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NASA Technical Library

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NASA ContractorReport 168304

FURTHER•DEVELOPMENTOF A METHOD FOR COMPUTINGTHREE-DIMENSIONAL

• SUBSONICVISCOUSFLOWS IN TURBOFANLOBE MIXERS

Shyi-JangLin, J. P. Kreskovsky,W. R. Briley,and H. McDonald

ScientificResearchAssociates

Glastonbury,Connecticut --

(tlASA-CR-16830q) FIIt{_BT.B DEIEI.O£BEST CF A 1_8_-I_q£2

EETHCD FOR COBPUTING _EI£E-£IB£_-cICEIL

CgBSOHIC VISCOUS _'LC_ I_ 'IUEEC_I_ LOBE

BIIEBS Final gepoEt (_=cientifBe- ic =erach Dnclas

_ssociates,Inc.) 66 F HC _O_/_I AC! G3/3@ 42_3 ?

November1983

Preparedfor " •

NATIONALAERONAUTICSAND SPACEAI_INISTRATION

Lewis ResearchCenter

Under ContractNAS3-22758
_j

LIST OF SYMBOLS

area

c specific heat !

P

total pressure loss coefficient

CpT

turbulent damping function

I': total enthalpy

F viscous stress

C viscost,stress

.o

h metric coefficient ./

k turbulentkineticenergy iq. turbulent length scale

p turbulenceproduction

p pressure

Q,q magnitude of velocity vector

R radius

turbulent Reynolds numbers

Rt. R

1" temperature

t, velocity

v rndia I velocity

w azimuthal velocity

ratio of specific heats Y

f. turbulent dissipation

K yon Karman' s constant

. viscosity

E. vortlcity ' --

i, density

._calar potential !

stream function [

It vorticity

i
LIST OF SYMBOLS

I inviscid

n normal direction

p primary

s secondary

t turbulent

v viscous
I

t

TABLE OF CONTENTS 1

J

• o • • • ° • • • • 1

[NTRODUCT 10N ...............

. . 3

ANALYSIS .........................

• • . • 3

General Approach ................. 3

Primary-Secondary Velocity Decomposition ....... 4

Surface Potential Equations .............. "

Primary Momentum and Pressure Approximation ....... • . • • • e 56

Secondary Vor ticity ............ 8 ° . • • • ° . • • • . •

Energy Equat _on ......... . 8

Compressibility Relations .............. 9 L

Governing System of Equations .............. 9

The Turbulence Models .................. i

APPLICATION TO FLOW IN A LOBE MIXER ............ " 12 !

12

°

Coordinate System ................. . 13 i

The Imposed Pressure Field ............... 14

Governing Equations in Orthogonal Coordinates . ..... . • • ° . ° 19

Specifjcatlon of Initial Conditions .... . . 22 11

Boundary Conditions ................. 22

Nonswirling Flows .................. 23

Swirling Flows ...................

• . 25

COMPUTED RESULTS .....................

26

Effects of Secondary Flows ............ 26

Effects of Turbulence Models ............ _ _ . 27

• . . • . ° • . . .

Effect of Swirl ........

....... 29

SUMMARY AND CONCLUSIONS ............

30

• • . • . . . ° • .

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

....... 32

TABLES ....................

. ° . ° • ° . . ........... 35

FIGURES .......

. . . . . . . • . ....... 45

APPENDIX - USER'S-GUIDE" - "
INTRODUCTION

Lobe mixers are devices which are currently being used for thrust augmentation on a variety of turbofan engines. The mixer is designed to provide a means

of generating mixing between the turbine and fan streams before they enter the

exhaust nozzle. Through careful design, the mixing can result in more uniform

energy distribution at the nozzle exit. Provided that the losses incurred in the

mixing process are minimized relative to the amount of mixing, small but significant performance gains may be realized.

Obviously, due to the sensitivity of mixer performance to changes in lobe

shape and number, inlet velocity and temperature ratios, and secondary flows

generated by the lobes, a computation procedure which could be used to sift

through a matrix of possible design configurations would be of significant value.

Such a procedure was previously developed by Kreskovsky, Briley and McDonald

(Ref. i), based on the velocity decomposition approach of Briley and McDonald

(Ref. 2). The analysis synthesizes concepts from potential flow theory,

secondary flow theory, and from an extension of three-dimensional boundary

layer theory in a manner which allows efficient numerical solution by forward

marching in space. The resulting procedure was used extensively in Refs. 3-5 !m

to examine the effects of lobe shape, lobe generated secondary flows, and :

turbulence effects on mixing. The results obtained in Refs. 3-5 were, in

general, found to be in excellent agreement with available experimental data.

Through interpretation of the results obtained using the analysis of

Ref. i, a more detailed understanding of the mixing processes in lobe mixers

was obtained. As a result, it became apparent that it would be beneficial to

extend the analysis of Ref. 1 and further develop and improve its capabilities.

These extended capabilities, which are the subject of the present investigation,

allow the current version of the analysis to be applied to mixer flows with

swirl, and to allow more general specification of initial condition with

nonzero secondary flow vorticity. In addition, the procedure for determining

the required __compressible potential flow pressure gradients from an

[ncompresslble flow solution, has been improved. Finally, the formulation of

the K-_ turbulence model used in the previous version of the procedure was

somewhat unreliable, leading in some instances to predictions of negative

values of turbulence kinetic energy. Under the current investigation, the

reliability of this model was improved by making minor changes in the nt,merical

formulation, and optional turbulence models were added to the procedure.
As a result, the present version of the mlxer analysis and computer code

represents a reliable,lmproved analysis with extended capabilities which can be

used to further the understanding of lobe mixer flow fields, and to aid in

mixer design.

_. _ i
ANALYSIS

General Approach

The present analysis is described in detail in Ref. i, and is outlined

only briefly here. The analysis is based on approximations made relative to

a eurvilinear but not necessarily orthogonal coordinate system fitted to and

a]igned with the flow geometry under consideration (el. Fig. i). The coordinate

system is coosen such that the streamwlse or marching coordinate downstream of

the lobes either coincides with or is at least approximately aligned with a

known inviscid primary flow direction, as determined for example by a potential

flow for the given geometry. Transverse coordinate surfaces must be either

perpendicular or nearly perpendicular to solid walls or bounding surfaces,

since diffusion is permitted only in these transverse coordinate surfaces.

Equations governing streamwise vorticity and a scalar viscous correction

u to a known inviseld primary flow velocity UI are derived utilizing assumpv

tions which permit forward-marching solution, provided reversal of the composite

streamwise velocity does not occur. Terms representing diffusion normal to

transverse coordinate surfaces are neglected. Approximate pressure gradients

are derived from the inviscidprimaryflow and imposedin the streamwisemomentum .i

equation. Secondaryflow velocitiesare determinedby scalarand vector poten- ,

rials in the transversecoordinatesurfaces,to suppressstreamsriseelliptic

behavior requiring downstream boundary conditions. Finally, the use of streamwise vorticity in obtaining secondary flow velocities avoids the explicit use of

transversemomentumequationsin predictingthe secondaryflow velocities.

Primary-Secondary Velocity Decomposition

The analysis is based on decomposition of the overall velocity vector

and a secondary flow velocity U The

field U into a primary flow velocity Up s"

overall or composite velocity is determined from the super-p_sition

6 -0p+0, (2.1)

For lobe mixer flows the primary flow velocity is represented as

(2.2}

_p -D,uv

where UI is a known inviscid primary flow velocity satisfying slip conditio,s

and determined from an a priori potential flow solution in the geometry under

./

3
consideration. The (non-dimensional) scalar quantity Uv is a viscous veloclty

profile factor which introduces viscous shear layers and may also correct for

internal flow blockage effects. In the present case of the lobe mixer uv accounts

for both boundary layers and the free sbear layer which exists between the fan

and turbine streams. The viscous velocity correction uv is determined from

solution of a primary flow momentum equation. As will become apparent, for the

lobe mixer it is conventient to solve for the component of Up in the primary

flow direction rather than uvTo suppress streamwise elliptic behavior, the secondary flow velocity Us

is defined by, and presumed derivable from, scalar and vector surface potentials

denoted _ and _, respectively. If iI denotes the unit vector normal to transverse coordinate surfaces (also presumed here to be in the direction of the

marching coordinate), if p is density, and if Po is an arbitrary constant

reference density, then Us is defined by

_s _ v,_+(po/p)vx_,_ (2.3)

where V is the surface gradient operator defined by

S

V s m V-_I(_I'V) (2.4)

!

