -0.38 (not shown), which corresponds

well with K. Lee's value of O,Vu = -0.375 for rectangular geometry in the limit of large PrVa.

The effect of Fo shown in figure 13 is to shift the phase of qw ahead so that now qw leads

D O. A curious observation is that Nu is independent of Fo. This is proven mathematically in

appendix F. A scaling of equation (68) shows that l_u ~ _ Va. The thermal penetration in the gas

is only affected by _ Va, whereas Fo only affects the tube-wall temperature boundary condition.

Figure 14 shows plots for the same phasors as shown in figure 13 with the same values of

PrVa and Fo for an OPT with D L = 1.0 and CU = -0.1, where D L = D L e iov . Comparing figure 14

with figure 13 for corresponding PrVa and Fo shows that all phasors are shifted forward, except for

Nu, which remains with the same relative phase angle. That is, l_u is independent of D L. This is

shown mathematically in appendix F. Thus i_lu is only a function of PrVa and is independent of Fo

and D L. F Nu versus PrVa is plotted in figure 15; the figure can be used for either BPT or OPT

systems, independent ofFo, D L, and OU.

The complex Nusselt number may be used for one-dimensional linear oscillating flow in a

tube to correct for radial heat transfer,

i1"os c = i _l + l_u(PrVa)l"osc (69)

where 7"osc is the local oscillating temperature and Nu = ,4 e i¢.

Complex wall shear factor for oscillating shear

A similar relation can also be defined for oscillating shear and oscillating bulk velocity.

Gedeon termed this the "complex wall shear factor," F (ref. 66). It is defmed as

# =

fib

where

1

_2w= UO, r r=l and fib = 2[rfio, r dr

0

(70)

(71)

43
>

II

>,

II

C3

1

+

"4

o

I

_ / ",4

•

! I

o!

j'

/

/

z

/ +

/

/

• N /,_2 -

I _ /

..;/ i

/ ",4

// c.

",4

8_

4-

"4

- I

!1 <_ N

II

",4

[/ "4

0

_ < °

+

N

o

r,

/

/

+ /

',4 /

I

I

I c,4

c_

N

c_

Ce_

N

c_

+ '_

N

I

I

I

+ /

N

,_ I

/

I

",4

p

s

"4

+

N

_ I

,Z cp

I

°,,_,
II

It

L,.

\ +

"N + |

e..i "' e.4

"\_ .

\

e

so e

/

• ° N

oS I

I ,"

/

,os

,i

/

I N \ ,z _.

._. ,4 \ ,4

g')

o*

oI

ss ._

o° _

oe "_ I

i "NI

ot

,e

s•

oo

<I._ 1

",4

-IN

+

"4

i"

/

,o

oe "_.

o* • N

/

/

s

,,o

N

O

e,l •

I I oe

I N /"

oo

I ,

I ot

N

"4-

",4

•,,,. °

÷ +

N N

°

oI

/

o°° .,.I "_

/ N

/

#i

x

: \

+

"4

| oe

<f... Xl /

_ ,,

/

oo

1o

o o

<Z _ "4

I

r,

I

II

II

-4

g;

,a,

%

Itl
These relations are of the same form as the complex Nusselt number, and so figure 15 may be used

in a similar manner with the scaling length characterized by Va. The complex wall-shear factor may

be used as a lumped-parameter approximation of shear for one-dimensional linear oscillating flow,

ifiosc = -P2,z + [:(Va) Uosc (72)

where fiosc is the area-averaged oscillating velocity and _: = A e i4).

25!

20

15

i0

5

-0.38

-0.4

-0.42

-0.44

-0.46

-0.48

5 i0 15 20 25 30

._ or

5 10 15 20 25 30

Figure 15. Amplitude and phase of the complex Nusselt number, lqu(PrVa) = A e i¢, or complex wall

shear factor, F(Va) = A e i4_.

Mean-Steady Secondary Flow

The mean-steady flows are composed of the time-averaged product of two oscillating

quantities. The first is velocity; the second may be density for mass flux streaming, temperature for

enthalpy streaming, pressure for work streaming, entropy for heat streaming, or velocity itself for

momentum streaming. In all cases, each oscillating quantity depends on the axial velocity amplitude,

as seen in table 5. Thus, all mean-steady fluxes have a quadratic dependence on axial velocity.

Details for the mean-steady secondary flow solutions for To, z = 0 are found in appendix F. The

results are summarized here.

Eulerian Velocity, Ul(r,z )

The Eulerian mean-steady velocity results from two nonlinear flow components. These

components are the Reynolds stresses fil and the nonlinear product of the oscillating first-order

density and leading-order velocity, pluo . Although the two components are not themselves

independently measured, they do constitute separate components of the mean-steady Eulerian

velocity U--/(r, z),which is measured. U(x,t) is defined as the observed velocity component of the

mass flux vector j where j(x, t) = pu; hence,

46
U(x,t) andj areexpandablein e,

U(x,t) U 0 + eUl + O(e 2)

=pouo+ (pou, +pl,,o)+ )

Substituting equations (74) and (75) into equation (73) and equating like orders in e results in

(74)

(75)

U 0 = u 0 and U 1 = u 1 + plu----& (76)

Po

where PO = 1 for the strong-temperature case.

the radial component Vl

and

Ul(r,z ) is composed of the axial component Ul and

U--1= _1 + PlUO (77)

Vl = _1 + PlVO (78)

Figure 16 shows calculated mean-steady Eulerian velocities for a BPT with an isothermal

wall, Fo --->0. The left column shows the velocity field Ul(r,z ), and the right column plots gl(r)

and _l(r) at z = 0. The z-dependencies are such that fi-1 is linear in z, and Vl is constant in z. This

can be seen by examining equations (b) and (d) in table 5. For the viscous case in which Va = 1, the

mean- steady velocity components of order fi-1 = O(10 -2 ) and _1 = O(10 -3) due to the domi- are

nance of diffusion throughout the gas domain. Flow in the vicinity of the centerline (r = 0) moves

toward the closed end, and flow near the wall is toward the oscillating end. There is a radial

component of flow at z = 1, because no-slip at the tube ends is not enforced in the equations. The

which contains axial diffusion of the radial velocity, is of O('e 2)" and so is r-momentum equation,

not included in the O(e) mean-steady problem.

As Va increases to Va = 10, the mean-steady flux increases by the same order, and the

steady flows remain in the same directions. Upon further increasing to Va = 50, flow at the centerline decreases and flow near the wall increases. For Va < 50, the radial flow component (dashedline plot) is seen to be always outward toward the walls, that is, streaming flows down the center of

the tube toward the closed end and radially outward toward the tube wall, and then back to the

oscillating end along the tube wall. At Va = 60, the flow at the centerline decreases to the point

where the flow reverses itself and begins moving toward the oscillating end. At this point, a double

boundary layer develops. This is consistent with the analysis of Stuart (ref. 67). Stuart refers to the

diffusion layer near the wall as the "inner layer," and the transition layer between the inner layer and

the centerline as the "outer layer." His terminology is adopted here. For Va > 60, the development

of the double boundary layer requires streaming to now flow from the outer layer to either the inner

layer or to the centerline region, as shown by the negative and positive radial velocities in the plot

for Va = 100. The Va = 100 plot distinctly shows the inner and outer layers, and the centerline

region. For Va > 1O0 (not shown) the centerline and outer layer flows increase in magnitude, and the

inner layer flow remains relatively constant.

47
Va=l

Steady Velocity Fields Ul(r,z)

r=l

r=O

z=l

O. 015

0.01

0.005

-O.0O5

_l(r) at Z = 0. ; _l(r) - -

0.2 0.4

V a=lO

r=l

222222222:::::

z=O _ _ _ _ _ _ _

..... __EE

r=O

z=l

0.15

0.i

0.05

-0.0

V a =50

r=l

_ __ _ _ _ _ _ z=l

r=O

0.1

-0.

-0.

Va_O

r=t

...........!!

r=O

0.I

z=l

-0.

-0.

V a=lO0

r=l

r=O

0.2

0.I

-0.

-0.2

Figure 16. Effect of Va on the Eulerian mean-steady velociW for a BPT." e = 0.002.

48
Theresultsshownin figure 16correspondwell with theresultsof Grotberg(ref.3) for

mean-steadymassflux of oscillatingincompressibleflow in adivergingtube.In Grotberg's

analysis,thelargerendof thetubecorrespondsto loweramplitudeoscillations,whereasfor this

analysistheclosedendof thetubecorrespondsto loweramplitudeoscillations.Thepresentresults

for comparablee and Va are consistent with those of Grotberg in both direction and magnitude of

the component velocities, including the predicted double boundary layer.

Figure 17 presents the mean-steady velocity fields "Ul(r,z ), and velocity plots _l(r) at

z = 0, 0.5, and 1, respectively, for an OPT-type system (oscillating flow at both ends). Fixed

parameters are UL = 1, _U = -O. 25, and Fo ---) 0. The range of Va examined is from Va = 10 to

Va = 100. For Va = 10, the velocity enters the tube at the centerline from both ends, reverses flow

within the tube, and then exits in the diffusion layer near the wall. The plot for Va = 10 shows that

14"-/= O(10-1)which is ofthe same order as the previous BPT case for Va= 10. For Va< 10, fi-1

scales with Va as with the BPT. Increasing to Va = 30, the centerline velocity at z = 0 is seen to

decrease to zero, while the velocities at z = 0.5 and z = 1 decrease further in the negative direction.

For 30 < Va < 50, the centerline velocities at all three z-locations become negative, and a double

boundary layer develops at z = 0. Continued increase in Va results in decreasing the inner layer at

z = 0. At Va = 50, the remnants of the inner layer at z = 0 can be seen. Further increasing Va, the

centerline velocity for z = 0 and z = 0.5 continues to decrease and the diffusion layers near the wall

increase. At Va = 60, the centerline velocities have all converged at about Ul = -O. 7, and the inner

layer at z = 0 has almost disappeared. Further increasing to Va = 1O0 shows that the velocities for

z = 0, 0.5, and 1 have inverted, with the centerline velocity at z = 0 still decreasing while at z = 1 the

centerline velocity is now increasing, and the inner layer now completely disappears. For Va = 100

there is very strong streaming of the order of the oscillating velocity.

The results shown in figure 17 give a sense of how the second oscillating velocity (at z = 1)

of an OPT can lead to streaming patterns that are much more complex than those of a BPT. For

small Va < 10 streaming is small and it scales with Va. For large Va = 100, streaming is very strong

and is of the order of the oscillating velocity.

The effect of phase angle on mean- steady flow is given in figure 18. The calculations are

based on Va = 100, UL = 1, and Fo --) 0. The plots are for _l(r) at z = 0, 0.5, and 1. For

0 > ¢U > -0.25 there is only a single boundary layer at all three z- locations. 2 For ¢U < -0.30 at

z = 0 (not shown) a double boundary layer begins to develop. For OU < -0.40 at z = 1 a double

boundary layer develops with a direction opposite to that at z = 0. At ¢U = -0.50, the flow is

mirrored across z = 0.5, and at Ou = -0.55 the flow is mirrored to that of Cu = -0.45 (not shown).

2The case CU= 0 and UL = 1 is incompressible flow, hence mean-steady flow is zero since uo,z = 0, that is, the

Reynolds stresses are zero.

49
V a=10

Steady Velocity Fields U/(r, z)

r=l

z=O _ _ % _ _ _ z=l

r=O

V a =30

r=l

z:o _££_ ::i

r=O

V a =50

r=l

r=0

V a=60

r=l

iiii222_2 '""*::

r=O

V a =100

r=l

.......

r=O

_(r) atz=0--; z=0.5--; z= 1-.-

O.1

0.05

-0.0

-O.1

-0.15

-0.2

/

/

/

/

/

0.2 / \

o'.2 o._ i -'o_'6 _'*.E.__.._

/ /

I

-0.i t /"

-0.4 /

/

-0.6 ."

0.4

0.2

-0.

-0.4

-0.6

/

\

/

-__7_._. _

0.4

0.2

-0.

-0.4

-0.6

/ \

0.2 0.4/o46/ o.8

1 ./

0.5

0.25

-0.25

-0.5

-0.75

-i

P,

"_.\

0.2 0.4 _i r

Figure 17. Effect of Va on steady velocity for OPT. e = 0.002, 0 L = 1, (_U= - 0.25.

50
@U = - 0.10

Steady Velocity Fields _l(r,z )

r=l

...... ___

r=O

@U = - 0.25

r=l

_ggggggfgg_ggff

r=O

@U = - 0.40

r=l

_.o_$$-._._:-:_-_-_-"_ _:_

_==-__=__== ==-==:--- : :: _ _

r=O

@U = - 0.50

r=l

•° 55q i i

r=0

SU = - 0.55

r=l

r=O

__(r) atz=0--; z=0.5--; z= 1-.-

0.2

-0.

-0.4

0.5

0.25

-0.2

-0.5

-0.75

-i

0.6

0.4

0.2

-0.

-0.4

-0.6

-0.8

0.4

0.2

-0.

-0.4,

0.6

0.4,

0.2

-0.

-0.4

02 O4, _._0:8

• ;I"/ /

0.2 0.4 _i r

/ ./

/'\

- --_._.. o.4/ .o._ /o.,

".,/ \.,.,

_" ./°

.... "_. ,.,'.,

_ 0.4,.. o.6-.og_,7 r

Figure 18. Effect of @u on the steady velocity for OPT." e = 0.002, Va = 100, (JL = 1.

51
Themean-steadyvelocityfield isacomplicatedfunctionof velocityphaseangle.In general,

for q_U < -0.40 the centerline and diffusion-layer regions flow in opposite directions. For

_0U < -0.40, flow reversals between the centerline and diffusion-layer regions become more

pronounced because radial flow components become stronger. For q)U = -0.25, streaming is

strongest, implying that at this phase angle, mean-steady flow quantities such as enthalpy flow, will

be largest. Later, the mean-steady enthalpy flow will be shown to be a maximum at q_U = -0.25.

Particle Velocity, _p

The mean-steady particle velocity, Up, is used to compute the particle path, or pathline. It is

the velocity in Lagrangian coordinates and it is different from the observed mean-steady Eulerian

velocity given by U--). Appendix B provides details on the formulation of _p, and appendix F

outlines its computation. For e << 1 the components of Up are

-ffp(r,z) = Ul + UO,rjO vod'c + UO,z_oUOdZ (79)

and

_p(r,z)= 71 + 1)O,r__ 1)od'c + l)O,z_ouOd'C (80)

Equations (79) and (80) are simply the steady velocity conversions between Eulerian and

Lagrangian coordinates. Interpreted physically, the oscillating components of velocity u0 push

particles across mean-steady streamlines U/, which results in an additional drift quantified by the

quadraticterm(_ouod'r'Vuo).

Figure 19 plots gp(r) and U--](r) at z = 0 for a BPT, with Va = 1, 10, and 100. The Eulerian

and Lagrangian velocities are seen to be very similar for both axial and radial velocities. The BPT is

a standing wave device, hence all local velocities generally are in-phase. The additional quadratic

terms of equations (79) and (80) each contain a time integral of velocity, which when integrated,

results in a 90 ° phase shift. The quadratic product thus becomes nearly zero since the product of two

phasors that are 90 ° out of phase is zero. (The mean-steady product of two standing wave phasors is

proportional to the cosine of their relative phase angle, according to eq. (7).)

Figure 20 shows the fields for U/(r,z) and gp(r,z) for an OPT with _rL = 1, OU = -0.1,

and Va = 1, 10, and 100. Also shown are plots for ul(r) and gp(r) at z = 0, and Vl(r ) and gp(r)

at z = 0. For Va = 1 and 10 there is a significant difference in the field plots between U 1 and gp.

For Va = 1, U--] is seen to flow toward the middle of the tube at the centerline and then reverse flow

near the walls, with u-1 = 0 near z = 0.45. The corresponding flow for gp shows a centerline flow

that is continuous from left to right, with a reverse flow near the walls. The plot shows that u1(r) is

significantly smaller than _p(r) for Va = 1 and 10, and the radial flows vl(r) and _p(r) are nearly

equal. For Va = 100, U 1 and gp are nearly the same.

52
axial velocities radial velocities

0.015

0.01

O. 005

-0.00

Va =1

0.002

0.0015

0.001

0.000!

0.2 0.4 0.6 0.8 1

0.15

0.i

0.05

-0.0

0.2

0.I

Va =10

0.02

0.015

0.01

r 0.005

Va = 100

0.02

0.01

r

-0.0

-0.02

-0,03

0.2 0.4 0.6 0.8 1

0.8 1

Figure 19. Comparison of Eulerian and Lagrangian particle velocities for the BPT at z = 0 and with

Fo --9 0." "l.l,(r) --; _p(r) - -.

53
II

t_

1

I

I

I

I

==

I

co

/J

/

I

/ 1,0

J

/ o

/

i

I

/

j o

/

/ /

I o

I

I

/

J

,=,

>

i

o= _ _= g

o o o o

o o o o

N

'"""qil

i!!!i!::::ill

"_llil

'"'"'tlilil

'""_"tOliil _:_

II

N

_ttttt_.t

_ttt_tttttlf

,_._,tqt4441t t

......_....... l.....,ttt1111

,,..,.__.. I....,ttttltTT

N N

ttt.

fit.

tth_

tth,

ttt,,

ttt_

ttt_

it1,

#tt_

Hi,

ttt,

i,t,,

i_th

I_th

_D

i

N

ttt,.,

'*illllii Jttt,'t_Hilli

'*_|lllll ittt,'"_llllll

"_lillii I_tt,,'_liilil

'*_lillii Ittt,i'_liilil

'tllllJiJ _ Jttt,.,_6111111

tT _t "_lillil J,tt,.._lllll

*_'_liilli /,tt,,,,_lllll

'*llllil i,tt,. _1111[1 /iltt] _"_liliil _th,, J

I

II

II

II

c_

:,,,.,.
In general, for Va of about 1, the axial Eulerian and particle velocities have similar profiles;

however, the Eulerian velocities are much more restrained. As Va increases to 10, the axial velocity

profiles are no longer similar, being quite different in both direction and magnitude. For Va of 100,

the fields become nearly identical in both speed and direction. The results reflect the large influence

of the progressive wave component contained in the additional quadratic term of the particle

velocity.

Effect of Heat Transfer on Particle Velocity

Figure 21 illustrates how heat transfer between the gas and the tube wall affects the meansteady particle velocity for the BPT. The figure compares _p at Fo --->0 and Fo = 100 for

PrVa = 0.7, 7, and 70. For PrVa = 0.7, the axial velocity _p for isothermal wall conditions (Fo --->0)

is nearly the same as for the near adiabatic case (Fo = 100). Upon increasing to PrVa = 7, the

adiabatic condition increases the axial velocity until it is about 30% greater than for the isothermal

condition. For PrVa = 70, the adiabatic condition now reduces the axial velocity to about 50% less

than that for the isothermal condition at the centerline; they are about equal in the outer layer; and

the adiabatic case is greater by about 30% in the inner layer. For the radial velocity Up, the

difference between the adiabatic and isothermal conditions increases with increasing PrVa.

Figure 22 illustrates the affect of heat transfer for the OPT on particle velocity and plots the

same conditions for Fo and PrVa as for the BPT of figure 21. In general, Fo and PrVa have the same

affect on the OPT velocities as for the BPT. One exception is that for PrVa = 70 the double boundary layer is not present in the OPT so that the streaming in the diffusion layer near the tube wall is in

the opposite direction compared with the inner layer of the BPT. Also for the OPT at PrVa - 70, the

streaming in the diffusion layer near the tube wall for adiabatic conditions is less than it is for

isothermal conditions, whereas for the BPT, the streaming in the inner layer is greater for the

adiabatic conditions than it is for the isothermal conditions.

The mean-steady velocity reduces pulse tube performance because it directly transports gas

from the hot end to the cold end, and vice versa, thereby destroying the temperature gradient. The

mean-steady velocity is rather a complicated function of Fo and its effect on velocities is not easily

determined by simple examination of the solutions. The benefit of figures 21 and 22 is to give a

general understanding of how PrVa and Fo influence mass streaming for the BPT and OPT.

For the BPT, the difference in viscous and thermal diffusion lengths provides the phaseshifting mechanism between velocity and temperature to produce enthalpy flow. In this case it is

best that the PrVa be sized so that diffusion fills the entire gas domain and that the tube wall be sized

for Fo --->0 (isothermal wall condition). A good value of PrVa is about 7, as indicated in figure 21.

These conditions allow diffusion over the gas domain while at the same time reducing mean-steady

streaming (relative to the adiabatic wall condition).

55
II

i

i

II

==

I

C_

?

C

r_

I

I

C3

O

I

C_

1"

t_

E

@

II

C_

r.l.

==

@

C_

I=

/

//

u_

?,

¥YFFPL_bk&&&&&

l_VFP.*_ekkk_li8

VWfpp_._LLltlt

ttff qlt_F_.bk6 II11

..,_,L.HH u_

,,,,,_.,,ttttt

.,,,ttttt

,,,,,,,,_tttlt

,,o,,,,,_1111T

N

N

4

TV_yF,,q_LLtt_

11111

,.,,,...L,Itttt _

,,,,,.,,,_ttt?!

N

>

ii//

_ d

V

..... ? ?

,_ I, iii

iii I

11'/

= _ o c; o

n

¢.

N

Ft_VSk&

,,,,,_L,ttttt _

,.L_tttt

,,,,,.,_tttll

,,,,,.,,_ttttl

N

N

PFPA4*_bbLLi _p_&&LmL_tt_

_FFPA_bLLkil _t_&iLl_5111

t_pP_vl_kiki4 t_ i&ii.'_ 519_9

vv.t_A "_Ittt t _L¢i_,_ttl t

,,.,,..,,mttt i._t.,.,,_[

I,,,,,.,,ttttt ....

ht_Tlt*'_t /1/

I,,,,,,,,_1111 h,,t!t,,,_[,,, ' _'"

I,,,,,.,,,t1111I' ,Ht,,, '***

el bl

N

I II I I I .

lF_LLL&L_-1511

_ _AkLL&&if_q 5 I_

t_LkLlii_5_l

i,"!!'"",,

!:_ttt,,.,,,',

I_l,_t|h,,,,t II '

._l|h,.,.,

Hh";

H

N

e_

t¢'5
r_

I

I

II

r.z.

I

$

0

L,T.

i

/

co

o

o

c_

_ c3 c_

c_

I

I

C,

r_.

==

I

$

0

r_

/

,,,.._tttttt

....tttttt

,,,,.ttttt

8 ,,.,.tttt

,,.,.tttt

= ,,,,,*tttJt

_ ,,.,.tttt

_ ,,,,,.tilt

,,...,tlttl

,,,,,,ttttt

..,.tttt

,,.,.tttl

=_" ,,.,ottttt

N

,.,.._tttttlt

,,,..tttttll

,,..,ttttl

,,.,.ttttl

,,.,.ttttl

,,.,.ttttl

,,..tttttl

,,,,,,otttt!

_ ..,.tttt!

..,,tttttl

,,,,._ttJt _

,,,,,,_tttl'

,,,,,,.tfl

,,.ottttt'

,,,,,,.fit'

N

c_

tl ii

//_.s **s s SS_ __i

o c_ o

o

o c3 c_ ¢_

u_

"1!

/

Q

5, /

/

IIII

I

!

o \

????

N _ N

lith _NIIi|_ ITh *_lllill

!.tit....,,, ,,,..,,._,, tth' "illliI _tt_,,°_IilIII

,,,,,,,,,,.,,,,,,,,.,,,,,,,I,,,,:;:I"**_"""..... dh,,;: 'l!

,,,0.m..0 "_ i,,,,""""" _ I_tt*,, °*l_I|[] _ _th,,, °ill

,,.,m_tttttt ,,,,,mtttttt Idh'"_l_]][[ -- ill

I,tt,'_'*_llll ,ttt,'"""'_li

..,.,,,,,,,,tt ..,,,,.tttlttt I,,....ttttlt ....,,it,iti.,,,iitt dth, t ,ttt,..,,,, **

,.,,,m_I1111I,,..,,.t1111T,tt,.i *ill ,ttt,,,*_li

N N N N

I

li

II

II

r_

;=

.L'

> > >
For the OPT, the velocity boundary conditions at the tube ends provide the phase-shifting

mechanism. Diffusion is not required and, in fact, diffusion will reduce enthalpy flow by reducing

the velocity and temperature amplitudes near the tube wall. Operation at large PrVa and large Fo

(adiabatic wall conditions) is desirable because the diffusion layer now is confined to a thin layer

near the tube wall. From figure 22 for PrVa = 70, the mean-steady axial velocity plot shows that for

Fo = 100, the steady axial particle velocity is reduced relative to Fo --->0. The combination of large

PrVa and large Fo confines the diffusion layer to be thin while at the same time it reduces meansteady streaming (relative to the isothermal wall condition).

Mean-Steady Temperature, T2

The mean- steady temperature T2 is found by a double quadrature of equation (m) in table 3

with To, z = O. In conservative form it is

l (rT2'r),r=(-_l+-_),z+l(_lr+pll)or)r (81)

Pr Va r r - ,

which shows that T 2 is a result of the axial and radial work flows. That is, gradients in conductive

heat transfer are a result of gradients in work flow.

Figure 23 shows T2(r ) for the BPT and OPT with Fo --_ 0 and Fo = 100, and PrVa = 0.7, 7,

and 70. The mean-steady temperature is constant in z. The plots show that for PrVa = 0.7 and 7, heat

is continuously being transferred to the wall. For PrVa = 70, heat is also being transferred to the

centerline region. Physically, this would result in the centerline region heating. However, for the

present thermally strong approximation ( To, z = 0), there is an "imposed" thermal sink that

maintains T o constant.

D

Axial Enthalpy Flow, H 1

The first-order steady enthalpy flux is given by hj = ToPo_ 1 + ToPlU---o+ Po Tlu---o,which after

rearrangement becomes

-hl = TO(POUl + _luO) + PO_lUO (82)

When integrated over the cross-sectional area, equation (82) is the enthalpy flow. It is important

because it quantifies cooling,

1

HI = I[To(Pog 1 + PlUO ) + PO_lUo]rdr (83)

0

From the zero-net-mass-flow constraint (eq. (48)) and since To(z ) is independent of r, the first term

in equation (83) is zero. Recalling that Po = 1, the enthalpy flow becomes

1

Hl = _lUO rdr (84)

o

58
0.0012

0.001

0.0008

0.0006

0.0004

0.0002

BPT

PrVa = 0.7

o = I00

Fo_O

0.2 0.4 0.6 0.8 1

0.0005

0.0004

0.0003

0.0002

0.0001

OPT, U L = 1 and $u =-0.1

_o = I00

Fo_O _ r

0.2 0.4 0.6 0.8 1

0.08

0.06

O.04

0.02

0.35

0.3

0.25

0.2

0.15

0.i

0.05

PrVa =7

_00 0.03

0.025

0.02

0.015

0.01

0.005

r

0.2 0.4 0.6 0.8 1

PrVa =70

0.12

0.I

0.08

0.06

0.04[

0.02 ,'>--..\ r

0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 l

r

0.2 0.4 0.6 0.8 1

Figure 23. Comparison of T2(r ) for BPT and OPT at z = 0 and z = l forFo --+ 0 and Fo = 100.

which shows that the mean-steady enthalpy flow at O(e) is due to the time-averaged product of the

oscillating temperature and the oscillating velocity.

Using the equation of state, equation (84) can be rewritten in terms of the work flow pu,

1

H--1 = _(U 1 + PlUo)rdr (85)

0

59
which showsthattheenthalpyhastwowork flow components:theproductof theoscillating

pressureandoscillatingvelocity,andwork flow dueto mean-steadystreaming,po_l, where Po - 1.

Figure 24 shows enthalpy transport for the BPT. Shown are enthalpy flux fields -hl(r, z) for

PrVa = 0.7, 7, and 70 and Fo _ 0 and Fo = 100. Corresponding plots for hl(r) are for z = 0 and

z = 0.5, with h-/= 0 at z = 1. For the isothermal wall condition ofFo --_ 0, axial enthalpy flux

decreases from z - 0 to z = 1.

For Fo = 100, there is a reversal of enthalpy flux in the vicinity of the wall. The enthalpy

reversal results from the inability of radial temperature gradients to be generated because of the

near-adiabatic wall conditions for Fo - 100, that is, sufficient heat cannot be transferred to the wall.

Operating a BPT with high Fo is undesirable because of the large enthalpy flow reversal near the

tube wall.

Figure 25 shows the corresponding enthalpy transport for the OPT with UL = 1 and

CU = -0.1. The plots of "hl(r) are shown for z = 0, 0.5, and 1. The OPT allows velocity oscillations

at z = 1, which enable enthalpy to flow out of the tube, whereas in the BPT enthalpy flow goes to

zero at z = 1 since the velocity goes to zero. For the case ofPrVa = 0.7 and Fo --_ 0, enthalpy flows

in the reverse direction from z -- 1 to about z = 0.6, showing that an OPT operating with isothermal

walls and small PrVa is not desirable.

In general, a BPT should be operated with the thermal diffusion length and tube radius sized

so that PrVa = l0 and the tube wall is near isothermal, Fo --_ 0. The Fo _ 0 condition allows good

heat transfer between the gas and wall so that the phase angles between the velocity and temperature

are advantageous for enthalpy flow. The PrVa -- 10 condition sizes the gas domain so that all of the

gas is efficiently transporting enthalpy. Figure 24 illustrates the enthalpy flux for these conditions.

The OPT should be operated with the thermal diffusion region confined to a thin layer near

the tube wall and the tube wall near adiabatic. These conditions correspond to large PrVa and large

Fo. For the OPT, diffusion is no longer necessary in order to supply the correct phase angle between

velocity and temperature. The phase angle is supplied by the velocity boundary conditions at the

tube ends. An example of the enthalpy flux for these conditions is shown in figure 25 for PrVa- 70

and Fo - 100.

In figure 26, enthalpy flow ff11(z ) is plotted for the BPT and OPT with UL = 1 and

OU = -0.1 for PrVa = 0.7, 7, and 70; and for Fo ---) 0 and Fo = 100. For the BPT (first column),

adiabatic wall conditions (Fo = 100, dashed line) reduce enthalpy flow, particularly for large PrVa.

For the OPT (second column), enthalpy flow is greater and more constant for adiabatic wall

conditions. This is very apparent for large values of PrVa.

The plots of figure 26 reiterate the previous assertion regarding heat transfer to the tube wall.

For the BPT, heat transfer between the gas and the tube wall is necessary and desirable to allow for

the proper phase angle between velocity and temperature, whereas for the OPT, heat transfer is

undesirable. For the OPT, the phase shift is supplied by the velocity boundary conditions at the tube

ends.

60
II

0

t

t

It

t_

C_

It

r_

c_

I I

o

iI

/ /

._o o

i,

I

I

II

It

,;.,,)

c_

I I c_

o

/

• o • o

0 N

'q_qp_b_bkkkkk6A

_k_kkbkklAlAl

II

0 kbbkl&klll&&i&

T_ kkkilkkil6161

$

._.,.,tttttt

_- ,_,,,.ttttttt

.,.,,tttttttt

,,,,,,ttttttll

=_ .,,,,_ttttTlTT

_ ,.,,,_ttttTTTT

_.bbbbbkkkkk6&

_kkbkkkkk&LS4_

!_kIILIIILIlIII

,_l,kLlkl&llllt_

...,_tttttt

.,,,.ttttttt

.,,._tttttlt

.,,,,ttttttTl

.,,.,,tttt?ITT

..,.,,ttttTTTi

ooooo

//

/, :

i /

o

bbbkkkkkkkk&&L l;_,bbkkkl&&|

kbkkkkkl&LiA && 1; F wbkkk&k&_&i

kkkkkkkki&kll6 l_P_kki&l&ii6

,I , I ,

o o o o

bbbk_bkbkkkk&i

bbkktbkkkkkkii

kLbkkkkkLilLil

kLkbkkkikllil_

,,,,,,tttttttl l,,,,_,,_ttttlt l_ll_ttttttt

,,,,,_ttttttlf I.,,,,,,ttttJl I,_tllttttttttt

.,,,_tttttflT I.,,,,,,_tttT11I,_t1111ttttttt

,,,,,ttttffTlT I,....,,,tttlTTT I,_11111ttttttt

N

ffP_bVbbkk4

ff/_ikL|i|i|ll

_1_ttttttt

,_ttttttt

,_tttttttt

,,_J|tttttt

,,_ll|tttttt

,_|lJlttttt
I

'z"

I

++,

"l"

?,

3_

I

r

I

"l"

II

_E

2

m_

II

c+

1.

O

L:-,

..m

+E

•" ,_

/ I' /,'/

I+ i .:.,

c_ c:+

,,..i

\

<:; "\

\

, <_ 't

I '

i

o. . _. o.o _ g,

7,

N

,.++ttttttt|tl

..,+tttttltttt

,.0..ttttttt

,.,,tttttttttt

,.,mtttttttt <=+++

,,,,tttttttttl"_

.,,mttttttlt

..,,mttttttt

..,mttttttIt

..,,mtlttttt

..,mmttttt

..,,tttttttttT

i '//,J

Ii +

///

[,1_ . ,

°

iii !I

// , l _

?,

N

|_,Ll+llblll tilt I

_lLl_illltlt+t

....++_ttttttt

N

..,,ttt_ttttttt ,,ttttt_+ttttttt

I.,,+ttttttttlt ,+tttltttttttt

t..++tttttttttt ,+tttttttttttt

I..,,tttttttttt ,ttttttttttttt

I.,++.tttttltt ,ttttttttttttt

I,,+mtttttttl ,+fllttttttttt

I..++tttttttttt ,ttttttttttttt

,,,+ttttttttt_ ,tt' 'ttttttt

,,+vtttttttlt_ ,it' ttttttt_

.,,tttttttttt ,it lttttttt

,,,tttttttttt ,tt ttttlttt

.,,ttttttlTTT ,it !ttttttt

.,mtttttttl ,tt |ttltttt

.,++tttttltll ,tt tttttttt

,,+vtttttTTTt.+t tttttttt

ttttt[tttlttt

gtt111ttttttt

gtltlttttlttt

ttltltttltltt

rtltltttttttt

gilt11111111

glTll_tttttlt

trill+ttttttt,_ trillPTIIIIII

ttl 11 ,Itttti,II

I

H

II
BPT OPT, (]L = 1 and _U = -0.1

0.005

0.004

0.003

0.002

0.001

0.04

0.03

0.02

0.01

0.06

0.05

0.04

0.03

O. 02

0.01

_Va=0.7

0.2 0.4 0.6 0.8 1

PrVa =7

0.2 0.4 0.6 0.8 1

0.008

0.006

0.004

0.002

0.04

0.035

0.03

0 •025

0.02

0.015

PrVa =70

0.145

0.14

0.135

0.13

0.125

0.ii

Z

0.2 0.4 0.6 0.8 1

0.2 0.4 0.6_

012 04 0.6

Figure 26. The effect of heat transfer on enthatpyflow, Hl(z), for the BPT and OPT for Fo _ 0 --,"

Fo = 100--.

The amplitude-squared dependence of the mean-steady quantities on leading-order velocity

was previously mentioned at the beginning of this section. Although it has not been investigated

here, Storch et al. found refrigeration to depend on the square of the pressure ratio, where the pressure ratio is defined as the maximum pressure divided by the minimum pressure (ref. 35). This is

consistent with the results of the present study since, for a given frequency, the oscillating pressure

depends on fluid displacement at the tube ends, which is the integrated velocity at leading order.

63
In figure 27, the area-normalized enthalpy flux versus OU is plotted for the conditions of

e= 0.1, Va = 250, Pr = 0.7, 0 L = 1.0, Fo = 20, and I/= 5/3. The figure shows a peak in the areaaveraged enthalpy flux at CU = -0.25. This is reasonable considering that maximum mean-steady

velocity streaming was seen in figure 18 to be at ¢U = -0.25. Kasuya et al. (ref. 68) measured

optimum phase angles, and Radebaugh reported that for a typical OPT, OU -- -0.1 (ref. 11). However, increasing the phase angle leads to increased velocities in the regenerator, which results in

larger regenerator losses. Thus OU = -0.1 would be relevant for use as a system optimum.

1.4

1.2

Enthalpy 1

Flux ^

(w/cm_') 0.8

0.6

0.4

0.2

0

........... _,.......... $........ .:........... _.......... _,........... -..........

........... $........ _............ ; ........... _........... _,........... -..........

" "'i

' ._! .... I .... I .... I .... I .... I ....

0.05 0.1 0.15 0.2 0.25 0.3 0.;

Figure 27. Area-normalized enthalpy flux versus velocity phase angle. Maximum occurs at

_U =- 0.25.

Discussion

The calculated leading-order quantities for pressure, temperature, velocity, and heat transfer,

the mean-steady velocity and enthalpy flux fields, and the mean-steady temperature give an

understanding of the transport mechanisms for pulse tubes.

The BPT is essentially a standing wave device, because there are no phase-angle gradients

along the tube length. Phase shifts between velocity and temperature--required for enthalpy

flow--are obtained through differences in the viscous and thermal diffusion lengths. The ratio is

quantified in the Prandtl number. A BPT is able to operate when Pr < 1; the lower the Prandtl

number the better. A Prandtl number of Pr = 1 results in zero enthalpy flow.

The BPT should be operated with an isothermal tube wall to enable a large heat transfer

between the gas and tube wall. This requires the Fourier number to be near zero, Fo _ 0. Operating

a BPT with a large value for Fo (adiabatic wall condition) would result in enthalpy flow reversals

(enthalpy flow from hot to cold ends) near the tube wall, a result of the inability of the work flow to

convert to heat flow. Calculations also show that an isothermal tube wall reduces mass streaming

relative to an adiabatic wall. Mass streaming has a negative affect on performance, because streaming directly transports hot gas to the cold end, and vice versa. This tends to destroy the axial

temperature gradient. The tube radius of the BPT should also be sized to the thermal diffusion length

64
sothat all of thegasis efficiently transportingenthalpy.Thecalculationsshowthatto accomplish

this, thePrandtlnumbertimestheValensinumbershouldbePrVa- 10fortheBPT.

TheOPTis moreof a progressivewavedeviceandsoit doesnotrely ondiffusion to obtain

theappropriatephaseanglesbetweenvelocityandtemperature.Phaseanglesareobtainedthrough

thevelocity boundaryconditions.TheOPTshouldbeoperatedwith thethermaldiffusionregion

confinedto athin layernearthetubewall. ThisconditionrequiresPrVaandFo to bothbelarge.The

calculatedplotsof mean-steadyvelocity showthatlargeForeducesmassstreamingrelativeto

Fo --, 0. However,largeVa tendsto increasemassstreaming.

