> Paper presented at the 8th International Cryocooler Conference,
> June-28-30, 1994, Vail, CO.

**Steady Secondary Momentum and Enthalpy Streaming in the Pulse Tube
Refrigerator**

> **J. M. Lee** **and** **P. Kittel**
>
> NASA Ames Research Center
>
> Moffett Field, CA 94035
>
> **K. D. Timmerhaus**
>
> University of Colorado, Boulder
>
> Department of Chemical Engineering
>
> Boulder, CO 80303
>
> **R. Radebaugh**
>
> National Institute of Standards and Technology
>
> Boulder, CO 80303

ABSTRACT

Our study investigates the steady secondary momentum and enthalpy
streaming that occurs in the pulse tube refrigerator. The linearized
mass, momentum and energy conservation laws that are described by N.
Rott1 are applied to a pulse tube, with the phase and amplitude of the
axial velocity boundary conditions treated as independent parameters.
Heat transfer between the gas and the tube wall is included. Heat
transfer is shown to affect enthalpy flow by modifying the dynamic
temperature amplitude and the temperature phase angle of the gas. We
also calculate the steady mass flow circulation due to Reynolds stresses
in the pulse tube. The length scale of the circulation is shown to be of
the order of the tube length. Mass flow circulation is a loss mechanism
because it results in a direct convection of enthalpy between the cold
and hot ends.

INTRODUCTION

The one-dimensional enthalpy flow model for the pulse tube refrigerator
(PTR) has been extensively developed2. It is applicable in the limit of
negligible heat transfer between the gas and the tube wall. For a system
with transport in the axial z-direction parallel to the tube length, the
steady enthalpy flow is calculated from the time-averaged quantity \<H\>
= \<\\O(m,˙)CpTd\> where the quantities in brackets are time averages.
The time-averaged enthalpy flow is \<H\>, the heat capacity at constant
pressure is Cp, the mass flow is \\O(m,˙), and the dynamic temperature
is Td. Though the one-dimensional enthalpy flow model has been very
successful, it does not include the effects of momentum and thermal
diffusion.

Including momentum and thermal diffusion into the analysis will provide
a greater understanding of the flow physics within a pulse tube. With
thermal diffusion, one can examine how heat transfer between the gas and
the tube wall affects enthalpy flow. One can then determine whether it
helps or hinders. With momentum diffusion, one can understand how the
Reynolds stresses in the viscous layer produce the forces that drive
steady momentum streaming3. Momentum streaming is important for the PTR
because it is a loss mechanism, i.e., it results in steady mass flow
circulation between the cold and warm ends.

The present work addresses how thermal and momentum diffusion affect
enthalpy flow and momentum streaming in the PTR. A two-dimensional
axisymetric model -- based on the mass, momentum and energy conservation
laws -- is the foundation of this analysis. The model is used to
understand how heat transfer between the gas and the tube wall changes
the temperature amplitude and phase angle of the gas. The changes are
then assessed in terms of their impact on enthalpy flow. Steady momentum
streaming is then addressed. The resulting analysis gives a general
understanding of the transport mechanisms within the pulse tube.

**SCALING PARAMETERS**

The model is based upon the set of linearized conservation equations
examined by Rott1 and used by Swift4 and by Merkli and Thomann5 to
describe thermoacoustic effects. The equations are recast in
dimensionless form and applied to a cylindrical tube geometry with
boundary conditions representative of the PTR. The basic state problem
is then solved for the case of negligible axial temperature gradient. A
complete description of the governing equations and the solution method
would leave little room for the main purpose of this paper -- that of
showing calculated results. In the Appendix we give a brief description
of the equations. For full details, the reader is referred to
elsewhere6. In this section we define the system and outline the
problem.