= 0 and thus U lies entirely within transverse

It follows that iI " Us s

coordinate surfaces. Equation (2.3) is a general form permitting both rotational

and irrotational secondary flows and will lead to governing equations which are

elliptic in transverse coordinate surfaces and which are, therefore, solvable

within a forward-marching context. The overall velocity decomposition (2.1)

can be written 0 . DXo v + V$_ + (polp)VX_l_

(2.5)

• Up+v$_+ (po/p)vxT,_

Surface Potential Equations

Equations relating _ and _ with Uv, p, and the streamwise vorticity

component _ can be derived from Eq. (2.5) as follows: #

V-pU o 0 - V-PUxU v + V'pVs_) + PoV'VX[,_ (2.6)

(2.7)

{,.v_x-a,- _,-vx_,uv+ T-vXlpo/PlVx{+,_T,vxv, - _

4
,_in,-c thl' last term in each of Eqs. (2.6, 2.7) is zero by vector identity,

}'{is. (2.6, 2.7) can be written as

v.pvs. -V.p zu" (2.85

a (2.95

X e

[ .VxCpoip) v ,,_. N -i .VXOxUv

' which are elliptic for _ and _ in transverse coordinate surfaces. The last

term in Eq. (2.9) is identically zero in a potential flow coordinate system

for which iI and UI have the same direction, and is small if iI and UI are

approx_Itely aligned. Given a knowledge of uv, _, and p, the surface potentials

and _ can be determined by a two-dimensional elliptic calculation in transverse coordinate surfaces at each streamwise location. In turn, Us can be

computed from Eq. (2.3), and the composite velocity U will satisfy continuity.

Equations for uv and _i are obtained from the equations governing momentum

and vorticity, respectively.

PrimaryMomentnmand Pressure Approximation

The streamwise momentum equation is given by

+cvp ;p]-t, (210)

where p i_ pressure and pF is force due to viscous stress. Terms in F representing strea=wise diffusion are neglected; however, since the viscous terms are

complex for compressible flow, the modified viscous force is temporarily denoted

F', and further consideration of viscous terms is deferred to a later section.

_q_e remaining assumption for Eq. (2.10) concerns the pressure gradient

term and is designed to permit numerical solution as an initial value problem.

An obvious pressure approximation for curved flow geometries is to impose

pressure gradients from an inviscid potential flow (Briley [615, and for

internal flows, to correct these with a mean pressure gradient term depending

only on the xI coordinate (Pantankar & Spalding [715. _nis approximation is

appropriate both for flows consisting of an irrotational core region with

thin shear layers on bounding surfaces, and also for some fully viscous flows.

If pressure gradients are derived from the inviscid potential velocity

UI' and if a mean viscous pressure correction pv(xl5 is introduced, the pressure approximation can be written as

5 i

i

J

f.
O_!_"fl_ALPAGEIg

OF POORQUALFI'Y

;,.vp • 7,.[Vp,,(x,)-

(2.11)

=- ti-V(pz + pv)

where Pl is the imposed pressure. Typically, for internal flows, Pv is

determined to ensure that an integral mass flux condition is satisfied, such

as

SA[I- pUdA - constant (2.12)

Combining Eqs. (2.10)and (2.11) and setting F _ F' provides an equation

nominally governing Uv:

_u"[(O-V)O + (VPv)/P -V(Oz-Oz)/Z ] " ;1" F' (2.13)

where U is to be written as

0 " ;,uzu v + _z(VzUv +v.) + _3(WzUv+ ws) (2.14)

U I , ViJ WI VSJ ,% . I

where are components of UI" and are components of Us T

+

Secondary Vorticity

An equation governing fll in compressible flow can be obtained from an !

approximate application of secondary flow theory. The equation governing the !

growth of vorticity _s along a streamline for compressible flow with constant

viscosity is given by Lakshmlnarayana & Horloek [8] and may be written for t

variable viscosity as

I I

as (1"l._) " 2opqRn pqI _ _-Vx--Vpp + _ _-_ (2.15) :

+

where q is velocity magnitude, s is the unit vector along a streamline, s is

distance along a streamline, fin is vort_Ity in the direction of the unit

d

principal normal vector n of the stream!llne, for which R is the principal

radius of curvature. These quantities are related by the Frenet formula

6

_, I......................
ORIGINALPAGEItl

OF POOR QUALITY

,/R = Ds/Ds = s - Vs. !.n Eq. (2.15), G = Y x F, and the term €:ont_i.i. K p

vanishes if P is constant. Since an intrinsic coordinate system forrmtlatJon

as in Eq. (2.15) provides an "inverse" coordinate system if used to compute

U, a,d since intrinsic coordinates are degenerate on no-slip surfaces and

nonorthogonal for general rotational flows, intrinsic coordinates are not

nttr;nctive for numerical computation. However, if the coordinate system used

for c'omp,tation is approximately aligned with the flow direction, then an

approximate equation governing fl, can be derived from Eq. (2.15) by replacing

s by i, as in the following development:

ns _'_ . II'_ . _I (2.16a)

= "_. = -1[

q 0 I I" 0 uI _.;

._

a n (_.v_)._ . (;,-v;,)-_ _In, (2116b)

qR _ -U +

I1"0 UlR n

where u 1 = UiUv; R1 is the principal radius of curvature of the x I coordinate,

nnd fin] is vorticity in the direction of nl, the principal normal of the xI

I

coordinatel_ne. The quantitiesnI and RI are defined by the Frenet formula

nl/R l = iI Vi I. To illustrate, in an orthogonal coordinate system,

b

R--_ hih z 0x z + hlh 3 ax_

where hl, h2, h3 denote metric coefficients. If p varies, p is rep]aced by

"7

rlte impo:;ed,pressure Pl as defined in Eq. (2.11). Finally, taking q- --"Ul2

and s - i L in the last two terms in Eq. (2.15), and neglecting streamwi_e

diffusion, Eq. (2.15) becomes

-

where G' does not contain streamwise diffusion.

_le transverse vorticity _ . in Eq. (2.18) contains components _'_ic||, itt n_.

orthogonal coordinates and assuming an irrotational UI" are given by

t

7 ..
ORIGINALPA_# tq'

OF POORQUALFi'Y

nz _z.vxO. u, au, w, auv i

" _ _I3 - _ hlh3 _X, (h3w'_) (2.19a)

vz _uv I

n 3.%.vxO -- UZh--__a"% 4-__h, _z, + h/hz _l (h.,v$) (2.19b)

b

In the applications contemplated here, the first term on the right-hand side of

each of Eqs. (2.19a-b) is expected to dominate, and the remaining terms may be

neglected as a convenience.

Energy Equation

Since the flow being considered is nonadiabatic, solution of an equation !

governing energy is required. The energy equation can be written in a variety

of forms, one of which is '-

pU-VE - V-kVT + U-F + • (2.20)

where E is total _thalpy, T is temperature, k is thermal conductivity, and

€ is the dissipation function. Solution of Eq. (2.20) by forward marching

integration requires only that terms representing streamwise conduction of heat

and also streamwise viscous diffusion in F be neglected. _

Compressibility Relations

i

In the imposition of streamwise pressure gradients, compressibility effects i

are represented by introducing the perfect gas equation of state p = pRT.

No otherassumptionsarenecessaryforconsiderationof compressibleflow.

For moderate subsonicMach numbers,inviscidpressuregradientswould ideally

be obtained from a compressible potential flow calculation or otherwise from an

incompressible potential flow corrected for compressibility uning a suitable

compressibilitycorrection. Replacingp in Eq. (2.11)by the state equationand

eliminating temperature using the temperature-enthalpy relation

u.u (2.21)

E " CpT + 2
ORIGINALPArlE Itl

OF POORQU'ALITY

where ,: denotes sped:if it heat, t_.e followil;g auxiliary equation relatSng

P

the lmpo_,_d nressur_ l-radients with density, velocity, and total enthaipy

i:; obtai_ed:

wi,ere "/ is sp:.cific _eat ratic. A slight simplification results if U " U is

replaced by (i I " _)2 in Eq. (2.22).

Governing System of Equations

A complete sTstem of six coupled equations governing Uv, "ql" ¢' _" E,

and O is given by Eqs. (2.8), (2.9), (2.13), (2.20), (2.22}, and (2.]8).

Ancillary relations _re given . ' Eq. (2.5) for composite velocity, Eq. (2.12)

for mass flux, and Eqs. (2.19a-b) for transverse vorticity.