Operatingat smallPrVaandsmallFois detrimentalto anOPTbecauseheattransferbetween

thegasandthetubewall (1) reducestheoscillatingtemperatureamplitudenearthetubewall, and

(2) createsunwantedphaseanglesbetweenvelocityandtemperature.Bothof theseeffectstendto

reduceenthalpyflow. Thereis apracticallimitationto havingbothPrVaandFo very large,for these

requirementsleadto a systemthatmustcontainhighpressureswith alargediameter,thin-walled

tube.Also, thecompressormustbelargeto drivealargersystemwith a largetubediameter.

An additionaladvantageof theOPTovertheBPTistheability to alsohaveindependent

controlof thevelocityamplitudes.Thelargevelocity amplitudeof theOPTatthehot endof pulse

tube(z = 1)allowsmuchmoreenthalpyflow relativeto theBPTwhosevelocity goesto zero.For

theBPT,becauseof thesteepvelocity gradientsalongthetube,enthalpyflow is continuouslybeing

convertedto heatflow alongthetubeat ahighrate.Theheatthenflowsbackto thecold endasheat

conduction.As aconsequence,only a smallamountof enthalpyflow arrivesnearthehotheat

exchangerforrejectionto theenvironment.FortheOPT,becausethevelocityis finite at thehotend

of thetube,moreenthalpyflow canarrivenearthehotheatexchanger.Largeramountsof enthalpy

flow canthenberejectedat thehotheatexchanger.

Heattransferbetweenthegasandthetubewall hasanimportanteffectonthepressureand

temperaturephasors.Whenthereis significantheattransferbetweenthegasandtubewall,

Fo = O(1), the pressure and temperature phasors move out of phase relative to each other for both

the BPT and OPT; calculations indicate this difference to be as much as 20 °. This is important,

because 1-D models often assume adiabatic conditions on the gas and so there is a presumption that

the temperature is always in phase with pressure. Most pulse tubes operate at Fo = O(1), which is

closer to isothermal wall conditions.

At the tube ends, the complex Nusselt number is found to be independent of Fo, of the

velocity amplitude ratio _rL, and of the velocity phase angle CU. When written in the form

Nu(PrVa) = A e i¢), f_ is about 4 for PrVa < 3 and is linear with PrVa for PrVa > 25. The phase

angle for PrVa < 0.5 is _ --_ -0.5 and for PrVa > 500, ¢ ---) -0.38. A similar relation for the

complex shear wall factor exists, using only Va as the independent parameter. The complex Nusselt

number and shear wall factor can be used for one-dimensional linear oscillating flow in a tube to

account for radial heat transfer or shear at the tube wall.

The axial velocity was found to be a complicated function of Va and OU. The mean-steady

velocity increases linearly with Va for Va < 10, and can be of O(1) for Va > 100. It is strongest

when ¢v = -0.25.

65
In general,anOPTshouldbeoperatedwith largePrVaand OU--"-0.25. This maximizes

mean-steadyenthalpyflow. However,sincemassstreamingis of thesamemechanismasthemeansteadyenthalpyflow, lossesowingto massstreamingandthedestructionof theaxialtemperature

gradientwill alsobemaximum.Understandingthetrade-offbetweenmean-steadyenthalpyflow,

massstreaming,andaxialtemperaturegradientrequiresasolutionto thecoupledzeroth-, ftrst-, and

second-order equations. That is left for future work.

66
5. EXPERIMENTAL MEASUREMENTS OF THE

MEAN-STEADY PARTICLE VELOCITY

A flow-visualization system was constructed to permit observation of particle flow patterns

and measurement of particle velocities. In this section, the observed patterns are presented and

interpreted within the framework of the anelastic solution.

Experimental System

A schematic of the experimental flow-visualization system is illustrated in figure 28. A clear

polycarbonate tube (38.9 cm long, 2.22 cm i.d., and 2.54 cm o.d.) is filled with air at 1 atm mean

pressure. Diaphragm compressors are attached to each end of the tube and are sealed from the

ambient. The compressors are each driven by separate stepper motors capable of 25,000 steps per

revolution. The compressor/motor assemblies can be independently controlled in order to adjust the

relative phase angle between them. Three 0.00- cm stainless steel wires are strung across the tube

diameter at the indicated positions, and a light oil is applied to the wires before operation. The

surface tension of the oil is sufficient to hold a thin film on the wires. During operation, when the

compressors are producing an oscillating flow, a short pulse of electrical current (-0.1 sec duration)

is applied to a wire. This quickly heats and vaporizes the oil from the wire. There is no combustion

of the oil-- only vaporization. The vaporized oil quickly cools and condenses into an oil fog or

"smoke" that stretches across the tube diameter. One can now observe the leading-order oscillating

flow and the secondary mean-steady flow of the smoke, which represent particle paths of the gas.1

A CCD video camera (30 frames/sec) is used to record the smoke flow patterns.

Figure 28 also shows the physical dimensions of the system. The volume displacement of

the two compressors is 13.5 cm 3 each, and the total system volume is 155.2 cm 3, resulting in

e = 0.0435. The volumes of the connections between the compressors and tubes are converted into

equivalent lengths by dividing the volumes by the connector cross-sectional area (3.88 cm2). The

range of speed of the compressors is 5 to 20 Hz. Smoke-wires are positioned at z = 11.0 cm,

20.4 cm, and 32.1 cm. The tube was oriented with the gravity force acting in the positive

z-direction. During operation, the smoke is generally neutrally buoyant.

When operating, the speed and the relative phase angle between the two compressors are

fixed. The compressors are then started and allowed to come up to the final operating speed, at

which time the video camera begins recording. After about 10 sec have elapsed (to ensure quasisteady flow), the smoke-wire is pulsed with electrical current, and the smoke is seen to immediately

leave the wire. Initially, when the smoke comes off the wire, it rises against the gravitational-force

vector because of buoyancy, that is, the smoke is warmer (less dense) than the surrounding air. This

condition exists for about 2 sec until the smoke temperature and the surrounding air temperature

equilibrate. The video camera records the particle paths of the smoke at 30 frames/sec, and records a

time tag on each frame. After the smoke dissipates, which takes from 20 to 80 sec, the system is shut

down and reset for the next run.

1 The "particles" tracked were more like "blobs," diffuse yet distinguishable.

67
stepper

motor

driver

diaphragm

compressor

(11.3 cm3 )

total system volume

(155.2 crn° )

F z=o

smoke

wire

smoke

wire

diaphragm smoke

compressor _ wire

(11.3cm 3) _. [

stepper _

motor L_

driver z=40.4 cm

-- z=11.0cm

- z=32.1 cm

Figure 28. Schematic of the smoke- wire flow-visualization experiment.

Coordinate positions of the smoke for each time were determined by digitizing each

individual video fi-ame. Particle velocities were determined from the coordinates and incremental

times. The measured particle velocities were then compared with predicted values.

Comparison with Theory

Several runs were conducted to compare the experimentally measured mean-steady axial

particle velocities with those predicted by equation (79). Table 6 summarizes the dimensionless

quantities investigated. The BPT configuration was tested for varying Va, and the OPT configuration was tested for varying Va and Ou. There was no independent variation of _]L, because the

velocity boundary conditions defined by UL should have no unusual effects on Ep(r, z) since

Ep(r,z) is simply linear along z. There was also no variation in Fo since Fo << 1 for any of the runs

68
(isothermal wall conditions). The following section presents and compares the experimental results

against calculations. The code for computing the solutions is given in appendix G, and the data are

tabulated in appendix H.

Table 6. Range of dimensionless numbers investigated

Run 1 2 3 4 5 6

Configuration BPT BPT OPT OPT OPT OPT

Evaluation Velocity Velocity Observation Observation Velocity Velocity

e 0.0434 0.0434 0.0434 0.0434 0.0434 0.0434

Va 34 103 137 137 68 68

M 1.66 x 10-3 5.06 x 10-3 6.7x 10-3 6.7x 10-3 3.14x 10 -3 3.14x 10 -3

_, 4.5 x 10-5 4.13x 10 -4 7.3x 10 -4 7.3x 10 -4 1.62 x 10 -4 1.62x 10 -4

UL 0 0 1 1 1 1

CU n/a n/a - 0.5 - 0.94 - 0.254 - 0.125

Basic Pulse Tube Configuration

Axial particle velocities _p(r, z) were measured near the centerline of a BPT configured

system for Va = 34 and Va = 103. The measured velocities were in the vicinity of the centerline

region.

Run 1: BPT, Va = 34

Figure 29 shows the mean-steady particle path and particle velocity of a smoke particle. This

is shown by plotting the axial particle velocity fi-p(r, z) at the specified r and z positions and comparing this with the calculated _p(r,z) given by equation (79). Figure 29(a) shows the calculated flow

field _p(r,z), figure 29(b) plots -ffp(r) and _p(r) at z = 0, and figure 29(c) compares the measured

69
valuesof _p(r,z) with those predicted by theory. The ordinate of figure 29(c) gives the measured

and calculated values for _p(r,z), and the corresponding r and z coordinates. The abscissa identifies

the corresponding time for each _p (r, z). Figure 29(c) shows that the calculated velocities are in

general agreement with the measured values in terms of speed and direction. Velocities are positive

for the particle coordinate range r ---0.2 to 0.24 and z = 0.67 to 0.72. As time progresses, the particle

moves toward z = 1, as shown by the flow field in figure 29(a). Since the particle is at r - 0, there is

Fig. (a) Steady velocity fields _p(r,z) Fig. (b) _p(r) -- at z = 0: _p(r) -'-

r=l

q

r=0

z=l O.

0.3!

0._

-0.

-0.2

-----''7.'--'--',--'--" ,'--'_',_'_ r

Fig. (c) Comparison of measured and calculated _p(r,z)

0.800 [

0.700 _ ............................... _ :'/IX" • X X'M'sl 'x_x"w u'_r _tt X XK ]IX X-XX _:Xl

F--KXXX X 31.-_tx x ,,-x It x xx

m i F

[--'--u_ (measured, 3pt moving avg.)

0.500 " r .....................................................

z

o,oo.............................................. ..........................

,,/Vk/_

0.300 ................ _ )_-_---_-_-/_-------1- -_,j_....................... i..__A, ......... ,,

o.=oo ;-_'---..="r ?...... L-':._:.___. ........ ._.. -..--...--- _-;--: .-._: _.'_ -:-:--'--" :---:-:-:-

ooo...................................... y 0.000 ." .......... .........

-o.1 O0 -_

elasped time (0.2 second increments)

Figure 29. Particle velocity for BPT, e = O.0435, Va = 34, UL = O: (a) particle velocity field,

_p(r,z)," (b) calculated component velocities p(r) and p(r) at z = O; (c) plot of measured

particle coordinates, and measured and calculated axial particle velocity _p( r, z) for corresponding

elapsed times.

70
noradialflow componentandsotheparticleremainsatr = 0. It is unclear why the measured

velocities decrease to zero for elapsed times greater than 35 see although it may be a result of

inaccurate particle tracking, since the smoke dissipates as time progresses.

Run 2: BPT, Va = 103

The particle flow field shown in figure 30(a) shows that for higher Va a double boundary

layer is formed. Figure 30(b) shows that the velocity in the inner layer (r = 0.8 to 1) is negative, then

reverses in the outer layer (r = 0.38 to 0.8) to positive, then again reverses itself in the centerline

region (r = 0 to 0.38) back to negative. A positive and negative radial flow maintains mass

conservation.

The measured _p(r,z), the calculated _p(r,z),. and r and z coordinates at the given time

intervals are plotted in figure 30(c). The measured negative velocities are in the centerline region

and are comparable to those predicted. The larger negative velocities measured at the initial points

are a result of the buoyancy forces present after the initial pulse. As time progresses, the temperature

equilibrates between the smoke and the air, and so the velocity levels off.

The presence of the inner and outer layers of the double boundary layer predicted by theory

could not be confirmed experimentally. This is because the smoke did not distribute itself in either

of these two layers after pulsing. The simplicity of these experiments did not allow for observation

of velocities throughout the entire field. Future work--perhaps with laser Doppler or anemometer

velocity measuring instruments--will allow for further validation.

Orifice Pulse Tube Configuration

Orifice pulse tube observations

Mean-steady secondary flow observations for an OPT configuration axe described for

Va = 137, 0 L = 1.0 with Cu = - 0.5 and Cu = - 0.94. The observed smoke flow is in qualitative

agreement with the predicted particle velocity fields.

Run 3: smoke flow observations for ¢U = -0.5

Figure 31 shows the observed steady flow for Va = 137, 0 L = 1, and ¢U = - 0.5.

Figures 3 l(a) and 31 (b) predict large negative radial velocities at r -- 0.4 and large axial velocities in

both the negative and positive directions symmetric about z = 0.5. This is verified from the smoke

observation data shown in figure 31(c) where the two smoke lines are seen to "stretch" in the axial

direction and "compress" together in the negative radial direction. This is predicted by the particle

velocity field of figure 31 (a), and is an indication of the presence of the outer layer and centerline

flow regions. Unfortunately, smoke did not distribute near the tube wall, so the presence of the inner

layer could not be directly confirmed.

71
Fig. (a) Steady velocity fields_p(r,z)

r=l 0.2

........ _MAMA_8_ ?iii r=O -0.1

-0.2

0,7000

Fig. (b) $p(r) -- atz = 0; _p(r) -'-

r

0.6000

0.5000

0.4000

0.3000

0.2000

0.1000

0.0000

-0.1000

-0.2000

-0.3000

Fig. (c) Comparison of measured and calculated _p(r,z)

......_.....-.....=......=....._....._......:....._.....-......_.....:_......=.....

4- 5;, (measured) L---j_ 1............................................................

L,z ! ...........................................

.......... _ .... _._ ............... __ ....... _ ................................................

elapsed time (0.2 second increments)

Figure 30. Particle velocity for BPT, e = 0.0435, Va = 103, UL = 0." (a) particle velocityfield,

gp(r,z); (b) calculated component velocities _p(r) and gp(r) at z = O; (c) plot of measured

particle coordinates, and measured andcalculated axial particle velocity gp (r, z) for corresponding

elapsed times.

72
Fig. (a)Steady velocity fields_p(r,z) Fig. (b) _p(r) atz = 0--;

_p(r) atz = 1-- ; _p(r) -or=l

r=0

0.4

i

= = = ....

-0.:

°O.i

Fig. (c) Time elapsed observed mean -steady particle flow

/ /---- smoke wire

smoke 0:53:15:01 • _ 0:53:21:01

,a

• 7 0:53:17:01 • ! I 0:53:23:01

• _ 0:53:19:01 • l 0:53:25:01

Figure 31. Particle velocity for OPT, e = 0.0434, Ira = 137, _]L = 1, OU = - 0.5: (a) particle velocity

field, _p(r,z); (b) calculated component velocities _p(r) at z = 0 and z = 1 and _p(r); (c) observed

smoke flow at indicated times. The observed flow is seen to stretch axially and compress radially as

predicted by the flow field of(a).

Though the smoke-wire is positioned at z -- 0.5, the observed smoke flow shows that at

t = 53:15:01, the smoke is skewed at a location z < 0.5. This is because of the buoyancy effect

immediately after the smoke is pulsed. Also, we would expect the flow to be symmetric about

z = 0.5, hence, we would not expect the smoke to cross the z = 0.5 plane. However, at t > 53:21:01,

smoke can be seen at locations z > 0.5, possibly because the smoke was of a slightly higher density

than the air, or because of the instability of a purely zero velocity plane at z = 0.5. This second

73
possibility would allow smoke to cross at z = 0.5, allowing the smoke to get caught in the positive

direction centerline flow fields for z > 0.5.

The smoke observations are seen to "curve" at the ends of the tube. This is because the

connections from the compressors to the visualization tube are at right angles. The curved flow is

generally restricted to z < 0.3 and z > 0.7. Flow measurements are taken within the range of

0.3 <z<0.7.

Run 4: smoke flow observations for Cu = - 0.94

Figure 32 shows the observed steady flow for Va = 137, 0 L = 1, and _l_U=- 0.94.

Figures 32(a) and 32(b) show strong positive velocities in the centerline region and negative

velocities in the viscous layer near the tube wall. Radial velocities are very small. Figure 32(c)

shows that the observed smoke flow qualitatively confrere the model. The observed flows are in

the centerline region and are seen to quickly flow in the positive direction with little radial displacement, except near the positive end (z = 1) where the curved flow due to end effects begins to

compress the pathlines.

Orifice pulse tube measurements

Mean-steady particle velocity measurements were taken near the centerline and within the

viscous layer near the tube wall for an OPT configuration for Va = 68, 0 L = 1 with CU = -0.254,

and _u = -0.125. Particles were observed flowing with the directional sense and speed as predicted

by equation (79). Circulating flow from the centerline region to the viscous layer near the tube wall

was observed; it also was in general agreement with predictions.

Run 5: particle velocity measurements for _U = - 0.254

Centerline and viscous layer flow near tube wall

Figure 33(a) shows the measured and predicted velocities for flow in the centerline region.

Calculations correspond well with measured values. The measured velocities at times greater than

18 time-increments (1.8 sec) are due to the curved flow end-effects as described in the observation

section. The inlets to the tubes from the compressors are at right angles to the tube, and so the

flow is curved at the tube ends. This results in a significant particle position change in r as z

approaches 0.3.

Figure 33(b) shows the five-point moving average of axial particle velocity in the viscous

layer near the tube wall. Although there is scatter in the data, the results are still in general

agreement with prediction.

74
Fig. (a) Steady velocity fields _p(r,z) Fig. (b) _p(r) atz = 0--;

_p(r) at Z = 1-- ; Vp(r) -'-

r=l

r=O

0.2

0.1

-0.1

-0.2

Oa r

Fig. (c) Time elapsed observed mean -steady particle flow

I_ _ 0:48:21:25 I_ _ 0:48:24:25

I_ _ 0:48:22:25 I_ P 0:48:25:25

I_ _ 0:48:23:25 I_ _ 0:48::26:2S

Figure 32. Particle velocity for OPT, e = 0.0435, Va = 137, UL = 1, qrO = - 0.94: (a) particle

velocityfield, _p(r,z)," (b) calculated component velocities _p(r) at z = 0 and z = I and _p(r);

(c) observed smoke flow at indicated times. The observed flow is seen to stretch axially in the

positive direction with little radial displacement, as predicted by the flow field of(a).

75
0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

.._,_----_ i,._--.----, i , ..... _, ..... , ---'--" 119 210 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1

i.-_-z

iii ._viiiiiiiiiiiiiii

elasped time (0.1 second increments)

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

--e- ffp (Eq. 4.2.7) 1

--_ rO'p (measured, 5pt moving avg.) .................................................

L''_'z 1

l I i I I ) I I I _ I I ] I I I I I I I I _ I I I I I

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2627 28

elasped time (0.1 second increments)

Figure 33. Particle velocities for OPT at indicated r and z positions for correspondin E elapsed

times.'E = 0.0435, Va = 58, 0 L = 1, #PU= -0.254,'(a) is measured and calculated axial velocities

Ep (r, Z) at centerline; (b) plot is measured and calculated axial velocities Ep (r, z) in the viscous

layer

76
Circulating flow between centerline region and viscous layer near tube wall

Figure 34 shows the flow for an OPT configuration with Va = 68, UL = 1, and Cu = - 0.254.

The flow field of figure 34(a) shows negative flow in the centerline region and positive flow near the

wall. Figure 34(b) predicts a radial component of flow that is significant between r 0.5 and r 0.8.

The radial velocity allows for flow reversal where fluid particles can move from the negative axial

flow region to the positive axial flow region.

Fig. (a)Steady velocity fields ip(r,z) Fig. (b) _p(r) at z = 0-- ;

"_p(r) at z = 1 -- ; _p(r)-'-

r=&

==o

_ _ ,m--, .up,,,-.o-,. ,inb_,i,-,. ,i,_ 4o=-,-,,ap,--,_

r--O

0.4

0.2

-0.

-0.4

-0._

ss_

s %

012 -'"o/ o.e o.a v x r

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

Fig. (c) Comparison of measuredand calculal_d "ffp(r,z)

_'p (Eq. 4.2.7) 1 1

•.-=- TO (measured, 5pt moving avg.)

_z....... ' ............. i........................................... !

elapsed time (0.1 second increments)

Figure 34. Particle velocity for OPT, e = 0.0435, Va = 68, _]L = 1, t_U = - 0.254: (a) particle

velocity field, _p(r,z); (b) calculated component velocities _p(r) at z = 0 and z = I and _p(r) ;

(c) plot of measured particle coordinates, and measured and calculated axial particle velocity

_p (r, z) for corresponding elapsed times.

77
Figure 34(c) plots the measured arid calculated velocities of equation (79). The data confirm

flow reversal between the negative-flow centerline region and the positive-flow viscous layer near

the tube wall. Particles, initially at r = 0.42 and z = 0.45 have negative velocities. As time progresses, the positive radial component of flow moves the particles outward toward the viscous layer

which has positive velocities. The particles enter the viscous layer region and so reverse from

negative to positive flow. Equation (79) predicts reasonably well both the magnitude and direction

of the particle velocities, and the location at which the particles flow from the negative to the

positive velocity regions.

Run 6: particle velocity measurements for @U = -0.125

Figure 35(a) shows the mean-steady particle velocity field and figure 35(b) shows a plot of

the axial and radial particle velocities for an OPT with Va = 68, UL = 1, and ¢U = - 0.125. The flow

field is negative in the centerline region and positive in the viscous layer near the tube wall, with

radial flow significant between r = 0.5 and r = 0.8. This is similar to the previous case in which

@u = --0.254. However, the magn!tude of the flows is about 30% less than for @U= --0.254. The plots

of figure 35 are the predicted fields for the data presented in figures 36 and 37.

Fig. (a) Steady velocity fields _p(r, z) Fig. (b) _p(r) at z = 0 -- ;

_p(r)atz=l -- ; _p(r)

r=l

5_-02 2 2 2 2 _:""

........

r=0

0.3 /_'_ %

0.2 --_1

0.1

0'2'"o'_"

-0. _/

-0.2 ,'/'

-0.3 "s/

-0.4

Figure 35. Particle velocity for OPT, e = 0.0435, Fa = 68, 0 L = 1, OU = - 0.125: (a) particle

velocityfield, Bp(r,z); (b)calculatedcomponent velocities _p(r) atz = 0 and z = I and _p(r)

Flow near centerline

Figure 36 shows the measured and predicted velocities for the centerline region. The data for

elapsed times greater than 15 time-increments are again due to the flow end-effects as described

previously. The measured results are in general agreement with prediction.

78
0.6

0.4 '-_ , - ............................................................

I"- ' easureO av'' ....ii ii-7

02 r i ....................................................

--,ll-- Z

! 1

-0.2 t

-0.8

elasped time (0.1 second increments)

Figure 36. Plots of measured and calculated axial particle velocity _p (r, z) at the indicated r and z

coordinates and given elapsed times for OPT in the centerline region, e = 0.0435, Va = 68, (-]L = 1,

¢_U = - O. 125.

Circulating flow between centerline region and viscous layer near tube wall

Figure 37 shows the measured axial velocity and position corresponding to the calculated

results of figure 35. Figure 37(a) shows a plot of the results for one-half of the radial domain and

figure 37Co) is a plot of the results for the other half. 2 Figure 37 shows flow reversal between the

centedine region and the viscous layer near the tube wall. The calculated velocity using equation (79) shows good prediction of axial particle speed and direction. There is also good prediction

in the transition region where flow goes from being negative to being positive. The transition from

negative to positive flow is not as steep as that for the case of #u = - 0.254, figure 34(c). This is due

to the much smaller radial velocity in the flow-reversal region between r = 0.5 to r = 0.6 of

figure 37.

2The smoke comes off the wire nearly axisymmetric within a single plane perpendicular to the camera view.

Subsequently, two "smoke blobs" can be tracked, one on either side of the centerline.

79
-9- __p (Eq. 4.2.7) I

o.s _up (measured, 5pt relying avg.) ................................................. .., _.

t

L-.N-z

0.6 ........................................................................... : ....................

02......................... L ilb ....

-0.2

-0.4

elasped time (0.1 second increments)

°"I"4_E p (Eq. 4.2.7) ! "

|

|/"L u'p (measured, 5pt moving avg.)

o.e .... r !....................... ....................................

0.4

-0.4

elasped time (0.1 second increments)

Figure 37. Plots of measured and calculated axial particle velocity Ep(r, z) at given elapsed times

for OPT, e = 0.0435, Va = 68, (JL = 1, and C_u = - 0.125: (a) is data from half the tube," (b) is data

from the other half of the tube, symmetric about r = O.

8O
Discussion

The measured and observed axial velocities of smoke particles were well predicted by the

particle velocities given by equation (79). These results provide confidence that the present theory is

valid and useful.

The measured direction and magnitude of the axial centerline velocities were predicted for a

BPT configuration in which the only variable parameter is Va. For Va = 34, the axial velocities in

the centerline region were positive. For Va = 103, the axial centerline velocities were negative,

which is in agreement with prediction. For Va = 103, a double boundary layer is predicted. Unfortunately, this could not be confirmed with the present system. Future work using more precise

methods of measuring velocities over the entire flow field should be conducted to further validate

the theory.

OPT operation adds UL and OU to Va as variable parameters. For the OPT experiments, UL

was set to 1, and Va and Ou were allowed to vary. OPT observations of the secondary streaming for

Va = 137 and CU = -0.5 were in good qualitative agreement with predictions. The smoke was

observed to stretch axially owing to opposite flow between the outer layer and centerline regions.

The smoke also was seen to compress radially because of a radial flow component between the outer

layer and centerline region. This stretching and compressing was predicted by theory. Observations

for a change in velocity phase angle, tPU= -0.94 while retaining Va = 137 showed strong positive

streaming in the centerline region with little radial flow; this was also predicted.

Measurements of the axial velocity were taken for an OPT configuration, and compared with

theory. Once again, the predictions were in good agreement with measurements. For Va = 68 and

CU = -0.254, smoke that began in the centerline region was observed to flow in the negative axial

direction and positive radial direction. Upon entering the viscous layer near the tube wall, the smoke

reversed itself and flowed in the positive axial direction. The speed and direction of this axial flow

were well predicted. Also predicted were the coordinates at which the flow reversal occurred. At the

point of flow reversal, a strong radial velocity was predicted and verified.

A change in the phase angle ¢U = -0.125 showed the same type of flow reversal as for

q_u = -0.254. In this case a weaker radial velocity at the point of flow reversal was observed, and the

axial speed and direction were again well predicted by equation (79).

In general, the measured secondary mean-steady particle velocities obtained from the flowvisualization experiments indicate that the present linear theory is applicable, and can be cautiously

extended to eva 1. Though this is beyond the eva << 1 constraint for linearization of the momentum equation, a case can be made--from the present flow-visualization experiments--that the theory

is useful for engineering calculations in the region eVa = O(1), and that the theory can be used to

help understand the transport mechanisms, particularly mass streaming and enthalpy flows, in

pulse tubes.

81

6. CONCLUDING REMARKS AND FUTURE WORK

The present study formulates a set of two-dimensional, axisymmetric differential equations

for describing mean-steady secondary flows generated by periodic compression and expansion of an

ideal gas in a pulse tube. An anelastic approximation of the fluid equations--mass and energy

conservation, and the equation of motion--is used to construct a consistent set of linear differential

equations amenable to a series-expansion solution in the small parameter e, where e is the inverse

Strouhal number and is the ratio of gas displacement length to tube length. The anelastic approach

applies when shock and acoustic energies are small compared with the energy needed to compress

and expand the gas. Other parameters resulting from the formulation are the Valensi number, Va,

relating system transverse length to viscous diffusion length; the Prandtl number, Pr; the Mach

number, M; the velocity amplitude ratio at the tube ends; and the velocity phase angle at the tube

ends. Additionally, heat transfer to a tube wall that has thin but finite thickness is considered, thus

introducing the tube wall Fourier number, Fo.

The elasticity parameter _ relates e and M and is used to order the pressure gradient in the

momentum equation. It is the product of Mach number and the ratio of the oscillation frequency to

the system acoustic resonance frequency, and is useful for identifying the distinguished limit

between e and M. A linear acoustic set of equations results when _ = O(1) and e << 1. This

corresponds to a distinguished limit of M = e 0'5. An anelastic set of equations results when

< e << 1. This corresponds to a distinguished limit of M < e. For pulse tubes, A << e << 1, which

is the limit used in this study. Additionally, a linear approximation of the momentum equation is

taken. This requires the added constraint eVa << 1.

The derived set of leading-order relations requires simultaneous solution of the zeroth-,

first-, and second-order set of equations. The zeroth-order temperature--which is the equilibrium

bulk temperature--is coupled to the zeroth-order equation of state, the first-order oscillating

advection of enthalpy, and the second-order mean-steady conversion of work flow to heat flow. The

full problem is nonlinear and requires solving 15 equations in 15 unknowns--an ambitious task left

for future work.

In determining the lower-order mechanisms for the mean-steady transport of momentum and

enthalpy, a solution to the equations is obtained for the strong temperature case To, z = 0. This

effectively decouples the three separate orders of equations, leaving for the basic state problem the

first-order oscillating equations, which are completely self-contained.

The oscillating solution is used to compute oscillating heat transfer and shear at the tube

wall. It is found that the complex Nusselt number, relating the ratio of conduction heat flux at the

wall to bulk temperature difference between the gas and tube wall, is independent of the Fourier

number and of the velocity boundary conditions at the tube ends. This also applies to the wall shear

factor, which is similarly defined as the ratio of momentum flux to bulk velocity difference. The

complex Nusselt number and shear wall factor are simply scaled by the Prandtl number. The

usefulness of these two relations is to correct for transverse heat transfer and wall shear when used

with a 1-D model.

83
The leading-order oscillating solution is taken to the next higher order to arrive at the meansteady solutions for the Eulerian and Lagrangian velocity fields, the enthalpy flow field, and the

mean-steady radial temperature profile. These are each examined as functions of Va, UL, q)U,

and Fo.

Plots of the velocity fields for the basic pulse tube (BPT) and orifice pulse tube (OPT)

configurations show complicated flow patterns. The flows are highly dependent on Va and Ou.

Examination of the higher-order mean-steady velocity equation shows that the Reynolds stress and

the quadratic product of oscillating density and oscillating velocity produce the steady secondary

streaming. The streaming depends on gradients of the velocity amplitude and phase angle. In

general, the leading-order solutions for all properties (temperature, pressure, velocity) depend on the

velocity amplitude; hence, for any mean-steady quantity the magnitude will depend on the square of

the velocity amplitude.

Plots of the enthalpy flux field show the effect of heat transfer with the tube wall. For a near

isothermal wall (Fo --> 0), gradients in the enthalpy flux result in mean-steady heat transfer to the

tube wall. This can be seen in the mean-steady enthalpy equation where the work flux is balanced

by heat flux. Under these isothermal wall conditions, the thermal and viscous diffusion layers are of

the same form, both pinned at the wall, and scaled only by the Prandtl number, which for most cases

is of O(1) but not equal to 1. This allows a unidirectional flow of enthalpy for the BPT, because the

difference in viscous and thermal diffusion lengths enables a favorable phase angle between velocity

and temperature. Alternatively, for a BPT with Pr = l, the phase angle between velocity and

temperature is 90 °, hence enthalpy flow is locally zero everywhere.

The OPT overcomes the necessity of heat transfer by obtaining appropriate phase angles

through the axial velocity boundary conditions. This allows favorable phase angles to be present

throughout the entire transverse domain, whereas for the BPT, phase angles are favorable only in the

diffusion layer. Also, the inclusion of a finite velocity at the hot end of the tube allows a finite

enthalpy flow at this end, whereas for the BPT the enthalpy flow goes to zero. The velocity

boundary conditions also allow a more constant enthalpy flux throughout the axial domain (when

compared with the BPT). Hence, there is less conversion of work flow into heat flow along the tube

length, and so more of the work flow is available for enthalpy flow to the hot heat exchanger for

rejection. (For the BPT, the steep decrease in work flow results in a large heat flow that is conducted

back to the cold end.)

The experimental smoke-wire flow-visualization experiments confirmed the predictions of

the present theory. The system was configured for BPT and OPT operation, with Va varied for both,

and _0Uvaried with UL = 1 for the OPT. The experiments were conducted in the range

1.5 < eVa < 6, which is not strictly within the constraint eVa << 1 required for linearization of the

momentum equation. However, for all cases examined, the calculated particle velocities

satisfactorily described the observed and measured experimental particle velocities, including flow

reversals between the centerline regions and diffusion layers, and the locations at which flow

reversal occurred.

The major points of this study can be summarized as follows.

1. Basic pulse tubes should be operated at Fo _ 0 and PrVa = 10. The first condition represents an

isothermal tube wall, thereby allowing good heat transfer between the gas and the tube wall, and

84
the second condition ensures that most of the gas is efficiently used to transport enthalpy.

Calculations also show that an isothermal wall reduces mass streaming relative to an adiabatic

wall for a BPT.

2. Orifice pulse tubes should be operated for large Fo and large PrVa. The first condition represents

an adiabatic tube, thereby reducing heat transfer between the gas and the tube wall, and the

second confines the diffusion layer to a thin layer near the wall. Calculations also show that an

adiabatic wall reduces mass streaming relative to an isothermal wall at large PrVa. However,

large Va tends to increase mass streaming. Maximum enthalpy flow for orifice pulse tubes was

shown to be at a phase angle of 0u -- -0.25.

3. The mean-steady velocity increases linearly with Va for Va < 10 and can be of O(1) for

Va > 100. It is strongest when 0U ----0.25.

4. The complex Nusselt number can beused for one-dimensional linear oscillating flow in a tube to

correct for radial heat transfer. It is of the form l_u(PrVa) = ,4 e i¢ and is independent of Fo,

velocity amplitude ratio /-)L, and velocity phase angle ¢U at the tube ends. A is about 4 for

PrVa < 3 and is linear with PrVa for PrVa > 25. The phase angle, for PrVa < 0.5 is ¢ _ -0.5,

and for PrVa > 500 is _ _ -0.38. A similar relation for the complex shear wall factor exists

using Va as the independent parameter.

5. Heat transfer between the gas and the tube wall can shift the phase between the pressure and

temperature phasors. This is important because 1-D models often assume adiabatic conditions on

the gas and so there is a presumption that temperature is always in phase with pressure. Most

pulse tubes operate at Fo = O(1), which is closer to isothermal wall conditions. Calculations

indicate the phase shift between pressure and temperature to be as much as 20 °.

6. The calculated particle velocities using the linear anelastic approach were supported by the

measured velocities from the flow-visualization experiment for 1.5 < eVa < 6, even though this

is not strictly within the linear constraint eVa << 1.

Though this first attempt to verify the theory has proved relatively successful, there is

considerable room for further validation and improvement. Possible areas for future study include

the following.

1. A numerical solution to the full coupled set of equations. This would yield the leading-order

equilibrium temperature TO, and allow a better understanding of how mass streaming affects the

TO temperature gradient, the trade-offs between enthalpy flow and heat flow, and the ability to

optimize pulse tube operation by calculating entropy generation due to temperature gradients.

2. Extending the theory to individual components, such as regenerators, heat exchangers, and

inertance tubes; and adding lumped-parameter boundary conditions to model an orifice and

reservoir. Once accomplished, a modular approach for constructing a pulse tube engineering

model from component level modules would be available.

3. A solution of the equations with different boundary conditions would yield some interesting

insights. For example, a solution for the case of a slowly varying tube diameter could be used to

reduce streaming losses, since streaming is a function of velocity gradients. Another example

would be to relax the no-penetration condition at the tube wall and probe how this might affect

85
enthalpyandmassstreaming;andwhethersuchresultscouldbeusedto designa"continuousstage"pulsetube.Finally,theisothermaltemperatureconditionsatthetubeendscanbere-cast

asanoscillatingtemperatureboundaryconditionto determinehow thetemperaturephasor

wouldbeaffected.Thisboundaryconditionwouldreflectheatexchangerineffectiveness.

4. Theuseof laser-Dopplervelocimetersor hot-wire anemometervelocimetersto measurelocal

velocitiesovertheentiredomain.Thiswouldsubstantiallyvalidatetheaccuracyof thelinearized

approachandthelimits atwhich it breaksdown.

5. Thetheory canbeextendedto othertypesof oscillatingsystemsin which oneis interestedin the

mean-steadystreaminggeneratedby oscillatingflows. Forexample,anexpansionof the

conservationequationsfor individualspeciesof multi-componentmixturescanbeperformedto

determinespeciesstreaming.Its usefulnesswouldbein understandingspeciesseparation.

Theformulationof thepresentsetof anelasticequations,andtherelativesuccessof the

strong-temperaturesolutionin predictingmean-steadyparticlestreamingprovideameasureof

confidencethatthepresenttheorycanbeusedto designpulsetubesandto predicttheir

performance.

86
APPENDIX A

SCALING

The governing fluid equations that describe the flow dynamics of the tube are scaled. A

sketch of the system is shown in figure 38. Two problem domains are taken: the gas domain extends

from r* = 0 to r* = rw and z* = 0 to z" = L* (starred variables are dimensional quantities); and the

tube-wall domain extends from y" 0 to y = and from z = 0 to z L', where I* is the tubewall thickness. Adiabatic conditions exist for the outer wall surface, and continuity of temperature

and heat flux must exist between the gas and the tube-wall interface. The velocity boundary conditions are of small amplitude and periodic so that time can be represented using complex notation: at

(...)

z " =O,u= 0_ eiW *;andatz'=ZY, u= Cr[e _ 09 t +Ou where o9' = 2trf* is the angular frequency

and f" is the frequency, t" is the time, _U is the velocity phase angle between the tube ends, and 0_

and 0[ are the velocity amplitudes at each end. The energy boundary, conditions at the outer tube

wall are adiabatic, with the temperature at z* = 0 and z" = L" taken as T* = Tc and T* = T_,

respectively. We will refer to the z-direction as the axial direction, and the r-direction as the

transverse direction, and use Z, o = Oz/Orl as the notation for partial derivatives.

at r*= 0,

_10" ei o) *t*

Y*_" l*

r =0

"ff //f ff f _Ff JJ//f /Jl/J/////J/////////f Jf Jf /////f _

Z =0

ADIABATIC AT WALL

OUTER SURFACE

J TUBE WALL

DOMAIN

GAS DOMAIN

z * =L* at r*= 0,

Th* IJL* ei (w't* + _U)

Figure 38. Two-dimensional axisymmetric system for rw / E << 1.