The axisymetric system under study is shown in Figure 1. All variables
are dimensionless and are scaled from 0 to 1. The system is an ideal gas
inside a tube that has a finite wall thickness. The problem is divided
into two domains. The gas domain, which considers only the gas inside
the tube, extends from *r*=0 to *r*=1 and *z*=0 to *z*=1. The tube wall
domain, which considers only the tube wall thickness, extends from *y*=0
to *y*=1 and *z*=0 to *z*=1. A planar geometry is used for the tube wall
domain. This is valid in the limit of a very thin-walled tube relative
to the inner tube radius. The two problems are linked together through
the boundary conditions between the gas and the tube wall interface;
temperature and heat flux must be continuous across the interface.

For the gas domain, the velocity field and the thermodynamic variables
of temperature, *T*, and density, *ρ*, are functions of spatial
coordinates *r* and *z*, and of time, *t*. The pressure, *p*, depends
only on *z* and *t* . This approximation is valid when the ratio of the
tube length to tube radius is large. For the case of the PTR, it is
usually greater than 10. For the tube wall domain, the temperature is a
function of *y* and *t*.

The normalized velocity field for the gas domain is composed of axial
velocity, *u*, and radial velocity, *υ *. The boundary conditions for
*u* are at *r*=0 and *z=0, u=Uo* ei2π*t* where *Uo*=1; and at *r*=0 and
*z*=1, *u*=*UL*ei2π(*t*+*φU*) where *UL* is the normalized velocity
amplitude. The imaginary number \\R(--1) is given by i. The boundary
conditions for temperature are at the centerline (*r*=0) the radial heat
flux is zero; at the inner surface of the tube wall (*r*=1, *y*=0), the
temperature and heat flux are continuous between the gas and the wall;
and at the outer surface of the tube (*y*=1) the radial heat flux is
zero (adiabatic).

![](media/image1.pct){width="6.347222222222222in"
height="1.9305555555555556in"}

**Figure 1.** Axisymetric system studied. The gas domain extends from
*r*=0 to *r*=1 and *z*=0 to *z*=1. The tube wall domain extends from
*y*=0 to *y*=1 and *z*=0 to *z*=1.

> **Table 1**. Definitions of scaling parameters

  ---- --------------- ------------ ---------------------------------------
       Name            Definition   Physical Meaning

  ---- --------------- ------------ ---------------------------------------

  ----- --------------- ---------------------------------------------------- ---------------------------------------
  *ε*   expansion       ![](media/image2.pct){width="0.6527777777777778in"   ratio of displacement length to tube
        parameter       height="0.2638888888888889in"}                       length

  ----- --------------- ---------------------------------------------------- ---------------------------------------

  ---- --------------- ---------------------------------------------------- ---------------------------------------
  Va   Valensi Number  ![](media/image3.pct){width="0.8055555555555556in"   ratio of tube inner radius to viscous
                       height="0.3472222222222222in"}                       diffusion length

  ---- --------------- ---------------------------------------------------- ---------------------------------------

  ---- --------------- ------------------------------------- ---------------------------------------
  Pr   Prandtl Number  ![](media/image4.pct){width="0.5in"   ratio of viscous to thermal diffusion
                       height="0.2916666666666667in"}        length scales

  ---- --------------- ------------------------------------- ---------------------------------------

  ---- --------------- --------------------------------------- ---------------------------------------
  M    Mach Number     ![](media/image5.pct){width="0.875in"   ratio of velocity at r=0 and z=0 to
                       height="0.3055555555555556in"}          speed of sound

  ---- --------------- --------------------------------------- ---------------------------------------

  ---- --------------- ---------------------------------------------------- ---------------------------------------
  Fo   Fourier Number  ![](media/image6.pct){width="0.7083333333333334in"   ratio of thermal diffusion length to
                       height="0.3055555555555556in"}                       tube wall thickness

  ---- --------------- ---------------------------------------------------- ---------------------------------------

  ------ --------------- ---------------------------------------------------- ---------------------------------------
  *UL*   velocity ratio  ![](media/image7.pct){width="0.5277777777777778in"   ratio of velocity at r=0 and z=1 to
                         height="0.2638888888888889in"}                       velocity at r=0 and z=0

  ------ --------------- ---------------------------------------------------- ---------------------------------------