The Turbulence Models

The pre._ent analysis is to be used to predict a turbuleng flow field,

thus the stress teran_ F' and G' in Eqs. (2.13), (2.18), and (2.20) contain

turbulent shear term_ in the form of fluctuating velocity and velocitytemperature correlations. These terms may be modeled using any one of a number !

of turbulence models _hich have been included in the present analysis. In

general these turbuleuat shear terms are modeled through the introduction of a

turbulent eddy viscogity, and thi_ eddy viscosity may be computed from one of

three optional turbulence models. The most complex and general turbulence

mod(:1 used in the pre._ent analysis is the _-_ tarbulence model deserebed

by lamnder and Spaldi_ag (Ref. 9).

_e transport eq_mr:'ons governing k and c are given in vector form

(Ref. 9) for steade: fJ_'.-" as

pu'Vk = _7 (/_+pT/Crk] Vk + P- p_ (2.23)

pU -Vc:V -(]z+pt T/_r_|V k . Cl_--'P-CZ Pc---z

k k (2.24)

In Eqs. (2.23, 2.24), P is the turbulence Froduction. l'he quantities Ok,

oc , CI, C2 are empirical constants which, as recommended by l;_under & Spaldi,g

(Ref. 9) are taken a_ 1.0, 1.3 1.44, and 1.92, respectively- Upon soluth,:t

o
v__ - .... •

I

ORIQ_@,LF'.rlO_|_

OF POOR QUAL'r¢

of Eqs. (2.23 and 2.24) the turbulent viscosity and length scale may be

i computed as

/_r-- c_p_/E

(2.25)

and

/ .2 = C/jU4 kbrz/_ (2.26)

If low Reynolds number flows are considered (i.e. wall boundary layers are

resolved) the constants C and C2 are adjusted from their high Reynolds number

yalues. As suggested by Launder and Spalding (Ref. 9), C2 is given as

C2 = C2,,(I.O--0.3e-R_) (2.27)

where C2= is the high Reynolds number value of C2 given above, and Rt is a

turbulence Reynolds number

R! = pk_/.LE (2.28)

The constant C is adjusted following Sbamroth and Gibeling (Ref. I0)

- z (2.29)

; CF= 40 I

": where aI is a function of a different turbulent Reynolds number Rx, and is

: given by HcDonald and Fish (Ref. ii) as

.'-. °, =°o --i j/I t.o +6.66oo ,oo l (2.30)

t_" In Eq. (2.30) aO is taken as 0.0115 and f(RT) is given as (Ref. II)

o.zz (2.31a)

f{R r) = I00 R r R r_< I

:' (2.31b)

"f f(R r) = 68 I gr "1"614.:5 R r >_40

i

f

Between the limits i < R < 40 a cubic is used in the two functional forms

"• T

_ :_ given by Eq. (2.31). For these functions R is defined as

' -- (2.32)

:_ Rr = F .

i Although the k-€ turbulence model given above provides a general

turbulence model in which both the velocity and length scales of turbulence

"4

| are predicted, the formulation has proven to be unreliable and unduly

sensitive to the initial profiles at times leading to the prediction of

_i

negative values of k and/or c. To provide a means of obtaining meaningful

| I0
, .... • .. . _ •_

.t

OR'C-INALPAGEIS __

OF POOR0UALITY

r_-._ultswhen such a failure occurs an alternative turbulence model may be used. .

]n this turbulence model a free stream length scale is specified and the

turhu]ence dissipation equation (Eq. 2.24) is not solved. Instead, the

turbulence dissipation appearing in (Eq. 2.23) is eliminated using (Eq. 2.26)

in conjunction with the specified length scale. In near wall regions the

length scale is damped following McDonald and Fish (Ref. ll)

: .19_ootonh( _<dl_oo) (2.33)

h

where £= is the freestream ]ength scale, K is the yon Karman constant

(=0.43), and _ is a sublayer damping function given as

(d"- (,.34)

= is the

Here, P is the normal probability function and d+ duT/_)where u

friction velocity.

For situations where it might be desirable to investigate the role of

turbulence models, a third and very simple turbulence model has been included

in the analysis. This turbulence model is referred to as a wake turbulence

model and is derived from the definition of turbulent viscosity Pt' and

Prandtl's suggestion that the diffusional flux of some property _ = _ + @'

may be expressed -- --

m k,,Zz a_ /"'r a_

- =7 -rr (2.35)

hence one obtains the Prandtl-Kolmogorov relationship

I k''zZ p (2.36) PT -

The wake turbulence model is obtained by assuming that the turbulent length

scale and velocity scales are constant throughout the flow field. From

equations 2.25 and 2.26 it follows that the constant I/o_ is simply _3/4

_me velocity scale is taken as the square root of the initial]y specified

turbulence kinetic energy k. The length scale is specified using the

geometric constraints of the mixer as a guide, for instance assuming that the

initial length scale wot:Idnot be smaller than the inlet boundary layer or

larger than the duct height. As a consequence of this formulation, the

turbulent viscosity is dependent only upon the initial choices of k and £,

and the loca] density which is determined as the solution progressesIi
...... • .

I

8

APPLICATION TO FLOW IN A LOBE MIXER

In this section, specific details arising in the application of the

foregoing analysis to lobe mixer problems are given. A cross section of a

typical lobe mixer geometry is shown in Fig. 2. The area immediately downstream of the nozzle plug is faired in with an assumed streamline to model

the reversed flow region expected in this region. Since the flow area thus t

excluded from consideration is very small, this treatment is not believed

to introduce significant error. A more detailed analysis of this separated

flow region could be performed following a zone embedding approach using the

Navier-Stokes equations (cf. [12]). Although the mixer geometry is axisymmetric, the flow is three-dimensional due to the aximuthal variation of the

hot and cold streams. However, due to observed symmetry, only a ple-shaped _

region of the transverse coordinate surface need be considered. The shape

of this region and the extent of typical hot and cold streams at the mixer

exit surface are shown in Fig. 3.

Coordinate System

Curvilinear orthogonal coordinates x, y, z are constructed to fit the --

flow passage boundaries downstream of the lobes as shown in Fig. 2. Metric

coefficients hI, ho, h3 are defined su2h that in2rementaldistances is J

determined by (6x)_ = (hl6X) + (h26Y) + (h3_z) . In planes of constant z,

(azimuth), orthogonal streamline and velocity potential lines from a twodimensional planar incompressible potential flow analysis are utilized as the

coordinate lines for constant y and x, respectively. This x-y coordinate

system is then rotated about an axis representing the centerline of the lobe

mixer, to form an axlsymmetric coordinate system. Thus, the z direction can

be regarded as cylindrical (i.e., 0) and h3 = r. The x coordinate is taken

as the primary flow or axial coordinate, and is associated with surfaces for

which the two-dimensional potential is constant. The y and z coordinates

define transverse secondary flow planes at any given x location. In this

coordinate system, the normals _n to the transverse planes coincide with il

the unit vector in the x direction. A two-dlmenslonal incompressible

potential flow analysis and computer program developed by Anderson [12] was

employed in the present investigation, without modification, to compute the

necessary coordinate data.

12
.....

° j

OF POOP,._Ug.LITY

The Impcsed Pressure Field

As discussed in Ref. 1, the axis),mmetric coordinate directions generated

by rotating planar two-dimensional potential flow coordinates about an axis

provides a good approximation for the primary and secondary flow directions,

however, the planar two-dimensional potential flow pressure gradients are .,.

not suitable for use in approximating the required imposed pressure gradients.

C,,nsequently, an incompressible axisymmetric potential flow is computed in

tilt"given coordinated system and then scaled for compressibility effects as

part of the forward marching solution- In Ref. I the compressibility correction

w;,s performed using Laitone's rule (Ref. 14), and as a result, inaccuracy at

high subsonic _Lach number was expected in the correction. This inaccuracy in

the imposed pressure field was accepted in Ref. I since the main interest in

mixing was the immediate region downstream of the lobes and upstream of the

throat, and consequently the transonic problem was not a major issue. Under

the present effort a more appropriate compressibility scaling based on the

work of l.ieblein and Stockman (Ref. 15) was developed. Given the incompressible

velocity field UI, Lieblein and Stockman (Ref. 15) demonstrated that an

accurate estimate of the compressible flow velocity field could be obtained

fromtherelatio.shipU )("VOi) (3.])