The fluid equations of Bird et al. (ref. 64) are reduced for our system using the following

simplifying assumptions: (1) two-dimensional, axisymmetric cylindrical geometry; (2) ideal

gas; (3) constant transport properties; (4) Stokes assumption for the second viscosity; and

*2 _ ,2

(5) rw/L << 1 (implying that c_p*/Or'= 0 so that the r-momentum equation can be decoupled

from the rest of the problem and axial viscous transport is negligible). The reduced fluid equations

for mass conservation, equation of motion, energy conservation, and equation of state become,

respectively,

(p*l)'r') *

Pt.. + . ,r + (p'.') • =0

' r ,z

jO.[u,t*..... + l) U r*+UU z, ]=--p *+ r Ur* ,

(A2)

87
" "r "] (:t " " ) ff---(r*T', ) + _:'T', • +# U,r

p .+I)'T , .+u*T ,'. • =- p .+up, z. + r" \ ,r ,r * ,Z Z

(A3)

p" =p'R'T"

The energy conservation equation for the tube-wall domain for/'<< rw is

(A4)

' W o;• =k "(0;.v. w. _

+0".

,Z Z

• )

(A5)

where 0 is the temperature of the tube wall. The mean-steady (time-averaged) enthalpy flow is of

primary interest since it represents refrigeration,

4 dC (16)

l/O) °

where the overbar represents time-averaged over a cycle. The kinematic velocity components in the

z" and r" direction are u" and v'; the thermodynamic gas variables p', p', and T" are pressure,

density, and temperature; the density of the tube wall material is P_v; gas properties, _', k', and Cp,

are the dynamic viscosity, thermal conductivity, and heat capacity; and the tube-wall properties, k_,

and C'pw, are the thermal conductivity and heat capacity of the tube wall.

The above dimensional equations are scaled (normalized) resulting in dimensionless

variables ranging from 0 to O(1) (order 1). The variables are scaled as follows: r" is scaled with r_.,

z" is scaled with L', y" is scaled with the tube-wall thickness l*, and t" is scaled with the angular

frequency o) * ; U* is scaled with the axial boundary condition velocity U_)" v o is scaled with

(Uor w/L ); and p', p', and T', are scaled with mean pressure PO and reference density and

temperature p_ and T_. The transport properties _', k*, C_, k[, and C_w, and the tube-wall density

Pw are taken as constant and so they are in themselves the scaling factors. These scaling parameters

are substituted into the dimensional equations (A 1)-(A6) and rearranged to give the corresponding

dimensionless form (unstarred variables are dimensionless) for mass conservation, equation of

motion, energy conservation, equation of state, tube-wall energy consen, ation, and mean-steady

enthalpy flow,

['P_-rr)'r ;=0 (17) p,, + e +(pu):

p[u,t+E(VUr+UUz)] = 1 1 (rU, r) r

•

---£ P.z + Va r

(A8)

p[Tt +E(I)Tr+uT.)]=, ,_ T(p, t_/-1+EUpz) +' (r ).r t._2 Tz z +(¥_ 1,",Va M2 u'r (A9)

p = pT (A 10)

88
ot=Fo(oss+ Ozz)

H- = COa puTdt dr (A12)

o Ll/o 

The mean-steady enthalpy flow of the gas, equation (A12), is scaled by the leading-order

oscillating enthalpy flow, H_ = n: r_2p'oCpUbT_. The above set of equations identifies six dimensionless scaling groups, five of which are independent. Two additional dimensionless groups, ¢U

and _'L, enter through the boundary conditions. The dimensionless groups are listed in table 7 along

with their physical interpretations. The relative magnitudes of the groups provide an understanding

of the importance of the various effects (friction, heat transfer, compressibility, etc.).

Table 7. Dimensionless scaling groups.

Name

_. Inverse Strouhal

number

Acoustic

resonance

parameter

Va Valensi number

Pr Prandtl number

M 2 Mach number

squared

Fo Fourier number

Definition

yM 2 _ L'co"

E a a

•2 ,/ *

rwCO /v

V'/a*

~ *2 * Vo/TRro

Length scales

Displacement length to

tube length

Mach number times ratio

of sound wavelength to

system length

Transverse system length

to viscous diffusion length

Viscous to thermal

diffusion lengths

Ratio of velocity

amplitude at z = 0 to speed

of sound

Thermal diffusion to tube

wall thickness

Time scales

Period of oscillation to

residence time

Mach number times

ratio oscillation to

resonance frequency

Viscous diffusion time

to oscillation time

Thermal diffusion to

viscous diffusion times

Period of oscillation to

acoustic time

Oscillation time to

thermal diffusion time

89
The numbers also represent time ratios. If the residence-time scale is L° / U0 (time in which

a particle of velocity U_) travels the length of the tube L') then e is the ratio of the velocity timescale to the residence time-scale. Va is the ratio of viscous diffusion time-scale to velocity timescale, Pr is the ratio of thermal-to-viscous diffusion time-scales, and M is the ratio of acoustic

time-to-residence-time scales.

9O
APPENDIX B

PARTICLE VELOCITY

For a two-dimensional system, the two directional components of the mass flux vector pu

can be combined into a single scalar function--the stream function o/--which exactly satisfies the

differential equation for mass conservation. For the case of axisymmetric cylindrical geometry, the

stream function is defined as - o/,r = put, and + o/,z = put where r and z are the radial and axial

directions and 1) and u are the corresponding directional velocities.

The stream function represents lines of mass flow. For truly steady-state flow, the loci

of points where the derivative of the stream function is zero (do- 0) represents lines of

constant mass flow (do/= pudA = dth = 0) or streamlines. A parametric plot for the condition

do/= purdr = pvrdz = 0 over the domain gives the streamlines, with the difference between any

two streamlines being numerically equal to the difference in mass flow between those two lines.

Streamlines in the Eulerian reference frame for steady-state flow also represent particle

paths. However, for the oscillating flow investigated in the experiments, Eulerian mean- steady

streamlines are not equivalent to mean-steady particle paths. The mean-steady particle paths are

best represented in the Lagrangian reference frame because--as in the experiments--it is the visual

tracking of a distinguishable particle over time. Here we derive the equations for the particle path.

The exercise here is to determine the particle velocity field Up(X,t), given the Eulerian

velocity field U (x,t). Consider a particle at position x and time t that initially was at position x0 at

time t : 0. Its velocity is

Up(X,t;Xo):U p /X0+ IUp(X,'_)dT, t / (B1)

0

It is understood that for any given position and time, the velocity of a particle is equal to the

Eulerian velocity,

hence equation (B1) is

Up(X,t)=U(x,t) (B2)

Up(X,t;Xo)=U (tXo+IUp(X,z)dT, t /

0

(B3)

For small particle displacements (small time increments) of O(e), equation (B3) can be expanded in

a Taylor series, about x0,

Up(X,t;Xo)=U(xo,t)+e Up(X,v)d'r.VUp(X,t) + ... (134)

xo

91
Now U and up are both expandable in e,

Substituting equations (B5) and (B6) into equation (B4), and equating like orders in e,

Upo(X,t;Xo) = U_(xo,t ) (BY)

t

=U,(xo,,)+ (B8)

0

Recalling the relation of equation (B2), equation (B8) becomes

t

Up1(X,t;Xo)= Ul(xo,,) + I Uo(xo,t) d_c. VUo(xo,t) (B9)

0

The mean-steady observed velocities of equations (B7) and (B9) can be rewritten in terms of the

mean-steady velocity produced by the Reynolds stresses. The applicable relations are Uo = uo and

U I = u I + plU° and they are given by equation (76). Using these relations, equations (B7) and (B9)

PO

become

and

u po(X,t;Xo) = Uo(Xo,t) (BIO)

Upl(X't;Xo)= Ul PO Jxo Xo

Equations (B 10) and (B11) are now time-averaged over a cycle to arrive at the particle velocity,

_pO(X;Xo) = ¢}upodt = 0 (B 12)

and

Upl(X;Xo) = /Ul + PlUO + l!Uo d_c. Vuo ) (B13)

PO -IXo Xo

where the overbars represent time-averaged quantities. The quantity fipl is the mean-steady particle

velocity. It is composed of the observed mean-steady field velocity U I = u 1 + plu-----_°and the

Po

quadratic product of the mean-steady velocity produced by the Reynolds stresses as the particle

transverses across the O(e) streamlines defined by U 1.

92
APPENDIX C

ANELASTIC APPROXIMATION

The definition of anelastic flow is somewhat vague and unsettled, possibly because it is so

seldom used. In general though, it can be thought of as the "filtering of sound" from the fluid

equations (ref. 2); its effect is "to remove acoustic phenomena from theoretical considerations"

(ref. 1). Mathematically, it is an approximation of the fluid equations where pressure gradients are

ignored in the mass conservation equation, but are retained in the momentum equation. This allows

decoupling of the pressure gradient between the two equations and results in density variations

owing to bulk pressure changes in time only, ignoring density variations caused by pressure changes

in space.

To further illuminate the meaning of anelastic flow, consider once again the problem in

which an ideal gas is enclosed in a cylindrical tube. Now take the scaled equation of motion in

which there are no body forces given by the conservative form of equation (10):

(pu), t + eV-(puu)= -+Vp-7_ --1,V.x (C1)

A va

where ,;t - 7M2 . Consider 1 a series expansion for small values of e,

E

Substitute equations (C2) into the equation of motion and collect leading-order terms. For a typical

pulse tube, the leading-order pressure term is Vpo = 0(_) = 0(10-7), a very small number, hence

Vpo = 0 (c3)

Although the leading-order pressure is not spatially dependent, it can still be temporally dependent,

Po = po(t) • Now, take the scaled mass conservation equation of equation (9)

O= 1 Dp _-eV-u (C4)

p Dt

I Equation (C I) can be rearranged into a more familiar form by multiplying through by Va,

EVa

Va (pu) t + eVa V. (puu)= -T-M- T Vp - V.'_ where the quantity eVa is the dynamic Reynolds number, Re,

F"w U_

multiplied by the length ratio --, so eVa -

r'_Z_c°" - U°r'w r'w - Re r_' . This arrangement gives the momentum

L" L'w" v" v" L" L"

equationintheform Va(pu) t + Re r_ v.(puu) - Re r'w vp_ V. z.

L" yM 2 L"

93
where DZ

D----t= Z,t + eu. V Z. The equation of state for a single-phase, single-component system allows

the density to be written in terms of pressure and temperature,

1 Dp_ Dr _cDp (C5)

p Dt /3--_+ Dt

where/3 = -(1/p)(Op/OT)p is the thermal expansion coefficient and r= p(Op/Op) T is the bulk

modulus. The equation of state is substituted into the mass conservation relation, equation (C4),

giving

eV. u =-/3(T t + ell" VT)+ x'(p,t + eu •Vp) (C6)

Substituting the series expansion into equation (C6) and expanding to O(1) with 13and _cconstant

gives

0 =-3To, t + K'po, t (C7)

which states that leading-order temperature and pressure are not, in general, time-dependent. This

implies that the density is not time-dependent, but it can still be spatially dependent. Equating O(e)

terms of equation (C6) gives

V "U 0 =-j_(T1, t + u 0 • VTo)+ K(Pl,t + Uo Vpo) (08)

or, since Vpo -- 0 from equation (C3),

V.u0 =-/3(Tl,t +u0' VT0)+ rpl,t (C9)

Equation (C9) shows how the problem becomes anelastic in the limit for Vpo <<< 1 where

pressure gradients do not significantly contribute to the divergence of the velocity. The divergence

of the velocity is primarily a result of bulk-pressure and bulk-temperature changes, and advection

through temperature gradients. However, PO may still have significant temperature dependence,

which will be reflected in the energy equation.

94
APPENDIX D

TWO-DIMENSIONAL ANELASTIC EXPANSION AND ORDERING

Expansion Series

Consider the conservation equations for mass, momentum, and energy conservation, and the

equation of state for an ideal gas:

O=p, t + eV •(pu) (D1)

= -Vp- iv. z (D2)

Va

where

M 2

r-1 1 V.q +(r+ 1)__a x:Vu (D3) (pT) t + eV .(puT)= --_--(P,t + eu. Vp) Pr Va

p = pT (D4)

A << e << 1 (D5)

Also take Va = O(1), Pr = O(1). Assume an expansion of the variables for pressure, temperature,

density, and velocity in terms of the unknown functions f(e,_,), g(t,X), h(e,A), and j(e,X),

P = PO + fJ(e,;t)Pl + f2(e,A)P2 +" (D6a)

T = TO + gl(e.,_,)Tl + g2(e,_)T2 +-.. (D6b)

P = Po + hl(e.,X)Pl + h2(e,X)P2 +" (D6c)

u = u 0 +jl(e,2)Ul + jz(e,&)u2 +-.- (D6d)

The task is to determine f, g, h, andj so that a consistent set of equations is obtained that will

describe the oscillating flow problem. Substitute equations (D6a)-(D6d) into equations (D 1),

(D2), and (D3) and order. At O(1), mass conservation becomes

Po, t = 0 (D7)

and momentum becomes

or

which implies

energy equation, (D3), we have

Vpo = 0 (D8)

Po,t = 0 from equation (D7) and the use of the equation of state, (D4). From the

TO,t = 0 (D9)

95
The leading-orderequationof stateisthus Po = Po(z)To(z)"

At next order, take the energy equation, (D3), and order,

glPoTl,t + hiToPl, t + eV. (PouoTo)

M 2

- gl V kVT I +(y+ (D10)

- _ l(flPl't +EIIo'VPo)- Pr Wa _- _ . 1)_a zo:Vu 0

Note that Vpo = 0 from equation (D8). Equation (D 10) is an order e relation by virtue of the

advection term. Viscous dissipation is negligible at O(e), M 2 << e. For equation (D10) to balance,

g l = h l =fl = e (except for the condition that 0 = V-(pouoTo)= V-(pouo)= poV.uo which, in this

case, requires that the flow be incompressible, which does not apply to pulse tubes). Equation (D 10)

becomes

7-1 1

poT l,t + TOOl, t + V . (PouoTo ) = -- P l,t V . kV T I (D11)

?, Pr Va

which represents a balance between oscillating bulk temperature, pressure, bulk advection, and

conduction. The next-order mass conservation is

Pl,t + V. (poUo) = 0 (D12)

which is consistent with O(e). The next-order momentum equation is

eVp I =O(e 2)

or

Vpl = 0 (D13)

which implies that Pl = Pl (t). This is consistent with the ordering for energy given in

equation (D 11). Equations (D 11) and (D 12) are the leading-order oscillating equations for energy

and mass conservation, but we still require the leading-order relation for momentum.

Proceeding, at next order, the energy equation (D3) is

g2PoT2, t + h2ToPl, t + E2TlPl,t + eV. (anaop 1 + jlulPo)

M 2

_ ?"- 1 f2P2,t + _- 1 e(eUo. Vp I + jlUl "Vpo) Prg2Va V.kVT2+(Y+I)__._a,_:Vu (D14)

7 Y

Note that Vpo = 0 and Vpl = 0 from equations (D8) and (D13), respectively. Equation (D14) is an

O(e 2) relation by virtue of the advection and acceleration terms. As long as M 2 << _2, viscous

dissipation is negligible. For equation (A14) to balance, g2 = h2 -f2 = _, and j2 = e. Equation (D14)

becomes after time-averaging

1 V.kVT 2 (D15) V.(u0p l+ulp0)= PrVa

96
which is a conversion of work to heat conduction. The next-order mass conservation is

P2,t + V. (PoUl + PlUo) = 0 (D16)

which is consistent with O(e2). The next-order momentum equation is

2"(PoUo)t = _e2Vp2 _ X_.____ 7.1:0 (D17)

Va

which from equation (D5) requires that 2, = O(e 2 ) if equation (D 17) is to balance. Equation (D 17)

becomes

-1-._-V.'ro (D18) (PoUo),t=-Vp2 Va

which implies that P2 = P2 (x, t). Hence, momentum is driven by pressure gradients of O(2,),

whereas the lower-order pressure Po represents the mean pressure and pl(t) represents the bulk

oscillating pressure. This separation of pressure into time and space functions is the anelastic

approach. Equations (D11), (D12), and (D18) are the oscillating equations for energy, mass, and

momentum conservation, respectively.

Carrying out the expansion to the next order shows that the next terms are simply of O(e3).

The expansion is thus

P = PO + gpl(t) + e2p2(x,t) + E3p3(x,t) +""

T= To(x)+ 8Tl(X,t)+ t_2T2(x,t) + g3T3(x,t)+--.

p = Po(X) + _pl(x,t) + g2p2 (x,t) + _3p3(x,/) +...

U = Uo(X,t) + eUl(X,t) + e2Uz(X,t) + e3U3(X,t) + "'"

(D19a)

(D19b)

(D19c)

(D19d)

Equations (D 19a)-(D 19d) are series expansion in e. This would seem to be physically correct,

because e is a displacement length of the gas at the tube ends. One would reasonably expect that an

oscillating displacement at the tube ends of order e would result in oscillating-pressure, temperature,

and density changes of order e. However, the pressure gradient that drives the oscillating velocity

can be much less than the speed of sound and much less than the resonance frequency of the system;

hence, the pressure gradient is of higher order and an anelastic approximation is applicable.

Equations (D19) apply for the specific case of _, = O(e2). In general, however, 2, can be of

higher order, that is, A, = O(e3), 2' = O(e4), 2' = O(e5), etc. To represent these more general cases,

the pressure is written as

P = PO + epl(t)+ 2P2(x,t) + t_2P3(x'/)+"" (D20)

where 2' << e << 1.

97
Zeroth-Order Equations

_, 1 In. To, r dS = O

Po,t - "V Pr Va

S

From mass conservation, the leading-order density is not a function of time:

POx = 0 (D21)

From the zeroth-order momentum equation, pressure is not a function of space:

PO,z = 0 (D22)

Integrating the energy equation over the total system volume and using the divergence theorem, the

leading-order pressure is found not to be a function of time since the temperature boundary

conditions are steady and there is no accumulation of energy within the system:

(D23)

The above condition, Po,t = 0, combined with the previous condition, Po,z = 0, requires that

PO = constant. The energy equation becomes

0=

Pr Va r

This is coupled to the zeroth-order energy equation for the tube wall

0 = Fo Oo,yy

through the appropriate ordering Fo = o(l) _ . This ordering condition requires that

(D24)

(D25)

-_2 = O << 1, allowing the tube wall to be approximated as a thin flat plate with the use of

r w

rectangular coordinates in equation (D25).

The boundary conditions for equations (D24) and (D25) are at r = 0, TO,r = 0; aty = 1,

00,y= 0; and at the interface between the gas and the tube wall, the temperature and heat fluxes are

continuous. These conditions require that TO be independent of r. Finally, pressure, temperature, and

density are related through the equation of state,

Po = Po(z)To(z) = constant (D26)

First-Order Equations

The next-order momentum equation is

C

0 = _ Pl,z

which implies that Pl,z = 0 and hence,

(D27)

98
Pl=PI(t)

The next-order mass conservation and equation of state are

Pl,t + V. (pouo) = 0

Pl = PoT1 + PlTo

The first-order energy equation for the gas is

t

P___L+V.(pOUO)=

1 (rTl,r),r

Pr Va r

This is coupled to the first-order energy equation for the tube wall,

(D28)

(D29)

(D30)

(D31)

Ol,t = Fo Ol, yy (D32)

rUo,r ),r

The next-order momentum equation is

1

(PoUo),t =--P2,z +_/a

(D33)

An additional relation is needed to find pl(t). This relation is obtained from the volume integral of

the energy equation

f_( 1----_fn dS _n.pouodS ) (D34) P)= V_PrVa_ "Tl'r -S

which states that the periodic nature of pl(t) results from the forced oscillations from the tube ends

and periodic radial heat conduction at the tube walls.

99

APPENDIX E

REDUCING THE FIRST- ORDER EQUATIONS

USING COMPLEX EMBEDDING

Take the nonconservative form of the energy equation in terms of enthalpy

0_-

1 (rTl,r)r

Pr Va r

PoTl,t + _y- 1Pl,t -PoUo "VaTO (El)

Y

_9 a

Expand u 0 •VaT 0 where u 0 •V a = 1)0 _r + uo--q- and To, r = 0 clz

, y-1

0 = 1 rTl, r) r PoTI,t +--Pl,t -PouoTo (E2)

Pr Va r y

Equation (a) in table 4 gives the explicit r-dependence of uo(r, z, t) as

uo(r,z,t ) : @(r,z)e it : i _)(z) [1- _o(r,z; _f-V--a)]e it (E3)

Here complex embedding is used to separate out the time dependence, Z = _[f((x)eit], where Z

represents the real part of the complex function _,(x)e it . In general, 2(x)is spatially dependent and

is itself complex. Substituting equation (E3) into (E2) and eliminating the time-dependence results

in

O: 1 (r']l'r)r ^.i_/-l_,-iT_[1-_o(r,z;_/-Q--a)] (E4)

PrVa r ' -iPoT1 y

This can be solved in terms of r using the boundary conditions at r - 0, ']l,r = 0, and at r - 1,

']1 = Tw eir_r , where l"w is the temperature amplitude and _T is the temperature phase angle at the

interface between the gas and the tube wall. Using the equation of state, 1 = po(z)To(z ),

equation (E4) becomes

"]1- _ZylplTo[1-_o(r,z;_)]+ l"wei¢T_o(r,z; P_f-P-_-_)

_ ,^, {I1- o(r, z; Pr r z;_)]}

(E5)

The quantities irw and OT are determined by solving the equation for the tube wall,

01 = Fo 01,yy (E6)

whose solution is

101
 ': eiOT[ei Y e' lisin Y lcos JJ (E7)

where Z =

We now know 7_i in terms of the unknown pressure p2(z), the unknown oscillating bulk

pressure amplitude /31, the zeroth-order temperature To, the interface temperature amplitude 7"w,

and the temperature phase angle, OT- Now substitute 7_1 into the first-order equation of state (eq. (h)

in table 3) and use the zeroth-order equation of state to obtain _31 in terms of/32(z).

[_l poT! (E8)

_)1= -_0 - " TO

_l= _._j_ 1 pO_.weiCTfo(r,z; p_-_-_) - _1/31[l_fo(r,z;.x/-p--r_----aa)] 7-0 7"0

+ TTO/3_ Ill- fo(r,z; P_fP--r-_)]- (P---_- 1_ fO(r'z;_/_a)- fro(r,

Jo V

(E9)

From the first-order mass conservation equation

(PO_)Or),r _ -(PO_O),z + iPl

F

(El0)

substitute for Pl and _o

(Po{)or)'r -r i( D_[1-fO(r'z;_(Q--a)]

(Ell)

P lTo ]/ -1_ /31[1- f o (r, z; "qf-P-r---Vaa )] - PTO° TTowei OTf o ( r, z; "/-P-r_ ))

Integrating equation (El 1), and using the boundary condition v - 0 at r - 0, results in the unknown

constant being zero and a relation for v0 in terms of r,

102
where

f9o = i(ml(r, z; _/-_)To[_'2'

+ mj(r,z;P_-_--a)-_-p--_)[ 3t, ,z;_-Q-_)-m3(r,z;P'_-_-_ To(lnTo)P2

+m_(r,z; Pr_V_)p, +m_(r,z; P_TV-_)_o e'_/

(El2)

(El3) ml(r'z; °) = -[ 2- m3(r'z; °)]

m 2 r,z; =- + m 3 r,z;

m3(r,z;O)= _l(r'z;O)

o_(z)

(El4)

(El5)

(El6)

The no-penetration condition uO = 0 at r = 1 is used to obtain a second-order ordinary differential

equation for P2(z) in terms of the unknown bulk pressure amplitude /31,

^,, [ m/(1,z;) Pr [m_(,,z;_) ,

O= p2+].ml_,Z.,_aa ) p-__l[ml(1,z;_-_-_)- ml(1,z;_/-V---a) ]., (lnT0) /3}

m2(l'z; P_-_--a) Pl m3(I'z; P_-_-_) 7"wei(_T

"_ m l (X,z;_-V_ ) TO f _ T2

This is the pressure equation to be solved for P2(z) with unknowns To(z ), and Pl.

103

APPENDIX F

SOLUTION FOR THE THERMALLY STRONG CASE, VT 0 = 0

Leading-Order Results

Oscillating First-Order Solutions

For VT 0 = O, we have PO = TO = Po = 1 and so the pressure equation given by equation (e)

in table 4 reduces to

rnl(l;_/-V-a) /31-_ rnl(1;_V--a) l'weL_r (F1)

The general solution is

rm2(1; r-,_P-_-_ P ) m3(1;P_-_-a)_wei_r]Z2_t_Ciz+C 2

P2=-L p'+ j2

This is substituted into the axial velocity relation given by equation (a) in table 4 to find uo

The boundary conditions on axial velocity

at r=0, z=0, fi0=l

at r = O, z = 1, Uo = (JL eit = (]L ei(Ou+t)

are used to determine the unknown constants C1 and/31,

Cl _

/31 =- i 1-ULeicpv m/(1;-4V--ff) m3(1; P_--_-a ) _..weirPr

1- _0(0;_--a ) m2 (1;x/-P--_) - _

Substitution and simplification gives a simple relation for uo,

Uo=[1-(1-(JLeiCV)z] 1 _1

which is linear in z as would be expected in the anelastic limit. After substituting for/31 in the

temperature equation given by equation (b) in table 4, and recalling T D - O, the temperature

becomes

(F2)

(F3)

(F4)

(FS)

(F6)

(F7)

(F8)

105
7"1 -iY--1 m/(1;-,_)[1-go(r; P -Vff)]:

y rn2(1;-_-_-'a){ 1-_o(0;-,_ ) }[-Ogei_U)

+(,o(r; '- i,_,o(r;]) we O,

and for the radial velocity given by equation (d) in table 4,

(F9)

(FIO)

iqu Independence on Tw and 0L

Here we show that lqu is independent of both Tw and UL- Consider the definition of Iqu

= OWtr=!

1

TI r=l - 2I r_'l dr

0

where 0w =-Tl,r, hence

lqU =

-Tl, r r=l

Tllr=l-2_r_'l dr

From equation (c) in table 5, Zl is of the form

7"1= f l + f 27"w

where

f! = fl(r)=-A + A(o(r;_ ) and f2 = f2(r) = -B+(1 + B)(o(r;',J-P-7_ )

with

A=iY-1 m/(l;_ V/--V--_a)l_(fLeiOU y-lm3(1;"f-P-7_)

Y m2(1; P"f-P-_) 1-_'O(O;_Va) and B= Y m2(1; p_-_)

From equation (F 13), the derivative and integral are

(Fll)

(F12)

(F13)

(F14)

G15)

106
1

7"l,rr=l ' ^ I

= f'l + f 2Tw and 2 r]'l dr = f°I + f_°^T w (V 16)

0

where the superscript '°' indicates the definite integral from r = 0 to r = 1. Equation (F 12) becomes

- - f2Tw (F 17) f 2Tw - f_ ' ^ Nu = ^ -f'l '^

f_Tw°^ =0-f2)Tw- ° f°l

Taking the derivative of equation (F 17) with respect to 7"w, we obtain

^ (f:w-f°1-f2f_w)(-f2)+(f'l+f2f:w)(l-f2) (FI8)

Nu,#.. = (fw_f__f)_.w)2

and after simplifying this becomes

f_If_+ f_(1- f_) (F19) ^ _-

Taking equation (F14), we obtain for f_ and f_

f_ =-Aa/-PTQ-a (l(1;a/-P-_--a) and f_ =-A + 2A _1 (1; Pa/-P-_-a) (F20)

and for f._ and f_

f}(r)=-(l+B) Pa/-P-_-a_1(r; Pa/-P-_--a)and f_=-B+2(I+B) _1(1; P4fi_--_)- " (F21)

Substituting equations (F20) and (F21) into equation (F 16) and then evaluating the numerator of

equation (F 12) results in

lqu,p.. = 0 (F22)

showing that l_u is independent of Tw- Similarly, it can be shown that 1Qu is independent of g)LTake f/to be of the form

f l = gl(1-(JL) (F23)

where

g, :a(-l + _o(r;_-)--Va)) and a: i T-1 m/(1;a/-V--a) 1

Y m2(1;PrxI-P-r_)l-_o(O;xf-V--ff )

Substitute equation (F24) into (F23) and substitute the result into equation (F 17)

(F24)

107
^ t ^

_u= -_'lO- V') - I:rw

l"w-g°l(1-OL)- f_Tw

Now differentiate equation (F25) to give

^ • ^ o

and simplify the result as

• ^ o •^ • o ^

glTw - f2glTw + f2glTw

o^ 2

(Tw-g}(1-OL)-f2Tw)

For Nu to be independent of 7_w, the numerator must be equal to zero,

0 = g)(1- f_)+ f_g° I

Substituting for f_ and f_ of equations (F21), equation (F28) becomes

g'I[-B + 2(I + B) -i)=-(I + B ) EPrV-a P_ _1( r; P_P--_--a )g°l

or after rearranging

g)(1 + B) -1 =-(1+ B)_/PrVa_i(r;',]-P--_a)gl

Now from equations (F24)

g_:-_ P_;V_/(r_P_-_) and g} = -a +

a_l(r;_--a )

_/]_rVa

Substituting equations (F31) into equation (F30) gives

____. ___. .(_,O;,mv-_)_] -a.,/PrVa_ltr;_/PrVa)(l + B)_ _ -

= -(1 + B) P'x/-P__l(r;_]-P--_)( -a4

which reduces to an identity.

(F25)

(F26)

(F27)

(F28)

(F29)

(F30)

(F31)

(F32)

108
Mean-Steady Solutions

Eulerian Mean-Steady Velocity Vector, 'U1

The mean-steady velocity components produced by the Reynolds stresses are determined by

equations (46) and (47). Recalling that PO = 1, they are respectively,

1(r l,r)r

(_OUO) r t-(u---_) z =-fi3,z + " (F33)

r , Va r

1

O= f(Ul+PlUo)rdr (F34)

0

The associated boundary conditions are gl,r = 0 at r = 0 (symmetry) and _-1= 0 at r -- 1 (no slip).

Equations (F33) and (F34) and boundary conditions are used to solve for _1 and P3,z. The meansteady quadratic components are determined by recalling that for X1 = _[21 eit ] and Z2 = 9"_[X2eit ]

-- 1 ^ ^ CC

X? ='_XlZl

1_[ ^ ^cc ] 1_[ ^ ^ cc ]

Z1Z2 =_?)/[Z1Z2 ]= ,tz2zl J

(F35)

(F36)

where the superscript 'cc' represents the complex conjugate. Carrying out the integrations of

equations (F33) and (F34) gives fil and fizz in table 8. with the axial Eulerian mean-steady

velocity given by U-l,

12---1= Ul + PluO (F37)

Equation (48) gives the relation for the mean-steady mass flux

O : (-_lr + p-_r),r +(U1+/91Uo) 7 (F38)

r

The observed radial Eulerian mean-steady velocity is given by Vl,

v/= _I +/91l)0 (F39)

Using equations (F37) and (F39), equation (F38) can be rewritten as

0 (_lr) r - (F40)

- ' b 121,z

r

A single quadrature of equation (F40) and using the boundary condition v-1 = 0 at r - 0 gives

_l(r,z)=-l _ r_l,zdr (F41)

109
Table8.Mean-steadysolutionsfor g1(r,z).

fi__L= Q2(r,z ) + J2(r,z ) + (r 2 - 1)K(z)

Va

(a)

_3,z = 16[q3(1,z)-2 q2(1,z)+ g3(1,z)-lg2(1,z)+ k3]

where

Q2(r,z) = q2(r,z)- q2(1,z)

G2(r,z ) = g2(r,z)- g2(1,z)

K(z) = 4[k3(z) + q3(z) + g3(z)]- 2[q2(1,z) + g2(1,z)]

and

(b)

(c)

(d)

(e)

q2(r,z)= Ill Uo cr rl - drldr q3(z) = _ q2(r,z) rdr

,Z 0

(f, g)

g2(r'z) = I _[ _ _cc l

uO 0 ]dr

2

1 1 ^ ^cc

k3(z)=-_a!91[-_] dr

g3(z)=Ij2(r,z)rdr (h, i)

(J)

We also note that equation (F40) defines the mean-steady stream function,

I[J'r - U--1 and + g.z = VI (F42)

y r

Lagrangian Mean-Steady Particle Velocity. Vector, Up

As previously discussed, the mean-steady particle velocity is used in describing the meansteady particle path. The mean-steady particle velocities were presented in equations (79) and (80),

and are reiterated here,

110
and

ffp(r,z) = Ul + UO,rI£ vod'r + UO,zIO uod'C

Op(r,z)= vl + l)O,rI£1)od'_ + l)O,z louod_2

To compute, first recall that Z = 91[_feit] =1/^.2 _zeit + f_CCe-it), thus

t

X, rl = l (fQrleit . X,__cc_t_ rle-it _) and I x d72 = 2 (- fceit + fcCCe-it )

so that

t

Zl, r/IZ2 i^^ 2it __cc_cc-2it ^cc ^ _ 22_icc0) d'c=-_(-Z2Zl, rle +Z2 Zl,r/e +Z2 Zl, r/

hence,

t

I [ ^cc ^ ^ ^cc

Zl, r/IZ2 d_'= -_tZ2 Zl,r/-Z2Zl,r/)

(F43)

(F44)

(F45)

(F46)

(F47)

Using equation (F47), the particle velocities given by equations (F43) and (F44) become

i [f)ccfi _ ficc ^cc ^ fi ^cc

gp(Z,r) = _1 +-_ 00,r - uO O,r + Uo UO,z - oUO,z] (F48)

and

_,(z,r)=_l+4(_)_c^ 1)O,r -- DO^ _)ccO,r] , (F49)

^ ^ CC

since Vo, z = OO,z = O.

Mean-Steady Temperature, T 2

The mean-steady T 2 temperature is defined by the energy equation given by equation (m) in

table 3,

1 (rT2,r),r

PrVa r

For computational purposes it is found from

F 2

=(u'+pluO),z+l(°lr+p-'ll-_°r)rr- " PrVa _O'zz

PrVa ,z

(FSO)

F2 To,zz (F51) l(glr+91[blf)SCr]),r prVa

111
I

Integrating twice and applying the boundary conditions at r = 0, TZr = 0 and at r = 1, T 2 = 0

leads to

F 2

T2(r,z)=PrVa[wl(r,z)_wl(X,z)+w2(r,z)-w2(X,z)]--_--TO, zz(1-r 2) (F52)

or since VT 0 = 0 for the thermally strong problem,

T2(r,z ) = PrVa[wl(r,z)_ We(1,z ) + w2(r,z ) - w2(1,z)] (F53)

where

and

w,(r,z)= I 1 Ircr(ul + _[pl_t_gc]),z dCrdr

w2(r,z)=I(_l+_[PllA_C])dr

(F54)

ff55)

112
APPENDIX G

COMPUTATIONAL PROGRAM

The symbolic application used to compute the solutions was Wolfram Research

Mathematica v. 2.2.2. for the Macintosh. The following program code directly translates from

the text.

The user must specify the variables in the cell titled 'Input system parameters and number of

terms in Bessel function for 10-digit accuracy in ber and bei'. The units are noted. Each variable is

commented for clarity. The variable 'TERM' is specified to provide the accuracy required of the

Bessel function that is computed with a series expansion. The number of terms depends on the value

for Va, and a commented field suggests the number of terms for various Va.

Immediately after running the program, a shaded cell block prints all relevant parameters,

including dimensional specifications and nondimensional numbers. The program continues

computing leading-order oscillating solutions:

TW = amplitude of gas/tube wall interface temperature

PHIT = phase angle of gas/tube wall interface temperature

u0Re = oscillating axial velocity

v0Re = oscillating radial velocity

p 1Re = oscillating pressure

T 1Re = oscillating temperature

rho 1Re = oscillating density

ql = oscillating heat flux

NuC = complex Nusselt number

and mean-steady quantities:

These

u0T1ReAvg = axial enthalpy flux

u0pl ReAvg = axial work flux

rhoSuSAvg -- axial Eulerian velocity

rhoSvSAvg -- radial Eulerian velocity

upAvg = axial particle velocity

vpAvg = radial particle velocity

T2 = temperature

quantities are plotted in phasor diagrams and field plots.

113
;:(" FileNsme and prereq ")

<<Calculus "VeCt orA_alysis"

_<Graphic S"Graphics"

Needs [*Graph/¢s' PlotField" "] :

Share [ ] ;

(. conversion factors and constants *)

conver tZnToCm= (2.54 c=)/in;

conver tlnToCmSqrt=Sqrt [2.54*2 c=_2 ]/in:

conver tCgsToDyne=dyne/{g cm/sec^2 };

conveztAtmtoMPaz0.1 O133MPa/atm;

J:W sec;

meterz (100cm) ;

GasCons_antstar=8. 3143 J/(nol K) ['82.05cs'3 arm/(mol'K'MW)" I ;

(* Bounds on vector fields ")

jAxialRMSBoundI so :: (_AxialRMSFIow[ 0,0,0 ], JAxiaiI_LgFlow[ I, 0,01 )

jAxialRMSBounclHX fr := (jAxialRMSFIOW[0, 0,TW], JAxialRMSFIow[ i, 0. TW] }

rhoS_S Bo uncLI SO : =

{ [rhoSu$ [0, 0,0]. rhoSuS [.67,0, 0] .rhoSuS [.89,0, 0] ), {rhoSuS [0,1,0] ,rhoSuS [.67,1,0], rhoSu${ .89, i, 0] } ):

h_xial F luxBoundl so: =

{ (bAx_alF1ux [0,0,0,0 ].hAxialFlux [. 67,0,0,0 ], hAxialFlux [.89,0,0.0 }),

{bAxialPlux [0,1,0.0 ].h/ix/aiFlux [. 67, I, 0,0 ]. hAxialFlux [.89, i, 0,0 ]}) ;

hAxialFluxDoundHXf r :s

[{hAx/alF1ux(0,O,TW. 0] .hAxielFlux[ .67,0,TW, 0] ,hAxialFlux[.89,0,TW, 0] },

(hAxialFlux [0,1, TW, 0 ), hAxialFlux [.67,1, TW. 0 ],hAxiaiF1ux [. 89, I, TW, 0 ]});

(- pcedefined si_n_ifica_It digits ")

da_eLockxDace [ ]

da_e : zPrependIDrop[Drop [dateLock, 1 ], -2 ], S_ringJoin [kindSys_,N(Va, 3 ] ] ] ;

epsilon: =N[eps, 3 ]

valansi: =NIVa, 4]

prandl t :=N{Pr. 3 ]

fourier: =N [FO, 5 ]

v_f:=valensi ° prandlt ° fourier

Strouhal :=N [i/eps, 4]

ZsoPR: =N [IsoPressRatio. 4 ]

Adi PR :=N [AdiVol PR, 4 J

PXIUPrint :=N [PHIU, 3 ]

PHITPrint :=N [PHIT, 4 ]

{1997, 3, 15, 19, 24, 57}

(* MABTI_ Lineaz & Secon4_'_ *)

(" Maetez Li_" ")

_* Z/O _ FOIICTI(:Ill "_

(- pARA_R_'ERS * )

Ir_ut Paran_eters in E_glish or CgS tU:itS

(" InpU_ CgS Par_J_eters "_

(* OFT ")

(* _ syston parameters ar_ number of terms in Bessel function for i0 digi_ accuracy in bet and _ei ->

ty_e=''; (" CC or Thesis *)

type2='dT/dz=O'; {* dT/dz<>0 ")

freqStar= ._/(secl; (" a freq baseline of IHZ corresponds tO Va=50. *)

rwS_ar= .81649 cm; (o baseline=.04442c=: 2 ".00254 (-$qrt[.1Power[.053.2]]')cm')

(- Va adjustment; .053cm corresponds to VaPr-l: .04442c_ to Va=l *)

LStar= 10 C_; {" baseline=10_; e_s adjus_t *)

pis_onDynAmpStar_0,0]= .02 cm; {. baseOPT=l.0c=, BPT=I.4142_: M adj:')

pis_onDynAmpStar[0,LStar]= 1.0"pistonDynAmpStar[0.01; (* as a percentage of pistonDynAmpCm[O,0] ")

:_HIU =-.I0: (* velocity phase angle of UL wrt _0; PHIU<0 means UL lags UC *)

[* .'::._i:__.... 1 Tube wall d_aain ")

rho0TubeStar= 7.820 g/cm^3;

Cp0TubeStar= 0.46 J/(g K) ;

kTubeStar= 0.163 W/(ca K) ;

tubeThickneamS_a_= .004903 c_;

(" _ gas domain °)

tempLoys;as Star= 50K;

_Re fGasStar= 300K:

pressureRef$_ar= 9.866 arm:

C_0GasStar= 5.2 J/(g K);

kGas$_ar= 0.00149 w/(_ K) ;

mustar= 2. 0134"10^-4 g/(c_ sec}:

_WStar= ¢.2"10_-3 ky/mol;

GAMMA= 5/3;

{" .IHz is Va=5->TERM=7:.6MZ is Va=30->T_RM=11: IHZ is Va=50->TERM=13.