  ------ --------------- ---------- ---------------------------------------
  *φU*   velocity phase             phase angle at r=0 and z=1 relative to
         angle                      velocity at r=0 and z=0

  ------ --------------- ---------- ---------------------------------------

The quasi-steady, periodic solution for temperature, pressure, density
and velocity field is analytic in *r*, *z*, *t*, and has non-dimensional
scaling parameters *ε*, Va, Pr, M, Fo, *UL* and *φU*. Table 1 defines
the scaling parameters in terms of *dimensional* (starred) quantities,
and describes their physical meaning. *U*\\O( *\**,*o*) is the velocity
at *z*=0, *U*\\O( *\**,*l*) is the velocity at *z*=1, *T*\\O( *\**,*o*)
is the reference temperature, *l*\\O( *\**,*t*) is the tube thickness,
*τ∗* is the period of oscillation, *rw*\* is the tube inner radius,
*L\** is the tube length, *ν∗* is the kinematic viscosity, *αt*\* is the
tube wall thermal diffusivity, *αg\** is the gas thermal diffusivity, γ
is the heat capacity ratio and *R* is the ideal gas constant.

RESULTS

The Effect of Heat Transfer on the Periodic Gas Temperature

Heat transfer affects enthalpy flow by changing the dynamic temperature
amplitude and phase angle of the gas. The quasi-steady periodic solution
to the basic state problem for the gas temperature has the form

![](media/image8.pct){width="2.6805555555555554in"
height="0.4027777777777778in"} (1)

where *T1* is the oscillating temperature of the gas, *Tw* is the
temperature at the interface between the gas and the tube wall and *φT*
is the phase angle between the temperature at the interface and *Uo*.
The temperatures and phase angle are dimensionless and range from 0
to 1. Note that only the real part of Eq. (1) has physical meaning. The
first term of Eq. (1) represents isothermal wall conditions. The second
term is due to heat transfer coupling between the gas and the tube wall.
Heat transfer adds a lagging component that is strongest in the thermal
diffusion layer. Figures 2a and 2b plot *Tw* and *φT* as functions of Fo
for the given parameters *ε*, M, *UL* , *φU* and VaPr (product of Va and
Pr). The parameter values are representative of an OPT. For Fo\<\<1,
*Tw*→0 which means that isothermal wall conditions are approached. For
Fo→∞ adiabatic conditions are approached. Figure 2a shows that near
isothermal conditions are reached when Fo≤10.

A typical operating point for an OPT is at VaPr≈10 and Fo≈10. In this
case, Fig. 2a shows that *Tw* is ***O***(10--2). The symbol ***O*** is
used to mean \"of order\". This is about one order less than the dynamic
temperature amplitude, which is ***O***(10--1) relative to the reference
temperature. This means that for OPTs, the temperature at the interface
between the gas and tube wall is close to

![](media/image9.pct){width="3.25in"
height="2.6944444444444446in"}![](media/image10.pct){width="3.25in"
height="2.6666666666666665in"}

**Figure 2.** The effect of heat transfer on the dynamic temperature
amplitude *Tw* , and the phase angle *φT*, at the tube wall inner
surface (at *r*=1). Fo is the Fourier Number for the tube wall (defined
in Table 1). Other parameters; *UL*=0.3, *φU*= -- 0.25, *ε*=0.667, M=5.1
x 10--4.

being isothermal, i.e., heat transfer is present. This is important
because for one-dimensional models, it is assumed that the conditions at
the interface are adiabatic. Figure 2b plots Fo vs. *φT* . The plot
shows that *φT* is less than zero which means that the temperature at
the interface lags the reference velocity at *z*=0.

The BPT which operates at *UL*=0 and VaPr≈1 ideally operates at the
isothermal limit *Tw*→0. In this case, heat transfer between the gas and
tube wall produce the proper phase shift between velocity and
temperature for enthalpy to flow. Calculations show that *φT* remains
relatively constant at *φT* ≈-- 0.25 even for large Fo. However, since
proper phasing takes place only in the diffusion layer, maximum
efficiency is obtained when thermal diffusion fills the entire tube,
i.e., VaPr\~***O***(1). Stack arrangements are then necessary to
increase the total enthalpy flow, such as used in acoustic
refrigerators.