,_here O is the compressible velocity, PI is the incompressible (or stagnation)

c

density, Pc is the local compressible density, and UI is the area averaged

in(-ompressible velocity at a given axial location determined as

u_dA (3.2)

Lieblein and Stock-man t;uggest that the density ratio, PI/Pc be computed using

one-dimensional isentropic flow relationships with the Math number determined

by the area ratio. In the present application the density ratio is computed

by using the isentropic relationships in conjunction with the local Math number

computed as part of the forward marching solution- In this manner, a fully

three-dimensional compressible velocity field can be approximated from the

two-dimensional incompressible potential flow. Knowledge of the compressible

velocity, and assuming constant total temperature and pressure in the potet_tial

flow the static temperature is obtained from

13
_F POORQUALfTY

TO=T --I/2U_ (3.3)

The corrected pressure is then found from the relationship

p = Po(T/To ) T-, (3.4)

from which the compressible imposed pressure gradients are determined.

=

Governing Equations in Orthogonal Coordinates

As in Ref. I, the governing equations reviewed in the previous section,

and the turbulence model equations must be expressed in the orthogonal

coordinate system used to represent the mixer passage geometry. Although these

equations are fundamentally the same as those presented in Ref. I, the approximations to the stress terms in the primary momentum equat_on, and the production

term in the turbulence kinetic energy and dissipation rate equations have been

a]tered in an effort to improve the reliability of the k-_ turbulence model.

With the exception of these approximations, the equation in orthogonal coordinates

follows directly from Eqs. (2.8, 2.9, 2.13, 2.18, 2.20, 2.22, 2.23 and 2.24) by

performing the vector operations in the given coordinate system.

Throughout the remaining discussion, all variables in the governing equations are nondimensional, having been normalized by the _"_wing reference

quantities" distance, L ; velocity, U ; density, Pr; temperature, Tr; total

" r 2 r

enthalpy, U2;r pressure, PrUr; viscosity, Br" Here the subscript r denotes a

reference quantity. This normalization lends to the following nondimensional

parameters: Mach number, M; Reynolds number, Re; Prandtl number, Pr; and

specific heat ratio, y. These parameters are defined by

M =Uric , Re -prUrLr//Lr , Pr - CpPr/k, )" = Cp/c v (3.5)

where Pr is the molecular viscosity, k _; thermal conductivity, and Cp and cv

are the specific heats at constant pressure and volume. The reference speed

2 is the gas constant. Since of sound, c, is defined by c = yRgTr, where Rg

the flow is assumed to be turbulent, the dependent variables are taken to he

the time-averaged quantities in the usual sense; however, third order correlation as well as second order correlations including the fluctuating density

have been neglected. The decomposition of the velocity vector thus can he

written

14 -"
It

ORIG!_,I_LPAGE 13

OF POORQUALITY

0 - ]',u+;zv+]'=w

(3.6)

-7

i,qJere _1" 12" and i3 are the unit vectors in the x, y, and z coordinate

directions, respectively. Since 11 is closely aligned with tile primary flow

. direction, vI is small and is neglected as a minor convenience. This approxinmtion is of little consequence since the neglected quantity vI effectively

re.appears as part of vS and is thus determined from solution of the continuity

equat ion during the forward-marching solution process, instead of being imposed

a. priori_ from the potential flow. Since wI is zero by definition, the velocity

decomposition can be written as

- ;,Up +i=V I +I3w a (3.7)

Since u is the onlv streamwise component of U the subscript "p" is

p

dropped. Under the stated assumptions the streamwise momentum equation can be

written as

_u au c)u

puh2h3"_-x + p(vs)hnh3-_'Y + p(ws)h'h2 -_z + plvslUh3 a_-_l -- P(Vsl2h3 _Ox i

--peW%s_)Oh_+ __O+Ph'h._x) , 0 [( h,_ OCu_,)] Ox h2h3 Ox 23 dx -- Re Oy /"L+/"LI) h2 -_y 'J

R_._o_ _+_')-Gf R_N 0z h2-Ez+v-G-Jy R_2h0x -_ _._

_2/J-_, 'L[h_2/a_h3_T2;, +c,-_ ah2_ ;,h__2] u

15

r_
OF POORQU;_,L.'_-¢

The energy equation is approximated by

aE aE _I£

puhzh3--_'- + p(v s) hlh3"- _- +p(ws)hlhz" _

,_!_2_1-_(_+ _.\,,h, _ (3.9)

hlh3 /Jr. prT)_y 1_2+ + h'----"_ 2 (/J" +_r'T er

+ R,---e-" h3 aZ _-r + -_"

+ I/2 +_T pf PrT a"Zwith the Prandtl number Pr and turbulent Prandtl number PrT assumed to be 1.0.

To obtain the rotational secondary flow components of the velocity, the

streamwise vorticity equation is written as

! --

+ _ hzh3 _'_ hlh% P _'y j + he _-z _" ]1 .!

- -,,.

where Q2 = u 2 4- v2

S + w2S

and the vorticity is related to velocity by .

i

(3.ll)

, (ah3w. ah,,v. ) _ .-_

16

b
|_

.°

OR! vn ,Gfr,.::LPJ_zEI_

OF POOR QUPL_f ,

_%e vector potential _s is determined from

%%

'_'I(e",--h. _ +",--'_%'°I( _- 7 _')j--€ (3.12)

and the rot_tlonal secondary flow velocities are determined as

• I 8(h,_'s) i

v_ -phjh_ 8z (3.13a)

I _(hi_s) (3.135)

w_, " phlh z _y

The scalarpotentialis governedby

i

aay [ phihhe3 aay(_s] + -_8" [ Phlhh3e a'as] z " 8[h'hh8x pu) (3.14) :i: "

The velocity components v and w_ are found to be

I a_s (3.15a) --

I a_s (3.15b)

w_- h3 _z

" The secondary flow velocities are thus

• = V_ + V_ (3.16a)

,• = w_ + w_ (3.16b)

The turbu]ence model equations must also be expressed in orthogonal coordinates.

It is assumed that the turbulence equations are tensor invardant [9], and thus

may be expressed directly in orthogonal coordinates by performing vector operations in such a coordinate system. With streamwise diffusion neglected to allow J

• forward-marching solution, the result for the turbulence kinetic energy equation

in steady flow is

17

}

/
t4

r"

II

ORI_NALPACEIS

OF pOORQUALIW

ak ak ak (3.]7)

plhzh3_-- 4-pvshih3_- + pWlhlhz_-z"

I I a r hlhs I 4-/_X ak

• _. _L h, _e _-.)_] /

4- - h3 _'z F + - hzhzh3P( + p

and similarly the turbulence dissipation equation is given by

a( _€ ae

puhzh3._._ + pvshlhs._- 4- pwshjhz"_- . (3.18)

l [ a I" h,h, / , __s_a. 14.h.h. a rl . P, \ a•ll

"-_-.i_ L ,,, u".. J_,J -%-3_LLP"-_-,IEJ_

- hlhzh_Czp-_- + Cl-_-p

The production term is approximated as

r r,a,., , l _ [, a. ua_(, )+ ah,,, _l" (3.19)

., acu_,)+] __ + ,

+ h2 _yy "J _ ely h3 az h, a Z

The turbulent viscosity is then

• R, (3.20)

/J'T •

and the length scale may be determined as

3,4 "Jrz (3.21)

C/_ k

£

_{hen the k-£ turbulence model is used, Eqs. (3.19) and (3.20) are combined

to eliminate the appearance of the turbulent dissipation from Eq. (3.19).

E = CF_s14k3/:'/.._ (3.22) -

18

.

:.: . .

1__
IS

OF POOaQu TY

and it follows that

Ft' = C'p.'/4 h'/°-'_ Re (3.23)

Specification of Initial Condition._

The initial conditions for a lobe _xer calculation may be specified by

using either an automated starting routine or by reading in experimental data.

The fully automated procedure is considered first.

' To obtain initial conditions for a lobe mixer calculation, it is necessary

to specify the velocity and total temperature of the respective hot and cold

streams in addition to a mean value of static pressure. The lobe shape is

specified, and based on the specified lobe shape, a decision is made as to

whether a grid point lies in the hot or cold stream. The appropriate values

for velocity, temperature and density are then assigned to the grid point. -"

The total energy is assumed to be constant but may differ in both streams.

The velocity in the two respective streams is assigned its nondimensional

reference value which is then corrected to account for normal pressure

gradients present at the initial plane as determined from the axisymmetric

potential flow. To account for boundary layers on the lobe, hub and shroud

surfaces, the free stream velocity profiles are further scaled in accordance

with an assumed turbulent boundary layer velocity profile, and the distance

from the surface. Radial secondary flow velocities at the initial station

may be specified parametrically in each stream as a fraction of the streamwise

velocity- This in effect sets the flow angle in the fan and turbine streams,

respectively. Finally, a swirl velocity component may be specified, when

appropriate.