2HZ is Va=I00->TERM=IS, 5Hz is Va=25_->TERM=19,

6Hz is Va=300->TE_=22 ")

{" for PV a_d V-I then TER_=6; for PV ar_ V-10 then TERM=IS;

for V'2=30, TERM=I0; for V_2=300, TERM=22 ")

(- stainless steel *}

(" s_ainless s_eel ")

{" s_ainless steel *)

('FO ad)ustmen_: _.026853 cm corresponds to Fo=I ")

(.pressureMea11Star=pressureRefStar')

('air at STP 300K: 1.005 J/(g K)') (*He: 5.2 J/(g K)')

('air at STP 300K:0.000267 W/{C_ K)*} (*He st STP: 0.00149 W/(Cm K);

(no_e: for kGas=.0010471552, Pr=l.000000000)')

('air at 300K:I.843"10"-4 g/{cm sec); He:2.0134"i0_-4 g/Ice s)

0 any resonable pressure')

(*air at STP 300K: 32"i0^-3 kgl_ol; He:4.2"i0^-3 kg/mol*)

{" 7/5 for air *)(* 5/3 for He *)

(- o_tp_t _l_ c_Lculat_ons -I

rhoRefGasStar=pressur_RefStar.(_OOK/teaRpRefGas$_ar)*l,624*lO^-4 g/(a_ ca^3):

l'air at STP 300K: 1.1_0"10^-3 g/(aum c_^3) ")

('He d_sitT/atm _ 1 ate: 1.624"10^-4 g/(a_ ca^3): °)

nuStar--_S_ar/rhoRefGasStar; ('air at STP 300K: 0.1566 ca_2/sec: He:0.124 Cm^2/sec *)

a0Star=$qrt[GAa£_A Simplify[

GasConst_a_tS_ar/MWS_ar t_RefC, asS_ar J/(sec W) (kg meter^2/sec^2}/J {sec^2/ca^2)]] ca/sec:

(*air at STP Anderson p_.55:

He:$qrt[GA_G4A 2.08"I0^_ 300It m/s;= I01,9800=/S *J

(- Cha_ut convert from Star=cgs to English units ")

_Inch=rwS_ar/convertInTo<3_;

LI_ch=LStar/convertInToCm;

tubeThicknessZnch=tu_eThick_ess$_ar/co_vert_ToCm;

pi StonDyT_P _nch [0,0 ]=_is tonDynAmpStar[ 0, 0]/convertInToCm;

pistonDynAmplnch[O,LInch]=pistonDynAm_$tar[O.LStar)/convertlnToCm;

pressureRefStarMPa=pressureRefStar convertAtmtoMPa:

114
_* I.-_ut 51n.gli;h Par_-neter. _ and convert to C_s ")

ty_e='CC'; (° CC or Theli$ ")

type2='dT/dz=0" : (" dT/dz.>0 "}

freqSter= 15/8e¢: (" set *)

rwlnch= .5.7/8 in: (" Ve adjustment .) (" value2orV'Sqtc[nu/omegaBar]/convertInToC_Sqrt allows

specified value for V °}

LXnch= 5 in; (° ep$ ad_ ") (" 15"_5/8" is _he undasplaced tube leith *)

piston_%,nAmpInch(0,0]= 1 in ('5Inch/(2Pi)°); (* M edj ") ('--piston dynamac a_litude;

(1.09)/2 in is 13cc confessor DietonDynAmplnch;

for spec ep$, pistonDyn_Inch[0,0]=_s LZnch/f2Pi).)

piatonDyr_Inch[0,LInch|= 0.2 plstonDynAn_Xnch(0,0]_

PH_U =-0.25: (° velocity phase angle of UL wrt U0; PHIU_0 means UL lags U0 °)

TEP/_=IS; (" for PV and V-I then TERM=6; for PV and V-10 _hen TERM=IS ")

(" '..n:>,,_ Tube wall d_main "I

rho0_h_b4Star= 7.820 g/_^3;

Cp0T%LbeStax© 0._6 Jl(g K);

kTu_tar= 0.163 W/(cm K);

t%tbe_%ickneasInch= 0.020 in;

(" i_._u. _ gas dora, rain ")

t _apRe fGasStar= 300K;

pressureRefStar= i0 a_m;

C_0GasSter= 5.2 J/(g K) :

kGasSterz 0.00149 W/(cm K1 ;

muStaz= 2.0134"I0^-4 g/(_ sac);

MWStaz= 4.2"10^-3 kS/tool;

GA_4A. 5/3:

(" Output gas calculations "}

[" stainless steel *)

(° stainless steel °}

[" stainless steel °}

(° Fo adJul_aent °}

('pressureMeanStar-presauraRefStar-)

('air at STP 300K: 1.005 J/(g K) °) ('He: 5.2 Jl(g K)')

{°air at STP 300K:0.080267 W/(cm K} °) ('Me at STP: 0.00149 W/(cm K);

(note: for kGa$=.0010471552, pr=l.000000000) °)

('air a_ 300K:1.843"10^-4 g/Icm sac);

He:2.0134=i0^-4 g/(cm $) @ any resonable pressure')

I°air at STP 300K: 32"10"-3 kg/_ol: Ke:4.2.10^-3 kg/mml'}

{" 7/5 for air -)(- 5/3 for He °)

rhoRefGasStar=preasureRefStar'(_eE_RefGasS_ar/(300K))-l.624.10*-4 g/(a_m cm^3);

('air at STp 300K: 1.170°10^-3 g/(a_m cm^3) ,)

('Me dansity/a_m _ i a_m: 1.624"10^-4 g/(a_m cm_3}; ")

nuStar=muStar/rhoRefGesStar; ('air at STP 300K: 0.1566 cm^2/$ec: Me:0.124 ¢m^2/$e¢ °)

a0StarsSqrt(GA_4A GaIConstantS_ar/MWatar tem_RefGesStar] cm/sec;

{°air at STP Anderson p_.55:He:Sqrt[_ 2.08"10"7 300]c m/s;= 101,980cm/s "I

(" Ouput convert from _gllsh to Star _its ")

rvStar =rwlnch°convertInTo_m:

5Star _LInch'convertlnToQm:

tubeThickne_sS_ar =tuheThicknessInch.conver_InToCn;

p_stonDyn_Star[0,0] =pistonDynAm_Inch[0.0l'convertlnToCm:

pistonDy_$tar(0,LS_ar] =pis=onDynA_Inch[0,LInch]°conver_InToOm;

p0=l:

If[ty_e2=='dT/dz=O',TO[z]=l,TO[z]=.]:

If[type2=='dT/dz=O',TO[O]=l.TO[O]=.]:

rho0[z_):=i/T0[z]:

o_aBarStar

UStar[O,0]

UStar[0,LStar]

UStar[0,0,t_]

UStarI0,SStar,t_]

alpha0TubeStar

alpha0GasStar

hOscFlux0Star

xSectAreaStar

h0$cFlow0Star

jO$cFlux0Star

jOscFlow0Star

_GasRadial0S_ar

wallaheazStress0Star_1=uStar _Star[0,0]/rwStar;

tubelnnerArea =2 Pi rwStar LStar:

drag0Star =wallShearStre$$0Star tube_nnerAres convertCgsToDyne//N;

deltaTS_ar =tet_RefGasStar-_empLowGasS_ar:

deltaT ={te_RafGasStar-t_Lov_asStar)/tem_RefGasStar;

volOfTubeStar :- Pi rwS_ar^2 5$_ar

(°velocityCoc_bo[t_]:=U5_Cos[2Pi t]/Co$[2Pi(t*PHIU)]')

periodStar=i/freqStar:

tauStar=periods_ar: (" characteristic time is the period °)

M =USter[0,01/a0Star;

Va =rwStar^2 omegaBarSrar/nuStar:

Pr =muSter C_0GasStar/kGasStar:

FO =alpha0TubeStar/(omegaBarStar tubeThicknessStar^2);

e9_ =pis_on/>_D_Star[0,0]/LS_ar:

S =i/aDs:

L4_DA =M^2/(e_$ GA_):

CAPGA_K_A-rwStar/(LStar°ep$);

[!C =UStar ( 0,0]/UStar [0, O] :

UL =UStar[O,LStar]/UStar[O,O];

KGAS =kGasStar/kT%_beStar ;

K'[_!BE =kTubeStar/kTubeS_ar;

EL =tubeThickneasStar/zvStar//N;

Re0 =jOscFlux0Star 2 rwStar/muStar:

v =Sqrt|VaI:

P =Sqrt[Pr]:

_'v=Sqr_[Pr]'v;

F =SgrtIl/Fo]:

UO=.;UIz.;(°FO=.°)

d[0]--plstonDynA_Star[0,_] /5Star;

d[l]-_Is_onDynAmpStar(0,LStar]/LStar;

volSys_[__) := 1 - d[01Sin[2Pi t] _ _ [I_ Sin[2P_(t ° PHIU)];

= 2Pi freqStar; {° characteristic inverse _ime period °)

= omegaBarStar-pi|tonDynAm_Ster[0,0]; (° U0=Amplitude of d/dt([plstonDy_[0,0]

S_n[omega_aratar t])')

= c¢_egaBarStar-pistor_)ynA_Star[0,LStar]; (- U0-A_litude O5 d/dt([pistonDynA_[0,0]

Sin[o_egaBarStar t ÷ phiu]} "I

:= UStar[C.0 ] Cos[t ] (" t.ome_aBarStar tatar °}

:= UStar[O, Later] Cos[t * PHIU]

=kTubeS_ar/(rho0TubeStar C_0Tu_eStar):

-kGasS_ar/(rhoRefGasStar Cp0GasStar);

=rhoRefGasStar°Cp0GasStar'_Star[O,0]'tem_RefGasS_ar;

=Pi-rwS_ar^2//N;

=xSectAreaStar°hOscFlux0Star;

=rhoRefGasStar°UStar[0, 0);

=xSe°tAxesStar'jOscFlux0Star;

=kGa_Star'teE©RefGaeStar/rwStar;

ll5
FindM1nimum (volSystlm I t ). ( t, . 5 } ] ;

t _evolMinst I. % ( [2,1 ] | ;

I f [t _meVolMin<. 5, _ imevo iMax = t imeVolNin-. 5, t imeVolMax= cimeVolMin -. 5, t _msVo iMax= t lmeVo 1 M in-. 5] :

isoPressRa_io= volSyst_[timevolMax]/volSystem(timeVoiMin] I/N;

AdiVol PR= (volSySt_ (t imeVo IMax )/volSy$_ [timeVolM_n] } ^GA_//N;

plCLhat[Uo_,Ul_,phiU_] = (-! GAMMA)'(UO-UI Ex_[I 2Pi phiU])//C_lex_d;

plCL[UO_,UI_,phiU_.t_] = plCLhat|Uo. Ul.phiU)'Ex_[I 2Pi t}//Com_lexEx_u_:

plCLRe[Uo_,UI_.DhiU_,__] = piCL[Uo,Ul,phiU,t]-I Coeffi¢iemt|plCL[Uo,Ul.phIU, t].I]//N;

FindMinimum [e_$'plCLRe [U0. UL, PHIU, t ], { _, .5 ] ] ;

pICLAdiaA_l=Abs [% [ [ 1 ] ] ] ;

_ICLAdi_PR- (l÷plCLAdiaA_l I / (1 -p 1 CLAdi_A_I ) ;

(- HfluxlDSt&r*(AdiVolPR-l)°pressureRefstar/(AdiVoIPR÷I) ;°)

HFlowRayStar [_ OscWarmInG tamper See_, TMotInK_, PdynI nA_m_. PmeanlnA_/__ ] :=

(i/2) GasC_stantS_ar/MWStar THotInK jOscwarmXnGramPerSec {PdynXnAtm/Pmean/nA_m) kg/(i000 g)//N

HFlowRayStar2 (J OsCCol_nGramPerSec_, TcoldlnK_, pdynlnA_m_. PmeanlnA_m_, phiRadians_] : =

(1/2) jOscColdInGramPexSeC GasCons_antStar/MWStar TCOIdInK (PdynInA_m/PmeanInA_m} CosIphiRadlan$] kg/ (1000 g)//N

texter=Graphic| [(Te_t ["r=0", (0.5, -. 14), (0,I) ] , Text [ "r=l", {0.5,1.13}. {0,-I)] ,

Text| "z=0", (-.06,0.5] , (i,0}] .Text ['z=l", (i.06, 0.5}, (-i, 0} ] }I ;

liner| (xl_,yl_}, (x2_,y2_}] : =GraphicsILine ( ( {xl ,yl }, {x2,y2] }I ] ;

_f[d[0]==0 II d Il ] =*O ,ki _Y St em=BPT ,ki ndSyStem=OPT " kindS_$ tem=OPT ] ;

! f [ kindSyste_==BPT,

fileNameBPTZsoList= { (.type, °) "BPT", "epm", "va", "Pr" , "vaPr •, "UL", • _MIU", "M". •LAMBDA", "IsoPressRatio",

•KdiPressRatio". "CLAdiPRatio" ) ;

f ileName.BPTHX f r L_ $ t = { file_ameBPTI $oList, ( "Po', ( "_HZT", "TW" ), "vaPrFo • },

• (totalDragAmp0, totalDragRMS/tu_e, HOscFlow0Star] ", "date" } ;

f 1 owNamesEPT := ( ( ° _ype," ) "BPT", e_$i lon, val_$i, prandl c, N [valensi prandl t, 5 ], N [UL, 2 ] .

• n/aPHIU", N [M. 4 ] ,N[LAMBDA, 3 ] ) ;

PRNa_S := { I$OPR, AdiPR, plCL_iaI_R ) :

MxferNames:=(fo_rler. {PHITPrin_,TW}, vpf}:

dragNames:z(dxagSs_ar (A_) . to_alDragRM_Star" (tube^-l) ",hOscFlow0Stax/IN) ;

f il e_u_e_ _TI so: = ( f 1 _%_u,esBPT. PRName$ ) ;

f il eName.BP_KX fr := {fi i eNameBPTI SO, _ferNames. dragNames, da_e ] ] ;

If [kindSyst e_= =OPT.

file/_eOPTXsoList* { ("rye)e, ° I "OPT" , "e_s" , "va" .• Pr ° , °VaPr", "UL ° , •PHIU", "M" . -LAMBDA" , "IsoPressRatio ° ,

•AdiPressRa_io • , "CLAdiPRa_io ° } ;

fileNameOPTMXfr_ist= (fileNameOpTIso_ist, { "FO • . ( "PHIT". "TW" }, "VaPrFo ° },

• {totalDragAmp0, totalDragRMS/tube, MOscFlow0Star}','date']:

f 1 owNamesopT: = { ( • type, ° ) "OPT", eDsi ion, val_si, prandl t, N [valensi prandl t, 5 ], N | UL, 2 ], PHIUPr_nt, N [M, 4 ], N _LAMBDA, 3 ] } :

pRNa_eS := (I $oPR, AdiPR, _ICLAdiaPR ) ;

KXferNames:=[fourier, (PHITPrin_,TW), _f):

dragNames := { drag0S_ar (Amp) , t o_alETagRMSSCar • ( tube ^ - 1 ) ",hOscFlow0S tax //N } ;

f il eNameOpTI so : = ( f 1 owNamesOpT, _$ ) ;

f il eNameOFTHX f r := {fi i eNameOPT_ so, HX f erNames, dragNames, date } ] ;

I"

If [kindSyste_==BpT,

file/4a_PTIsoList= (_y_e, "BPT", "e_s". "V", "P° , *PV", "UL", •PHIU" , "M', ° _$oPressRa_io" , "AdiVoIPR", "CLAdiPRa_iO" ) ;

fileName38PTHXfrList= ( fileNa_eBpTIsoList. { "F". { "PHIT", "TW • ] ), ("HOSCFIOW0" ) ,date] ;

f i i eNameBpTI $o := { type, •BPT". ®psi lon, V//N, P//N, N [PV, 5 ] ,N [UL, 2 ] , "n/aPHIU • , N [M, 4 ], _ SoPR. Adi DR, plCLAdiaPR ) :

fileNameBPTHXfr := (fileNameBPTI$o, (F//N, |PMITPrint ,TW) ], {h0scFlo_0Star//N} } ] :

I f I kindSy $ t_==OPT,

file2_a_eOPTIsoList= { t_1_, "OPT" . "ep8" , •V • , "P" , "PV" , *UL', "PMIU" , •M', •isoPressRatio" , •AdiVolPR" , "CLAdiPRatio" } ;

fileNameOPTMXfrList= {fileNameOPTXsoLi&t, ( "F", | • PHIT". "TW" ) ], ( °MOSCFIOW0" ) ,date}

fi i eNameOPTI SO : = ( type. "OPT • , e_$il on, V//N. P//N, N I PV, 5 ] .N [UL, 2 ] , PHIUPrint, N [M, 4 ] , I$oPR, AdiPR, plCLAdiaPR } ;

fileNameOPTMXfr : = (fileName0PTIso, (F//N, (PHITPrint, TW} ), {hO$cPlowOStar//N] } ] ;

"I

• DIMENSIONAL VARIABLES SPECIFIED:

({OPT. e_s, Va. Dr, VaPr, UL, PHIU, M, LAMBDA, IsoPressRatio, AdiPressPu_cio, CLAdiPP_atio], {Fo, {PH_T, TW}, VaPrFO),

(toualDragAmp0, totaiDragRMS/tube, HOscFlow0Star), date}

-6 -9

(((OPT, 0.002, 100., 0.703. 70.266, i., -0.I. 3.788 10 , 4.31 10 }, (1.002, 1.004. 1.00413}), |i00., |PHIT, TW), _026.6}.

(0.00476915 Amp dyne, (tube'-l) totalDragRMSStar, 1.97349 W}. {OPT<>100., 3, 15, 19}]

- - inches :

z%v =0.321453 in; L =3.93701 in; t_beThickness =0.00193031 in; pistonDyn_InchAt0 =0.00787402 in; piston_InchAtL =0.00787402 in

-- CgS: 3 2

rw=0.81649cm: L =i0 cm: tubeThickness =0.004903 cm; volOfTu_e=20.9436cm ; CrossSectArea=2.09436cm

3

pisconDSnlA_At0 =0.02 c_n: piston_AtL =0.0_ cm: fre_ =---

sec

0.376991 _m 0.376891 cm 0. cm 0.00160224 g

pistonSpeedu0 = ........... ; Pistor_K _-_edUL = ........... ; _pistonSpeed = ..... ; pRef =0.999722 MPa: tearer =300 K: rhoRef = ............

sec sec " sec 3

cm

0.045313 cm 2 7.82 g 0.46 sec w 0.163 W 0.045313 cm 2

alphaTube = ............ ; rho0Tube = ...... ; Cp0Tub_ : .......... ; kTube = ....... ; alphaTube = ............

sec 3 g K cm K $ec

cm 2

0.178836 cm 2 0.0016022_ g 5.2 SeC W 0.00149 W 99488_6 C_* 0_125662 cm 0.00020134 g

alpha0Gas = ............ ; rho0Gas = ............ : Cp0Ga$ = ......... : kGas = ......... : a0 = .......... ; nu = ............ ; mu = ............

sec 3 9 K cm K sec sec cm $e¢

cm

0.942286 W 0.00060403 g 0.00126506 g 0.547465 W

hOscFlux0 = .......... ; hOscFlow0 =1.97}49 W; jUscFlux0 = ............ ; jOscFlow0 = ............ ; _GasKadial0 = .......... ; HFlowRay =

2 2 sec 2

cm cm sec cm

116
0.000d0268 g Pi Pi

NFiowRayStar[ ............... , 300 K, 0.0203251 arm, 9.866 arm. --]

sec 6

totalDragA_10 =0.00476915 dyne; totalDragRMS/tube = (tube ^-I) totalDragRMSStar

• NON-DIMENSIONAL SCALING VARIABLES CALCULATED:

-6

Va*I00.; Pr=0._03; Pr*Va=70.2663: S sS00.; For100.; EL=0.00600497; M=3.78929 I0 ; Re0 -4.89902

p0-1; T0[z]=I : rho0[z]=l

• NON-DIMEWSIONAL VARIAJLES USED IN CO_LEX SOLUTIC_:

-6 -9

VffilO.: Pffi0.83825; PVffiS.3825; eps-0.002; F-0.1:Mffi3._8929 I0 ; 5AMBDAffi4.30762 i0 ; eps*Va-0.2

5

U0=I. ; U5=1. : PMZU--0.1 ; TERM=f6; KGAS =0.0091411; KTUBE=I. ; GAR_4Am-: d0-0. 002; dL=0. 002

3

Archive

ber[order_,r_.z_,ligwa_,TEP/f]=(r $i_ $qrt{rho0[z}]/2)_order °

Sum|Col{ (3orderl4_k/2)Pi] / (Factorial|k| _[order÷k_l ])* (( (r sigma $qrt [rho0 [zl ] }^2)/4) "k, {k, 0,TERM} ];

bei [order_, r_. z_, sigma_, TERM] = (r slgma $qrt [rhoO [z]]/2 )"or4ar °

S_m[Sin[ (3orderl4*k/2) Pi ]/ (Factorial|k| G4umal order÷k÷l ])* (( (r sigma $qr_ [rho0 [z) ] )^2) 14) ^k, {k, C,TERM} ];

bet [0,r_. z_, V.TERM]=ber [0, r. z,V, TER_) ;

bei [0, r_. z_, V.TERM]-bei [0, r. z, V, TE_M] :

bet [i, r_. z_, V, T_M] _ber [i, r. z, V, TERM| :

bei [I, r_. z_, V. TERM]-bel [1, r. z, V, TERM| ;

bet [0. r_, z_, PV, TERM) zber [0, r, z, PV,TERM] ;

bei [O ,r_, z_. PV .TERM] =bei (O +r. z .PV ,TERM] :

bet |i. r_. z_, PV, TZRM}-ber (1, r, z, PV,TERM] ;

bei [i. r_, z_. F4. TERM]-bei [1, r. z. PV.TERM] :

ModBesselJHold[order_, r_, z_. sigma_| :-Hold|bet |order, r, z,si_Da.TZRM] ]*I Hold[bei |order, r, z, sigma,TERM| ]:

(* BesmelJ[0._rt[-I]si_ma|=ber[0,l,sigma,TERM]÷I bei[0,1,$igma,TERM) *)

(" Bessei3, ber-I bei. large argument ap_roxmmauion °)

Print['V = ",V//N." TERM -',TERM]

beJselAccuracytist='{ {V} ;BesJelJ; (ber.x bel); ModBesselJHold; Argument->infinlty a_roximatio_}';

besselAccuracyHi=N [Abs I({V ],Bessel J [0, $qrt [-X]V| .bet [0,1 .z,V,TERM]_I h_i[O, 1 ,z,V, TERM] .

MO4_essel JMold [0. l,z,V]//Releasei_old, Sqrt[2/(Pi*S_rt[-I) V] )Co$[ $qrt [-I]V-Pi/4] }], 9]

besselAccuracySozN [Abs [({0. IV) ,BesselJ[0, 0. iSqrt [°Z]V| ,ber [0,0. I.z,V,TERM| *I bei[0.0 .i. z, V. TERM) ,

M_42e8se1JM_d[_._.1_z_]//Re_easeM_1d_Sqrt[2/(Pi_qrt[-I_.1v)_C_s[_._qr_[-I_V-Pi/4_)_.9]

V = 10. TERM =16

{(9.99999925), 1&9.84_528. 149.8_7526, 149.849526, 148.537427)

{(0.999999925), 1.01552483, 1.01552483, 1.01552483, 1.00386903)

kseTem_late=C_lexEx_and[ (avarKse*Z bVarKse] / (cVarKse*I dVarKse) );

k_eZmTem_ iat e=Coe f fIclen t[kseT_la_e, I ];

kseReTem_ 1at e- kseTem_ let e- I kseImTem_late;

kseReHo id [order_. r_, z_, sigma_| =kseReT_late/.

(aVerKse- >Ho142orm |her (Order, r, z, sigma, TERM } ],

bverKse- >MoldForm [bei [order, r, z, sigma, TERM| ],

cvarKse- >MO idForm |her [0,1, z, sigma. TERM] ],

dVarKse- >Sol,Porto|be1 [0, I, z, sigma. TERM] ]};

kselmMold [order_, r_, z_, sigma_| =kselmTem_late/.

{aVa=Kse- >HoldPorm |bet [order, r, z. sienna, TERM ] ],

bVarKse- >HoldForm [bei [order, r, z, sigma, TERM ]),

cvarKse->HoldForm |bet [0, I, z, sigma, T_%M] |,

_VarKse->Hol_Form [bei [0, i, z, sigma, TERM) ]):

kseHold [order_, r_, z_, si_a_] =kseReHold Iorder, r, z, sigma ]_I kseImMold[order, r. z, sigma| ;

m3Te_lateC_IeXE_[ (aVarM3-Z bvarM3) (ovarM3*I dvarM3 )];

m3 ImT_m_late=Coef fici_c [m3Tem_late. Z ] ;

m3 ReTem_l at e=m3 T_I a_e -I m3 ImTem_la_e;

mReMold [3, r_. z_. sigma_ )=m3ReTem_lat e/.

(aVarM3 ->HoldForm {-i / (sigma S_rt [2 rho0 [z ]])]

bvarM3->HoldFor_[-I/(si_%a Sqrt [2 rho0 [z] ])]

cVarM3 ->kseReMold [i, r, z, sigma ],

dVarM3->kseZmHold [1, r, z, 8i_z_a] |:

mZmMold [3, r_, z_. sigma_] _a3I_late/.

{avarM3->HoldForm[-I/(si_m_a S_rt [2 rho0[z] ])]

bVarM3->Hol_Form[-i/(si_a $_rt [2 rho0[z] ])]

cVarM3->kseReSold [1. r, z, sigma],

dVarM3-)kseImHold |1, r, z. siS_a] );

mReMold [i, r_, z_. sigma_] =- (i/p0) (r12 - mReMold[ 3, r, z, sigma) ):

mImHold [1, r_, z_, $i_m_a_] -- (I/_0) ( - mImHold[ 3, r, z, sigma| ):

mReMold|2,r_,z_.s_ma_]=-(1/_0) (r/(2 GAMMA) _ ((GA_-I) /GAR_A) mReHold[3,r,z,si_ma] );

mlmHold[R,r_, z_, sig_a_| =- (llp0) ( ÷ ((GA_-I)/GA_) mZmMold[3, r, z, sigma| I;

(°

mSold[ I, r_. z_, sigma_] =mReHold [i, r, z, sigma) ÷ I m/mHold[1, r, z. sigma] :

resold|2, r_, z_. s_ma_] =mReHold [2, r, z. sigma| _ Z m_mRold[2, r. z. sig_m] ;

mMold[3,r_,z_,sigma )=mReHold[3,r,z,$igma] _ I m/mMold[3,r,z.sigma]_

*)

m_a_oTe_la_e=COm_lexEx_d [(avarMRatio°I bVarMRatio) / (cVarMRatio_I dVarMRatlo) ];

mRa_ioImT_late=Coef ficient [mRatioT_mDlate, I] :

mRa t ioReTem_l a te=mRa tioTem_ la te -I mRatioZmT_late:

mRat ioReHold |(kindNum_, rNum_, sigmaNum_ ), (kind_, rDem_, sigmaDem_ ),z_ ]=mRatioReTe_lat e/.

{aVarMRat io->mReHold [kindNum, rNum, z, S igmaNum ].

bVerMRat io ->m/m/_o id [kindNum, rNum, z, si-gmaNum I,

cVarMRatio°>mReMold [kind_, rDem, z. sigmaDem] ,

117
o'VarMRa_io ->mIllllold [kindDan, rDell, z+ sigmaDlm I };

mRatiolmHold [{kindNum_, rN_m_, sigmlNum_ }. (kindDlu_. rDe__, si_aDem_}, z_] _LRa: ioImT_la:e/.

{avarllRat io -_,_ello id _klncl_u,, rNum. z. sil_Num].

bV_rllRat i o->ml_old [kin_. rNum. z+ si_llum ].

cVa:llRatio- >mReHold IkinllDem, rDan. z. $i_DaDan ).

dVarllRat i o->mlmHold [k_ndl_, rD_. z. sig_aDen )_;

mRatioHold [(kin_um_. INure_. sigmaNum_ ). (klndDe__. rD__. $_gmaDe_m_ I. z_] =

mRatioReHold |{kindNum. rNum. siglulRum]. (kindDem, rD_n. $_). z] +

I mRatiolmJ_old [{kin_lum, rNum. sigmaNull}. {klndDIm, rDe_. sigmaDe_ }. z] ;

_imeHold[t_)zHoldFormltimeRe[t] )+I HoldForm[timelm[_] ];

timeRe[_.|-Co$ [2Pi t] ;

tilleIll(__]=Si_[2Pi t] ;

tlmeTwHold[t_, phiT_]=HoldForm[ti_eTwReCt] .phiT]+I HoldFor_|timeTwRe(t] .phiT] ;

CimeTwRe|t_.phiT_|=Co$i2Pi (t+phiT) ] ;

_il0eTwIm[__.phiT_l=Si_[2Pi (_$phiT) ] ;

Timing [

mRe[3, l.z_.V] _IReHold[ 3.1. z. V]I/RelealeHold//N;

mlm[3, i. z_.V] *mIm_old[3. i. Z.V] //RelealeHold//N:

rare [3, i, z_. IV] =mReHold [3. i. z. PV] IIReleaseHold/IN;

mira t3. I. z_. _V] =mZmHold [3. i. z. PV] //ReleaseHoldl IN:

IRe [i. I. I_.V] _IRIHOId [1. i. z .V]//Releal_dold//N:

mIm[ 1. i. z_.V) _old[1. I. z .VI//RelealeHold/]N:

rare[2, i+ z_. PV] =mReHold 12+ i. z. PV) //Rele&seHold//N:

mira[2. i. z_. PVI -mImHold [2. I. z. PV] //RelealeHold/IN ;

kseRe [0. r_. z_. V] =kseReHold [0. r. z. V] IIReleaseHold/IN;

kselm [0. r_. z_. V] =kleRe_old [0. r. z. V] /IReleaseHold/IN:

kseRe [0. r_. z_. I_] =kleReHold [0. r. z. PV] / /ReleaseHold//N;

kselm [0, r_. z_. FV] =kleReHold [0. r. z. PV] /IReleall_old/IN;

kseRe [I, r_. z_. V] =kseReHold | i. r. z. V] I IRelealeHold/IN;

kseZm [i. r_. z_. V] -k.leReHol d [I. r. z. V] //RelealeHold//N:

kseRe [i. r_. z_. PV] =kleReHold I I. r. z. PV] /IRelelseHold/IN;

kseIm[ i. r_. z_. PV] =kseReHold [I. r. z. PV] /IReleaseHoldl IN ;

mRati_Re_{1_1_V___2___PV_.z-]=mRati_Re14__d[{__1_V___2___PV}.z]__Re_ea$eH__d__N:

mRatio$m [ {1. I.V)+ (2.1. _V}. z_] =mRatiolmMold [{1. i+ V}. (2. I. PV}. z ]IIRelealeHoldl IN;

mRatioRe t {3.1. PV}. {2. I. PV}. z_l _RatioReHo1d |{5. I. PV}. (2. I. PV}. zII/ReleaseHoldl IN;

mP_tioZm[ {3.1. IV}. {2.1. PV). z_] =mR_io_mHold [{3.1. PV}. {2.1. PV} .z)I/ReleaseHoldl IN:

mRatioRe[ {I. r_.V}. Ii. 1.V}. z_] --llRatio_eHold[ (1. r.v}. {1.1.V}. zl//RelealeHoldl/N:

mRatiolm[ [i. r_.V), li. 1.v}. z_] _IRatioZ_IHold[ {1. r.v} + {i. 1.V}. z]/IReleaseHoldl/N:

mR_ioRe I(_. r_. PV) .{2. i. PV}. l_] _l_l_ioRel_old [(2. r. PV). (2.1. PV}. z] IIReleaseHoldllN;

mRatiolm [{2. r_. PV]. {2. I. PV). z_] =mRa_io_mHold [{2. _. PV). {2. i+ PV}. z] llReleaseHoldllN;

mRa_ioRe [(3. r_. pV}. {3.1. PV}. z_] =mRa_ioReHold [{3 +r. PV). {3.1. PV}. z ]//ReleaseHold//N:

mRatiolm[ {3. r_. PV). {3. i. _}. z_) _Ratio_old [(3. r. _V). (3. i. PV}+ z ]//ReleaseRold/IN;

(12.4_33 Second. Null]

Dk_eRe_t 0. r_. z_.V] =D [kseRe [0. r. z .V] .r)//N;

Dkselm_[ 0. r_. z_.V] =D (kseIm [0. r. z. V] .r]//N;

DkseReR [0. r_+ z_. _V] =D(kseRe [0. r. z. PV] .r]//N;

DkseImR [0. r_. z_. PV] =D(kseIm[0. r. z. PV] .r]//N:

_Ra_ioReR ((1. r_.V}. (i. 1 .V}. z_]=D[mRatioRe[ {i. r.v). (i. 1.V) •z] •r] //N;

DmRatioImR [{i. r_.v) .{i. 1 .V}. z_] =D II_satioZm [{i. r.v). (i. 1.V) .z] .r)//N;

_RatioReR [{2.r_.pV}. {2.1. PV) .z_] =D[mRatioRel {2.r. PV} .{2.1. PV) .z] .r l//N:

DmRatioImR[ {2. r .PV). {2.1.PV). z_) _D[mRa_iolm[ {2.r. PV}. {2. I.PV) .z| .r]//N;

_RatioReEI {3. r_. PV). {3.1._V). z_] =DlmRa_ioRe[ {3. r. PV}. {3.1.PV].z] .r]//N;

_atioImR[ {3.r_. PV) + (3.1. PV}. z_) =D[mRatioIm[ (3. r. PV}. {3.1+_V}. z] .r|//N;

(o

DkseRe_[0.r_.z_.sigma_)=DlkseReHold[0.r.z.si_ma]//Rel_aseHold.rll/N;

DkseI_R{O.r_.z_.sigma_]=D[kseZmHold[O.r.z.$igma]//ReleaseHold.r]//N;

DkseReZ_(0.r_.z_.V)=D[kSeReHold[0.r.z.V)//ReleaseHold. r3//N:

DkseZm_[O.r_.z_.V]=D(kseI_4old[O.r.z.v]//Releas_4old.rl//N;

DtimeReT[__]=D_time.Re[_]._]:

_timeZ_2[t_]=D[timelm[t].t];

D_imeTwReT[t_.phiT_)=D[tL_eTwRe[t.phiT]._;

_imeTwImTlt_.phiT_}=D[timeTwlm[t.phiT]._];

(*

Du0hat_.-'Tem_la_e = IUO-(UO-_I EX_II 2Pi phiU])z}(1-(DkseRe0rzV_ ÷ I Dkselm0rzV_l)/II-{k_eRe00zV + I kselm0+mV});

Du0hat_Tem_late = _l(oo-(Uo-Ul Expll 2Pi phi_l)z)Ii-(kseRe0rzV ÷ I kse_m0rzV) I/(1o(kseRe00zV + I kselmOOzV)).:_;

_v0_at_empla_e = ((mRellzV + _ mImllzV)/I1-(kseRe00zV+l kselmG0zV)))*

<IBmPatioRe2r?V21PVz_ + _ _IRatioIm2_PV21PVz_>-(_mRatioRelrV11Vz_ + I E_Rat_oImlrVlIVz_>l IUO-UI EX_[I 2Pi _h_U]_ -

io(mRe31zPv+_ mIm31z?V_'((B_atioRe2r_21_Vz_+I _Ratiolm2rP_21PVz_>-

(:_RatioRe3rPV31PVz_+I _mRa_o_m3rPV31PVz_I>+Tw*Exp[ _ 2Pi phiTl;

_0ha_ZTem_late = llmRellzV + I mlmllzV)/(1-1kseRe00zV÷I kseIm00zV) l) °

l(mRatioRe2rPV21PVz ÷ I mRa_ioZm2rPV21_Vz)-(m_at_oRelrVllVz + I mP_tio_ml_V11Vz)l (UO+UI EX_[I 2Pi ph_U]) -

iolmRe31z_Vol m/m31_PV)ol(mRa_ioRe2rPV21Pvz+I mRatio_2rPV21Pvz>-

(mRatioRe3_Pv31_z+I mRa_ioIm3rPV31_Vz)l°Tw°ExP[ _ 2Pi _hiT];

_0hat_Template = 0; I" _or a_ least the grafT0=0 case *)

°I

_'_ r._i ', _+:+r::++_: s_i', + :_'.;_+_+:_,_ :_t: _ ̧_ ̧_+:_ ̧c:" _:j,_r:,_ '_,v +;,+".?i_L:_: "h_

118
(. _.,,,:_,,,_T:i'e_:c ̧ _:._._c ̧ _,_ ._,_r _r,_ _p c',_:._ CL,',T_,..!_;_,= _c: t._.:_J:,_,_. t_

si=F Sqr_I-I]

the_alFindTw_Co_l_[Tw'Ex_{I 2Pi phiT_ Ex_[I 2Pi _l(Ex_[_ $i y]-Ex_{i li] I Sin[$i yl/Cos[si])];

_he_alReF indTw [y_, t_, F_, _hi T_. Tw_] ._he_al FindTw- I Coef f icien_ [thetalFindTw, I ];

DthetaIReFindTw_ _y_, t_,F_,_hiT_,Tw_]=Collect [D[_he_alReFindTw[y. _, F._hiT,Tw] ,y),TW] //N:

DTIRT_Ia_eFi_dTw = Coo_l_{(-I (((_-I)/GA_4A) {UO-U1 Ex_{I 2Pi phil]}

(mRatioRel 1V2 IPVz* I mRa_iolmllV21 PVz )"

(0-(_kleReR0rzPv*I Dkse_mR0rzPV) ) / (i- {kseRe00zV*I kselm00zV) )) -

{ (Dk_eReR0rz_V÷_ DklelmR0rzPV) - (( (GAR_A-II /GAMMA) (mRa_io_e31Pv21PVz÷I o_a_io_m31PV21PVz) °

(0- [Dk_eRe_0rzPV_I DkseImR0rz_V) ))) Tw Exp[I 2Pi phiT] )"Exp{_ 2Pi _) ];

DTIImRTe_la_eFindTw = Coef fici_1 [DTiRTem_lateFindTw, I]:

DTIReRTem_lateFindTw = DTiRTe_lateFindTw-I DTII_RTe_la_eFindTw;

DTI ReFindTwRHo IdTe_ [r_, z_, t_, V, PV, UO_, UI_, phil_, _hiT_, Tw_] •DTI ReRT_m_la teFindTw/.