Figure 3 shows an example of how heat transfer affects radial
temperature distribution and phase angle relative to an isothermal wall.
The parameter values are the same as those in Fig. 2 with VaPr=10 and
Fo=100. Three times are shown: *t*=0.2, *t*=0.35 and *t*=0.5. The plot
shows how the oscillating wall temperature modifies the temperature
primarily in the thermal diffusion layer (*r*\>0.5). At the core
(*r*\<0.5) the temperatures are similar, both in amplitude and phase. At
*r*=0, the temperature amplitude and phase are close to one-dimensional
predictions. For one-dimensional models, the temperature is not a
function of *r*. Figure 3 clearly shows that when diffusion is
considered, the temperature profiles are very different from idealized
one-dimensional models.

> ![](media/image11.pct){width="3.75in" height="2.0555555555555554in"}

**Figure. 3.** The effect of heat transfer on the oscillating gas
temperature as a function of radial distance for Fo=100 and for an
isothermal tube wall; *UL*=0.3, *φU*=--0.25, *ε*=0.667, M=5.1 x 10--4,
VaPr =10.

> ![](media/image12.pct){width="3.986111111111111in"
> height="2.138888888888889in"}

**Figure 4.** Normalize enthalpy flow vs. *z* for an OPT configuration:
*ε*=0.55, VaPr=10, *UL*=0.3, *φU*=--0.25 and M=2.0 x 10--4; The
reference enthalpy flow is 58.7W.

The Effect of Heat Transfer on Enthalpy Flow

Enthalpy flow is calculated from the quantity

![](media/image13.pct){width="2.013888888888889in"
height="0.6111111111111112in"} (2)

where the overbar means time-averaged. The reference boundary condition
on \<*H*\> at *z*=0 is \<*H*\>=0. The heat capacity and density are not
explicit in Eq. (2) because the variables are dimensionless, i.e., the
dimensionless heat capacity and dimensionless density are both 1.

Figure 4 plots the normalized enthalpy flow vs. *z* for an OPT
configured system. The strict interpretation of Figure 4 is that it
shows the enthalpy flow at a time immediately after a PTR is started
from isothermal conditions. Figure 4 shows that the effect of heat
transfer is to steepen the enthalpy flow gradient; enthalpy flow
decreases as *z* increases. This means that enthalpy is continuously
rejected to the wall. This reduces the amount of enthalpy that can be
rejected out of the system at the hot heat exchanger, thereby reducing
refrigeration. Isothermal wall conditions are effectively attained for
Fo≤10. OPTs typically operate at Fo≈10. Figure 4 shows that when there
is heat transfer, enthalpy flow near the hot heat exchanger (*z*=1)
decreases when compared to the more adiabatic case of Fo\>\>\>1.

Enthalpy Flux Fields

Figures 5a and 5b show an OPT enthalpy flux field for both an isothermal
tube wall and for Fo=1000, respectively. The scaling parameters are the
same as in Fig. 4 for VaPr=10. For the

![](media/image14.pct){width="6.430555555555555in"
height="1.7777777777777777in"}

Fig. 5a. Isothermal tube wall Fig. 5b. Fo=1000

**Figure. 5** Enthalpy flux field for an OPT. The enthalpy flux in the
thermal diffusion layer is seen to reverse for Fo=1000. System
parameters: *ε*=0.55, VaPr=10, *UL*=0.3, *φU*=--0.25, M=2.0x10--4. For
Fo=1000, *φT*=--0.187, *Tw*=0.049.