The initial velocity profile with boundary layers is given by

u " U(UsTRU , BaL' Y/BaL '$'*' H) (3.24)

where USTRM is the velocity of the appropriate stream, 6BL is the boundary

layer thickness, y/6 represents the shortest distance from a surface (hub, BL *

shroud or lobe) to the grid point in question, _ is the local displacement

thickness and H is the shape factor. The assumed boundary layer profile used

in the present code is a Coles-_ype profile modified as suggested by Walz [16].

. u=Ux" [(- 3y'- 5)e + IC mEi+y+)+c+-;-,I (3.25)

In Eq. (3.25) C is a constant taken as 5.0, _ is a parameter w_ch determine the !

strength of the wake component of the velocity profile, and w is Coles wake functio_ _ _

i

[16]. 19 4i

I
Following [4], the radial component of velocity in the fan or turbine strean is then

given as (3.26)

v = u Cfan

for the fan stream and

v = u a (3.27)

core

for the turbine stream where c is defined as a mean flow angle.

Finally, the swirl velocity is specified in terms of a mean swirl velocity

magnitude and by specifying the radial limits over which there is to be swirl

as an initial condition. The swirl velocity at the initial station (but only

at the initial station) is assumed constant in the azimuthal direction. This

condition could be changed at an arbitrary inlet swirl profile, if required,

but is adopted here for convenience.

The majority of calculations to be performed with the present

code are expected to be run witheut resolving the hub and casing boundary layers.

For these cases, the initial velocity field in the immediate vicinity of sol_d surfaces

must be treated in an approximate manner. This is accomplished by using wail

functions which assume that locally between the wall and the first grid poiut

away from the wall the velocity profile is logarithmic such that

I I k_ury +C) (3.28)

where u is the friction velocity, K = 0.43 is the yon Karman constant, and

c is taken as 5.0. It follows that

au ur (3.29)

&y Ky

Equation (3.28) is used to determine a wall slip velocity such that the finite

differenced form of the velocity gradient one point off the wall is €_nsist_t

with the assumed law of the wall profiles. When wall functions are used to

represent the primary flow velocity, the secondary flow velocities are still

required to obey no slip conditions. Thus in the near wall regions the

secondary flow velocity is assumed to vary from zero to its free stream

value in a cubic manner with distance from the nearest wall across s_me

specified secondary flow boundary layer thickness. Once the velocity and

temperature distribution are set up, the density is computed to be co=siste_t

with the _emperature and local static pressure as determined from the mean

static pressure corrected for the normal pressure gradients.

20
ORIC4_;,LP_?,2 _; -i

OF POORQUALITY

Initial conditions for the turbulence model are determined through

specification of a reference length scale, 9ref and free stream turbulence

levels. Near the hub and shroud the length scale is determined from the

McDonald-Camarata [17] relationship

t. t - tanh( _--'_---_ (3.30)

= fel \ Zre f /

where y is the distance from the surface in question. In the region about the

lobe surface, the distribution of Eq. (3.30) may be used wl rh y representing

the distance to the lobe from the grid point in question- Pm_ever, sis_ce the

lobe may pass arbitrarily close to a grid point the length scale determined

from Eq. (3.30) will approach 0.0 as y approaches 0.0- This results in a low

Reynolds number region in the initial flow field which can create problems

with the turbulence model. This problem is circumvented through use of an

option which renders the length scale constant in the region near the lobe, ,

and the distribution of Eq. (3.30) is used only near the hub or ca=-ing if

boundary layers are specified there. With the length scale distril_tion known,

the initial turbulent viscosity is obtained from the generalized mlxi_g length

relationship

+ ay• + az, + * (3.31.)

The initial turbulence kinetic energy and dissipation are then determined from

simultaneous solution of Eqs. (2.25) and (2.26). The kinetic energy is

modified slightly following McDonald and Kreskovsky [18] to account for the

free stream turbulence, and the turbulent viscosity is then recompn_ed from

k and c to be consistentUnder certain conditions, neither the wall length scale option Eq- (3.30)

nor the wake length scale option (constant length scale) will provide satisfactory initial values of k, £ and PT" This may occur when the mesh is coarse

and the velocity gradients in Eq. (3.31) vary greatly betwee= adjaoent grid

points. Under these circumstances, it is useful to initiate the turbulence

quantities using an option which specifies the turbulent k£netlc emery,

dissipation and viscosity as essentially constant throughout the flow field.

Such an approach has also been used previously by Launder, M_rse, Zodi and i

Spalding [II] in the computation of free shear flows. i

21 |
If experimental profiles are read in, the specification ef initial conditions

is similar. The velocity components and static temperature are input at each

grid point. The lobe shape is specified as In the automated starting procedure,

and a reference length scale and free stream turbulence intensities must be

provided. Additionally, estimates of the boundary layer thickness on the hub,

shroud and lobe must be specified for use _n setting up the initial turbulence

quantities. Initialization of the turbulence quantities is similar to that

used in the automated procedure.

Boundary Conditions

To march the solution downstream, it is necessary to apply boundary

conditions in each transverse plane. Boundary conditions are needed for the

energy equation, the primary flow momentum equation, the streamwlse vorticity

equation, the _ector and scalar potential eqnations, and the turbulence model

equations. The boundary conditions will in general depend on whether swirling

or nonswirllng flow is considered, and whe=her or not wall boundary layers _re

considered. Referring to Fig. 4, the boundaries at 8mi n and 8max represent

either symmetry planes or periodic surfaces. Boundaries at the hub (or centerline) and shroud surfaces require conditions appropriate for solid surfaces

(hub or shroud) or a centerline (symmetry).

Nonswirlin_ Flows

For nonswirling flows boundary eonditior.s along Omin and Omax are

specifiedas follows:

E, k and _ are set to zero I) aximuthalgradientsof Up,

2) the vorticityis set to zero, and

3) the normal velocity component is zero, thus the vector

potentialis zeroand the azimuthalgradientof the

scalar potential is zero.

Boundaryconditionson the hub and shroudsurfaceare specifiedas follows:

i) the normal velocity is zero along these surfaces, and the vector

potential and the normal gradient of the scalar potential are

thus set to zero,and

2) these surfaces are taken as adiabatic, and thus the normal

gradient of enthalpy is set to zero. i

4

For the majority of calculations it is not expected that the wall boundary i

layer will be resolved. Thus, the appropriate boundary condition for =he I
OF POORQ_J_.LrI'Y

primary flow velocity, Up, is the gradient condition of Eq. (3.29). For the

turbulence quantities k and c boundary conditions consistent with the log

law are

U2

k=

1/2 (3.32) C

M

which is used to replaceEq. (3.31) one point off a walland then

Bk

= o (s.ss)

By

t

is specified on the wall. For _ the condition

u3

€ = T (3.34)

Ky

is used. The use of Eqs. (3.32 anJ 3.34) effectively sets the turbulent viscosity one point off a wall consistent with the log law. If boundary layers are

resolved, then Up = k = _ = 0 on solid walls is specified.

With regard to the boundary conditions for the secondary vortlcity,

in Ref. i it was argued that a zero vorticity boundary condition would be

suitable for use with the wall function approach to the primary flow.

Although this was satisfactory in many cases, in some instances it gave rise

to large tangential velocity components on the hub or shroud surfaces. As a

result, under the present modifications of the code the secondary flow vector

potential and vorticity equations are solved using a coupled solution procedure.

This procedure allows specification of the correct coupled stream functionvortlcity boundary condition which ensures that the secondary flow velocity

components obey no slip on solid walls.

Swirling Flows

With the exception of the vector potential, boundary conditions on the

hub and shroud surfaces remain unchanged for _irling flow. Special treatment must be given to vector potential, howzver, because flow may pass in and

out of the computational region along ghe radial lines at B . and O •LIL_ _X

The amount of flow crossing these radial lines is due to the swirl velocity,

and a means of determining the growth or decay of the swirl _Jst be accounted

for. This is accomplished by solving a 0-momentum equation along one of the

radial coordinate lines prior to solving the secondary flow vorticity-vector

potential system. The resulting prediction of the 8 velocity component can

be integrated to obtain the difference in the stream function between hub and

shroud boundaries due to swirl,

23
\

I / ........

f

!

A_ - w_) hI h2 dy (3.1_

which provides the necessary information for the boundary conditions. In

Eq. (3.35), w_ is subtracted from w, consistent with Eq. (2.3), since the

velocity predicted from solution of the O momentum equation contains both

- rotational and irrotatlonal components. With regard to the boundary comlit_s

"- in the azimuthal e direction, all flow variables are assumed to be periodic.