(mRatioRellV21FVz->Nol_Form[mRatioRe[ {1, I,V), [2,1,PV},Z] ).

mRatioIml 1V2 I_Vz ->Nol_Form {_Rat iolm [{1. i, V}, {2, I. PV}. z] ],

DkleReR 0rz PV- >HoldFon_ [DkleReK [0, r, z, _V] ],

DklelmROrz_V->Hol_Form [DklelmR [0, r, z, PV] ],

kseRe00zV->HoldForm (kleRe [0, 0, I,V] ].

kleIm00zV->HoldForm [k_eIm[0, 0, z, V] ],

mRItiORe31FV21PVz->HoldForm ImRatioRe [(3, i, PV), {2, i,PV), z] ],

mRa_ioIm31FV21PVz->Hol_orm[mRatioIm[ (3, i, PV}, (2, l,PV}, z] ]) ;

DTI ReFindTwRHo id [r_, z_, __, V, PV, Uo_, UI_, ph/u_, phiT_, T__] -

Col lect [Expand [DTIR_FindTwRHoldT_0_ {r, z ot, V, PV, UO, U1, phiU, phiT, Tw] I.TW) ;

DT1 ImFindTw_o IdTe_ [r_, z_, t_. V, PV, Uo_, UI_, _hiU_, _iT_, Tw_ ]=DTI ImRTe_q_l a teFindTw/.

{mRati ore IIV21 PVz -_No l_Form [mRatloRe [{1,1, V}, {2, I, PV). z ]],

mRa_iolmllV21PVz->Hol_Fo_[mRatloIm[ (I, I,V}, {2, i, PV) ,z) ],

DkleReR0 rzPV- >HOI dFor_ [DkseReR [0, r, z, PV] ].

DkleImR0r zPV ->Nol _Fo rm [_k_eImR [0, r, z, PV] ],

kseNe00zV->HoldForm [kseRe [0,0, z, V] ],

kle_m00zV->HoldForm Ik_e_m [0, 0, z,V] ],

mRa_ioRe31PV21PVz->HoI_Fo_[mRa_IORe [{3,1, FV), {2, i, PV), z] ],

mRa_ioIm31PV21PVz->HoldFo_m(mRa_ioZm[ {3, I, _V}. {2, i, PV), I] ] };

DTlI_FindTwF'4old [r_, z_, __, V, PV, _o_, UI_, phiU_,phiT_, Tw_] =

Col lec_ (EA_a_d [DTI ZmPir_Tw_Mol dT_ {r, z, _, V, PV, Uo, UI, _hiu, phiT, Tw) ),Tw );

condD_taS_ar-((.8, 20,.02830),(.8, 25,.03108],(.8, 30,.03518),{.8, _0,.0_160), {.8, 50,.04766),{.8,60,.053_2],

(.8, 8_._6426)_.8'_?439}_.8.12_._8396)'{.8"_4_93_{.8_16_19}_.8_8_`_1_3_

(.8,200,.1185) ,{.8,220,.1265) .[.8,240, .1343),{.8,260,.I_18) ,(.8,280,.1492),{.8,300,.1565],

(.8,350..1740} ,{.8,400,.1909}}:

condFi_$_ar[p_,T_]=Fit[con_Da_aS_ar,[l._,T,p^2,T^2,_*3,T*3},{p, T)];

======.=======m==================================

l- NOl_ £0: Tlll_ hexe °_

For FO = 99.9996

TWha_ = 0.3102_ - 0.0779612 I

TW = 0.319915

PHIT - -0.0391786

(* OLD Solve for TW o)

_" _-_t_ :_ut _.:.z _ t_e w_i_ _oLv_ _o: _ _:*,_ _k_ t_e _._-r.. _ _.,;

Timing [

_11 ( 1, z_, __, V, E_r,uo_, UI_, phiU_,phiT_, _, F. _BE ) =_/. _var ( [ 1,1 ) ] / ! Rel_seMold//N;

deltaS(1, z_, [ _ 1_, _2_ ], V. _, UO_, UI_, phiU_, p_iT_, K_, F, _UE_) =

_all [ 1, z, t2,V, _V. UO, U1, phiU,_iT, _, F, k_u_BE)-_all [ 1, z, _I,V. _,Uo, UI,_iU, _iT, _, F,_];]

(1_.6667 Second, Null]

te_V_r_N[FindR_[de_aTw_1_{._._5_V`PV_L_PH_hiT_KGA_,F_KTUBE]_{_hiT`.3_I];

_empPHIT = 0.260_85; :_TW = -0.516552

PHZT=-0.239215

Tw_ii:{0.516552, 0.51_552, 0.516552, 0.516552. 0.516552, [0.516552}}

(" check z (in)dependence of Twall ")

Pio_ [Twall [1, z, 0, V, PV, U0 ,UL, PHIU, PHIT, KGA$, F, KTUBEI, {z, 0,1 ))

-GraphicsTimiog IBlock[ [:! =. _, _=. 45 },Plot [del_aTw[l, 0, {_i. t2 }, V, pV, U0, _L, PHIU, phiT, KGAS, F, KTUBE},

{phiT, 0,1 ],(-Plo_Range-> (-. I, .1 }T-) Axesbabel--> ["phiT ° ,"ATw" )]]]

119
(3.05 Second. -Graphics-}

Timing [B1_ck I _ _ L= _4 5 _ -_2._ .55 } _ P1_ _de1taT_.|1_ _ _ {_ _ _2 _ __ .PV _ _L_ PHIU _phi T _ KGAS _ F _ KTUBE] _

{phiT, . 35, . 45], (*PlonRange-){-. 1, . i}, -) AxesLabel-> {'phiT", "_Tw" } ] ] ]

{0._,83333 Second. -Gra_hlcs-)

FindRoot (deluaTw[1,0. { .45, .55),V, PV, UC,UL, PHIU._hiT,KGAS,F,KTUBE)==0,

(phiT, {._6, .3_} )]

('PHITffiphiT/.%[ [I] ] ;*I

I f [Less ( _ _TW, 01, PHIT= _empPRI T-. 5. E_IT = _e_qpPH_T ) ;

(-if (Greacer[_e_PKIT, *0.5) ,PHIT=tempPHIT- I,PHIT=tempPHIT) ; ° )

Print ( "PHIT=*, PH_T) ;

Print [ °Twal 1 = • , {Twall [ i. 0, . 0, V, PV, U0, UL, PHIU, PHIT, KGAS, F, _BE |,

TW = TwalI(I,0,Random[].V,PV,U0,UL.PHIU,PHIT.KGA_,F,KTUBE];

Twall (1,0, .125, V, PV. U0, UL, PHIl, PHIT. _GA_. F, KTUBE|,

Twall [1, D, .25, V, PV, U0, UL, PHIU, P_IT, _GAS, F, KTUBE ) .

Twall [I, 0, .50, V, PV, U0. UL, PHIU, PMIT, KGAS. F, KTUBE ] .

T_II [i, 0, Random( ], V, PV, U_, UL, PHIU. PHIT, KGAS, F_ KTUBE] } ] ;

Print('( ° INT_TE SAVING OF C_CUf_ TwalIII_0r_VIPV,U0,ULIPHIU_phIT, KGAS.F,KTUBE| *)'_;

Print I " [_ FOR parame_erLii_('.pLi$_var, ") *, " ") "] ;

Prin_ [ -TW_II [i, 0, __,V, PV, U0, UL. PHIU.phiT_. KGAS, F, KTUBE] =", Twall [ i, 0, _, V, PV, U0, UL, PHIU, P h/T. KGAS ,F, KTUBE 1 ,- ; • _ :

(" INTERMEDIATE SAVING OF CALCULATED lambdaAtl and TwalI[1,0,C,V, FV,U0,UL,PHIU,DhiT, KGAS,F,KTUBEI ")

[° FOR parame_erLis_|pLis_var] °)

lambdaAt1=0. 00394773 - 0. 0000298093 _; lamb_aAclRe_0. 00394773 ; lambdaA_iIm=-0. _000298093 ;

T_al I [ i, 0, t_, V, _V, U0, UL, PHIU, phiT_, KGAS, F, KTU_E) =

(-0.00807124 (??.2504 COS(6.28319 C) _ 88.2443 Sin(6.28315 C])) /

(15.1675 C0s(1.3376 ° 6.28319 phiT - 6.28319 t| - 6.39891 Cos(6.2B319 (phiT * t|) -

84.1828 COS16.28319 phiT * 6.28319 _] - 8.66834 $_n(1.33_6 - 6.28319 phiT - 6.28319 t| ÷

0.526a23 $_n[6.2B319 (ph/T _ t)] o 84.1828 $in[6.28319 phiT * 6.28319 t]);

,==,.,m,,..=_,= (* AIYI"OBR.EAK *} _"='===ffi==='==

• pARAMETER LIST {_o= u_e in exporting to plo_ rou_inss)

¥_r_eCecLI_C. {V, _V ,KGAS ._. UO ,UL. _. ep_ ,GA_, PHIU __, TE_, N [TW ._ ] ,N [THIT. $ ] )

-6 5

Paz_eterLis_=(5.641g Sqrt[Pi], 4.72932 _t(Pi]. 0.0091411, I_, i., I., 1.20617 I0 Pi, 0_002, -, -0.I. 0.0564191 Sqr_[Pil, 16, 0,31991, -0.039181

3

• SOL_'T_ON OF T_ AT PHIT FOR CONTINUITY OF MEAT FLUX AT GAS/TUBE BOUNDARY

T_0.319915 a_ PMIT.-0.0391796

• RANDOM TI_4E CHECK, ALL VALUES FOR _ SHOULD BE THE SAME

_Rando_= 0. 215549

THITffi - 0.0391796

Twall.{Twall(1., 0. 0., i0.. 8.3825, I,, I.. -0 I, -0_0393796, 0.0091411, 0.I, i,].

T_all[l., 0, 0.125, 1_,, 8.3825. 1,, I,, -0.1, -0.0391796, 0.0091411, 0.i, I.].

TW_II[I., 0, 0.25. 10., 6,3825, i., _.. -0.i, -0,0391_96, 0.0091411, 0,1, I.].

Tw_ll|l., 0. 0_5, i0., 8.3825, i., I.. -0.1. -0.039179_. 0.0091411. 0.1, i.I,

Twall[l., 0, 0.215549, I0., 8._B25, i., i., °0.I. -0.0391796, 0_00g1411, 0.I, I_]]

Besiel F_ Ac_u_acy_ { (V} ;_selJ; (h_°I bel): Mo_Bes_elJHold; Ar0u_t.>in|ini_y ap_zoxi_tio_}

B_SS_I AcCUracy Sl = {{9.9999_925|. 149.847528, 149,847526, 14g. O47525, 148.537427}

Bessel AcCuracy Lo- {{0.999999925|. 1.01552483_ 1.01552463, 1.01552453, 1.00386903)

• NON-DIM_IO_J, SCALZN_ VARIABLES CALCULATE_: -5

Vail00.; Pr=0.703: Pr°Veffi?0.2663; $ I500.; Fo=IO0.; _i0.0060049_: MI3._0929 i0 ; Re0 .4.8_002

pO.l; TO[z]=l : rhoO[z]-l

• NON-DIM_IO_L VARIA_ USED IN COMPLEX SOLUTION:

-6 -9

V=I0.; P.0._3825; PV:8.3825; spoi0,002; _=0.1; H:5.78529 i0 : LAMBDA:4.30762 I0 ; eps*va=0.2

5

U0=I, ; UL=I.; PHIU=-0,1: TE_=I6: KGAS =0.0091411; KTUBE=I.; GA_--; d0=0.002: _LL=0.002; E_IT=-0.0391796; T_=0.31951_

3

AR_IVE Yazite= List

t• NE-'; " ;

_" re: 2 /,'?:_ "}

{" ?'_.r 'F'._i:. el,: ":

=================================================

(" uO, TO, T%. _l. _hol ")

plhatTez_la_e = ( I-I/(I- IkseRe00zV*_ kselm00zv) I )°

(mRa_ioRellV21PVz * I mRa_io_mllV21PVz )IUo-ul Ex_(I 2Pi phiUI )-

(mRa_ioRe31PV21PVz * I mRa_io_m31PV21P_z) Exp[I 2Pi phiT]Tw);

plhatTe_qPlateTwha_ = ( ( -If Ii ° (kseRe00zV÷_ kseIm0Dzv) ) ) "

120
(mRa_IoRelIV21PVz * I mRa_iolmlIV21PVz )(Uo-UI Exp[Z 2Pi phiU] )-

(mRa_ioRe31PV21PVz + I mRa_ioIm31PV21PVz> Twha_);

p2ha:Te_pla_e = Z ((Uo-01 Exp[I 2Pl phiu] )/ (i- (kseRe00zV*I kselm00zV) ) ) (z^2/2] - I z/(I- (kseRe00zV÷I kselm00zv) );

u0ha_Te_la_e = [Uo o (Uo-UI Exp[l 2Pi phiu] )z } 11- (kseRe0rzv _ I kseZm0rzV) 1/ (1- (kJeRe00zV _ I ksezm00zv) ) ;

Tlha=Te_lace = (-I (((GAMMA-I)/GAMMA) (Uo-UI Exp[I 2Pi phiU]) (mRa_ioReliV21Pvz*I mRa=ioZmllV21Pvz) °

(l-(kseRe0rzPV*I kseIm0rzPv) ) / (i- (kseRe00zv÷I kseIm00zV) )) ÷

((kseRe0rzPV_I kleZm0rzPV) -( ((GA_A*I)/GAMMA) (mRatioRe31PV21PVz_I mRa_ioim31PV21PVz) °

(l-(kseRe0rzPV÷I kseIm0rzPV} )}) Tw Exp[l 2Pi phiT] ):

v0haCTez_la_e = ( |_ellzV + I mlmllzV) / 11- (kseRe00zV÷_ kseIm00zv) })-

{ (mRacioRe2rPV21PVz _ I mRacioIm2rPV21PVzl-(mRacioRelrVllvz + I mRaCiolmlrVllVz)} {UO-UI E:Cp[Z 2Pi phiU]) -

I'(mRe31zPV÷I mIm31zPV) ° ( (mRa_ioRe2rPV21PVz_I mRa_ioIm2rPV21PVz)-(mRacioRe3rPV31PVz*I mRa_io1m3rPV31PVz) )°

Tw'Exp[Z 2Pi ;>hiT] ;

rholhatTea_late = plha_Templace/T0[z] - rho0[z_ Tlha_T_la_e/T0[z]:

par[icleTraceha_Tem_la_e = -I (do- (do-dl Ex_ [I 2Pi phiU] )z} (1- [aDOKe÷I aDOI_) )/(1- {bDORe*I bDOIm} ):

plYolha_Tem_la_e = l-I 2 GAMMA {r IDTlha_Re_.Var*l DT1ha_I_Var)/power[Pv,2] -

(mRellzV_I mlmllzV) (Uo-UI Exp[I 2Pi phiU])/(1-(kseRe00zv ÷ I kseIm00zV))l);

plTe_pla_e = Cos_lex_|Ex_[I 2Pi _] plha_Te=_la_e];

u0T_la_e = Co_lex_IExp[_ 2Pi _] u0ha_T_lace];

TIT_la_e = Co_l_[Ex_[I 2Pi _ Tlha_T_la_e];

v0Tem_la_e - C_lex_[Ex_II 2Pi _] v0ha_T_la_e]:

_aruicleTraceTem_la_e • Co_lex_[Ex_[I 2Pi _ par_icleTraceha_T_la_e]:

pIVoiT_la_e = Com_lexExpand[Exp[I 2Pi c] plVolha_T_la_e_:

rhoIT_la_e = Cm_l_[Exp[I 2Pi _] rholha_Tem_la_e_;

plReTem_la_e = plTempla_e-I Coefficie_[plT_la_e,I):

u0ReTem_late = u0Te_plate- I Coef ficienc (u0T_la_e, I] ;

TIReTem_lace = TIT_la_e-I Coeffici_[TiTem_la_e,I];

v0ReTe_la_e = v0T_la_e-I Coefficient[v0T_la_e,I];

par_icleTraceReT_la_e - par_icleTraceTempla_e-i Coefficient [particleTraceTe_la_e, I ];

plVolReTem_la_e = plVolT_la_e-I Coefficien_[plVolT_la_e, I ];

rholReTem_la_e = piReTe_pla_e/TO[zl - rho0[z] TiReTe=_la_e/TO[z]:

(-

plReha_Tem_late = plha_T_la_e-I Coefficien_[plha_T_lace,I] :

u0Reha_Te_la_e = u0ha_T_la_e- I Coef ficie_ [u0ha_Te_la_e, I ]:

TiReha_T_la_e = T1ha_Te_la_e-I Coefflci_[Tlha_Te_lace,I];

v0Reha_Tw_la_e = v0ha_T_la_e*l Coefficlen_[v0ha_T_la_e,I];

par_icleTraceReh_T_la_e = par_IcleTraceha_T_mpla_e-Z Coefficienn[_ar_icleTraceha_Templa_e,_]:

pivolReha_Te_pla_e = plVolha:Templa_e-i Coefflcien_IplVolha_Templane, I] ;

rholReha_T_)la_e = _IReha_Te_la_e/T0[z] * rho0[z] TIReha_Te_la_e/T0[z];

"I

u0hanImTempla_e=Coe f f£ci_ (Coveted [u0ha_Te_la_e], I ]:

u0ha_ReTempla_e-Co_lexEx_[uOha_T_la_e] -I u0ha_ImT_la_e;

u0ha _CCTem_I a_e-_ 0ha _ReTe_la _e- I u0ha_ImT_la_e:

v0ha_imTe_pla_e_Coef ficie_ (Co_I exEx_and [v0ha_Tem_la_e ] ,1 ];

v0ha_ReTa_la_e=Co_lexExpand[v0ha_Templa_e] -I v0ha:ImTe_la_e;

v 0ha_CCT_la_e_v0ha_ReTem_la_e- I v0ha_ImT_la_e:

Tlha_ ZmTem_lat e=Coef £ici_ ICo_lexEx_and [T lha _Tem_l a_e ],Z ];

Tlha_ReTe_la_e=C<_lexE_[Tlha_Tem_la_e] -I T1ha_ImT_la_e;

Tlha_CCTe_l a_esT1ha_ReTem_la_e- I Tlha_ImTem_la_e:

plha _ImTem_l a _esCoef fici_n_ [Co_l exExpand [pl ha _Template ], I];

plh_ReTem_la_e=Compl_[plha_Templa_e) -I plha_ImTem_la_e;

plha _CCTempla_ e=plha_ ReT_ la_e- I plha_ImTempla_e ;

p IVO lha_ lmT_l &_e=Coe ff ici_ [Com_l exExpand [piVolha_Te_pla_e J, I ];

plVolha_ReT_la_e=Co_plexE_[plVolha_Te_la_e] -I plVolha_ImT_la_e:

p ivo iha_CCT_N_I a_e=pivo lha_ReT_la _e -I plVolha_lmTempla_e;

rholhat ImTempla_e=Coe f ficien_ [Compl ex_ [rholha_Templa_e] ,I]:

rholhacReT_la_e=Cor_lexEx_a_d[rholha_Te_la_e] -I rholhatImTe_la_e:

rholha_CCTe_Ola_e=rholhatReTeE_la_e-Z rholha_ImTempla_e;

DTlhatRTempla_e = Coe_l_d[ (-_ [({GA_A-I)/GA/@L%) [UO-UI Exp[I 2Pi phiU] ) (mRatioRelIV21PVz÷I mRatiolmliV21PVz} °

|0- (_kseReR0rzPv+l DkseI_L_0rzPV) ) / (i- (kseRe00zV_I kseZmO0zV) )) ÷

[ (DkSeReR0rzPV*I DkseZ_R0rzPV) - I((GA_4A-I)/GAMMA) (mRa_ioRe31Pv21PVz÷I mRa_iolm31PV21PVz)"

(0- (D_seReR0rzPV*I DkselmR0rzPVl )}} Tw Exp(I 2Pi phiT] ) ]:

DTlha_ImRT_la_e = Coefficlen_[DTlha_RTem_la_e,II ;

DTlha_ReRT_m_la_e z DTlha_RTemplace-I DTlhaclmRTem_la_e;

DTIha_Re_HoIdT_ [r_. z_, __, V, PV, UO_, UI_, phiU_, phiT_, Tw_ ]=DTIha_ReRT_Ia_e/.

{mRa_ioRe11V21PVz->Holdrorm [mRacioae [(i, 1,V}, {2,1,PV) ,z} ),

_Ranlolmll_21PVz->HoldForm(mRa_iolm| {I, 1,v}, {2,1, PV),z] ],

DkseReR0 rz PV- _Ho idFo rm [DkseReR |0, r, z, PV ]],

Dks elmR,0 rzPV- >HO idForm IDkl eI=_ I0, r, z, PVI ],

kseRe00zY->HoldForm IkseRe [0, 0, z, V_ ],

kseIm00zV->HoldForm [kseZm (0, 0, z, v] ],

mRa_ioRe31PV21Pvz->HoldForm[mRa_ioRe( (3, i,PV}, {2,1, PV), z_ ],

mRa_ioIm3 iPv21Pvz->HoldForm[mRa_ioIm( {3. i,PV), {2,1, PV), z] I):

DTlha_Re_Hold |r_, z_, __, V, PV, UO_. UI_, _hiu_, 1_hiT_, Tw_] =

ColIec= [Expand[_TlhacReA_oldTemp[r. z, _, V, PV, Uo. UI, phiU, ph2T, Tw} ],Tw] :

_TI ha_ I_RHo idTe_ [r_, z .__, V, PV, Uo_, UI_, phiu_, phiT_, Tw_ 1=_Tlha_ImRT_l a_e/.

{mRa_ioRel 1V21 PVz->HoldYor_ [mRa_ioRe [{I, 1,V), {2. I, PV), z_],

mRa_iolmllV21PVz->HoldForm [mRa_iolm [{I, I,V) ,{2. I, PV), z] ],

DkseRe_ 0 rzPV- > HO idFo_m IDkseReK [O, r, z, PV] },

Dk_eImK0rzPV->HoldForm[DkseImR [0. r, z, PV] ].

kseRe00zv- >HoldForm [kseRe [0, 0, z.V} ),

kselm00zV->HoldForm (kseIm [0, 0, z,V] ],

mRa_ioRe3 IPV21PVz->HOIdFOrmImRa_ioRe [(3, i, PV}, {2, I, PV}, z_ ],

_R_ioI_31PV21PVz->HoIdForm [mRa_iolm [{3, i, PV}, {2, i, PV}, z ]]}:

DTlha_ ImRHold Ir_, z_. __. V. PV, UO_, UI_, phi0_, phiT_, Tw ]=

Coll ec_ [Expand [l_Tlha_ I_0_Hol dT_ (r, Z, _, V, PV. Uo, UI, phiU, phiT, Tw] ],Tw 3;

(*

v0ha _Te_1 a_e //Comet exExpand

% ==v0ha_ReTem_la_e* I v0ha_ImTempla_e

*)

(.

plha_CCT_la_e = plRehaCT(s_la_e-I Coefficien_(plha_Ten_la:e,I];

v0ha_CCTem_la_e = v0Reha_Te_la_e-_ Coefficien_ Iv0ha_Tem_la_e, I };

_ar_icleTraceha_CCTe_q_la_e = par_icleTraceReha_Tem_la_e-I Coefficien_ [par_icleTraceha_T_la_e, I} :

plVolha_CCTe_q_la_e = plVolReha_Tes_lace-I Coefficlen_[_iVolha:Tem_la_e,II ;

-)

121
plAssign= { kseRe00zV- )HoldForm I kseReHold [ 0.0, z, sigmaVl l,

kselm00zV->Sol_orm [ kseImHold [ 0, 0, z. sigmaV ] ],

mRatioRellV21PVz->HoldFom[mRa_ioReHeld [ {1,1. sigmav}, (2,1, eigmaPV), z] ],

tara Ciolml 1V21 PVz - >He l_Form [mRa_iolmHold [ (1,1. sigmav), { 2,1, sigmaPv ), z ] ],

m_acio_e31PV21 PVz->HoldFon_ [mRaCioReHold [ ( 3,1, SigmaPV), (2.1, SigmaPV}, z ] ].

mRa_iolm31PV21PVz->Holdrorm [mRa_olmHeld [ ( 3,1, sigmaPV ), (2.1, sigmaPV }, z ] ] ) ;

plKeHold[z_. __. ei_aV_, sigmaPV_. Uo_. UI_. phiu_, phiT_. Tw_] --plReTemplate/.plAseign;

_l R._ [ z_. t_. s igmaV_, sigma PV_. Uo_. UI_. phiU_, phiT_. TW ] :zplReHold [z. t. sigmaV. Sig_tPV. Uo. UI. phiU. _hiT. Tw ] / /

ReleaseHold//ReleaseHold//N;

plRe [ z_. t_. V. PV. U0. UL. PHIU. PHIT. TW] =plRe [z. t. V. PV. U0. UL. PHIU. PHIT. TW] ;

pIE_ (z_. t_.v. PV. U0. UL. PH_U. PH_T. 0]=_iRe[z. t.v. PV. U0. UL. PHIU. PH!T. 0] ;

_I _ _ He, d [ z_. sigmaV_, si_PV_. UO_. UI_. phiU_. ;)hiT_. TW ]=plhatT_lat e/. plAssign ;

_i_ _ Iz_. sigmav_, sigmaPV_. UO_. UI_. phiU_, phiT_. Tw ] :=plhatHold ( z. eigmav, ei_®_aPV. UO. UI. phiU. phiT. Tw] / /

ReleaseRoldl/ReleaseHold//N;

p!ha_ [z_. V. pV. U0. UL. PHIU. pHIT. Tw_l --_l_t [ z. V. PV. U0. UL. PHIU. PHIT. Tw] //Cow_lex_;

_l h_ _Rc /d [z_. ei;n®aV_, sigmaPV_. Uo_. UI_. phiU_. Twhat_ ]=plhatTe_lateTwhat/. DiAs sign:

_i_a_ [z_. sigmav_, eigmaPV .UO_. UI_. phiU_. Twhac_] : zplhacHeld [z. sigmav, sig_IPV. ?o. UI. phiU. Twhat ] / /

ReleaseHold//ReleaseHold//N;

plh_ [z_. V. PV. U0. UI_. _hiu_. re_| =p!hu_|z.V. PV. UO.Ul._hi_.Twhat_Ul.ph1_.fo]]:

p_old [ z_, sigmaV_, Uo_. UI_, phiU_] =__hatTemplate/.plAssign;

_2ha _ [ z_, sigmav_, Uo_, UI_. _hiU_ | : zp2_atHold [ z, |igmaV, UO, U1, phiU ] //KeleaseHold//Release_old//N

(-_._ [ z_. V. U0,UL, PH_U] =p_._,st[z,V.UO.UL. PH_U]//Co_I_;*)

Dp2ha_Z (z_. v. UO_. UI_. phiU_] =D [_2_mc Iz. V. Uo. Ul.i_iU]. z] :

D_2ha'.Z(z_.V.U0.UL. PHIU] =D[pl.ha_ |z. V. U0.UL. PHIU]. Z| ;

plVolAs$i_nz {mRel IzV- >HoldForm |raRe [ i. 1, z, V1 ] ,

mlmllzV->Hol_For_[mIm[ i. 1. z. V| ] .

kseRe00zV->HoldForm [kseReHold [0.0. z. V| ].

kselm00zV->NoldForm [ kseImHold [O, 0. z, V| ].

DTIhaCR_Var - >He idFor_ (DTlhatRePRold [i. z. _. V. PV. Uo. UI. phiU. DhiT. Tw] ].

DTlha_Im_Ver ->HoldForm [DTIhatlmRHeld[ I. z. _. V. PV. UO. UI. phiU. phiT. Tw ] ].

5->1} ;

plVolRe_old [ z_, t_. sigmaV_, $igmaPV_, UO_. _1_, ph_U_, pl_T_, T__I -plVolKeTer_laCe/. plVelAssign;

plVolRe [ z_. t_. sigmav_, sigmaPV_. UO_. UI_. _hi__. DhiT_. Tw_] :=_IVolReHol d [z. t. sig_av, sig_aPv. Uo. UI. phiU. phiT. Tw ] / /

Rele8 seHol d / / Rel eas e/4old //N;

pI VoIRe [z_. t_. V. PV . UO .UL. PHIU .PHIT . TW] =pI Vol Re [z .t. V. PV . _O. UL. PHIU . PHXT . TW I :

pivolRe [z_. c_. V. PV. U0. UL. PHIU. PHIT. 0 ]=plVolRe ( z. t. V. PV. U0. UL. PHIU. PHIT. 0 ] ;

TiAesign= ( mRaeioRel 1V21PVz- >HoldForm [mRatioReBold ( {1,1, sigmav }, [2.1. sigmaPV ), z ] ],

mRat ioIml IV2 iPVz- >HoldForm [mRatioz_old [ { I. i. sig_av ). (2. i. sigmaPV ). • ] ].

kseRe00zV->HoldForm | kseReHold [0.0. z. sigmav] ].

kselm00zV->HoldFo_ I kselmHold [0.0. z. sigmav] ] .

mRatioRe_IPV21PVz->HoldForm[mRac1oReHold | (3.1. sigmaPV). {2. I. sigmaPV] . z| ] .

mRatioIm_iPV21PVz->Hol_Form[mRatioIa_old | (3. i. sigmaPv). {2.1. si_maPvl, z| I.

kseRe0rzPV->HoldForm Ik_eReHo1d [0.5. z. sigmaPv ] ].

kselm0rzPV- >RoldForm [kseImHold [ 0.5. z. si_maPV] ] } :

T iReHold [r_. z_. t_. sig_aV_, si_maPV_. Uo_. UI_. _hi__. _hiT_. Tw_ )=TiReTe_late/. TIAssig_ ;

T 1 _e Jr_. z_. t_. sigmaV_, si_PV_. UO_. UI_. phil_, phiT_. Tw_] :=

TIReHoId [r. z. c. sigmaV, sigmaPY. UO. _i. phiU. phiT. Tw) //ReleaseHold//ReleaseHold//N:

TI._ [r_. z_. t_. V. PV. U0. UL. PH_U. PHIT. 0] =TIRe Jr. z • C .V. PV. U0. UL. PHIU, PHIT, 0] ;

TIA_ [r_. z_. t_. V. PV. U0. UL. PHIU. PHIT. TW] =TIRe i r. z. C. V. P_. U0. U5. PHIl. PHIT. TW | ;

_I[r_.z_.__.V.PV.U0.U_.PHIU.PHIT. 0| =.Ex_Ind[D[TII_Ir.z.t.V. PV.U0.UL.PHIU.PHIT.0]. 5]];

_I [r_. z_. C_. V. pV. U0. _L. PHIU. PHIT. TW] =-E_d [D[TI_ Jr. z. t.V. PV. U0.UL. PHIU. PHIT. TW] •r ] ]:

T !._._'_H 3 i _ | r_. z_. SigmaV_. sig_APV_. UO_. UI_. phil_, phiT_. Tw_] =TlhatTe_lace/. TiAssign;

Tlh_ [5_. z_. sigmaV_, sigma_v_. Uo_.Ul_.phiU_.phiT_. Tw_] :=

Tiha:Hold [r. z. sigmaV, si_aPV. _o. UI. _hiU. phiT. Tw ] / /ReleaseHold//ReleaseHold//N;

TI_ |r_. z_. V. PV. _0. UL. PR_U. PHIT. Tw_| =TID_[r.z.V. PV. UO.UL.PHIU.PHIT.Tw]//COm_I_;

(° Expand for some unknown reason keeps in a form for PTIhatR CO be in Re*I Zm form')

DTlhatRIr_.z_.V.PV. U0.UI_.phiU_.PHIT.Tw_] =D[TI-_a: It. z.V.PV.U0. Ul._hiu. PHIT. Twl .El ( * //Cou_lex_d ° ) ;

DTIhatR[r_.z_.V.PV.U0.UL.PMIU. PHIT.Tw_]=D(_Iha_ Ir.z.V. PV.U0._. PHIU.PHIT.Tw] .r] (- //Co_lexExpand*) ;

_lhat [r_. z_. V. PV. U0. _L. PHIU. PHIT. 0 ] =-DTIha_R [r. z. V. _V. U0. UL. PHIU. PHIT. 0 _ ;

qlha_ [ r_. z_.V. PV. U0._. PHIU. PHIT. TW] = - DTIhaCR [r. z.V. PV. U0. UL. PHIU. PHIT. TW] ;

In_Tlha_R01 I z_. V. PV. _0. UL. PHIl. PHIT. TW ] =

2 Integrace[r TIh_C|r.Z.V.PV.U0.UL. PH_U.PHIT.Tw] . (r._.l]] I*//Complex_d °) :

NuC [ z_. V. PV. U0. _L. _HIU. PHIT. TW_] =

-DTIhatR[1.z.V.PV. U0.UL.PHIU.PHIT.Tw]/(Tw Ex_II 2Pi PH_T_ - IntTIha_R01[z.V.PV.U0.UL.PHIU. PH_T.Tw])

(" //Cem_,l_") l/N:

(* CAN't DO _hls. _ot sl_ecify PH_T and TW indepe_de_ of _hiU _ UI!!!!