![](media/image15.pct){width="6.430555555555555in" height="1.375in"}

Fig. 6a. Isothermal tube wall Fig. 6b. Fo=1000

**Figure 6.** Enthalpy flux field for an OPT: *ε*=0.55, VaPr=1.0,
*UL*=1, *φU*=--0.25, M=5.1x10--4. For Fo=1000, *φT*=--0.143,
*Tw*=0.0086.

isothermal case the flux field is unidirectional. For the case in which
Fo=1000, the flux field reverses itself in the thermal diffusion layer
near *z*=0. Near *z*=1, however, the flux field has a larger positive
flow over the isothermal case, resulting in greater enthalpy flow near
the warm end as previously indicated in Fig. 4.

Figure 6 shows an OPT configuration in which VaPr=1 and *UL*=1. These
parameters illustrate a case in which the velocity amplitude at the tube
ends are equal, and diffusion fills the entire tube. For the isothermal
case in Fig. 6a there is a reversal of enthalpy flux in the vicinity of
*z*=1. This means that the highest temperature will be located at a
position where *z*\<1. For the case of Fo=1000, the entire enthalpy
field flows in one direction.

An interesting case is a comparison between isothermal and near
adiabatic conditions for a BPT. Figure 7 shows this comparison.
Isothermal conditions in Fig. 7a are seen to produce unidirectional
enthalpy flow. However, for adiabatic conditions in which there is
little heat transfer, there is significant enthalpy streaming in the
diffusion layer that is in a direction opposite to the core flow. For
the BPT then, adiabatic conditions can be detrimental.

Steady Secondary Mass Flux

The steady secondary mass flux due to the Reynolds stresses is a
quadratic quantity and therefore nonvanishing when time-averaged. Steady
mass flow is strongly dependent on the velocity amplitude, velocity
gradient, and on the velocity phase angle between the tube ends. It is
less dependent on heat transfer with the tube wall. Figure 8 plots the
steady *z*-component of mass flux as a function of *r* at *z*=0 for a
BPT configuration at different Va. The oscillating end of the tube is
located at *z*=0 and the closed end is located at *z*=1. The Reynolds
stresses, which are strongest in the momentum diffusion layer, are the
driving forces6. By mass conservation for a closed system the
time-averaged mass flow at any *z* location must be zero. This requires
that the gas in the core move in a direction counterflow to the gas in
the diffusion layer.

Figure 8 shows the axial component of velocity as a function of radial
distance for a BPT configured system for Va=10, 20, 40 and 80. For
Va\<35, the gas in the diffusion layer flows

![](media/image16.pct){width="6.430555555555555in" height="1.375in"}

Fig. 7a. Isothermal tube wall Fig. 7b. Fo=1000

**Figure. 7** Enthalpy flux field for an BPT configuration: *ε*=0.55,
VaPr=10, *UL*=0, M=5.1x10--4. For Fo=1000, *φT*=--0.262 and *Tw*=0.021.

> ![](media/image17.pct){width="3.9305555555555554in"
> height="2.513888888888889in"}

**Figure 8**. Normalized mass flux distribution for a BPT: *ε*=0.10,
M=7.7 x 10--4.

towards the oscillating end while the gas in the core flows towards the
closed end. As Va is increased, a double \"boundary layer\" is produced.
In the first layer very near the wall the gas continues to flow towards
the oscillating end. The gas in the core, however, has switched
direction and now it too flows towards the oscillating end. Between the
core and the first viscous layer is a second viscous layer -- a
transition layer -- which flows towards the closed end. This phenomena
has also been predicted for incompressible oscillating flow systems with
slightly varying tube diameters7.

Figure 9 shows the steady velocity field for an flow system configured
as an OPT for Va=80, *UL*=0.3 and *φU*=--0.25. This is an example of the
complexity of the steady flow patterns within a pulse tube. The radial
velocity component is very apparent and is a result of the Reynolds
stresses producing a body force on the fluid. We have experimentally
confirmed the direction of the core flow for both an OPT and BPT
configured system using smoke-wire flow visualization6 for values of
Va\>40.

The major consequences of mass streaming for the PTR are the losses
associated with a direct enthalpy convection between the cold and warm
ends. For axial temperature gradients of ***O***(1), convected enthalpy
flow is approximately one order of magnitude smaller than the enthalpy
flow used to do refrigeration as calculated from a one-dimensional
model.