/I

F

r:

t

io-

,?-

r.-

24
COMPUTED RESULTS OF POOR QUALITY /1

Several test calculations were performed to demonstrate the extended

capability of the modified computer code. The test cases were all performed

for the geometry shown in Fig. 2 immediately downstream of the lobes with the

lobe shape shown in Fig. 3. A summary of initial condition for each case is

given in Table I. As can be seen in Table I these test cases were run with a =_

_rariety of conditions to demonstrate the effect of secondary flows of various

magnitudes, the effect of swirl, and the effects of the various turbulence

models.

Before discussing results from these calculations, it is useful to

define average values of total pressure loss coefficient, ideal thrust

coefficient, and Mach number at each axial location, to aid in the evaluation

of flow properties. A mass-averaged total pressure loss coefficient _T is !

defined as 2(PTref_PT )

(pUZ)r_

pUdA

CpT " (4.1)

where PTref is a reference total pressure associated with the turbine strea_ .

reference conditions, and Pref and Ure f are the reference density and velocity

associated with the turbine stream. The value of CpT at the starting plane is

usually not zero due to the presence of shear layers in the initial conditions,

and for clarity this starting value is subtracted from CpT in the presentation

of results. The distributions of _T presented thus represent losses incurred

downstream of the starting plane. An ideal thrust coefficient is defined and

based on the thrust which would be obtained by isentroplc expansion from local

conditions to a predetermined exit pressure. The ratio of the local value of

this thrust T to the value T. based on the initial flow field is defined as i

the ideal thrust coefficient, T/T i. The th_ast is obtained from the relationship

T- _ _'PeMe2dAe (4.2)

ae

where Pe is the assumed exit pressure, He is the Math number based Oil

isentropic expansion from the local pressure to Pe' and Ae is the effective

exit area, again based on isentropic expansion to P .e

25
Effects of Secondary Flows

The effects of the magnitude of the initial secondary flows prescribed at the -:-

lobe exit was investigated by performing two calculations designated as case I and

case II in Table I, where the initial conditions are summarized. In the case I

calculation, a radial inward velocity equal to 13% of the local fan stream velocity

was specified. In the turbine stream, a radial outward velocity of 8% was specified.

The case II calculation had fan and turbine streams radial velocities specified as

20% and 25%, respectively. Both calculations used the new optional k-£ turbulence

model, thus the calculations also serve to demonstrate this capability. .

The relative effects of the different initial secondary flow on mixing are best I

examined by looking at the total temperature contours at the nozzle exit plane. As /

noted in Table I, the temperature ratio between the streams is different for cases I

and El, however, in the absence of initial secondary flows the thermal mixing between

the two streams results in a 'tear drop' total temperature pattern (c.f. Re-[. 3).

The departure from this 'tear drop' pattern thus indicates the degree to which initial secondary flows effect the mixing process, even though these two cases have different initial temperature ratios. The total temperature contours are shown in

Fig. 5. The case I results show a small but significant departure from the 'tear

drop' total temperature pattern observed when no secondary flow is specified at the

initial plane. However, the results for case II, where even larger secondary flows __

are specified at the initial plane, show a much more dramatic distortion of the total

temperature contours at the nozzle exit. Here the hot fluid in the upper portion of

the lobe is effectively pinched off from that in the lower portion of the same lobe,

and a classical mushroom-shaped pattern (c.f. Refs. 4 and 5), indicating the presence of strong counter-rotating vortices is observed. These results clearly demonstrate the important effects of vorticity or secondary flows in the initial conditions, and indicate the importance of lobe generated secondary flow on the tempera- __.

ture profile development.

Effects of Turbulence Models

The influence of the various turbulence models, k-c, k-£ or the wake model

were investigated by rerunning case II with the k-_ and wake turbulence models as

cases III and IV, respectively. With the exception of the turbulence model used,

cases II, III and IV are identical as can be observed in Table I. The results of

these calculations are summarized in Table II where area-averaged values of total

pressure ratio, and total temperature ratio are presented along with the previously

defined total pressure loss coefficient, and thrust ratio. Examination of the infor26 _2.
marion presented in Table II indicates that the computed results for this lobe mixer

flow show only very limited variation with the turbulence model. The results Indicated that the k-€ model gives the higher losses whereas the wake model predicts the

lowest. In view of this limited dependence on turbulence model, it would appear that

turbulence modeling in lobe mixers is not an area of primary importance. This is --

born out further by the comparisons presented in Fig. 6 where the total temperature

signatures resulting from each calculation are shown. This comparison shows that __

only small scale details are altered by the different turbulence models while the

large scale features remain unaffected. Additional evidence that turbulence modeling

may not be of primary importance in lobe mixer flows is provided in Fig. 7 where

the secondary flow vorticity contours at the nozzle exit are compared for the

three turbulence models. Here also, it is observed that the predicted vortlcity

contours for all three turbulence models are nearly identical. The pattern of

two large counter-rotating vorticities is observed for all three turbulence

models, and the strength and size of these vorticities remain uneffected by

the turbulence model used. Some small differences in the contours are observed,

however, particularly when using the k-_, turbulence model. These small scale _ __

differences appear as slight distortions of the vortex contours, but as indicated

in Table II, this effect does not apparently alter the mixing in any significant

manner.

Effect of Swirl

Perhaps the most significant modification to the computer code performed

under the present effort is the extension of the capability of the code to

swirling flow. To demonstrate this capability several computations were carried

out for swirling flows. The first of these calculations, case V, is identical to

case III discussed previously with the additional specification of an initial

swirl component of velocity, equal to 10% of the hot stream velocity, across the

entire mixing duct (hub to shroud). The pertinent integrated quantities for

these two calculatlons are presented in Table III.

As would be expected, case V with swirl exhibits higher losses reflected by

the area averaged tonal pressure ratio and the total pressure loss coefficient,

and a lower thrust ratio. The contour plots of the exit plane total temperature

distribution for these two cases is shown in Fig. 8. This comparison demonstrates

that the large scale features of the signature remain almost unaltered by swirl _-

and are simply convected in the azimuthal direction. Some distortion of the _I

pattern is observed, particularly as the center of the _ng duct is approached;
...... o 4

!4

however, these effects do not appear to enhance the mixing of the two streams

significantly. A comparison of the vorticity contours at the nozzle exit plane,

shown in Fig. 9, indicates a more substantial distortion of the counter rotating

vortex palr by swirl. This suggests that in longer mixing ducts, where swirl

effect may develop further, more significant effects on m_xingmay be observed.

However, in view of the higher losses incurred with swirl, and the absence of any

clear-cut benefits resulting from swirl, it would appear that swirl in lobe mixers

may not be beneficia! from a performance standpoint.

A final calculation with swirl, case Vl, was made using the wake turbulence

model, and with 10% swirl velocity confined to the outer half of the mlxing duct.

This calculation was made to determine if residual swirl in the fan stream would

alter the mixing significantly. The exit plane total temperature contours for

this calculation are presented in Fig. I0. Again, no significant effect on

mixing is observed. The large scale characteristics of the total temperature

• signature remain intact with some distortion observed in the outer region of the

flow.

Although swirl dld not appear to have a significant effect on the mixing

characteristics of the lobe mixer/exhaust nozzle used in the present calculation,

it should be noted that the exhaust nozzle was short. If a longer exhaust

•

nozzle were used, the effects of swirl would have more time to develop, and i 7

significant effects on mixing could result. However, as noted earlier, swirl

does appear to give rise to higher losses, and if performance is the objective,

swirl does not appear to be beneficial, i -

t

28

/
r

• |4

SUMMARY AND CONCLUSIONS

An analysis which was previously developed for detailed prediction and

computation of three-dlmensional subsonic turbulent flows in turbofan lobe

mixers has been extended and modified. An improved starting procedure which

allows specification of non-zero starting vorticity using a parametric representation of radial flow velocities at the initial plane has been developed.

The method of scaling an incompressible potential flow to obtain the compressible flow imposed pressure gradients has been improved. An optional k-£

turbulence model has been incorporated within the code, and the original k-€

turbulence model has been modified to provide improved reliability. Finally,

a swirling flow capability has been added to the code.