In_Tlha_R01 [z .V. PV. U0. UI_. phi3_. PH!T. Tw_ ]=

2 Integrate [r '._lha _ [r. z. V. PV. U 0. UI. phiU. PHZT. Tw] • (r. 0.1 } ] (" / /Co_pl exExpand" ) :

NuC [ z_. V. PV. U0. Ul_.phiU_. PH_T. Tw_] =

-DTIhatR[ 1. z.V. PV.U0.UI.pb_iU. PHIT.Tw] / (Tw Ex_(I 2Pi PH:T] - IntTlhatR01 [z .V. PV. U0.UI. _hiU. PH_T.Tw] )

{- //Com_l_d ° }/fN;

-)

I-DTlhatTw_at R [r_. z_. V. PV. U0. UI_. phiU_. Twha__] =

Collect [Ex_a_d[D[TlhatT_at [r. z.V. PV. U0._I .phiU.Twhat I ,r]] ,Twha_ ; ° )

Int TlhatR01 [ z_. V. PV. U0. UI_. PhiU_. Twhat-] =

Collect (2 Inte_rateIr T1hatTwha: [r. z.V. PV. U0. U1 .phiU.Twha_] . (r. 0.1) ]. Twha_] ( • //ComplexEx1_nd" ) ;

_C [z_. V. PV. U0. UI_. phiU_, re_| =

- (DT1hatTwha_R[l.z.V. PV.UO.Ui.phiU.Twha:[U1._h;U. fol]/ITwhat[Ul.phiU, f°] -

IntTlhacR01 [Z.V. PV. U0. U1. phiU. Twha_ (Ul.phi_. fol ] ] )

[" / ICon_l_d/IN" } :

D_I tat _a_ _ 0. V. PV. _0. UL. _HI_. PHIT. 0 ] =T ihat [i. 0.V. PV.U_.UL.PHI_. PHIT. 0 }-IntTlhatR01 | 0. V.PV.U_.U_. PHIl. PH_T. 0_ ;

Del taTlhat [ 0. V. PV. _0. UL. PHIU. PHIT. TW] =Tibet [i. O. V. FV. U0. UL. PHIU. PHIT. TW] -IntTlhatR01 [0. V. PV. U0. UL. PHIU. PHIT. TW ] :

DeltaTI[0.__.V.PV.U0.UL.PHIU.PHIT.0] =ReIDelcaTIhaA[0.V.PV.U0.UL.PHIU.PH_T.0] Ex_[I 2Pi t]]//N;

Defeat i [ 0. t_. V. PV. U0. _L. PHIU. PHIT. TW] =Re [Del tat !_ _ [0. V. _V. U0. UL. PH_. PHIT. TWI Exp I 1 2 Pit ] 1/ /N;

122
TI[z_,Z_,__.V, PV, U0,UL, PHIU,PHIT,0J =Re(T!hu_.(r,z.V, PV, U0,U_,PHIU, PHIT, 0] Ex_[I 2Pi t]]//N;

TIIr_,z_,__,V. PV, U0.UL,PHZU,PHIT,TW] =Re[Tlhat(r,Z,V. PV. UO,UL, PHIU,PHIT,TW] Exp{I 2Pi t)]//N;

rholReHold [r_, z_, t_, slaV_, sigmaPV_, UO_, UI_, phiu_, phiT_, _h_ _=

(plReHold (z, _, $i_p_av, Bi_PV, Uo, UI. DhiU, phiT, Tw ]-TIReMold [r, z. t, slgmaV, $igmaPv, Uo, UI, phiU, phiT. TwJ )

rholR_ [r_. z_. __, sigmav_. $i_PV_, Uo_. UI_, phiU_, phiT_, Tw ] :=

rh_ReH_d {r_ z. t_ si_maV_ si_maPV_U_ U_phiU_phiT_ Tw] //Re_ea$eH_d/ /Re_eaIe_1d/ /N;

_holReIr_,z_,t_,V,PV, UO,UL.PXIU, PHIT,TW]=rholRe[r,Z,_,V, PV,UO,UL,PHIU,PHIT,TW];

the 1Re [r_, z_, __, V. PV, U0, UL, PHIU, PHIT, 0 ]=rholRe [r, z, _,V, PV, U0, UL. PHIU, PHIT, 0 ];

thetal=C_le_[Tw'Ex_II 2Pi phiT] Ex_[I 2Pi t| (Ex_[I si y]-Exp(I si] I Sin[$i y]/Cos(si])];

_hetalRe[y_.t_.fo_.phiT_.Tw_l=N[thetal-I Coefflcie_t[_hetal. I]]:

_he_alR,_[y_,t_,fo_.PMIT.TWl:thstalRe{y, t,fO,PHIT,TW);

_he:_iRe[y_,t_.fo_,PMiT,O)=:he_alRe[y._,fo. PHIT, O]:

uOAls_gn={kseReOrzV-_HoldForm(kseReXold(O.r.z,sigmav]],

kseIm0rzV->HoldFormIkseIm_old[0, r,z,$i_mav_l.

kseRe00zV->HoldForm(kseRe_old[0,0,z,siOmaVl_,

kseIm00zV->xoldForm[kseImNold[0, 0,z.s_v]|);

uOReHold[r_,z_.__.si_av_.uo_,Ul_,phiU_|=uOReTem_lete/.uOA_slgn;

u_Re_r-_z-_t-_sigmaV-_U_-_U_-_PhiU-]:=u_Re_[r_z`t_si_T_aV_U_*U_hiU]//Re_easeH_d//Re_ease_4_d//N_

=ORe(r_.z_,t_.V, UO,UL,PHIU]=uORe[r,z,t,V. UO.UL,PHIU];

_i_=Held(r_,z_,sigmaV_,Ue_,Ul_,phiU_)=_OhatTem_la_e/.uOAssign;

u_h_={r_, z_.s_mav_.Uo_,Ul_,phiU_]:=uO_Hold[r,z,sigmaV,Uo, U1,phiU)//ReleaseHold//Release_old//N;

uChat[r_, z_,V, Uo_,Ul_,phiU_| =u0nat [r, z,V, Uo,Ul,phiU]//com_l_d:

In_uOha_ROl[z_,v, uo_, Ul,&hiU_]=2 Integra_e(r u0ha_[r,z,v, Uo,Ul.phiU], (r, O,1}]//C_lex_d:

Zn_u0ha_R[z_,V,Uo_,UI_,Dhiu_]=2 Zn_egra_e[r u0,ha_[r,z_V.Uo,UI,DhiU].r]//Co_lexEx_nd;

fC(z_,V,Uo_,UI_,phIU_]=D_!_Z[z,V, Uo,UI,_hiU]/(Z rho0{z] In_uOha£ROl[z.V,Uo,Ul.phiU])//Co_lex_and//N;

_{r_,Z_,V, U0.UL,PHIU] =u_ha_[r,z,V,_O,UL.PHIU]//CO_Ie_x_d;

In_u0ha_R01Iz_,V, U0,UL, PHIU]=2 In_egre_e(r u0ha_[r,z,V,U0,UL,PHIU].{r, 0,1)]//Co_lexExpand;

In_u0ha_R(r_,z_,V. U0,UL,PHIU]=2 Znuegra_elr uC_[r,Z,V,UO,UL. PHIU],r]//COn_Iex_d;

fC[z_.V]=Dp2ha_Z[z,V, U0,UL,PXIU|/(Z rho0|z] Intu0ha_R01 [z,V,U0,UL,PHIU])//Sim_Iify:

localWallShearS_resslz_,__,v, u0,uL, PHIU]=EXpand[2 Pi D{u0Re[r, z,_,V,U0,UL, PHIU].r]/.r->I];

(°2Pi i_ for integration around circumferenne °)

locelWallShears_ressS_uared[z_,__,v,u0,uL. PHIU]=Power[localWallShearS_ress[z,_,V, UO,UL._IU].2|//Ex_and;

localwellShearS_ressR_[z_,V, U0,US, PHIU]=_rtIIn_e_ra_eIlocalWallShearStres$S_uared(z,t,v,uo,uL.PMIU],{_.O,1}]]:

totalDragRM_[v,u0,UL, FHIU]=In_egrate{ZocalwallSheazS_reseRMS[z,V,U0,UL, PHIU]_{z,O,I))//N;

_ot_IDragR_Starz_o_alDragRMS[V, U0,UL,PHIU]°drag0Ster:

_otalDragRM_[t.V, U0,UL, PHIU)=In_e_reEe[locelWallShearS_ress[z,_,v, u0, UL. PHIU], {z,0,1}I//N;

_o ,,,_o_

v0Assign=[mRellzV-_Hold[mReHold{l,l,z,si_V]), m_mllzV->Hold[mZmHold[1,l,z,si_V]],

kseRe00zV-_Hold{kseReHold[0,0.z,sig_av]], kseZm00zV-_HoldIkse_mHold(0, 0,z,sigmav)).

mRa_ioReRrPV21PVz-_Hold[mRa_ioReHold({2,r,$igmaPV],{2,1,sigmaPv], Z]],mRa_ioL_rPV21PVz->

Hold[m_atloImHold[(2,r. sig_mPv},{2,1,slgmaPV),z)]_

mRatioRelrVliVz-_Hold{mRatioReHold[(l,r, sigmaV ).{1,1, si_aV },z]],mRa_ioImlrVliVzo_

Hold(m/_auiolm_oldI{l,r, si_maV },{l.l,si_maV ).z]l,

mRe31zPV->Mold[mReHold[3.1,z, sigmaPv)],mIm31zPv°_xold(m_mMold[3,1, z.s_maPv]l,

mRatioRe3rPV31PVz-_Hold[mRa_ioReHold({3, r,si_aPv},{3.1,si_a_V).z]],m]qa_ioIm3rPV31PVz->

Hold[mRa_ioIm_old[{3,z, sigmaPV},{3.1,sig_aPv),z]],

mRa_io_e2rPV21PVz->Hold[sRatioReHoldI{2.r,si_PV}.{2,1,_igmaPV},z)],mRatXoIm2rPV21FVz->

HoldJmRatloI_old([2,r,sigmaPV},{2,1,sigmaPV),z]l)_

vOReHold{r_,z_,__.si_m_V_,sig_aPV,Uo_,Ul_,_hiU_,phiT_,Tw_]= vOReTe_la_e/.vOAssign:

vO_e(r_,z_,__,si_m_V_,_igmaPV_,Uo_,Ul_,_hiu_,phiT_.Tw_]:=

v0ReHold[r,z, _,$i_mav,sigmaPv,ue,Ul,phiU,phiT,Tw]//Relea_eHold//ReleaseHold//N;

v_R_[r_,Z_.t_,V._V,U0,UL, PHIU, P_IT, 0]=v0Re[r,Z._,V.PV,U0,UL,PHIU,PHIT,0];

v_Re[r_,Z_,__,V, PV,UO,UL, PHIU. PHIT,_FW]=v0Re[r,z,_,V, PV, U0,UL,_IU, PHIT.TW];

i" _C °>

d0Assign={aDORe-_HoldFor_JkseReHold|0,r, z,sigma)l.

aD01m-_HoldForm(kseZm_old(0,r,z, sigmaS|,

_DORe-_Ho1dForm(k_eReHold[0,0, z.si_a) I,

bDOIm-_HoldFerm(k_elmHold(0,0,z,si_l]):

d0ReHold[r_.z_,t_, sxg_a_.do_,dl_,phiU_]--l_articleTraceReTem_lete/.d0Assign;

d0Re(r_.z_,__,$igmaV, do_,dl_,_hi__):=d0ReHold{r,z, _,$i_mav,do,d1,_hiu)//ReleaseHold//ReleaseHold//N

d¢_(r_,z_,__,V,dIOJ,d[l|,PHIU)=d0Relr, z.t,v,d(O].d(1].PHIU]:

Save_fileNameHXfr. _he=al,pl,rhoi. Tl.u0.v0_

(* {{eps, Va. Pr, U_, PHIU, M, $_PRa_io}, {Fo, {PHIT. TW}). Date) ")

(* {{pl[0; t=0.1, 0.2,0.3]}, pI[TW at t=.2], pisDyn_[0.0], v0[0,.8,.2], u0[0,.8,.2], {T(0.0,.1],T[0.0,.2],T{0,0,.3]}) °)

If(kind_yste_==OPT,Prmn_[EileNameOPTHX_r],. Print[fileName_THXfr]]:

Pr_n_(_pZRe_.._`V_PV_U_UL`PHIU_PHIT_0]_p_Re_.2_V_PV_U_L_PX_U_PXIT_0]_p1Re(0_.3_V._4`U0`UL`PHIU`PHIT._].

plRe[0..2,V. PV,U0,UL,PHIU, PH_T, TW],NIp£s_or_%_r_A_nch(0,0]/LInch].

vORe[.8.0,O,V. PV, UO,UL,PHIU,PHIT, O],uORs[.8,0,O,V, PV,UO,UL.PNIU],

(TIRe|0,0,.1, V, PV,U0.UL,_IU, PHIT. 0],TIRe{0,0,.2, V, PV, U0. UL,P_IU, PHIT. 01,

TIRe[0,D,.3,V, PV, U0.UL,PHIU.PHIT.0])]]

-6 -9

(({BPT, 0.002, i,, 0._03, 0.90266, 0., n/aPHIU, 3.789 I0 , 4.31 10 ), {I.004, 1.007, 1.00669}),

{10.001, (0.i02_, 0.000203363|, 7.02719), {0.00476915 Amp dyne, 0.00338034 (tu_^-ll dyne, 0.018_349 W|,

(BPT<>I., 2, 21, 18)}

123
{(0.317212, 0.485368, 0.46813}, 0.485313, 0.002, 0.0722032, u0Re[0.8, 0, 0, 0.56415 Sqr_[Pi],

0.472932 Sqr_[Pi], i.. 0., -0.3],

[0.0297984, 0.0133024, -0.00827462})

({CC. OPT, 0.698, 6.016 10E-?, 0.703, 0.66, -0.i, 0.003208, 1.427, 1.808}. {i.7559 10E6. {-0.125, 0.00407989)}. (199¢, 8, 24. 17, 7. 5))

([2.97577, 2.68448, 1.36781), 2.68702, 0.111111, 2.75102, 0.36, {1.25176 10E-7, -3.632?4 10E-7, -?.12965 10E-7)]

(" phasors ")

-6 -9

{([OPT, 0.002, 100., 0.703, ?0.266. I., -0.I, 3.?89 10 , 4.31 10 }, {1,002, 1,004, 1.00413}), (i00.. {-0.03918, 0.319915}. 7026.6},

{0.00476915 _ dyne, 0.159157 (Eube^-l} dyne, 1.97349 W). {0PT<>I00., 3. 15, 19})

TW_ PHITW Realm ReTW*ZmTW Phasors: Am_ PHI

U0 1. 0

u0(z=.5) 0.951 -0.05

uO(z-.7) 0.959 -0.0706

u0(z=.85) 0.975 -0.0856

UL 1. -0.1

pl 0.805 -0.0116

TIRe(0) 0.318 -0.0137

TIRe(l) 0.

Tlhat(0,z)0.318 -0.0137

Tlhat(1,z)O. 0

Tlbulk 0.272 0.0176

DTI 0.272 -0.482

uOpl(z=O) 0.403 -0.0058

v0lr=.8) 0.0743 -0,845

qltr=l) 2.59 -0.879

qlhaC(l.z)2.59 0.121

NuCSi_ple 9.51 0.603

NuC 9.51 -0.397

fC<0) 1.15 0.476

fC(z) 1.15 0.478

0.883 -0.026

0.353 -0.0266

0.32 -0.0392

0.353 -0.0264

0.32 -0.0392

0,343 -0.0256

0.0368 -0.393

0,401 -0.013

0.8504 -0.856

0.35 -0.79

0.35 0.21

9.51 0.603

9.51 -0.397

0.2706 ÷ 0.03006 I 0.339 - 0.05502 I

-0.271 - 0.0301 I -0.0268 - 0.0229 Z

1.87988 _ 1.78154 I

-7.584?6 - 5.?4094 I

-7.585 - 5.741 I

-1.138 * 0.1741 Z

-1.138 * 0.1741 Z

0.086417 ÷ 0.339135 1

-?.58476 - 5.74094 1

-7.585 - 5.741 I

============z============.=======_====s====-=====

(" Prelim *)

Calcs

U_T1

_woTimesu0TiAvgT_late = uOha_Tem_la_e * Tlha_CCT_late llCC_lex_d:

twoTimesu0TiReAvgTempla_e = cwoTimesu0TiAvgTemplate -I Coefficien_[_woTimesu0T1kvgTemplate,l];

t WOT imesu 0T 1 ReAvg F 1 uY.H 01 d [ r_. z_, $igmav_. sigmaPV_, UO_, UI_, phiU_, phiT_, Tw ] =

_woTimesu0TiReAvgTem_laCe/.Un/on[_0As$i_n,TiAssi_n]:

_0TiReAv_Fl%_x[r_,z_,si_P_aV_,si_PV_,Oo_,Ul_,_%i0_,phiT--,Tw-]:= i/2 "

_woTimesu0TiReAvgFluxHold[r, z,siG_aV, si_maPV,Uo. Ul,phiU,phiT,Tw]//ReleaseM°Id//ReleaseHoldl/ReleaseH°id//N;

u0TIReAVgFI_x[r_,z_,V, PV,UO,UL,PHIU,PHIT. 0] =uOTIReAvgFIux[r.z,V,PV,OO,US,PHIU,PHIT. O]//E_and;

u0TIEpsReAVgFIux[__,z_,V,PV,U0,UL.PMIUopHZT, C] =ep$ uOTIReAvgFIux[r.z,V,PV.OO,UL,PHIU.PMIT,O]//Ex_and:

u_T_ReAvgF_uX[r-_z-_V_V_U_.U_,PHIU_P_IT`TW]=u_T1ReAv_F_u_r.z.V.PV.U_UL_PHIU_PHIT`TW]//Ex_and_

u0TIEpsReAVgFI_x[r_.z_.V,PV,U0,O5,PHIU,PHIT,TW]=ePS uOTIReAvgFI_x[r,z,V.PV,UO,UL.PHIU, PHIT,TWI//Expand;

AxialHFIOWNOE_S[Z_,_] =2 Zn_egra_e[r u0TIReAvgFIux[r,z.V,PV,U0,UL,PHIU,PHIT,0], (r,0,1)];

Ax/alMFlowStarNoEp$[z_.0]=hO$cFiow0S_ar °AxialHFIowNOEps[z,0];

AxialH_lowS_arNormalizedNoEps[z_._]=A_ial_lowS_arNoEps[z, 0]/xSec_AreaStar;

A_i&IHFIowNoE_s{z_,TW]-2 Ince_rate{r u0TiReAvgFlux[r, z,V.PV, UO.UL,PHZU, PMIT,TW].(r,0,1]];

AxialHFlowStarNoEp$[z_, _]*hO$cFl Ow0Star°AxialHFI_NOEps[z'TW]:

AxialHFlowStarNormalizedNoEps[z_,TW] =AxialHFlOwStarNOE_s[z,TWl/xSectAreaS_ar;

AxialHFIow[z_,0]=2 In_egrate[r u0TIE_$ReAv_Flux[r,Z,V,_V,U0,OL,PHIU,PHIT,C3,(r,0,1}]:

AxlalHFlowS_ar[z_,C)-hO$cFlow0S_az*AxialHFlow[z.0];

AxlalHFlow$_arNormalized{z_,0]-AxialHYlowS_ar_z,03/xSec_AreaS_ar;

Ax/alHFlow{z_,Tv_3-2 In_e_rate[r u0TIEpsReAVgFlux{r,z,V, PV,U0,UL,PHIU,PHIT,_],{r,0,1)]:

Axi&IMFlowS_ar[z_,TW].hO$cFlowOStaX'AxialMFlow[z,T_;];

AxialHFlowStarNormalized[z_,T_4]=AxaalH?lowStar[z,_]/xSectAreaStar;

(" u0TI at r=0 ")

AxlalMFlowlD[z_,0]=2 Integrate[r uCTIEpsReAvgFIU_[0,z,V, PV,U0,UL,PHZU,PHZT,C],{r,0,1]]:

AxlalMFiowlDS_ar[z_.0]=hOscFiowOSta r°AxialHFIOWID{z.0]:

AxialHFlowlD$_arNormalized[z_,0]=AxialHPlowlDStar[z,0]/xSe¢_AreaS_ar;

AxialH_IowlD(z_,TW]=2 Integrate[r uOTIEpSReAvgFIux[O,z,V.PV,UO.UL, PHIU.PHIT,TW],{r,O,I}}:

AxialHFlowlDStar[z_, _).hO$¢FlowOStar-Ax£al_lowiD[z,."%_'|;

AxialHFlowlDStarNormalized[z_,_]=A_ialHFlowlDStar|z, T_]/xSectAreaStar:

twoTimes_0piAvgTe_plate = u0hatT_mplate ° plhatCCTe_plate I/ComplexExpand;

twoTimesu0plReAvgTe=pla_e = twoTimesu0plAv_Template -I Coefficien_[_woT_memu0plAvgTempla_e, I]:

twoTimesuOplReAvgFluxHoldlr_,z_.sigemV-.si_ aPV-.UO-,UI-,Ph£U-'PhiT-'Tw-]=

twoTimesu0plReAvgTe_plate/.Onion[u0Assign.piAs$ig_];

u0plReAvgFlux{0.z_,sigmaV_,sigm_PV--.Uo--.Ul--,phiU-,_hiT--,Tw-]:= 1/2 "

twoTimesu0plReAvgFlu_.Hold[0,z,$£_pmav, simaPV,Uo,Ul,phiU,phiT,Tw]//ReleaseH°id//ReleaseM°id//ReleaseH°Id//N:

u0pIEpsReAvgFIuX[0,z_,V,_V,U0,U5, PHIU,PHZT,0] =e_s uOplReAvgFlux[O,z°V. PV,UO,UL,PHIU,PHIT+O] //E_and:

u0plEpsReAv_Flux[0,z_,V,PV,U0,UL. PHIO,PHIT,TW]=e_S uOplReAv_Flux[O,z,V, PV,UO,UL,PHIU,PHIT,TW]I/EXpand;

Axialu0plFlow[z_,O]=2 In_gra_e[r u0pIE_sReAvgFIux[0,z,V,PV,U0,UL, pHZU,PHIT,0],(r,0,1}];

Axialu0_lFIowStar[z_,O]=hO$cFiOW0$tar*Axialu0piFl°W[Z,O];

Ax£aluOplFlowStarNormalized[z_,O)=AxialuOplFlowStar[z,u]/×$ec_AxeaS_ar;

Axialu0plFlow[z_,T;.]=2 Integrate[r uOplEpsReAvgFIuX[O,z,V, PV, UO,UL,PHIU,PHIT,'_:I,(r,O,I}];

Axialu0plFlowS_ar[z_,T_'|=hOscFlo W_S_ar°Axialu0plFIOw{z'y_;_;

Axialu0plFlowStarNormalized[z_,T_]=Axialu0plFlowS_ar[z,'rv_|/xSecCAreaStar;

MaxworkAe_Star{0. 3]=hOscFlow0$_ar °MaxWo_kAmp;

MaxWorkAmpStarNormalized[0,_3=MaxWorkAmP $:ar[0,_]/xSectAreaS_ar;

124
MaxWork_S=ar [i, O]=hO$oF1 ow0Suar oMaxWork_;

MaxWor k_S _arNormali zed [i, 0 ]=MaxWorkAm_S =ar [I, 0]/xSec=AreaS_ar

saxWorkA_pS_ex [ 0. T_J] =hO$ CF i ovOScar • MaxWor kJ_spTW;

MaxWor k_$ _arNo rmal ized {0. TW ]=MaxWork_S_ar [0, _'_ ]/xSec _Area$ _ar;

MaxWozk_S_ar [1, T_J =hOscF 1ow0 S_ar •Maxwor kAmpTW;

MaxWor kAmDS_& rNorm& i i z ed [ i. T_] =MaxWor kA_pS_a r 11, T%#]/xSeccAreaSr.ar:

_woTimesu0plVolAvgTem_late = u0ha_T_la_e - plVolha_CCT_la_e //C_lexE_q_nd;

_woTimelu0piVolReAvgTem_la_e = _woTimesuOplVolAvgTe_la_e -I Confflclem_ [_woTimesu0plVoIAvgT_la_e,I] ;

_woTimes u0_iVo 1ReAvgF iuxHold [__, z_, sigmaV_, si_ma_V_. Uo_, UI_, Dhiu_, _hiT_, Tw_] -

_woTime_u 0pIVol ReAvgTemDla_e/. union [u0A_ign, piVolAssign ];

u0p1volReAvgFlux[r_, z_. slgmaV_,sigmaPv_,Uo_,Ul_,p_/U_,phiT_,Tw_] :. 1/2 •

_woTime|u0pIvo 1ReAvgF IuxHold [r, z, slgmav, slg_u_Pv, UO, UI, _iU, p_iT, Tw] IIReleameHold/IReleaseHold//ReleaseHold/IN;

u0plVolE_sReAvgFlux [r_. z_, V. PV. 00, UL, PHZU, PHZT, 0] -eps u0p 1VolReA%,_F lux [r, z, V, PV, U0, UL, PHIU, PHZT. 0] //Zx_d;

u0pIVolEpsReAvgFlux [r_, z_, v. PV, 00, UL, PHIU, PHIT. TW] -e_s u0pIVoiReAvgFlux [r, z, V, _V. U0,05, PHIU, PHIT, TWI //Expand;

u0piVolFiow[z_. 0]-2 Zn_egra_eIr u0pivolEp_ReAvgFlux[r, z,v. PV.U0,UL, PHZU,PHIT, 0], (r, 0, i} ];

u0_IVolF1 owSuar [z_, 0 ]=hOscFl ow0$ _ar -u 0piVol Plow Iz, 0 J:

u0Dl Vo 1 F1 o_ _arNoz_al ized [z_. 0]=u0plVolFlowS_ar [z, 0 ]/xSectAxeaS_ar;

u0pIVolFI ow [z_, 'rw]_2 Zn_egra_e [r u 0DIVolE_sReAvgFlux [r. z, V, PV. U0,05, PHIU, PHIT, TW] .{r, 0,1 }] ;

U0_IVolFIowS_ar [z_, _'_]=hOscFlow0 S_ar'u0pIVolFI ow[ z, TW] ;

u0plVol F1 owS tarNormali zed [z_. Y_}]=u0plVolFlowS_ar [z, TW]/x_c_AreaS_ar;

(o

_0pivo iWor k/n_oTubeS taxznMinusO_ _Avu [z_, 0 ] =u0pivolFlowS_ar [z. 0 ]-u0plVolFlowS£ar [z, 0 ]//N;

u0piVo IWor kln_oTubeS _ar InMin_s0_ _Avg |z_, TW ]=U0pIVOI F1 owS_ar [z. TW] -u0_lVolFlo_S_ar [z, _"%_I//N:

U0_IVo iWor kZn_oTubeS tarNormal izedInMinusOutAvg [z_, 0 ] =uOplVolFlowS_arNormalized[z,O]-uO_iVolFlowS_arNormalized[z,CI//N:

U0DIVo IWO rk/n_oTu_ _arNormal ize_In_nusOu_Avg [z_, TW ]= (u0plVolFI OWSta r [z, Y&] -u0pIVolFlowS_ar [z, T_.'])/xSec_AreaScar//N:

°)

UwoTimesu0rholAvgTe_place = u0ha_Tem_la_e ° rholha_CqTempla_e //Com_lex_d;

t_T_mes_0zholReAvgTem_la_e = _woTimesu0rholAvgTe_la_e -I Coefficient[_oTima_u0rholAvgTem_pla_e, II :

_Timesu0 rho iReAvgFl uxHo id [r_, z_, si_V_, sig_PV_. UO_, UI_, Dh/U_, _hiT_, Tw_] =

_woTi_su0rho iReAvgT_pl a _e/. Union [u0Assi_, _IAssign, TIAssign ];

u0rholReAvgFluxI0,z_,si_V_,si_,maFV_.Uo_,Ul_,phiU_,phiT_.Tw_]:= 1/2 "

_woTi_su0rho IReAvuFIuxMold I0. z. SigmaV. $igmaPV, Uo. U1 •_hiu, phiT, Tw] //Relea|eMold//ReleaseHold//ReleaseHold //N;

u0rholE_$ReAvgFIUXI0,z_,V, PV, U0,UL,PHIU.PHIT, 0] =ep$ u0rholReAvgFIux[0,z.V, PV, U0,UL,PHIU,PHZT.0] //Expand;

u0 rhol _$ReA%_g F l_x [0, z_, V, PV, U0, UL, PHIU, PHIT, TW ]=e_$ u 0rho IReAvgFI%Lx [0, Z, V, PV, U0. UL, PHIU, PHIT, TW] //Expand;

Axialu0rholFlow [z_, O]=2 Integrate [r u0rholEpsReAvgFlux [_, z. V, PV. U0, UL, PHIU, PHIT, 0 ]. {r, 0,1 }];

Axialu 0rholFlowS_ar [z_, 0 ]=hOscF1 ow0S_a=°Axialu0rholF1ow [z, 0];

Axia lu0rho 1F1 owS_arNormalized [ Z_. D1=Axialu0rhol PlowScar [ Z, _]/xSec_AreaS_ar;

Ax_alu0rholFiow[:_.T._']=2 Int_ra_e[r u0rholEpsReAvgFIux[0,z,V, PV,U0,UL,PHIU,PMIT, T_I] .{r. 0,I}] ;

Ax_alu0rholFlowS_ar [z_, 'F_._]=hOscFl Ow0S_ar°Axialu0rhoIFiow [z. 'r_] :

Axialu0 rho iFlowS_arNormalized |z_, ?,_,]=_xialu0rholFlowStar [z. T_II/xSec_AzeaS_ar;

_Tl_u_sv0TiReAvgTe_pla_e = _Ti_sv0TIAv_e_la_e - I Coeffici_c[_oTi_esv0TiA_e_Dla_e,I];

_oT i_e sv0 T 1 _vgF lu_HoI_ [ r_, z_, slaV_, $t__, UO_. UI_, pt_tU_. _iT_, __] =

_woTiMs_0T1 ReAv_la_$/. union [v0A_sign, TIAssign 1 :

vOTiReAvgFl_x[r_.z_,si_V_,si__.Uo_,Ul_,_hiU_,_i__,I__] := 1/2 °

_woTi_esv0TiR_vgFl_old [ r, z, $i_n_av, $ig_aPV, UO, U1. g_iU, pb_T, _] / / Rele_$_old//Rel_u_seHold//ReleaseHold//N

v0TIReAv_Flu_ It_, z_.V, P_, U0, U_, _IU, PHIT, 01 -v0T1R_vgFlu_[ r, z,V. _,U0,t_, _ZU, _T. 0]//_d;

v0T1 Re_vgPl_ [ r_, z_, V, _, U0, UL, _BIU, PHIT, _] =v0T1ReAvgFI_ I r, z, V, _. U0, US. _IU •P_LIT, _ | / / _d;

P_rs

-6 -9

{{(OPT, 0.002, I00., 0.?03, _0.266. I., -0.1, 3.789 10 , 4.31 10 ). (1.002, 1.004, 1.00413}}, {100.,

[-0.05918, 0.319915}. ?026.6),

{0.00476915 Amp dyne, 0.199157 (_ube*-l] dyne. 1.97349 W}. [OPT<>100., 3, 15, 19}}

125
i;

If

0.05

/

/

/

/

\

\

\

-6 -9

({[OPT, 0.002, 100., 0._03, _0.266, 1., -0.1, 3._89 10 , 4.31 10 ), (1.002. 1.001, 1.00413}),

(100., (-0.03918, 0.319915}, ?026.6}.

{0.0C476915 Amp dyne, 0,199157 Icube^ol) dyne, 1.97349 W), (OPT<>100., 3, 15, 19}}

c* -> H [z=0 z=l] uP [z=0 z=ll KF1owRay [iso]

cTW" -> HTW[zffi0 zffil] uPTW[z=0 z=l] _lowRay[adia]

c'/Href" {i/ep$) -> 0.13_ 0.I13; 0.402 0.342: 0.118; 0,403

c'/Href" (i/eps) -> 0.145 0.143; 0.436 0.395; 0.196; 0.401

c ° (W/eps) -> 0.269568 0.222384; 0,793 Q.675; 0,232; 0_795;

cTW ° (W/_$) -> 0.285877 0.282791; 0.86 0,779: 0.387; 0._91:

C" /AXw(W/C_8) -> 0.128711 0.106182; 0,378 0.322: 0.iii; 0,379;

cTW°/AX Q {W/cm2eps} -> 0.136499 0.135025; 0,411 0.372; 0.185; 0.3_8;

c= /up" (w/we_$) -> 0.34 0.281: 1,000 0,852; 0.293; I.; I.

cTW°/u_ * (w/we_s) -> 0.361 0,357; 1.09 0.983; 0,488_ 0.999; 0,999

c" /max ° (W/W_S) -> 0.339 0.28; 0.99_ 0.85; 0.292: 1.000: i.

CTW°/BaX ° (W/Weps) -> 0.36 0.356; 1.08 0.98; 0,487; 0,986; 0.998

Href ° = hO/cFlow0S_ar =1.97349 W

MaxWorkAmp [z=0 z=l]

MaxWorkAmpTWiz=0 z=ll

0._95

0,791

0.3_9

0.378

CrolsSectionalArea =2.09436 cm^2

bl_e is iso, red is _h/n wall, gree_ i$ Radebaugh, purple ls local area integrated plu0 flow-work

=ffi====ffi=ffi=====

AxialHFlowil/ep$} vs z

Axial_low(i/eps) and RayAdiaCalc(I/epsl vs z

AxialMFlowStar(w/eps) vS Z

AxialHFlowStarNozc_alized{w/c_^2e_$) vs z

_ffiffiffi===_fffiffif=ffi

HFIO_/u0plFIO_WOrk{W/W) v$ z

u0Tl{1/eps) vs ¢ for isothermal wall

u0Tl{i/eps) vs _ for _hin-wall

0.145

0.14

0,135

0.13

0.125

0.115

126
0.18

0.16

0.14

",,,,i

0.2 0.4 0.6_z

0.28

0.27

0.26

0.25

0.24

0.13!

0.1:

0.125

0.12

0.115

0.42

0.31

0.36

0.34

0.3

0.25

0.2

0.15

0.I

0.05

0.35

0.3

0.25

0.2

0.15

0.1

0.05

o.2 Vo.4..

1

020@ _1

127
Plot[ ({deltaTStar) / (LStar) }° [condFicStar [pressure/%efstarMPa/NPa, (z'C_RefGa$Star*50K)/KI (w/(M K) ))°

(M/{100 cm)){°xSectAreaStar°)cm_21w,{z,O,1},AxesLabel->{'z','q" (w/cm^2)'}I;

q* {W/cm^2)

0.045

0.04

0.035

0.03

0.025

0.01:

Z,tlt/_ej

• DI_SIONAL VARIABLES SPECIFIED:

{{OPT, e_a, Va, Pr, vaPr, U5, PHZU, M, LAMBDA, IsoPressKatio, AdiPrassRa_io, CLAdiPRatio}, (FO, {PHIT, TW}, VaPrFo),

{totalDraAm_0, totalDragRMS/t_be, H0scFlow0$tar), date)scalingNames

-6 -9

({{OPT, 0.002, 100.. 0.703, 70.266, 1., -0.1, 3.789 i0 , 4.31 IQ ), {1.002, 1.004, 1.00413)), {100., {-0.03918, 0.319915}, 7026.6},

{G.00476915 A=_ dyne, 0.199157 {tube'-l) dyne, 1.97349 W), {OPT(>100., 3, 15, 19}}, scaling

-- inches:

rw =0.321453 in; L =3.93701 in: _ubeThitk:%ass =0.00193031 in: pistoD.Dy_InChAtG =0.00787402 in: pistonD%rn/_m_ZnchAtL =0.00787402 in

-- cgs:

3

zw =0.81648 cm; L =I0 ca: tubeThicknass =0.004903 cm; volOfTube =20.9436 ca : CrossSectArea=2.09¢36 ca

3

pistonDynAmpA_0 =0.02 ca: pistonDynAmpAtL =0.02 ca: freq =---

sec

0.376991 ca 0.379991 ca O. cm 0.00160224 g

pistonspeedu0 = ........... ; pisuonSpeedUL = ........... ; _pistonSpeed = ..... ; pRef =0.999722 MPa; te_@Eef =300 K: rhoRef = ............

sac sac sac 3

ca

2 2

0.045313 ca 7.82 g 0.46 sac W 0.163 w 0.045313 cm

alpbaTube ............. : rho0Tu_e ....... : Cp0Tube =.......... ; kTube = ....... ; alphaTube .............

sac 3 g K cm K sac

ca

2 2

0.178836 ca 0.00160224 g 5.2 sec W 0.00149 W 99488.6cm 0.125662 ca 0.00020134 g

alpha0Gas = ............ _ rho0Gas =............ ; C_0Gas = ......... ; kGas =......... ; a0 = .......... ; nu = ............ ; mu = ............

sac 3 g K ca K sac sac cm sac

ca

0.942286 W 0.00060403 g 0.00126506 g 0.547465 W

hO$cFlu.x0 =.......... ; HOicFlow0 =1.97349 W: jOscFlux0 = ............ : jOSCFIOw0 =............ : qGasPu_dial0 = .......... ; HFlowRay =

2 2 sec 2

ca cm set cm

0.00040268 g Pi Pi

HF1OwRayS_ar[ ............... • 300 K, 0.0203251 arm, 9.866 aca, --]

$ec 6

totalDragAm_10 =0.00476915 dyne; toualDragRMS/tube =0.199157 dyne

• PARAMETER LIST [for use in exporting to plot routines)

para_terList={v, FV, KGAS,KTUBE,U0,UL,M,eps,GA_, PHIU,F,_ERM,NITW, 5},N {PHZT,5])

-6 5

ParameterList=(5.6419 Sqrt[Pi], 4.72932 S_rt[Pi], 0.0091411, i., i.. 1.. 1.20817 10 Pi, 0.C02. -, -0.i, 0.0564191 Sqrt[Pi].

16, 0.31991, -0.03918}

• SOLUTION OF TW AT PHIT FOR CG_TZNUITY OF HEAT FLUX AT GAS/TUBE BOUNDARY

TW=0.319915 at PHIT=-0.0391798

• RANDOM T_ME C_CK. ALL VAL_/ES FOR TW SHOULD BE THE EAMZ

_Rand_=0.857914

PHIT=-0.0391796

Twall=(Twall[1., 0, 0., i0., 9.3825, i., 1., -0.1, -G.0381796, 0.0091411, 0.1, 1.],

Twall[l., 0, 0.125, 10., 8.3825, i., i., -0.i, -0.0391796, 0.0091411, 0.i, I.],

Twall[l., 0, 0_25, iD., 8.3825, I., i., -0.I, -0.0391_96, 0.0091411, 0.1, i.],

TwallI1., 0, 0.5, 10., 8.3825. 1., I., -0.1, -0.0391796. 0.0091411. 0.1, 1.],

Twall[l., 0, 0.857914, i0., 8.3825, I,, I.. -0.i, -0.0391_96, 0.0091411. 0.I, i.])

Be==el Fxn ACCUraCy Z { {V} ;BesselJ; (ber_I bei): ModSesselJHold; Argum_->in£ini_y a_roximation}

Be==el Accuracy Hi = {{9.99999925}. 149.847528. 149.847526, 149.847526, 148.537427]

Bessel Accuracy _o = {{0.999999925}, 1.01552483, 1.01552493, 1.01552483, 1.00399903}

• PHASORS:

A_ PHI TWAm_ pRZTW

uo i. 0

UL 1. -0.1

pl 0.805 -0.CI16 0.893 -0.026

Tl{r=0} 0.318 -0.013_ 0.353 -0.0264

Tl(r=l) 0. 0.32 -0.0392

v0(r=.8) _.0743 -0.845 0.0504 -0.856

- NON-DIM_SZONAL SCALING VARIABLES CALCULATED:

-6

Va=100.: Pr=0.703; Pr'Va=70.2663; $ =500.: FO=I00.; EL=0.00600497; M=3.78929 i0 ; Re0 =4.89902

p0=1:TO[z]=1 ; rho0[z]=l

• NON-DIMENSIONAL VARZABLES USED _N COMPLEX SOLUTION:

128
-6 -9

V=I0.; P=0.83825; FV-8.3825; eps=0.002; Fz0.1: Mz3.78929 10 ; LAMBDA-4.30762 10 ; eps°Va=0.2

5

U0sl.; UL=I.: PHIUz-0.1; TERM=16; KGAS =0.0091411_ KTUBE=I.; GAMMA=-; d0=0.002; d5=0.002; PHIT--0.0391796; TW=0.319915

3

ARCHIVE

-6 -9

({(OPT, 0.002, i00., 0.703, 70.266, i., -0.1, 3.789 I0 , 4.31 10 }, (1.002, 1.004, 1.00413}), {i00.,

(-0.03918, 0.319915), 7026.6),

(0.00476915 ;m_ dyne, 0.199157 {_be^-l) dyne, 1.97349 W}, {OPT<>100., 3, 15, 19]}

r-i

r=0

-Graphicsrsl

iiiiiii; ..... - __ - : _--,.

r-O

-GraphicsI$othem_al and thin wall comparison at z=0, .5, 1

-6 -9

{{{OPT, 0.002, 100., 0.703, 70.266, i., -0.1, 3.789 10 , 4._1 I0 }, {I.002, 1.004, 1.00413}), {i00.,

{-0.03918, 0.319915), 7026.6},

[0.00476915 Am_ dyne, 0.199157 (tube'-l) dyne. 1.97349 W}, (OPT<>100., 3, 15, 19}}

0.15 _%

0.125

0.1

0.075 %

0.05

0.025

r

0.2 0.4 0.6 0.8 1

0.175

0.15 •

0.125

0.1

0.075

0.05

0.025

0.2 0.4 0.6 0.8 1

0.175 _ _ --

o15

0.125

0.i

0.075

0.025 0.05

r

0.2 0.4 0.6 0.8 1

If [kindSyste_==OPT, Print (fileN_eOPTHXfr ] ,,Print [fileN_uaeBpTHXf r ]] ;

PIo t I(_J [i, 0, _, V. PV, U0, UL, PHIU, PHIT, 0 ]-,_ [1,0, t, V7 FV, U0, UL, PHIU, PMIT, TW ],TI;._ [0,0, t, V, PV, U0, UL, PHIU, PHIT, 0 ),

Del taTl [0, t, V, PV, U0, UL, PHIU, PHIT, 0] +Del taT i l0, t,V, PV, U_, UL. PMIU, PMIT. TW] }, (t_ 0.1 },

129
?ZocS_yle->[{]. {Dashing[{0.03,0.O3}}) ,{Dashing[{0.005,0.02,0.03.0.0R)I,RGBColor[1,0.0]),(),(RGBColor[0, 0,11,

Dashlng[{0.03,0.Q3}]] }, P1o_Range-,A11];

if[kindSys_c_n==OPT,Prin_[file_P_FHXfr],,Prin_[fileN_PTMXfr]_:

PIo_[{_I[1,0,_.V, PV, U0.UL, PHIU,PHIT,0].qlI1,0._.V,PV. U0,UL,PHIU,PHIT, TW],

DQI_aTI[0,_, V, PV.U0, UL,PHIU,PMIT,0],DeltaTI[0,_, V. PV,U0,UL.PHIU, PHIT,TW]},{_,0,1].