![](media/image18.pct){width="3.263888888888889in" height="1.375in"}

**Figure 9**. Normalized velocity field for an OPT with isothermal tube
wall conditions: *ε*=0.15, Va=80, Pr=0.7, *UL*=0.3 and *φU*=--0.25,
M=5.7 x 10--4.

SUMMARY

Thermal and momentum diffusion give rise to steady secondary flows that
can have important implications for the pulse tube refrigerator. We have
developed a two-dimensional mathematical model which can be used to
understand steady enthalpy and momentum flows. Included is the effect of
heat transfer between the gas and tube wall. Some insightful results
show that:

> • Heat transfer between the gas and the tube wall produces axial
> enthalpy flow gradients. This is a possible loss mechanism, since the
> enthalpy flow available to do refrigeration is rejected to the tube
> wall and not to the hot heat exchanger.
>
> • For real PT systems heat transfer is *not* negligible;
>
> • Convection of enthalpy due to mass streaming is a loss mechanism. It
> is approximately one order of magnitude smaller than the enthalpy flow
> used to do refrigeration;
>
> • The dimensionless parameters of *ε*, Va, Pr,, M, *UL*, *φU* and Fo
> allow for an extended range of scaling that can include modeling of
> heat exchangers and regenerators.

The model is being further developed to optimize PTR performance through
an entropy analysis that includes axial temperature gradients and
viscous dissipation for geometries in which Va\<\<1. Steady mass
streaming of enthalpy and the corresponding losses are also being
examined through velocity amplitude and velocity phase angle dependency,
and system geometry.

REFERENCES

1\. Rott, N., \"Damped and Thermally Driven Acoustic Oscillations in
Wide and Narrow Tubes\", Z. Angew Math. Phys., vol. 20 (1969), pp.
230-243.

2\. Radebaugh, R., \"Review of Pulse Tube Refrigeration\", Adv. Cryo.
Eng., vol. 35, (1990), pp. 1191-1205

3\. Stuart, J. T., [Laminar Boundary Layers]{.underline}, Oxford
University Press (1963), ch. 7.

4\. Swift, G. W., \"Thermoacoustic Engines\", J. Acoust. Sc. Am., vol.
84, no. 4, (1988), pp. 1145-1180.

5\. Merkli, P. and H. Thomann, \"Thermoacoustic Effects in a Resonance
Tube\", J. Fluid Mech., vol. 70 (1975), pp. 161-177.

6\. Lee, J. M., Ph.D. Thesis (to be published).

7\. Grotberg, J. B., \"Volume-cycled Oscillatory Flow in a Tapered
Channel\", J. Fluid Mech., vol. 141 (1984), pp. 249-264.

**APPENDIX**

Summary of Mathematical Formulation

A series expansion solution in the small parameter *ε* allows
linearization of the non-dimensional conservation equations. The
expansion parameter is equal to the inverse of the Strouhal Number which
is a frequency parameter. The resulting set of coupled linear partial
differential equations are derived in the limit of lubrication theory.
They are presented here as the non-dimensional equivalent to the
equations first examined comprehensively by Rott1. Periodic solutions in
the limit of negligible axial temperature gradients are sought for the
basic state problem. Axial velocity boundary conditions are parametric
in amplitude and phase angle. The resulting basic state solution is
extended to the time-averaged ***O***(*ε*) problem which describes the
steady secondary motion of interest. The steady momentum flow is found
to be driven by the Reynolds stresses in the viscous layer near the tube
wall. The steady enthalpy flow is generated by the conditions on axial
velocity boundary conditions and is influenced by thermal diffusion
whose temperature amplitude and temperature phase angle are intimately
connected with the tube wall Fourier Number, Fo.

Basic State Problem

The axisymetric system under examination is shown in Figure 1 of the
main paper. The normalized velocity field is composed of scalars axial
velocity, *u*, and radial velocity, *υ*. The velocity field and the
thermodynamic variables of temperature, *T*, and density, *ρ* are
functions of spatial coordinates *r* and *z*, and of time, t*.* The
thermodynamic pressure, *p*, depends only on *z* and *t*, in accordance
with lubrication theory.