Calculations have been performed for a number of test cases to verify

the modifications to the code, and to demonstrate the importance of the effects -

of secondary flows, turbulence model, and swirl on mixing. The results of these

calculations indicate that secondary flows generated within the lobes play the I

most dominant role in determining the degree of mixing. The effects of the

!

turbulence model appears to play a very limited role, altering only small scale i" °

details of the predicted flow field. Finally, calculations where a swirl component _

of velocity, equal to 10% of the turbine stream velocity, was specified across the

entire mixingduct indicated that swirl may not alter the degree of mixing substantially. In this case, the major effect was to simply rotate the total _

temperature pattern observed without swirl with little additional distortion.

However, specifying swirl over a limited portion of the duct radial extent may

have a more pronounced effect. More detailed investigations and experimental

confirmation will be required to fully assess these effects.

29

• .

f f jt
i,

i

REFERENCES

_...

./

1. Kreskovsky, J.P., Briley, W.R. and McDonald, H.: Turbofan Forced \

Mixer-Nozzle Internal Flow Field III-A Computer Code for 3-D Mixing

in AxisymmetriNcozzles. NASACR 3494,April1982.

I

2. Kreskovsky, J.P., Briley, W.R. and McDonald, H.: Prediction of Laminar

and TurbulentPrimary and SecondaryFlowsin S_ronglyCurvedDucts,

NASA CR3388,February 1981.

3. Povinelli, L.A., Anderson, B.H. and Gerstenmaier, W.H.: Computation

of Three-Dimensional Flow in Turbofan Mixers and Comparison with

Experimental Data, AIAA Paper No. 89-0227, January 1980.

4. Anderson, B.H., Povinelli, L.A. and Gerstenmaier, W.H.: Influence of

Pressure Driven Secondary Flows on the Behavior of Turbofan Forced

Mixers, AIAA Paper No. 89-I198, June 1980.

5. Anderson,B.H. and Povinelli,L.A.: FactorsWhich Influencethe -

Behaviorof TurbofanForced Mixer Nozzles. NASA TM81668, January 1981.

6. Briley, W.R.: Numerical Method for Predicting Three-Dimenslonal

SteadyViscous Flow in Ducts. J. Comp. Phys.,Vol. 14, 1974, p. 8.

. 4

7. Patankar, S.V. and Spalding, D.B.: A Calculation Procedure for Heat,

_ss, and Momentum Transferin Three-DimensionalParabolicFlows.

Int. J. Heat and Mass Transfer, Vol. 15, 1972, p. 1787. _ i qf

z

8. Lakshminarayana, B. and Horlock, J.H.: Generalized Expressions for

Secondary Vorticity Using Intrinsic Coordinates. J. Fluid Mech.,

Vol. 59, 1973, p. 97.

9. Launder, B.E. and Spalding, D.B.: The Numerical Computation of

Turbulent Flows. Computer Methods in Applied Mechanics and Engineering,

Vol. 3, 1974, p. 269.

I0. Shamroth, S.J. and Gibeling, H.J.: The Prediction of the Turbulent Flow

Field about an Isolated Airfoil, AIAA Paper 79-1543, 1979. -

]i. McDonald, H. and Fish, R-W.: Practical Calculations of Transitional

Boundary Layers, Int. J. Heat and Mass Transfer, Vol. 16, 1979, p. 1729.

12. McDonald, H. and Briley, W.R.: Computational Aspects of Internal Flows,

AIAA Paper No. 79-1445, 1979.

13. Anderson,O.L.: User's Manual for a Finite-DifferenceCalculationof

TurbulentSwirlingCompressibleFlow in AsixymmetricDucts with Struts

and Slot Cooled Walls. USA/OflIDL-TR-74-50, Vol. I, 1974.

%

14. Shapiro, A.H. : The Dynamics and Therodynamics of Compressible Fluids Flow, !:

Vol. I. Ronald Press, New York, 1955. ._

i

J •--i....
°

!

15. l.ieblein, S. and Stockman, N.O.: Compressibility Correction for Int{.rnal

Flow Solutions- j. Aircraft, Vol. 9, No. 4, 1972, P- 312-

16. Walz, A.: Boundary Layers of Flow and Temperature- HIT Press, 1969, /

p. 115.

An Extended Mixing Length Approach for

17. McDonald, H. and Camarata, F.J.:

Computing the Turbulent Boundary-Layer Development- In Proce.dingS,

Stanford Conference on Computation of Turbulent Boundary Layers, Vol. I,

published by Stanford UniversitY, 1969, PP- 83-98.

. .: Effect of Free-Stream Turbulence on I

.. _a H and Kreskovsky, 3 P ..... 1 I urnal of Heat and _lss 18. mcDona_u, " _ _.... 1=vet ]nternat_u-o_ _o

the Turbulent BounG--LY _- "

Transfer, Vol. 17, No. 7, 1974, P- 705. Prediction of Free

19. Launder, B.E., Horse, A., Rodi, W. and Spalding, D.B.:

Shear Flows - A Comparison of the performance of Six Turbulence ModelsNASA SP-321, Vol- I, 1972, o. 361-426.

H

31
i /

TABLE I

CASE # UF/UT TF/TT VF VT WS £ k TURB MOD Re

i .633 .4 -.13 .08 0 .006 .04 k-£ 1.13 x 106

2 .86 .74 -.20 .25 0 .006 .04 k-£ 1.13 x 106

L\ 3 .86 .74 -.20 .25 0 .006 .04 k-€ 1.13 x 106

4 .86 .74 -.20 .25 0 .006 .04 WAKE 1.13 x 106

5 .86 .74 -.20 .25 .i full .006 .04 k-c 1.13 x 106

outer

6 .86 .74 -.20 .25 .i half .O06 .04 WAKE 1.13x 106

: UF - Fanstreamvelocity Ws _ Swirlvelocity

UT m Turbinestreamvelocity £ m Turbulencelengthscale

k m Turbulencu kinetic energy

, TF - Fan scream temperature

Re = Reynolds number

: TT - Turbinestreamtemperature f

VF - Fan streamradialvelocity

VT - Turbinestreamradialvelocity

II_lr .............

\ '\
TABLE II

TURBULENCE TOTALPRESS_ TOTALTEMP. CpT 3_IRUST

CASE MODEL RATIO RATIO LOSS=(+)RATIO

II k-_ .9892 .7866 .0956 .9930

III k-_ .9890 .7863 .0975 .9926

IV WAKE .9896 .7866 .0923 .9933

!

I

33 j

4
' TABLE III

TURBULENCE TOTAL PRESSURE TOTALTEMP. CPT TH1_ST

CASE MODEL RATIO RATIO LOSS -- (+) RATIO

III k-€ .9890 .7863 .0975 .9926

V k-_ .973 .7884 .1.132 .9883

34
ORIC.4NALPAQ_ I_

OF POORQUALITY

/

/

/

/

/

/

/

..../

,_\

Transverse coordinates

| (Secondary flou plane)

%

\

Figure 1. - Sche_atlc of coordinate system for

three-dlmenslooal flo_ problem.

i

35 t

i

°

/
\

I,

II

:I

....

'tl

Centlf Plul

Calculation

'1A'-··· _ -..

'------------- t -L. -=~'_"~~L..,..I__..... ........_..

biB

Figure 2. - Typical lobe mixer geometry and potential flow coordinate 8Y8te~.

\
ORIGINAL PAGE IS

OF POORQUALITY

Casing

Bound

z, B/"-'_

Figure 3. - Typical cross section at mixer exit surface.

37

_° ° °
GOLD

(FAN)

SYMMEI"I_

BOUNDARY

_; HOT

CASING e

r_ln

GOLD

'- (FAN)

PERODIG

PERODIC BOUNDARY

BC

(

Fig. 4 - BoundarY conditions for flows with and without swirl.

3v"
Case II

Case I

/

|

Vfan 20 ,I

= "' = .25

Vfan 13 Vc0r e

= .08 'I

Vcore ,

\

oO

mS

rm

Fig. 5 - Effect of initialsecondaryflows on exit plane

total temperaturecontours.

\
Case II Case III Case IV ',

k-£ Model k-c Model Wake Model

/

\

"\.

\

\

\

;Of •

KD_

Fig. 6 - Effect of turbulence model on exlt plane total temperature contours.

\

f

I , \ •

!
Case II Case III Case IV ,

k-_ k-c Wake

\

. \

Flg. 7 - Effect of turbulence model on exit plane secondary flow vortlclty contours. '\

\

'\

'\

\

'\
Case III

No Swit'l

Case V

10% Swirl

Fig. 8 - Effect of swirl on exit plane total temperature contours.
'\

I

Case V

Case III

I No Swirl 10% Swirl

I

t

i

i.