Plo_S_yle->{(), {Dashing({0.03,0.03}]) ,{RG_olor{0,0,1]),{RG_¢olor[0,0.1_,Dashing[(0.03,0.03>l} },

Plo_Range-,All]:

(* {[Thesis_ OPT, _s, V, P, PV. UL. PHIU, M. I$oPressP_io. AdiVolPR. CLAdiPRatio), {F, {PHIT, TW}},

(HOscFIO_), (da_e)) °)

-6 -9

({(OPT. 0.002, 1O0., 0.703, 70.266, i., -0.I. 3.789 10 , 4.31 10 ], (1.002. 1.004, 1.00413}}, {i00.,

(-0.03918, 0.319915}, 7026.6),

{0.00476915 A_ dyne, 0.199157 (_u_^-l) dyne, 1,97349 W}, {OPT->100.. 3, 15, 19}}

-6 -9

{({OPT, 0.002, i00.. 0.703, 70.266, i.. -0.I, 3.789 i0 , 4.31 i0 ), (I.002. 1.004, 1.00413)), (100.,

(-0.03918, 0.319915}, 7026.6),

{0.00476915 A_ dyne, 0.199157 (uu_^-l) dyne, 1.97349 W}, {OPT<>100., 3, 15, 19))

-2

(_TWvs FO °)

i£[kin_y$_em=zOpT,Prin_[fileS_OPTHXfr],, PrintlfileName_fr]];

A_PIo_[V,_.U0,0. PHIU, fo|=L_b_PIo_[_[0._ iU, fo_,{fo,.1,150),Pio_R_ge-_Alll

-6 -9

({{OPT, 0.002, I00., 0.703, _0.266, i., -0.1, 3._89 i0 , 4.31 i0 ]. {1.002. 1.004, 1.00413}). {100.,

{-0.03918, 0.319915), 7026.6),

(0.004_6915A_dyne, 0.199157 (_ube^-l) dyne, 1.97349 w]. {OPT<>100.. 3, 15, 19])

0.5

0.2

0.1

0.05

0.02

0.1 0.5 1 5 10. 50.100.

-Gra_xcs-

(* PhiT vs FO *)

ArgPIo_(V._,U0,0,PH_U,_o]=_O_L_PIo_[-phiT[0,phiU. fo_,{fo,.l,150}. Plo_Range->hll]

0.3_

0.2"

0.i 0.5 I 5 10. 50.100.

130
-Graphics-

(t Code for Graphics Plots °)

If[kindSyste_n=sOPT, Print[file1_eOPTHXfr],, Print[file_BPTF_fr]];

-6 -9

({{OPT, 0.002, I00.. 0.703, 70.266, i., -0.i, 3.789 i0 , 4.31 i0 ]. {1.002. 1.004, 1.004131], {i00., {-0.03915. 0.319915}, 7026.6},

{0.00436915 A_ dyne, 0.199157 (tube^-1) dyne. 1.97349 W}, {OFT<>100., 3, 15, 19)}

?PhasorPlot

Global'PhasorPlot

Pha$orPlo_[All, 31.83098385134257"Pi, i.. -0.1, 314.1578982435413/Fi] =

Graphics[(Line[{(0, 01, {-7.584757533613621. -5.7409448947448131)],

(DashingI(0.03, 0.03)], Line[{{0, 0}, {0.3165449091259993, -0.0273577162062239611]),

(Dash/n_[(0.005, 0.02, 0.03, 0.021], Line{{{0. 0), (1.879881736290323, 1.781540135669653})]].

(Dashlng[(0.01, 0.01]], RGBColcr[0, i, 0], Line{{{0. 0), {_, 0))] }.

{Dashing{{0.01, 0.0111, RGBColor[0, i. 0], Line{({0. 0), {0.809016994374948, -0.5877852522924733))]}.

{Dashing[{]], RGBColor[0, 1, I], Line{{{0, 01, {0.803281420362929, -0.05861243832121366}}]),

{Dash/ng[{0.02, 0.02}I, R_olor{0, I. 1], LineI{{0. 01, {0.871707058032187, -0.1436918561245787)}]},

{Dashing{{0.03. 0.031], RGBCoIor[0, 0. 1], Line{{{0, 0}, {0.2706043229945105. 0.03006237550349716})]),

{Dashing{{0.03, 0.03}], RGBColor[I, 0. 1], Line{{{0, 0). {-0.2706043229%45105, -0.030062375503497161}]),

{D&shing[{0.02, 0.02}], RGBColor[0, 0, 1], Line[{{G, 0}, {0.3390299606637685, -0.055017042299847941111,

{Dashing[{0.03. 0.0311, RGBCoIor{I, 0, 0], Line{{{0, 0), {0.3484200046661029, -0.05826763675797807)}]],

{RGBColor[I. 0, 0], Line{({0, 01, {-7.58475753361361, -5.74094489474a81411]),

{Dashing{{0.005, 0.02, 0.03, 0.02)], RGBColor[I, 0, 01, Line{{(0, 01, (0.0864170124197421, 0.3391354590056961)}]}},

(Di_layFunc_ion -> {Display[$Displsy, #I] & ), As_ec_Ratio -> i, PlotRa_ge -> Automatic, A_Ra_io -> GoldenRatio^(-l),

Di_layFunction :> Zdentlty, ColorOut_t -> Au_aatic, Rxes -> AutOmatic, AxesOrigin -> Automatic, Flo_La_l -_ _ne. AxesLabel -> None.

Ticks -> Au_tic. GrldLines -> None, Prolog -> (), Epilog -> {}, AxesStyle o> Au_tlc. Backgro_Lnd-> AutO=m_ic, DefaultColor -> Automatic,

DefaultFon_ :> $Defaul_Font, Ro_steLa1>el -> Tr_e, Frame -> False, FromeStyle -> Aut_natio, FrameTicks -> Aut_ic, FrameLabel -> None,

FlotRegion -) Autcaa_ic} ]

FhasorPlot [NuC, 31.83098385134257°Pi, i.. -0.I, 0] =

Graphlcs{{Line{{{0, 01, {-7.584757533613621, -5.74094489474481311],

(Dashing{(0.005. 0.02, 0.03, 0.0211, Line{{{0, 0}, {1.879801736290323. 1.7815403356696831]]),

{Dashing{{0.01, 0.011], RGBColor[0, i, 0], Line{{{0, 0}. (i, 0)}]],

{Dashing{(0.01, 0.01)]. RGBColor{0, I, 0], Line{{{0, 01, {0.809016994374948, -0.5877852522924733)}I},

{Dashing{(0.03. 0.03)], RGBColor{0, 0, 11, Line{({0, 01, {0.2706043229945105, 0.03006237550349716}}]},

{Dashing{ {0.03. 0.03 } ], RGBColor[I, 0, 11, Line[ ( {0, 01 ; {-0.2706043229945105, -0.03006237550349716} } ] } ),

{Di_IsyFunction -> Identity, PlotRange -> A_tO=atic, As_ec_Ra_io -> GoldenRatlo ^ [-i) , DisplayFunction :> identlty, ColorOutput -> Automatic,

_es -> A_to_ic, Ru(esOrigin -> Auto=_atic, Plo_Label -> No_e. AxesLa_el -> No_, Ticks -> Auto, tic. Gri_LLines -> None, Prolog -> {), _ilog -> (}.

AxesStyl@ -> Autcaatic, Background -> AutOmatic, Da£aultColor -> Aut_aatlc, DefaultFont :> $DafaultFont, RotateLabel -> True. Frame -> False.

FrameS_yle -> Automatic, FrameTick8 -> A_tO=atic, Fr_Label -> NoDe, PlotRegion -> Auto_atlc)]

PhasorPlot[N_C, 31.83098385134257°Pi, 1., -0.i. 314.1575982435413/Pi] •

Gra_hics[{{RGDColor[l, 0, 0], Line{{{0, 0}, {-7.58475753361361, -5.7409448_4744814}]]),

{Dashln_{{0.005, 0.02, 0.03, 0.02]], RGBColor[1, 0. 0], Line{{(0, 0}, {0.0864170124197421, 0.33913545908569611)]],

{Dashlng[{0.01, 0._i)], RGBColor[0, i, 0], Line({{0, 0), {I, 011]),

{Dashing{{0.01, 0.011]. RGDColor[0, i, 0]. Line{{{0, 0], {0.809016994374948, -0.5877852522924733])]],

{Dashing{{0.02, 0.02]], RGBColor{0. 0, I], Line{{(0, 01, {0.33902996066376§5, -0.055017042299847941111,

{Dashln_[{0.03, 0.03)]. RGDColor[I. 0, i], Line{{{0, 0}. {-0.028760045103960_?, o0.02294412494308133})]]),

(Di_layF1;;Ictlon -> Identity, PlotR_ge -> A_tO_ic, AS;M%CtRAtiO -> Gold_tio^(-l), Dis_layFt_ction :> Ide_tlty, Color(_tput o> A_to_atIc.

Axes -> Auto, tic, AxesOrigin -> Automatic, PlotLabel -> None, AxesL4_el -> None, Ticks -> Autc,_atic, GridLines -> None. Prolo_ -> {}, E_ilo_ -> (),

AxesStyle -> AutoEatic, Background -> Automatic, DafaultColor -> Automatic. DafaultFont :> $DefaultFont, RotateLabel -> T_e, Frame -> False,

FrameStyle -> Au_aatic, FrameTicks -> AutOmatic, Fra_eLabel -> None, FlotRegion -> AutOma_ic}]

PhasorPlot_PT_, 31.83098385134257°Pi, I., -0.1, 314.1578982435413/Pi] =

Graphics[{{Dashing[{0.01, 0.011], RGBColor{0, i, 01. Line{{{0. 0], {i, 0})]],

(Dashing{{0.01, 0.01)], RGBColor[0, 1, 0], Line{({0, 01, {0.009016994374948, -0.5877852522924733}}]],

{Dashing{(}], RGBColor{0, i, 1]. Line{{{0, 0}. {0.803281420362929, -0.05861243832123366}]]],

{Dashlng{{0.02, 0.02}], RGBColor[0, i, i], Line{{{0, 01, {0.871_07058032187. -0.1436918561245787)|]},

{Dashing{(0.03. 0.03}I, RGBColor[0, 0, 1], Line{{{0. 0}, {0.2706043229945105, 0.03006237550349716}}]},

(Dashing{(0.02, 0.02}], RGBColor[0, 0, 11, Line{({0, 0), {0.3390299606637685, -0.055017042299847941]I)},

{DisplayFunction -> (Display[$Display, #1] & ), AspectRatio -> 1, PlotRange -> ((-0.2, 1), (-1, 0.21]. Plo_Range o> Automatic,

As1_tRatio -> C_id_tio^(-l}, Dis_layFll_ction :> Ide_Itity, ColorO_put -> A_to_tic, A_(e_ -_ Auto, tic, A_<esOrigin -> A_t_(_a_ic,

FlotLah_l-> None, AxesLa_l -> None, Ticks-> Automatic, GridLines-> Nc_e, Frolog-> {}. Epilog-> {}, AxesStyle-> Automatic,

Background -> Automatic, DefaultColor -> Automatic, DafaultFon_ :> $DefaultFont, Rota_eLabel -> True, Frame -> False, FrameStyle -> AutOmatic.

Fr_meTicks -> AutOmatic, FrameLabel -> None, PlotRegion -> Automatic)]

?FhasorPlotBlack

Global'PhasorPlo_Black

PhasorPlotBlack{NuC, 31.03098385134257°Pi, 1.. -0.i, 01 =

Graphics[{Line[((0, 0}, (-7.554757533613621, -5.74094489474481311],

{Dashing{{0.005. 0.02, 0.03, 0.02}], Line{{{0, 0}, {1.879881736290323, 1,78154033566968311]}, (Dashing{(0.01. 0.01]], Line{{{0, 0}, (i, 0]}]},

{Dashing{(0.01. 0.01]], Line{{{0, 01. {0.8090169943?4948, -0.5877892522924733}}]),

{DashingI[0.03, 0.03]], Line{{{0, 0}, {0.2706043229945105, 0.03006237550349716}}]},

{Dashing{(0.03, 0.03]], Line{{{0, 01, {-0.2706043229945105, -0.0300623755034_716)}]}},

{DisplayFL_tlon -> Id_Ity, FlotRa_ge -> AutO, tic, Aspec_Ratio -> Golde_IR_io^{-l), Di_lay_ction :> Identity, ColorO_ut -> Au_o_ic,

Axes -> AutO,silo, AxesOrigln -> Aut_tic. plo_Label o> None. AxesLah_l -> Nose, Ticks -> Aut_atic, GridLines -> None, Prolog -> {}, Epilog -> {},

AXSSS_yIe -> AutOmatic, Background -> A_t_atic, DafaultColor -> A_tomacic, Defa_itFont :> $Defaul_Font, RotateL_l -> True, Frome -> False.

FrameStyle -> Aut_aatic, Fra_eTicks -> AuCO=tatlc, Fra_NLa_l -> None, PlotRe_ion -> Autoa_cic}]

phasorPlotBlack[NuC, 31.83098385134257°Pi, i., -0.1. 314.1578902d35413/Pi] =

Grsphics({(Line[{{0. 0}, {-7.58475753361361, -5.?40944894?44814}}]}.

{DashingI{0.005, 0.02, 0.03. 0.02]], Line{{{0, 0), {0.0864170124197421, _.33913545908569611]I), {Dashin0[{0.01, 0.01]], Line{{(0, 0), {i, 01)]),

{Dashing{{0.01. 0.01}]. Line{{{0. 0}, {0.80901_994374948, -0.5577852522924733})]}.

{Dashing{{0.02, 0.02}], Line{({0, 01, {0.3390299606637685. -0.05501_04229984794)}]},

{Dashlng{{0.03, 0.03}]. Line{({0, 0}, {-0.02876004518396047. -0.02294412494308133])]]},

{DisplayFu_tion -> identity, FlocRa_ge -> A_tO_tic, A_pec_Raclo -> Gold_T_tio^(-l), Dis_layFIL_tion :> Identity, ColorO_tp_t -) AutOmatic,

Axes -> Aut(_natic, AxesOrigin -> AutO, tic, PlocL_I -> None, Ax_sLabel -> None, Ticks -> Autonetic, Gri_Lines -> NOne, Prolog -> {}, Epilog -> {},

AxesStyle -> Autom&tic, Background -> Auto, tic. Defaul_Color -> Autocratic, Da_e_IcFont :> SDe£sulcFont, RO_aCeL_I -> Tz'ue, Fra_ -> False,

FrameStyle -> AutCe_atic, Fr_eTicks -> Automatic, FrameLa_l -> None, PlotRegion -> Automatic]]

PhasorFlotBlack[PTU, 31. $3098385134257"PI, i.. -0.i, 314.1578982435413/Pi] =

Orsphics[{{Dashlng[{0.01, 0.011], Line[{{0, 01, {i, 0])]], {Dashing{{0.01, 0.01)], Line{{{0, 0). {0.8090169943_4948. -0.5877852522924733)}]}.

{Dashing[{)]. Line{{{0, 0), {0.803281420362929, -0.05861243832123366})]),

{Dashing[{0.C2, 0.02}], Line{{{0, 0}, {0.871707058032157. -0.14369185612457871111,

{Dashing[[0.03. 0.03}], Lins{{{0, 0), {0.270604322S945105, 0.03006237550349716}}]],

{Dashing{{0.02, 0.02}], Line[{{0, 0), {0.339029960_637685, -0.055017042299847941]])],

{DisplayF_u_ction -> (Dis_ley[SDi_lay, #i] & }, As1_ectBatio -> 1. PiotR_ge -> [{-0.2, 1}, {-1, 0.21}, PlotR_ge -> Auto(natic,

AspectRatio -> Golde_tio*(-l), Di_layF_ction :> I_e_ciCy. ColorOutput -> Auto, tic, Axes -> A_O_CiC. AxesOrigin -> A_tO_tic,

FlotLahel -> None. AxesLabel -> None, Ticks -> Automatic, GridLines -> None, Prolog -> {}. Epilo_ -> {), A,_esStyle -> AutOmatic,

Background -> Au_o_u_tic. DefaultColor -> A_toma_ic, Defs_l_Font :> SDeZaultFo_=, Ro_a_eLabel -> True, Frame -> False. FrameStyle -> Automatic,

Fra_Ticks -> Autoolatic, Fr_u_eLabel -> None, PlotRegion -> Automatic)]

A_PIo_[5.641895413009938"Pi'(I/2}, 4.729318913240664*Pi^( 1/21 , I., 0, -0.I, foJ =

131
Graphics[{{(Line[{{-1.. -1.927863365411739}, (0.802488661624436, -0.749019990592986), {1.1000832182914, -0.557442162235803},

1.275023265322688, -0.470051901193516], {1.399385249546218, -0.4196936259076566}. {1.495948843114022, -0.3869530852775928},

1.574898989258946, -0.3639536092490773), (1.641680632207505, -0.3469931894273173), {1.699548677948487. -0.3339242959739331].

1.750604a08323138. -0.3235644698694219}, {1.79628516981576, -0.3151533050972453}. (i.837614739188246, -0.3081900048869743),

1.8753506%6579289, -0.3023314621012289), {1.910068287208911, -0.2973348150100153}, {1.942214810694346, -0.293023383575%382],

1.972144959458221, -0.289265596918928}, {2.00014474070519, -0.285961481476084}, {2.026448140455147,-0.2830337261160283),

2.051249021610516. -0.2804216135760304), {2.074709803628545, -0.2780768007838752), {2.0969679150_8924, -0.2759603231885111}.

2.118140671487759, -0.2740404283127187}, {2.138329018246402,-0.2722909827598751), {2.157620441425085. -0.2706902832036441}.

2.176091259055681, -0.2692201567724531) ] ] } } ) , {DilplayFunc_ion -> (Display[$Display, #1 ] & ) ,

Plo_Range -> { (-1.079402281476392, 2.255493540532075}, {-1.969329445627721. -0.2277540765564709) }.

Axei0rlgin -> (-1.079402281476392, -1.969329445627721), PlocRange -> All, Aspec_Ra_io -> Golder_auio*(-i), Disp1ayFu/%crlon :, Identity,

ColozOu_Du_ -> Automatic, Axes -> Automatic, AxesOrigin -> AutOmatic. Plo_Label -> None. AxesLabel -> None, Ticks -> {LogScale, Lo_Scale),

GridLlne$-> None. Prolog -> {), Epilog -> {}, AxesS_yle -> AutOmatic, Backgro_%d-> Automatic, Defaul_Color -> Automatic,

DefaultFon_ :> $Defaul_Fon_, Ro_e_eLabel -> True, Frame -> False, FrameS_yle -> Automatic, Fra_eTicks -> {LogScale, LogScale, 5ogScale, LogScale).

FrameLabel -> None, Plo_Region -> Au_oma¢ic}]

Clear {PhasorPlot, phasorPlo_Black, AmpPIoC. Argplo_, AE_PlorTw]

=================================================

(e Nail FI_ P_eld e)

{" <'a!c_In_i.;l_._ "!

rhoSuSAvg: _xi3_ sece.nda_ .Tas_ f]ux

_woT1mesDrho0u0u_ZAvgTe_la_e = D[rho0[z] u0ha_Tem_la_e u0ha¢CCTe_le_e,z] //Co_l_d;

CwOT imesDrho0u 0_ 0 ZReAvgTemplar e =

{_woTimes_r_o0u0_0".AvgTe_pla_e - I Coefficien_[_wOTimesDrho0u0u0ZAvgTem_la_e,I]):

_woTimesDrho0u0u C ZReAvgF itu_old Jr_, z_, si_maV_, UO_, UI_, _hiU_ ] = _woTimesDrho0u0u0z ReAv_T_ i a_ •/. u0Assign;

Drho0u0u0ZReAvgFlux [ r_, z_, sig_V_, UO_, UI_, DhiU_] := 1 /2 _woTimesDrho0 _ 0u 0 ZReAvg Fl_xHold [r, z, si_maV, UO, UI, _hiu ]

llReleaseHold/IReleaseEcld//N;

DrhoCu0u0ZReAvgFlux [ r_, z_, V. U0, UL. PMIU ] = Drho0u0u0_NeAvgFl_x [r, z, V. U0, U5, PHZU] //Expand;

axialSr_ess [ r_, z_] = Integrate [ Incegra=e [ r * Drho0u0u0ZReAvgF lux [r, z, V, U0, UL. PHIU ] , r ]/r, r ] //Expand;

(* q3[r_,z_]=ln_e_ra_e[r " axialS_ress[r,z].r]//Ex;emd; ") (*don'_ really need r-dependence ...

coded here for c(_le_e_ess')

q3 [ z_] =Integrate [r ° axialS_ress[r,z],{r,O,l}]//Ex_and:

_woTimesrho0u0u0Av_Tem_le_e - IrhoO[z] u0ha_T_la_e u0ha_CCTem_la_e} //C_lexExpand;

_wOTimesrhoOuOuOReAv_Tem_le_e = (_woTimesrho0_Cu0Av_Tem_la_e - I Ccefficien_[cwoTimesrhoOuOuOAvgTem_la_e. II ):

_woTimes rho _U0U0ReAV_F i U_HOI d [r_. z_. si_m_V_. Oo_. UI_. _%iU_ ]= _woTimes rho0uC_ 0 ReAv_Tem_ I a_e / •U0Assign:

rho0u0_0ReAvgFluxlr_, z_. si_maV_.Uo_.Ul_._h£__] := 1/2 _wOT_mesrho0u0_ReA_FluxMold[r. Z. sigmaV. UO.UI. _hiU]

/ /Releasel4old/ /Releale_ol_/ /N;

rho0u0u0ReAv_Flux {r_. z_. V. U0. UL. PMIU ] s rho0u0u0ReAv_Flux [r. z. V. U0. UL. PH!U] //_d:

_° _vera_e ve_'.'c:._y _.i_.ude _.'.r ._he_=_ _

localReynoldsS_ress[z_].In_egra_e[r'r_oOuOuOReAvgFlux[r.z.V.UO._.PMIU]. {r.O.l}]//Expand:

RMSReMnoldsSt tess [z_] =Sqr_ [localReynolds$ _ress [z ] I :

twoTimesrho0v0u0Av_Tem_late = rho0[z] v0ha_Te_la_e u0hatCCTempla_e //Coa_l_d:

_woTimesrho0v0u0ReAvgTe_late = cwoTi_esrho0v0u0AvgTe_lace - I Coefficlenc [_woTimesrho0v0uOAvgTem_la_e, I } ;

_woT _me $ rho 0 v0u0ReAvgF lush old (r_. z_, $igmav_. $i_maPV_, UC_, UI_, DhiU_, DhiT_, Tw_] = _woTimesrho 0v 0u0 ReAv_Tem_l a ce/ .

Union [v0Assign, u0Assign] ;

rho0v0u0ReA%,_Flux[ r_, z_, sig_aV_, $igmaPV_,UC_, Ul_,phiU_,phiT_,Tw_] :- 1/2 °

t woTimesrho 0v0u0ReAvgF lu_Mcl d [r, z, si_mav, sigmaPv, Uo, UI, phiU, _hiT, Tw ] //ReleaseHold//Rel easeHold!/N ;

rho0v0u0ReAvgFl_x [r_, z_, V, PV. U0, UL, PHIU, PHIT. 0 ] = rho0v0u0ReAvgFl_x [r, z, V, PV, U0, UL. PHIU, PHIT. 0 ] //Expand ;

rho0v0u0Re.AvgFlu_x [r_, z_. V. PV, U0, U_. PHZU. PHIT. TW] = rho0v0u0ReAvgFlux [ r, z, V, PV. U0, UL, PHIU, P_IT, TW] //_d:

rad_a_ress[r-.z-._]=In_eg_a_e[rh_v_u_ReAv_F_ux[_.z_V_PV'U_U_P_IU.P_IT_]/_ExP_nd; (° radialS_ress= J 2°)

radialS_ress [r_, z_, TW ]= Integrate [rho0v0u0ReAvgFlux [r, z. V, PV, U0, UL, PHIU, PHIT, TW] ,r ] / / Expand :

j3(r_,z_,0]= In_e_rare[r ° r_dialS_resslr,z.O],r]//_d:

j3[z_,0]= In_e_ra_e[r ° zadialS_ress[r,z,0], {r,0,1)]//Ex_nd:

(* j3[r_,z_,TW]= Znce_race[r " radialStress(r,z.TW],r]//Ex_ and;°)

(.don't really need r-dependence ... coded here for completeness °}

j3[z_.TW]= In_e_rate[r ° radial$_ress[r.z.TW].{r.O.1)]//Expand;

_woTimesrholuOAvgTempl&_e = rholha_T_la_e u0_CCTem_l_e //Compl_d;

_woTimesrholu0ReAvgTem_la_e = _woTimesrholu0Av_Tem_late - I Coefficien_|_woTimesrhclu0Av_T_m©la_e.I]:

_wOT _mes rholu 0 ReAvgFlu)&Mo i d [ r_. z_. sigmaV_, si _maPV_. UO_. UI_. phiU_. _hiT_. Tw_ ] =_woTimesrholuOReAv_Tem_late/ .

U_%ion [plAssi_, TIASSig_, uOAssign I ;

rho lu0ReAv_Flux [r_. z_. sigmaV_, sigmaPV_. Uo_. _i_. _hiu_. _hiT_. TW ] :=

1 /2 rwoTi_es rho lu 0ReAr,F1 _w/_o i d [r. z. $i_av. $ig_aPv. Uo. UI. _)hiu. phIT. Tw ]

//ReleaseMold//ReleaseHold//N;

rholu0ReAvgFlux{r_,z_,V. FV,U0,U5,PHIU,_IT, O] = rholu0ReAvgFIUX[r,z,V, PV, U0,UL, PHZU.PHIT, 0]//Expand;

rholu0ReAv_Flux [ r_. z_. V. PV, U0, UL, PHIU, _MIT. TW] = r_olu0ReAvgFluX [r, z. V, PV, U0, UL, PHIU, PHIT, TW] / /Expand;

k3[z_,0] = (i/Power[V. 2]) Zn_e_ra_e[r°rholu0ReAvgFlux[ r,z.V.PV,U0,UL,PH_U,PHIT,0], {r,0,1}}//Ex_and//N;

k3 [z_,TW]= (I/POWer [V,2] ] Zncegra_e [rorh_lu0ReAvgFlux[r, Z,V, PV,U0, UL, PHIU. PHIT,TW], {r, 0_ i) ]//Ex_andI/N:

i" All!A-'. JEC{;_]AR'f !.'.AI5 F[A!X - rhoSuSAvg

_T_;AD_' {r._s"r_'_V'1 L'_CD_VA_;  ̧ ;'>_IAL '/:_;)C:Tv ulAV_ - _=_ <'_ _._yn_/.._:." _'_r_._._ ";

- 2 (ax_elS_ressll.z] _ radialS_ress{1.z.0]) //E_and: K[r_.z_.0| = 4 (k3[z.0] * q3[z] - j3[z.0])

K(r_.z_.TW]= _ (k3[z. TW] * _3[z] -j3[z. TW]) -2 (ax_alS_res$[l.z] ÷ radielStress[l.z.TW])//zx_and:

rho0ulAvg [ r_. z_. 0i =po_er [V_ 2 ) ( (axialS_ress [r. z] -axialS_r_$s [i. z] )- (radialStress Jr. z. 0] -

radialS_res$[l.z.0]) -(r ^2-I) K[r.z.0])!/Ex_%_d;

rho0ulAvg [ r_. z_. TW] =power [V. 2 ] { (axialS_re$$ [r. z ] -axialS_ress [i. z ] ) * (radialst tess [r. z. TW] -

rad_alSuress [I. z. TWl }o (r^2-1) K[r.z. TWl }//Expand;

rhoSuSAv_[r_.z_.0] = rho0ulAvg[Lz.0]- _ rholn0Re_vgFIuxlr.z.V. PV. U0.UL.PHIU. PHIT.0];

rhoSuSAv_ [r_. z__ TW I = rho0_iAvg [r. z. TW ] • rholu 0 ReAvgFlux I r. z. V. PV_ U0. U5. PHZU. PMIT. TW I ;

132
rhoSvSAvg: ramie3. _eco.ndarv ma._s fi*_

<- ;:A_AL eECOTC,_ARY _A,',_" ?L':_ - rhoSvSAvg "i

rhoSvSAVg[r_,z_,0] = - i/r ° Integrate[r*O[rhoSuSAvgEr.z,0], z],r) //Expand;

rhoSvSAvg [r_, z_, TW] = - i/r * IncegrateIr'D[rhoSu£Avg[r, z, TW] ,z] ,r] //Expand;

t_Timesrholv0AvgTem_late = rholhatT_lete v0hatCCTe_late //Co_lexExpand;

twoTime|rholv0ReAvgTem_late = twOTimesrholv0AvgTem_lete - I Coefficient[twoTlmesrholv0AvgT_late,Z] ;

twoTimesrhol v0ReAvgFluxHol d [r_, z_. sigmaV_. =igmaPV_, UO_, UI_, phio_, _hiT_, Tw_ 1=twOTimesrhoZ v0ReAv_T_lat e/.

Uni_ (plAseign, TIAssign, v0Assign] ;

the1 v0ReAvgFlux [r_. z_, simmer_, sigmaPV_, UO_. UI_, phiU_, _hiT_, Tw_] :=

1/2 twoTimeerholv0ReAvgFl_old [r, z. sigmav, siGm_aPV, UO, U1, phiO, phiT, Tw]

//ReleaseMold//ReleaseMold//N;

rholv0ReAvgFlux [r_. z_, 0 ] = rholvOReAvgFI_x[r,z,V, PV,UO,UL,PHIU,PHIT,O]//Ex_and;

rholv0ReAvgFlux[r_,z_,TW] = rhol vOReAv_Flux [r, z, V .PV ,UO ,UL, PHZU ,PHIT, TW] //Ea_Lnd;

rho0vlAvg [r_, z_, 0] = (rhoSveAvg[r. z. 0} -rholv0ReAvgFlux[r,z, 0] ];

rho0vlAvg [r_, z_, TW| = (rhoSvSAvg It, z ,TW] -rholv0ReAvgFlux {r, z,TW] );

(* _i.r-]Ja "}

I f[kindSys teD==OFT, Print [fileNameOPTMX frLie t ];Print [fil eNameOPTHX fr ]. ,Print [fileN_frLiet ] ;

Print [fileNameBPTHXfr ] ]:

Timing [maesF lux [1, meanSt eady, 0 ] =PlotVec_orField[ (rhoSuSAvg [r, z, 0], rhoSvSAvg It. z, 0] ), [z.0,1), {r,0,1),

As_ectRatio- •1/_, Displ ayFunc tion ->Identl ty ]];

Timing [maesFlux |1,meanSteady, TW] =PlotVeC_orField[ (rhoSuSAvg[ r, z, TW]. rhoSvSAvg [r. z,TW] }, {z, 0,1 }, (r, 0,1 },

A_ectRatio->l /3, Di_ iayFunc tion- >Ide_t i ty ]];

Show [(maeeFlux [1, meanSt eady, 0]. liner [(0,1 ), (1,1 )], texter), DieglayFunction-> $Di_leyFunction]

Show [[massF1ux [I, meanSteady. TW], llner [{0,1 }, [i, 1) ],text er ),DispleyFunction- >eDi_leyFunct ion]

[{OPT, e_s, V=, Pr, VaPr. UL, PMIU, M, LAMBDA, ZeoPre_sRatio, AdiPressRatio, CLAdiPRatio),

{FO, (PMIT, TW}, VaPrFo},

{totalDragA_0, totalDregRMS/tube, HO$cFlow0Star}. date|

-6 -9

[{{OPT, 0.002, i00., 0.703, 70.266, 1., -0.1, 3.789 10 , 4.31 10 ]. {1.002, 1.004. 1.00413}},

{100., [-0.03918, 0.319915), 7026.6],

{0.00476915 Am_ dyne, 0.199157 {tube*-1) dyne, 1.97349 W}, {OPT<>I00., 3, 15, 19}}

r=l

re0

-Gra_aicsr=l

....... - .---:-g

r=0

-Graphics -

Show[ {maseFlux [1,m_enSteady, 01 , liner[ {0,1 ), {I, 1}] ,texter), Dis_layFunction->$Dis_layFunctien]

Show ({maseFlux [1,meanSteady, TW] ,liner [{0,1 ), [I, 1}], texter }, DispleyFunc_ion-• $Dis_layFunct ian]

-Graphics -

r=l

f.i:_:f_f;i;;J.;$

Z=0 _ _ _ _" _ _" "_ _" _" _ Z=I

-Gra_icsi° i"i_',_:_"_, (° Of mass flux at a giv_ z, CheCk for zero net mass flow *}

I f [kindSyst_=OPT, ("Print [fileNam_DPTHX frLi_ t ];-)Print [fi IeNameOPTHX fr ], .[°Print [_iI_PT_ frLi st ];")

Print [f ileNameBPTMxf r]]

Block[ [z=. 5},

Plot [{rhoSuSAvg[r, z. 0] .:hoSuSAvg[r, z,TW] (*, rhoSuSAvg[r,l,0]*) }. {r,0,1].

PiotStyle-• {{ ]. {Dashin_ [{0.03,0.03 }], RGBColor [0,0,1 ]) , (Dashing[ {0. 005,0.02.0.03, 0.02 }].

RGBCoIor [1.0,0] ) },

PlotRa_ge-_All, AxeeLabel °• {•r- ,-•) ] {*rhoSuSAvg" ) ;

Timing [NZntegrate [rhoSuSAvg [r. z, 0] °r, (r, 0.i }] ]]

(.

{(BPT, e_s, Ve, Pr, vaPr, UL, PHIU, M, LAMBDA, IsoPressRatio, AdiPressRatio, CLAdiPRatio),

{Fo, {PHIT, TW)},

{totalDragAa_0, to_alDragRMS/t_be, MOecFlow0Star}, date}

-)

-$ -9

{{{OPT, 0.002, 100,. 0.703, 70.266, 1., -0.I, 3.789 10 , 4.31 10 }, {1.002. 1.004, 1.00413}),

133
(100., (-0.03%18, 0.329915), 7026.6),

{0.00476915 A_ dyne, 0.199157 (tube^-ll dyne. 1.97349 W), (OPT,>100., 3, 15, 19))

0.2

0.i

-0.1

-0.2

-0.3

-0.4

01;'"0...... 08 1

y-16

(51.%333 Second, 2.53812 10 }

O_her

P1o_ [{rhoSuSAvg [r, 0,0 ],rho0ulAvg [r, 0,0 ],rholu0ReAvgFlux [r, 0, V, PV, U0, UL, PHIU, PHIT, 0 ]}, {r, 0,1 ),

FlotScyle-,( {RGBColor [0,0_ 1 ]), (RGBCoIor [0,1,0] ].(RGBColor [1,0,0 ] ]},

pio_Range- TAIl, AxegLabel -• (°r" ,"rhoSus&vg" )] ;

rhoSuSAvg

0.2 • r

.... 8

-0.4

-0.6

(" Plot of Integrated over area Transverse RMS Reynolds Stress line,ration z °I

PZot [RMSReynoldsStress[zl, (z, 0. I), PlotScyle->{ IRGBColor [1, 0,01 }) ]:

0.46

0.455

',' ' ' 0.2_ _ D._ /0.8 1

0.44

ARCHIVE

OTHER FLOW COMP0_4E_TS

(° Steady rho0DiFlux Field "I

f [kindSystem==OPT, Print [fileNameOFTllX £rList ]_Print [fil _ameOPTHX£ r3,, Print [fi Ia_ImeBPTHX frList ]:Print [fi ieJ_am_PTHX fr ]];

T_mlng[rhoOUIFlux[1, quasisteady. 0] =PiotVeC_orField_ {rho0uIAvg[r, z. 0], rho0viAvg[r_ z, 0] ) ,{z, 0,1), It, 0_ 1) _Aspec_Ratio-•l/3

(-. xxesLabel-• ("r/rw", "z/L °}_;u<es- >True, ") ]]

Timing |rho 0U iFlux [1, quasiSteady. TH ]=PlotVectorField [(rho0ulAvg [r, z. TW |,rho0vlAvg [r, z.TW] }, (z, 0,1 }, (r, 0, I ),As_ec_Ratio-• i/3

{° AxesLabel-> (o r/rw •.•z/L" }, Axes°>True, °) I ]

((B_T, _g, Va, Pr, VaPr. UL. PHIU, M, LAMBDA, ZsoPressRatio, AdiPressRatio, C_AdiPRat_o}_ (FO, (PHIT, TWI},

(totalDragiu_0, cotalDragRMS/tube, HOscF1ow0S_ar}, dace)

-7

((|BPT, 0.1, 250., 0.703, 13.254. 0., n/aPH_, 0.0003158, 5.98 10 ), {1.222, 1.39_, 1.4)I. (20,, _-0.288669, 0.490179)},

(0.39_429 Amp dyne, 2.1_228 (tube^-1) dyne, 246.69 W), {BPT<>250._ 6, 7. 13)1

SAborted

(60.8833 Second, -Graphics-}

(" Steady rholUOFlux Field ")

if [kindSystem==OPT ,Pr_nc [fileNameOPTHXf _LisC 1;Prin_ |fileNameOPTHXf r], ,Prin_ [fileNameBPT_XfrL_st) :Print [fl le/_ameBPTHXfr ]];

T1minglrholODOFlux[1,_uasiSteady,O] =PlOtVeC_orFieldI{rholu0ReAvgFI_x[r,z,V*PV,U0*UL,PHZU,FH_T.0], rholv0ReAvgFlux[r,z,0]}

(Z, 0, i), (r, 0,1} _AspectRatio->l /3 {", AxesLa_el-> (-r/rw", "z/L" }, Axes->True, "}] ]

Timing [r_ol 0UOF l_x [1. quasi S ready, TW ]=P-I0tVec corFi eld [(rholu0ReAvgFlux [r. z, V, PV. U0, UL, PHIU, P_IT, TW ],rholv0ReAvgFlux [_, z, TW] }

{z, 0, I}, {r, 0.1 ), Aspet_Ratio->l/3 _*, AxesLab_l- >{"_/rw", "z/L" }, Axes->T_ue, ") _ I

134
({BPT, eps, Va, Pr, VaPr, UL, PHZU. M, LAMEDA, ZsoPressRac_o, AdiPressR_t_o, CLAdiPRa_iO), {FO, {PXZT. TW}},

{_o_alDragAmp0. ¢o_alDragRMS/tube, HOscFlow0Star). date}

-7

{{{BPT, 0.1, 250., 0.703, 13.254, 0., n/aPXIU, 0.0003158, 5.98 l0 ), {1.222, 1.39_. 1.4}), {20., {-0.288669, 0.L90179)),

{0.39_429 Amp dyne, 2.17228 (_ube^-l) dyne, 246.69 w}, (BPT<>250., 6, 7, 13})

"u."_--.%%%%% % ". ". ". _ : :

• f • • • • • • • • i • & *

{43.1333 Second, -Graphics-I

•s .. ......