The conservation equations for mass, momentum, energy and the equation
of state for an ideal gas are scaled to non-dimensional form, from which
the expansion parameter *ε* is identified. The series expansion solution
is of the form *χ*(*r*,*z*,*t*)=\[*χo*+*εχ1*(*r*, *z*,)+*ε*2*χ2*(*r*,
*z*,)+***O***(*ε*3)\] ei2π*t* where *t* is the dimensionless time
bounded between 0 and 1, and *χ* represents the dependent variables
*u*(*r*,*z*, *t*), *υ*(*r*, *z*, *t*), *T*(*r*, *z*, *t*), *ρ*(*r*, *z*,
*t*), and *p*(*z*, *t*). The symbol ***O*** means \"of order\". The
terms for the dimensionless quantities *po*, *ρo* and *To* are equal to
one while *uo* and *υo* are the basic state velocities to be solved. We
use the notation *χ,η* to mean the partial derivative of *χ* with
respect to *η*. The linearized equations for the basic state problem
are:

![](media/image19.pct){width="2.5555555555555554in"
height="0.6111111111111112in"} (A1)

![](media/image20.pct){width="2.6527777777777777in"
height="0.5555555555555556in"} (A2)

![](media/image21.pct){width="1.9166666666666667in"
height="0.5277777777777778in"} (A3)

![](media/image22.pct){width="0.8472222222222222in"
height="0.20833333333333334in"} . (A4)

The non-dimensional parameters written in terms of dimensional (starred)
quantities are

![](media/image23.pct){width="0.6944444444444444in"
height="0.4722222222222222in"};
![](media/image24.pct){width="0.7916666666666666in"
height="0.5694444444444444in"};
![](media/image25.pct){width="0.6111111111111112in"
height="0.5694444444444444in"} ;
![](media/image26.pct){width="0.9444444444444444in"
height="0.5833333333333334in"}

where *ε* is the expansion parameter and equal to the inverse of the
Strouhal Number , Str; Va is the Valensi Number, Pr is the Prandtl
Number, M is the Mach Number, *U*\\O( *\**,*o*) is the reference
velocity, and *τ∗* is the period; *L\** is the tube length and *rw*\* is
the inner radius of the tube; *νo∗* is the kinematic viscosity, *αg\**
is the gas thermal diffusivity and *T*\\O( *\**,*o*) is the reference
temperature; γ is the heat capacity ratio, and *R* is the ideal gas
constant.

The velocity boundary conditions are normalized with respect to *U*\\O(
*\**,*o*) which is the velocity at *r*=*z*=0. They are

*r*=0, *uo,r*=0, *υo*=0 (A5)

*r*=1, *uo*=*υo*=0

*r*=0 and *z*=0, *uo*=*Uo*=1

*r*=0 and *z*=1, *uo*=*UL* ei2π*φU*

where *UL* is the dynamic velocity amplitude ratio at *z*=1 (see Table
1). The conditions on velocity at the tube ends allow the velocity
amplitude *UL* and the velocity phase angle, *φU*, to be independent
parameters. The temperature boundary conditions are

*r*=0, *T1*,*r*=0 (A6)

*r*=1, *T1*=*Tw* ei2π*φT*

where *Tw* is the temperature amplitude, *φT* is the phase angle between
*T1* and the reference velocity *Uo*.

Periodic diffusion heat transfer applies over the tube wall domain while
temperature and heat flux are continuous between the gas and tube wall.
For the limit in which the tube wall thickness, *lt* \*, is much less
than the tube inner radius, *rw*\*, planar geometry can be used for the
tube wall. In this case, the unsteady thermal diffusion equation for the
tube wall domain is

![](media/image27.pct){width="0.9583333333333334in"
height="0.2222222222222222in"}*θ1*,*t*=Fo*θ1*,*yy* (A7)

where *θ1* is the dynamic temperature amplitude for the tube wall. The
Fourier Number, Fo, is defined as ratio of the square of the thermal
penetration length to the tube wall thickness:

![](media/image28.pct){width="0.7777777777777778in"
height="0.5555555555555556in"} (A8)

where *αt*\* is the thermal diffusivity of the tube wall and is equal to
the thermal conductivity divided by the product of density and heat
capacity. The boundary conditions are that temperature be continuous at
the inner surface of the tube wall (interface between the gas and the
tube) and that adiabatic conditions apply at the outer surface:

*y*=0, *θ1*=*T1*(at *r*=1)=*Tw* ei2π*φT* (A9)

*y*=1, *θ1*,*y*=0 .

The additional unknown of *φT* is found from the condition that heat
flux be continuous across the interface between the gas and the tube:

at *r*=1, *y*=0, *z*=0; {*ktθ1,y*}at y=0={*kgT1,r*}at r=1, z=0 (A10)

where *kt* and *kg* are the thermoconductivities of the tube wall and
gas, respectively. A numerical search for the value of *φT* and *Tw*
that satisfies Eq. (A10) is required. Equations (A1) to (A4), (A7) and
(A10) along with the boundary conditions comprise the basic state
problem for the unknowns *uo*, *υo*, *ρ1*, *T1*, *p1*, *θ1*, *Tw* and
*φT*. The equation set is solved yielding a complicated solution
contorted with analytic functions in *r*, *z*, *t*, and with
dimensionless scaling parameters *ε*, Va, Pr, M, Fo, *UL* and *φU*. In
general, the effect of decreasing heat transfer (increasing Fo, Fo\>1)
from the isothermal wall case (Fo\<\<1) is to add a lagging component to
*υo*, *ρ1*, *T1*, *p1*, while *uo* remains unaffected.

*O*(*ε*) Steady Secondary Flow

The steady secondary momentum streaming problem is obtained from the
***O***(*ε*) time-averaged momentum equation:

![](media/image29.pct){width="3.4583333333333335in"
height="0.5694444444444444in"} (A11)

where the overbars represent time-averaged quantities. The boundary
conditions are

at *r*=0, \\O\\AL(, *u*)*1,r* =0 (A12)

at *r*=1, \\O\\AL(, *u*)*1*=0

The terms on the left-hand-side of Eq. (A11) are the steady Reynolds
stresses. Mass flow is constrained within the tube so that the *axial*
net mass flow averaged over one cycle at any axial location must be
zero:

![](media/image30.pct){width="1.8888888888888888in"
height="0.5694444444444444in"}. (A13)

![](media/image31.pct){width="2.3055555555555554in"
height="0.4861111111111111in"}Equations (A11) and (A13) along with the
boundary conditions Eq. (A12) are used to find *u1* and the pressure
field *p2*. The quantity *u1* by itself is not of physical significance;
it represents only the streaming mass flux due to the Reynolds stresses.
The observed steady mass flux scalars -- a combination of the Reynolds
stresses and the counterflow pressure gradient -- are defined as
\\O\\AL(,(*ρsus*)) and \\O\\AL(,(*ρsυs*)) and are related
through the ***O***(*ε*) steady mass conservation equation:

![](media/image32.pct){width="1.5694444444444444in"
height="0.2777777777777778in"} (A14)

![](media/image33.pct){width="1.625in" height="0.2777777777777778in"}
(A15)

![](media/image34.pct){width="3.0972222222222223in" height="0.5in"}.
(A16)

The steady ***O***(*ε*) axial enthalpy flux is found by integrating the
*z*-component of the ***O***(*ε*) energy equation

![](media/image35.pct){width="2.486111111111111in"
height="0.5138888888888888in"}

![](media/image36.pct){width="1.3888888888888888in"
height="0.3472222222222222in"} (A17)

with the boundary condition at *z*=0, \\O\\AL(, *h*)*2*=0.

Equations (A14) to (A16) are solved for the steady mass flux field while
Eq. (A17) is solved for the axial enthalpy flux. The complete solution
to this problem is given in detail elsewhere6.