{ 45.1667 Second. -Graphics- }

(° Isothermal Flow movie o)

Block [{ tend=. 95 }. I f [kindSyst_==OPT, Print | f i IeNameOPTHX f rList ] ;Print [ f il e24ameOFTMX f r ] . . Print [ fileNameBPTHXf rList ] ;Print [f ileNamegPTHX f r ] ] ;

Timing [DO [velField [2, t, 0 ] =PlotVeCtorField [

{uORe[r,z.t.V,OO.UL,PHIU]+eps rhoSuSAvg[r,z.0| ,v0Re[r,z.t,V,PV,U0,UL.PBIU,PH_T,0]*e_s rhoSvSAvg[r,z,0] }, {z.0,1}. {r,0.1),

AspectRatio->113. ( OAxesLabel °> { • r I rw". • zl L" } , Axes->Tr_e, ° ) PlotLabel->t ]. { t, 0. tend.. 05 ) ] ] ]

(° _hin wall Flow movie °)

B1 ock [ { tend=. 95 ), I f [ kindSys tem==OPT, Print [f i IeNameOPTMX f rLi$ t ] _ Print [f ileNameOPTHX f r ] , ,Fr int [fi IeNameBPTHX f rLi st ] ;Print { f i IeNameBPTHX f r ] ] ;

Timing [DO [velField [2, _. TW] =PlotvettorField I

[u0Relr, z, _,V, U0,U_, PHIO ] *eps rhoSuSAvg It, z,TW] ,v0Re|r, z, _,V. PV. U0,UL. PHIU, PHIT, TW] *e_s rhoSvSAvg Jr, z,TW] ), (z, 0, i}, {r. 0,1 |,

Asgectgatio->ll3. ('Axe$Label-> ( "r Irw=, "z/5" } • Axes->True. ") PIOtLabel->t ] , {t. 0. tend. - 05 | ] ] ]

=================================================

(- lema-oEe_l_" B_'Z_=Ie Vel_i.ty ")

Du0ReR[r_,z_,t_,V, U0,UL,PMIU] = D[u0Re[r.z,t.V,U0.UL,PHI_].r];

Du0ReZ[r_,z_,t_,V,U0.UL,PHI_] = D[_0Re[r,z,t,V,U0,UL,PMIU|,_};

_'u0Re/_Iv0ReTAvgFl_x [r_, z_. 01 :=

Integrate [ Du0ReR [r, z, t, V. U0. OL, PHIU ] Integrate [v0Re [ r, z, t, V, _V, O0, _L. PH_U. PHIT, 0 ] , t ] , { t, 0, i } ]

D_0Re_Iv0ReTAVg_Iux [ r_. z_, TWI :=

inte_ra-.e [DU0Re8 [r, z, _ ,V, U0. UL, PBIU] inte_rate[v0ReIt, z. n,v, PV, U0, UL, PHIU, PHIT,TW}, t ] , { t, 0, % } ]

Du0ReZ _ U0ReTAVgFIux [ r_, z_] :=

Integrate| Du0ReZ [r, R, _, V. _0. O5. PMIUI Inte_raue[u0Relr, z, _,V,_0,UL. PHIU] , _], {t, 0,1 } |

u_Avg[r_.z_,0] := rhoSuSAvg[r,z,0] * Du0Re_v0ReTAv_Flux[r,z,0} * DuOReZ:uOReTAv_Flux[r,z]I/N

upAvglr_,z_,TWI := rhoSuSAvg[r,z,TW] ÷ L]u0ReRIV0ReTAvgFIt_x|r,z,TW] _ L_U0ReZIu0ReTAVgFIux[E, Z] //N

U'V0R_[t_,Z_,t_,V. PV,U0,UL,PXIU.P_IT,0] = D[v0Re[r,z,t,V, PV,U0,UL,PHIU,PHIT,0],r]:

Dv0ReZ[r_,z_,t_,V,_V,U0,UL,PXIU,PMIT.0] = D[v0Re[r,z,t,V, PV,U0,UL,FHXU.PH_T,0J,z];

Dv0ReR[r_,z_,t_,V,PV,U0,OL,PMIU,_RIT,TW) = D[V0Re[r.Z,t,V,PV,U0.UL,PHIU,PHIT,TW|,E];

Dv0 ReZ [ r_, z_, t_, V, PV. U0, UL, PHIU. PXIT, TW] = D [v0Re [r, z, t, V, PV, U0. UL, PHIU, PHIT, TW] ,z ] ;

Dv0ReRIv0R_TAvgFI_x [t_, z_, 0] :=

_nte_ra_e [ L_v0 Re/_ [r. z, t. V, PV, U0, UL, PHIU, PH!T, O ] Integraue (v0Re [r. z, U, V, PV, U0, UL, PHIU. PMIT, 0 ], t ] , { t, _, 1 } ]

Dv0ReZ ._0KeTAvg_l_ |r_, z_, 0 ] :=

Integrate [ E_0Rel [r. z, t, V, FV, U0, UL, PHIU, PHIT, 0 ] Integrate [u0Re [r, z. _, V, U0, OL, PHIU I , t ] . { t, 0,1 } ]

_v0ReRIv0ReTAvgFlux [ r_, z_. TW] ;=

In_egrate[Dv0ReK Jr, z, t, V, PV, U0, UL, PHZU. PHIT, _'_] integrate [v0Re[r, z, t,v, _V.U0.UL,PHIU,PR_T,TW] , t ], {t, 0.1} ]

Dv0ReZ ! u0ReTAvgFlux [ r_, z_. TW] :=

Inte_ra_e[Dv0ReZ[r,z,t,V,PV, U0,UL,PMIU,PHIT,TW] Integrate[u0Re[r,z,t,V.U0,UL,P_IU],t]. It.0.1)|

v_Avg[r_,z_,01 := rhoSvSAv_[r.z,0] * Dv0ReR!v0ReTAvgFIUX[r,Z.0] + Dv0ReZIu0ReTAVgFIUX[E,Z,O]//N

v_Avg [r_, z , T_] := rhoSvSAvg [r, z. TW] * Dv0ReR _v0ReTAvgF lux [r, z, TW] ÷ Dv0ReZ _u0ReTAvgFIux [r, z, TW] //N

uSPlot [{ r, .5 ) ,0 ]=Plot [rhoSuSAvg |r,. 5, 0), {r, 0,1} ] ;

uSPIo_ [{r, 1 ), 0 }=Plot [rhoS_SAv_[r, i, 0] , {r, 0,i} ] ;

Show| {uSPlot [ (r, 0 }, 0 ] , Grap_Lic$ [ {Dash/rig [ {_. 02,0.02 ) ] , RGBColor [ 0,0,1] , u_Av_Plot I0 • 0 _ } ] ), Axes->True]

( -Show [ {USPI ot [ (r, .5 ), 0 ], G_aphits [ { Dashing [ { 0.02,0.02 ] 1, RGBColor [0,0,1 ], u_Av_Plot [. 5, 0 ] ) ] ), Axes ->True ] * )

Show [ { uSPlot [ {r, 1 }, 0 ] , Graphics I [Dashing [ {0.02,0.02 ) ]. RGBColor [ 0,0,1 ], U_Av_P lot [i. 0 ) } I ), Axes->T_ue ]

Show|Graphics [ {upAvgPlot [0.0] , {Dashing[ {0.02.0.02} ] , RGBColor [0_ 0, i] ,u_AvgPlot [0.TW} } ) ) , Axes->True]

Show[ (uSPlot { { r, 0}, 0 ], Graphics| {Dashing| {0. _2.0.02 } |. RG_olor [ 0,0. I], upAv_Plot |0,0] } ] },

{uSPlot [ {r, i), 0] • Gra_hle$ [ {Dash/ng I {0.02,0.02} ] ,RGBColor 10.0, II ,upAv_Plot [i, 0] } ] }, Axes->True]

(*

Show|Graphics [ { { upAv_Plot [0, O] ],

{Dashing { {0.02,0,02) }. RGBColor [0.0, i], upAvgPiot [. 5,0] },

{Dashing I |0.005,0.02,0. _3.0.02 } _, RGBColor [ 1.0, O ], u_AvgPIot [ i. 0 ] }

> } ,Axes->True]

Show [GraphL cs [{ CuORe2: v0ReTAvgFluxPlo_ [0, 0 ] , ( Da shlng [ { 0,02. O. 02 } ] , RGBColor [C, G, 1 ],

Du0Re_Iv0ReTAvgFIUXPIot [ 0, TW} } ) ] ,AXeS ->TZ_e]

Show |Graphics [ Du0ReZ:u0ReTAvgFluxPlot [0 ) ] .AXeS ->True ]

°)

135
0.2

-0.:

-0.4

0.2 0.4 j/6 0.8 1

ys #asJ///

-Gr&_hlc$-

0.2

0.1

-0.

-0.2

-0.3

-0.4

0.2 _ 4f_.6 0.8 1

-Graphics*

Show{ (vSPlo_ [{r, 0), 0] , Graphics{lDashing[(O.O2,0.O2}],RGBColor[O,O,11,vpAvgPlot[O,O])]],Axes->True_

Show {Graphics [{vpAvgPlot [0, 0], (Dashing [(0.02,0.02 }],RGBColor [0,0,1 ],vpAvgP1ot [0 ,TW] ) }],Axes->True J

(-

Show{Graphics{ ({ v_AvgPlot [0, 0] },

{Dashing[ [0.02,0.02] }, PEBColor [0,0, i] .v_AvgPlo_ [. 5.0] },

{Dashing [ {0. 005.0.02,0.03,0.02 ]], RGBCoIor [I. 0,0 ],vpAvgPlot [i, 0] }

)| ,AXe$->TrUe]

Show {Graphics [{5_v0ReP_ v0ReTAvgF1uxPl ot [0,0 ], {Dashing {{0.02,0.02 } ],RGBColor [0,0,1 ],

Dv 0ReRI v 0ReTAvgF luxPl o_ (0, TW] )} ].Axes->True ]

Show (Graphics [{_v0ReZ: u0R_FA%_FluxPI ot _0.0 ],{Dashing [(0.02, 0.02 }],RG_olor [0,0,1 ],

Dv0Re_Iu0ReTAvgF IUXPIO _[0, TW] ] )],AXeS->True

-}

-Graphics0.015 j1 _

/

0.005 /' "

_0.0051 _,__, _., /.6 0 8 1

-0.01

-GraphicsIf [kindSys tem==OPT, Print [fi Ie_ameOPTHX frList ];Prln_ [fil eNameOPTMX fr ],,Print [fileNameBPTHX f rLis_ ];

Print [fi ieNameBPTHX fr ]];

Timing[parEiclevelField[1,meanSteJdy, 01 =PlotVectorField[{Re[upAvg[r.z,0]], Re[v_Avg[r,=,0]] },{z.O.l),

{r, 0, i] . AspectRatio->i/3, DisplayF_nction->Identiuy] ];

Timing [ particlevelField [ I, mea_Steady, TW] = Plo_Vec_orField [{u_AVg [r, z, TW], v_Avg |r, z, TW] ), {z, 0,1 ), {r, 0,1 },

A$_ectRatio->l /3, Di_layFu_c_ion->Identity] } ;

Show[{_articlevelField[1,meanSteady,0], liner{{0,1}, (i,i]] ,_exter).Dis_layFunction->$Dis_lay_n_nction]

Show [{par ticleVelField [i, means _eady, TW] ,liner [(0,1 }, (I, 1 }],_exter },DiaplayFunc _ion ->$DisD 1ay_unc_ion ]

{{OPT, eps, Va, Pr, VaPr, UL, PHIU. M, LAMBDA, IsoPressRatio, AdiPressRatio, CLAdiP_atio}, {Po, {PHIT, TW}, VaPrFo},

{_o_alDregAm_0, to_alDragP_4S/t_be, HO$cFIow0S_ar}, date}

-6 -9

{{{OPT, 0.002, i00., 0.703, 70.26_, i., -0.I. 3.759 i0 , 4.31 i0 ], (i.002_ 1.004, 1.00413)),

{i00., i-0.03918, 0.319915}, _026.6),

{0.00476915 Amp dyne, 0.19915_ (tube^-l) dyne. 1.9_49 w}, {OPT<,100., 3, 15, 19}}

(" Calculations ")

u!Avg[r_,z_,0] =rho0ulAvg[r,z,0]/ rho0[z]//Expand;

ulAvg [r_,z_,TW]=rho0ulAvg[r,z,TW]/rho0[z]//Ex_Lnd;

DulAvgZ[r_.z_,0] =D[ulAvg[r,z,0},z]:

_ulAvgZ[r,z_,TW}=D[ulAvg[r, z,TW],z]:

136
{.ooo°o .... .°.o°°.°...o..°o°oo°°o.o)

_vOTimesplu0AvgTe_plate = _lha_Templa_e ° u0hacCCTe_pla_e //Cae_lex_d;

_woTimesplu0ReAvgTempla_e = _woTimesplu0AvgTemplate - I Coefflcle_ l_woTimesplu0AvgTee_lece, I| :

_wOTime splu0ReAvgFl uxHo id (r_, z_, •ig_av_. •igmaPV_, Uo_. UI_. phiu_, phiT_, Tw_] =_woTimes_ iu0ReAv_Te=_ iat e/.

Union [plAssign, uOAssign] ;

pluOReAvgFlux(r_,z_,si_n_av_,sig_aPv_,uo_,Ul_,p_iU_,phiT_,Tw ]:z 1/2 °

zwoTimesplu 0ReAvgF iuxHol d {r, z, sigmaV, sig_PV, Uo, UI. phiU, phiT, Tw] /IReleaseHold//ReleaseHold//

ReleaseHold//N

plu0ReAvgFl_x (__, z_, V, PV, U0, UL, PHIU, PHIT, 0 ] =_lu0ReAvgFlux [r. g, V, PV. U0, UL, PRIU, PHIT •0 ]//Expand;

plu0ReAvgFlux [r_, z_, V, PV, U0, UL, PHIU, PHIT. TW] =plu0ReAvgFlux _r. z, V, PV. U0, UL, PHIU, PHIT. TW] //Expand;

Dplu0ReAvgFluxZ (r_, z_, 0] =D[plu0ReAvgFlux(r, z,V, PV,U0, UL, PHIU, PHIT, 0] , z| ;

Dplu0ReAvgFlux_ (r_. z_, TW ]=D [plu0ReAvgFlux [r, z,V. PV, U0, UL, PMIU, PHIT, TW] ,z ] ;

w1[r_,z_,0] =In_egra_e[1/r ° Integra_e[r'{DulAvgZtr,z.0] +Dplu0ReAvgFluxz[r,z,0]),r ],r]:

wl It_, z_, TW] =In_egra_e[I/r ° Integrate (r" (DulAvgZ[r, z,TW] _Dplu0ReAv_FluxZ [r. z, TW] ).r| .r| :

(..-°°° .°°°o.°.o.°°*°°.°.°.°o°°...t )

vIAvg[r_,z_,0] _rho0vlAv_[r,z,0]/ rhoO[z]//Expand:

vlAvg[r_,z_.T_]=rhoOviAvg[r,z,TW]/rhoO[z]//Expand;

twoT1mesplv0AvgTuvla_e = plha_Templa_e • v0ha_CCTem_la_e //COmplexExpand;

t_oTzmesplv0ReAvgTe_la_e = _woTimelplv0AvgTe_la_e - I Coefficient [twoTime_Iv0AvgTe_la_e, I| ;

_wOT_J_espl v0ReAvgFluxHo id Ir_, z_, sigmaV_, sigmaPV_, Uo_, UI_, phiu_, phiT_, Tw_] =_woTimesp lv0ReAvgTem_ iat e/.

Union [_iAssign, v0Assign ]:

_lv0ReAvgFlux_r_,z_.•igmaV_.sigmaPV_,Uo_,Ul_,_hiU_,&hiT_,Tw_l := 1/2 *

_woTimespl vOReAvgF iuxHol d [r, z, sigmaV, sigmaPv, Uo, UI, phiu, l_iT. Tw] //ReleaseHold/IReleaseHold//Rele&seHold//N

plv0ReAvgFlux _r_, z_,V, PV, U0,UL, PHIU, PHIT, 0 ] =plvOReAvgFI_x[r.z,V, PV. UO,UL,PHIU,PHIT,O]//F_q_and:

plv0ReAvgFlux [r_, z_, V, _V, U0, UL, PHIU, PHIT, TW] _plv0ReAvgFltu_ [r. z, V. PV. U0, UL, PMIU, _IT. TW ]/ /Ex_d;

w2[r_,z_,0_ =In_egra_e[viAvg[r,z.0| * plv0ReAvgFlux[r,z,V.PV, U0,UL, PHIU,PHIT, 0], r]//E_pand:

w2 [r_, z_, TW} =Integ rate [viAvg [r. z. TW ] * plv0ReAvgFlux [r, z. V, PV, U0, UL, PRIU, PHIT, TW] ,r]//Expand;

{°.°o.°...°..°.°..°o°°.o.°.°o.o.o,, )

T2[r_,z_,0] - {PV)^2 (wl[r,z,0] -wl[1,z,0} *w2[r,z.0] -w2|l,z,0]) -

CAPGAR_A^2 D(T0[z|,(Z.2}] 1/4 (l-r^2}//Expand;

T2[r_,z_,TW] = (PV)^2 (wl(r,z,TW]-w![1,z,TW]*w2[r,z,TW]-w2|l,z,TW]) -

_^2 D[TO[z),{z,2}) 1/4 (l-r*2}//Expand;

If (kindSy• _e_a==OPT, Print [fileNameOPTHX frLis_ 1:Prin_ [fil eNameOPTRX f r], .Prin_ [fileNameBPTHX frLi$_ ];

Print [fileName_PT_Xfrl ]:

T2I•o=Plot [(T2{r, 0.0, 0] ,T2 Jr..25,0|,

T2[r,. 50, 0] ,T2[r, .75,0] ,

T2[r,l.0, 0] }, {r, 0,1),

PlotStyle->{ {RGBColor (0, 0, .6} |, {RGBColor |0,0, .9| }. {RGBColor [0,0, .8] ), (RGBColor _0,0, ._] }, (RGBColor [0,0, I} ) },

PlotRange->All. Dis&layFu_c tion ->Iden_it y ]:

T2TW=Plot I{T2 (r. 0.0. TW], T2 [r, .25, TW],

T2[r,. 50, TW] ,T2 Jr, .75.TW_,

T2 [r,l. 0,TW] }, (r, 0,1},

PloUS_yIe->( {RGBCOIOr (.6, 0, 0) }, (RGBColor I.9.0,0] }, {RGBColor [. 8.0, 0] ), (RGBColor (. 7,0,0] }, (RGBColor (I, 0, 0]) ),

PlotRange->All, Displ ayFunc _ion ->Iden_i _y ];

Show[ {T2Ieo, T2T_} ,Di$_layFu_c_ion->SDis_layFu_c_ion, Axes Ldtbe I-> {"z", " "}

-GraphicsARCHIVE

ARCHIVE

{(Thesis, OPT, _ps, V. P, PV, UL, PHZU, M, IsoPressRa_io, AdiVOIPR, CLAdiPRa_io), IF, [PHIT, TW}), {HFIow0}, (1995, 12, 13, 13, 2, 59})

((Thesis, OPT, 0.1, 22.5122. _.83825, 18.871. 0.3, -0.5. 0.0006315, 1.299. 1.546, 1.55319], {0.000372373, (-0.253, 0.881585)), {500.081 W)}

-0.7 I

-0.75

-0.8

-0.85

0.2 0.4 0.6 0._ / 1

-0.9

0.9

0.8

0.7

0.6

0.S

0.2 0.4 0.6 0.8 I

137

APPENDIX H

EXPERIMENTAL DATA

The data presented are for Runs 1, 2, 5, and 6 discussed in section 5. The columns are

generally self-explanatory. The first two columns list the frequency and phase angle. The raw data

give position and calculate axial particle velocity, which is then converted into dimensionless

velocity. The system parameters _ and UO are then listed. The calculated axial particle velocity

corresponding to the measured position is in the next column. The final four shaded columns are

used in the plots of section 5. The first shaded column lists the single-point velocity, the second

column lists the 3-point nmning average of the single point velocity, and the third column lists the

5-point running average. The last column lists the corresponding calculated axial velocity.

139
I:I
qr

m_

oooooooooo

m_

oooooooooooo "',''_'''',','"


oooo

_o_o_ __o "00o0ooo
o. oo. ooo

_ _,

I_, ii

|

ol o! ̧

ol ol i i

oi ol

ol o_

o_ ol

ol ol 0 0

-I .I

ol ol

ro_o

Iolo

•REFERENCES

1. Sherman, F. S.: 1990 Viscous Flow, McGraw Hill.

2. Paolucci, S.: 1982 On the filtering of sound from the Navier-Stokes equations, SAND 82-8257,

Sandia National Labs, Livermore.

3. Grotberg, J. B.: 1984 Volume-cycled oscillatory flow in a tapered channel, J. Fluid Mech., 141,

249-264.

4. Hall, P.: 1974 Unsteady viscous flow in a pipe of slowly varying cross-section, ,1. Fluid Mech.,

64, 209-226.

5. Harris, H. G.; and Goren, S. L.: 1967 Axial diffusion in a cylinder with pulsed flow, Chem. Eng.

Sci., 22 1571-1576.

6. Watson, E. J.: 1983 Diffusion in oscillatory pipe flow, J. Fluid Mech., 133, 233-244.

7. Bowden, K. F.: 1965 Horizontal mixing in the sea due to a shearing current, ,1. Fluid Mech., 21,

84-96.

8. Schlichting, H.: 1932 Berechnung ebener periodischer Grenzschichtstr/Smungen, Physik.

Zeitschr., 33, 327-335.

9. Longsworth, R. C.: 1997 Early pulse tube refrigerator developments, Cryocoolers 9, Plenum

Press, 261-268.

10. K.ittel, P.: 1994 A short history of pulse tube refrigerators, http://ranier.oact.hq.nasa.gov/

Sensors__page/Cryo/CryoPT/CryoPTHist.html.

11. Radebaugh, R.: 1990 A review of pulse tube refrigeration, Adv Cryo. Eng., Plenum Press, 35B,

1191-1205.

12. Swift, G. W.: 1988 Thermoacoustic engines, Jr. Acoust. Soc. Am., 84 (4), 1145-1180.

13. Gifford, W. E.; and Longsworth, R. C.: 1964 Pulse tube refrigeration, Jr. Eng. Industry,

Transactions of the ASME, 63, 264-268.

14. Longsworth, R. C.: 1967 An experimental investigation of pulse tube refrigeration heat pumping

rates, Adv Cryo. Eng., Plenum Press, 12, 608-618.

15. Gifford, W. E.; and Longsworth, R. C.: 1966 Surface Heat Pumping, Adv Cryo. Eng., Plenum

Press, 11, 171-179.

16. de Boer, P. C. T.: 1994 Thermodynamic analysis of the basic pulse-tube refrigerator,

Cryogenics, 34, 699-711.

17. Wheatley, J.; Hoffler, J. H.; Swift, G. W.; and Migliori, A.: 1985 Understanding some simple

phenomena in thermoacoustics with applications to acoustical heat engines, Am. ,1. Phys., 53,

147-162.

18. Olsen, J. R.; and Swift, G.W.: 1994 Similitude in thermoacoustic, J. Acoust. Soc. Am., 95 (3),

1405-1412.

151
19.Homer,T. J.: 1988Conceptsforthermoacousticrefrigerationandapracticaldevice,Proc. Fifth

lntl. Cryocooler Conf., 93-101.

20. Rott, N.: 1969 Damped and Thermally Driven Acoustic Oscillations in Wide and Narrow Tubes,

J. Apt Math & Physics (ZAMP), 20, 230-243.

21. Rott, N.: 1980 Thermoacoustics, Adv. Appl. Mech., 20, 135-175.

22. Mfiller, U. A.: 1982 Thermoakustishe Gasschwingungen: Definition und Optimiertmg eines

Wirkungsgrades, Diss. ETHNr. 7014, Eidegenfssischen Technischen Hochschule, Zftrich.

23. Rott, N.: 1973 Thermally driven acoustic oscillations, Part II: Stability limit for helium, J. Apl

Math & Physics (ZAMP), 24, 54-72.

24. Rott, N.: 1976 Thermally driven acoustic oscillations, Part IV: Tubes with variable cross-section,

J. Apl Math & Physics (ZAMP), 27, 197-224.

25. Rott, N.: 1976 Thermally driven acoustic oscillations, Part V: Gas-liquid oscillations, J. Apl

Math & Physics (ZAMP), 27, 325-334.

26. Rot't, N.: 1983 Thermally driven acoustic oscillations, part VI: Excitation and power, .1. Apl Math

& Physics (ZAMP), 34, 609-626.

27. Rott, N.: 1974 The heating effect connected with non-linear oscillations in a resonance tube,

J. Apl Math & Physics (ZAMP), 25, 619-634.

28. Rott, N.: 1975 Thermally driven acoustic oscillations, Part III: Second-order heat flux, ,/. Apl

Math & Physics (ZAMP), 26, 43-49.

29. Rott, N.: 1984 Thermoacoustic heating at the closed end of an oscillating gas column, J. Fluid

Mech., 145, 1-9.

30. Merkli, P.; and Thomann, H.: 1975 Thermoacoustic effects in a resonance tube, J. Fluid Mech.,

70 (1), 161-177.

31. Mikulin, E. L.; Tarasov, A. A.; and Shkrebyonock, M. P.: 1984 Low-temperature expansion

pulse tubes, Adv Cryo. Eng., Plenum Press, 29, 629-637.

32. Radebaugh, R.; Zimmerman, J.; Smith, D. R.; and Louie, B.: 1986 A comparison of three types

of pulse tube refrigerators: New methods for reaching 60 K, Adv Cryo. Eng., Plenum Press, 31,

779-789.

33. Matsubara, Y.: 1994 Novel configuration of three-stage pulse tube refrigerator for temperatures

below 4 K, Cryogenics, 34, 259-262.

34. Tanida, K.; Gao, J. L.; Yoshimura, N.; and Matsubara, Y.: 1996 Three-staged pulse tube

refrigerator controlled by four-valve method, Adv Cryo. Eng., Plenum Press, 41B, 1503-1509.

35. Storch, P. J.; Radebaugh, R.; and Zimmerman, J. E.: 1990 Analytical model for the refrigeration

power of the orifice pulse tube refrigerator, NIST Tech Note 1343, National Institute for

Standards and Technology, Boulder, CO.

36. David, M.; Mare'chal, J. -C.; Simon, Y.; and Guilpin, C.: 1993 Theory of ideal orifice pulse tube

refrigerator, Cryogenics, 33 (2), 154-161.

152
37. Kittel, P.;Kashani,A.; Lee,J.M.; andRoach,P.R.: 1996Generalpulsetubetheory,

Cryogenics, 36 (10), 849-857.

38. Lee, K. P.: 1983 A simplistic model of cyclic heat transfer phenomena in closed spaces, Proc.

18 th Intersociety Energy Conversion Eng. Conf., 720-723.

39. Komhauser, A. A.; and Smith, J. L.: 1989 Heat transfer with oscillating pressure and oscillating

flow, Proc. 24 th Intersociety Energy Conversion Eng. Conf., 2347-2353.

40. Komhauser, A. A.; and Smith, J. L.: 1994 Application of a complex Nusselt number to heat

transfer during compression and expansion, ,1. Heat Transfer, 116, 536-542.

41. Roach, P.; and Kashani, A.: 1997 A simple modeling program for orifice pulse tube coolers,

Cryocoolers 9, Plenum Press, 327-334.

42. Jeong, E. S.: 1996 Secondary flow in basic pulse tube refrigerators, Cryogenics, 36 (5), 317-323.

43. Lee, J. M.; Kittel, P.; Timmerhaus, K. D.; and Radebaugh, R.: 1995 Steady secondary

momentum and enthalpy streaming in the pulse tube refrigerator, Cryocoolers 8, Plenum Press,

359-369.

44. Xiao, J. H.: 1995 Thermoacoustic heat transportation and energy transformation, Part 1:

formulation of the problem, Cryogenics 35 (1), 15-19.

45. Xiao, J. H.: 1995 Thermoacoustic heat transportation and energy transformation, Part 2:

isothermal wall thermoacoustic effects, Cryogenics 35 (1), 21-26.

46. Xiao, J. H.: 1995 Thermoacoustic heat transportation and energy transformation, Part 3:

adiabatic wall thermoacoustic effects, Cryogenics 35 (1), 27-29.

47. Bauwens, L.: 1996 Entropy balance and performance characterization of the narrow basic pulsetube refrigerator, J. Thermophysics and Heat Transfer, 10 (4), 663-671.

48. Bauwens, L.: 1996 Oscillating flow of a heat-conducting fluid in a narrow tube, J. Fluid Mech.,

342, 135-161.

49. Shiraishi, M.; Nakamura, N.; Kaquya, S.; and Murakami, M.: 1997 Visualization study of

velocity profiles and displacements of working gas inside a pulse tube refrigerator, Cryocoolers

9, Plenum Press, 355-364.

50. Nakamura, N.; Shiraishi, M.; Seo, K.; and Murakami, M.: 1997 Visualization study ofosciUating

flow inside a pulse tube refrigerator, Cryogenic Engineering Conf., Portland, OR, paper APG-1.

51. Hoffrnan, A.; Wild, S.; and Oellrich, L. R.: 1997 Evaluation of experimental pulse tube

refrigerator data with predictions of the thermoacoustic theory, Proc. 16 th International

Cryogenic Eng. Conf., Kitakyushu, Japan, 239-242.

52. Lee, J. M.; Kittel, P.; Timmerhaus, K. D.; and Radebaugh, R.: 1993 Flow patterns intrinsic to the

pulse tube refrigerator, Proceedings of the 7th International Cryocooler Conf., PL-CP-93-1001,

Santa Fe, 125-139.

53. Zhu, S.; Wu, P.; and Chen, Z.: 1990 Double inlet pulse tube refrigerators: an important

improvement, Cryogenics 30 (5), 514-520.

153
54.Lewis,M.; andRadebaugh, R.: 1997 Effect of regenerator geometry on pulse tube refrigerator

performance, Cryogenic Engineering Conf., Portland, OR, paper DPG-9.

55. Seo, K.; Shiraishi, M.; Nakamura, N.; and Murakami, M.: 1997 Investigation of radial

temperature and velocity profiles in oscillating flows inside a pulse tube refrigerator,

Cryocoolers 9, Plenum Press, 365-374.

56. Shiraishi, M.; Seo, K.; and Murakami, M.: 1997 Oscillating flow behavior in pulse tube

refrigerator under optimized conditions, Cryogenic Engineering Conf., Portland, OR,

paper APG-4.

57. Godshalk, K. M.; Jin, C.; Kwong, Y. K.; and Hershberg, E. L.: 1996 Characterization of 350 Hz

thermoacoustic driven orifice pulse tube refrigerator with measurements of the phase of the mass

flow and pressure, Adv Cryo. Eng., Plenum Press, 41B, 1411-1418.

58. Zhu, S. W.; Zhou, S.L.; Yoshimura, N.; and Matsubara, Y.: 1997 Phase shift effect of the long

neck tube for the pulse tube refrigerator, Cryocoolers 9, Plenum Press, 269-278.

59. Gardner, D. L., and Swift, G. W.: 1997 Use ofinertance in orifice pulse tube refrigerators,

submitted to Cryogenics.

60. Roach, P.; and Kashani, A.: 1997 Pulse tube coolers with an inertance tube: theory, modeling

and practice, Cryogenic Engineering Conf., Portland, OR, paper APE-6.

61. Batchelor, G. K." 1967 Introduction to Fluid Mechanics, Cambridge University Press.

62. Lee, J. M.; K.ittel, P.; Timmerhaus, K. D.; and Radebaugh, R.: 1995 Useful scaling parameters

for the pulse tube, Adv Cryo. Eng., Plenum Press, 41B, 1347-1356.

63. White, F. M.: 1991 Viscous FluidFlow, McGraw-Hill, Inc.

64. Bird, R. B.; Stewart, W. E.; and Lightfoot, E. N.: 1960 Transport Phenomena, John Wiley &

Sons, Inc.

65. Anderson, J. D.: 1990 Modern Compressible Flow, McGraw-Hill, Inc.

66. Gedeon, D.: 1986 Mean Parameter Models of Oscillating Flow. J. Heat Transfer, 108(3),

513-518.

67. Stuart, J. T.: 1966 Double boundary layers in oscillatory viscous flow, J. Fluid Mech., 24 (4),

673-687.

68. Kasuya, M.; Yuyama, J.; Geng, Q.; and Goto, E.: 1992 Optimum phase angle between pressure

and gas displacement oscillations in a pulse-tube refrigerator, Cryogenics, 32 (3), 303-338.

154
SUGGESTED READING

LighthiU, Sir James: 1978 Acoustic streaming, J. Soundand Vib., 61 (3), 391-418.

Mirels, H.: 1993 Linearized pulse tube cryocooler theory, Proc. 7th Int. Cryocooler Conf., 1,

222-232.

Ravikumar, K. V.; Yoshida, S.; Karlmann, P.; Myung, N. S.; and Frederking, T. H. K.: 1997 On

thermodynamic aspects of pulse tube operation, 11 th Intersoc. Cryogenic Sympos., Houston

ISBM 1-890277-05-3 (Book VI), Intersociety Cryogenics '97 28-35.

Richardson, R. N.: 1988 Development of a practical pulse tube refrigerator: coaxial designs and the

influence of viscosity, Cryogenics, 28, 516-520.

Riley, N.: 1965 Oscillating flows, Mathematika, 12, 161-175.

Riley, N.: 1967 Oscillatory viscous flows. Review and extension, ./. Inst. Maths Applics, 3, 419-434.

Schlichting, H.: 1968 Boundary Layer Theory, McGraw-Hill, Inc.

155
Form Approved

REPORT DOCUMENTATION PAGE OMSNoozo4o 88

Public reportingburdenfor this collectionof informationis estimatedto average 1hour perresponse,including the time forreviewinginstructions,searchingexistingdata sources,

gathering and maintainingthe data needed, and completingend reviewingthe co!lentionof information.Send co=mmentsregardingthisburdenes!imate orany other =esl_'t..of this

:ollection of information, including suggestions for reducing this burden, to Washington Heack:luarters SeNices, D=rectorate for mtormstion uperatJon$ eno Heports, 1 Z1 _>jQner$oi3

Davis Highway,Suite 1204, Arlington.VA 222024302. and to the Office of Management andBudget,PaperworkReductionPro act (0704-0188), Washington. DC 20503.

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

September 1999 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Steady Secondary Flows Generated By Periodic Compression and

Expansion of an Ideal Gas in a Pulse Tube

6. AUTHOR(S)

Jeffrey M. Lee

7. PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)

Ames Research Center

Moffett Field, CA 94035-1000

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

UPN 632-20

8. PERFORMING ORGANIZATION

REPORT NUMBER

A-9901168

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA/TM-1999-208769

11. SUPPLEMENTARY NOTES

Point of Contact: Jeffrey M. Lee, Ames Research Center, MS 244-10, Moffett Field, CA 94035-1000

(650) 604-5693

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified -- Unlimited

Subject Category 34

Availability: NASA CASI (301) 621-0390

Distribution: Standard

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

This study establishes a consistent set of differential equations for use in describing the steady secondary flows generated by periodic compression and expansion of an ideal gas in pulse tubes. Also considered

is heat transfer between the gas and the tube wall of finite thickness. A small-amplitude series expansion

solution in the inverse Strouhal number is proposed for the two-dimensional axisymmetric mass, momentum

and energy equations. The anelastic approach applies when shock and acoustic energies are small compared

with the energy needed to compress and expand the gas.

An analytic solution to the ordered series is obtained in the strong temperature limit where the zerothorder temperature is constant. The solution shows steady velocities increase linearly for small Valensi

number and can be of order 1 for large Valensi number. A conversion of steady work flow to heat flow

occurs whenever temperature, velocity or phase angle gradients are present. Steady enthalpy flow is reduced

by heat transfer and is scaled by the Prandtl times Valensi numbers.

Particle velocities from a smoke-wire experiment were compared with predictions for the basic and

orifice pulse tube configurations. The theory accurately predicted the observed steady streaming.

14. SUBJECTTERMS

Pulse tube, Anelastic, Steady secondary flows

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION

OF REPORT OF THIS PAGE

Unclassified Unclassified

NSN 7540-01-280-5500

19. SECURITY CLASSIFICATION

OF ABSTRACT

15. NUMBER OF PAGES

161

16. PRICE CODE

A08

20. LIMITATION OF ABSTRAC

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

9QR-1 _