\

b .'

i Fig. 9 - Effect of swirl on exit plane secondary flow vortlcity contours.

i \

/

\
Case VI

\, '\

, \

J\

".J

\

Fig. i0 - Distortion of exit plane total temperature contour

with 10% swirl confined to outer half of mixing duct.

I

I : . _i '._ /- .'"
./ .-

• .... ° .

J

O_C_.-_IALPAGEIS

APPENDIX - USER'S CUIDE OF POOR QUALITY j

Introduction

The Mixer computer code m,.a. y be run using initial conditions obtained

from experimental data and read in on a pointwise basis, or an initial flow

field may be generated by the code itself upon parametric specification of

fied flo_:

the proper input variables. In either case, the originally speci

fields are processed by the c-ode to assure that the initial flow field is -.

consistent with the finite-difference approximations to the governing equations.

kq_en parametric specification c'.f initial ,.;_nditions is used. it is also possible

to activate a s..,_rling flo,- option of the _ode. _q_en this option is utilized

the flow is as:;umed to be periodic, and t':;' initial conditions, with the

e:.:ception of the s,airL ve!ocity, are as,_u,,ncd to he sy-mmetrical about the lobe

centerline. Calculations performed with :;_:irl must, therefore, consider a full

lobe _.,ithin the computational tran..,verse !.';rues, ._hereas only a half lobe need

!.e considered for =lonswirling flows.

In an etfort to keep t.be initial condL'ion specification similar for both

...wirling and :-,t,n_;wirling f!ow C;_lculatious. the initial flow field for swirling !

r',.,,v¢, or other _aiculati(ms perf,_r,m, ed tmi,:_, petit:die boundary conditions, is set , --

:q, for half the lobe and t'.qen reflected. ,\:. a result, the user may use the same

_.lput r_:Ta strea:a for both per.;t,dic and na::q.triodi(" calculations by adjusting only

." ;:_" [111,.1_ -', C.-- of grid points, the axi.,muthal cc.-_putational limits, and calling for

ti.. ;,r.'i.,--r _,pti:m.q- Specification of the l,,be shape need not be altered; only

i_alf a lc.Se must he specified via the inp_,t in either case.

ThL° ,,:.t-r :,hc:zid read ,.,,.,er all inp,t; dea.-riptions before attempting to set up

a c ¢t.qe.

i

!

t

45

t

!
DESCRIPTION OF INPUT

Input to the MIXER code is accomplished by a combination of card input 1

and information which is stored onauxiliary files. Metric information

generated by the ADD code [13land restart data sets are stored on separate

auxiliary files. Program control is performed through card input, which is

divided into three categories:

i) Plot file information for NASA-Lewis plotting package,

2) NA_LIST input, and

3) Experimental flow field input.

Format of the required input is as follows:

Plot File Input

CARD # COLUMNS VARIABLE DESCRIPTION

Card I Col. 1-32 TITLE(l) Plot title, format 5A6,A2. (May be left blank).

Card 2 Col. 1-2 ISYM Twice the number of lobes in a 360 °

cross section of the mixer geometry.

Col. 3-12 SYSTEM Indicates the type of coordinate system

for plot routines. Format FIO.0,

Use SYSTEM=2. _-

46

!

~
NAMELIST INPUT

NAMELIST IN1

Namelist Description

$1NI

Gas constant,ft lhf/(slug°R), default 1716.3 (air)-

RG

Constant pressure specificheat, ft lhf/(slug OR),

CP default is 6012.384 (air).

Reference temperature,assumed to be nominal primary i

TZERO stream temperaturein degrees Rankine. !

Referencevelocity,assumed to be nominal.primary k

UZERO stream velocity, ft/sec.

Reference static pressure, assumed to be nominal

pZEROI pressure at initial plane, Ibf/ft2, default is 2116.8-

Reference length, ft.

-. YZERO

ReferenceViscosity,slugs_ft-sec.

VISCI

SEND

NAMELIST IN2

The desired computational options and mesh information are specified in

namelist $IN2.

" Namelist Description

$IN2

IAXI =i, cylindricalcoordinates.

=2, rotatedorthogonalcoordinatesgeneratedfrom

ADD code geometry file. Default.

If IAXI is input = 2 (or defaulted)the variables,

XENTR, ILAP, LX, LY, and IPCORmust also be input.

Dimensionlessdistance from initialADD code potential

XENTR line to -initialplane of lobe mlxer calculationUsed only for IAXI=2- Determinedfrom ADD code

_ parametersasXENTR=(JSTEPxENTR-I)*DSwhere JSTKPXENTR

is the number of the ADD code potentialsurface

representingthe MIXER code initialplane and DS is the

streamwisedelta used in the ADD code. See Fig. BI.

47

-- - -. :;, .... T

-

.
= i for IAX _=2

= 0 if IAXI=I

ILAP is defaulted = 1. If _ is set to I,

ILAP mus___tbe specified as O.

LX is the number of potential lines in the ADD

I_ code geometry file. LX must be input if ILAP=I"

LY is the number of streamlines in the ADD code

LY geometry file. LY must be Input if IL AP=IIPCOR is used with ILAP=I to identify the potential

IPCOR surface whose average dimensianless velocity is 1.0

.. in the axisymmetric potential calculation- IPCOR

should be input as IPCOR=JSTEPxENTR in general.

= -1, stop after input dumpIcOMP

_. = 0, normal run (default)-

l, stop after printing initial plane geometry-

_ 2, stop after printing initial flow field.

= 0, automated start (default)-

IDATA = 2, read initial velocity and temperature field

data at each grid point.

ILAM=I, Laminar flow.

ILAM ILAM=0, k-c turbulence model.

ILAM=-I, Wake turbulence model.

ILAM =-3 ,k-g turbulence model.

=I, specify uniform turbulence kinetic energy, at

IDIS the turbine stream value throughout the initial

flow field.

=0, initial turbulence kinetic energy assumes

,"_ turbulence production equal to dissipatlonIDIS=I is recommended-

=0, use hlghReynolds number form of turbulence

/ LOWRE model equations

"" se low Reynolds number turbulence model

: . =i, u - "-es IDI S=0, IL_=O or -3, and proper i

' uations- Kequxr

eq - -f wall bou_iarylayers- Also requires

specificatlon o

a high degree of resolution of near wall regions

using IGRID(1)=I or 2 optionwith appropriate

EPS(1)- This option is not gen_rally reco_ended

for mixer flows.

-I

4B

!
Namelist Description

YS(I,J) YS(I,J) sets the limits of the transverse computational

planes.

YS(I,1) = rmin (dimensionless, r/YZERO)

YS(2,1) = rmax (dimensionless,r/YZERO)

YS(1,2)= Omin (radians)

YS(2,2) = 8max (radians)

For mixer geometriesgeneratedby theADD code YS(I,I)must

be the dimensionlessplug radiusat the firstADD code station RMIN, and Y(2,1)must be 1.0. See Fig. B-I. For nonswirlingor non-periodicflows,YS(I,2)must correspondto

the 0 value at the maximumlobe penetrationand YS(2,2) to

the @ value midway betweenthe lobes. See Fig. B-2. For

periodicor swirlingflows,YS(2,1)and YS(2,2)correspond

to sequentialvalues of 0 midway betweenthe lobes. See

Fig. B-3.

NE(I) NE(1)= numberof gridpointsin the radialdirection.

NE(2) = numberof azimuthalgrid points. For periodic

calculationNE(2)must be odd.

Maximumvalues for .*[£(I)are 50. For periodiccalculation

NE(2)max= 49.

IGRID(I) IGRID=0,no grid stretching(default).

=i, Robertsstretchingat rmin or Omln.

=2, Robertsstretchingat both rmin and rmax

or @ . and 0

D-In max"

=4, hyperbolicsine stretchaboutXCTR(I).

I=l for radialdirection,I=2 for aximuthaldirection.

EPS(1) Used for IGRID(I)_0to controlthe grid stretchnear boundaries. For Robertsstretching0.0< EPS< f.0. For hyperbolic sine 1.0<F_PS<5.0.

XCTR(I) Used only with IGRID(I)=4to specifythe r(l=l)or

8(I=2)locationabout which grid pointsare clustered.

YS(I,I) < XCTR(I)< YS(2,I).

NS Numberof axial stationsto be computed.

NS=I is initialplane. Maximumvalue is 50.

AP If AP is input .GT.0a geometricstreamwisegrid is

set up with Ax(J)/Ax(J-l)= AP.

If AP is input .LT.0the user must specifythe axial

mesh. AP has a defaultof 1.05 but the AP.LT.0option

is recommended.

!

i

49

_ . _
X is an array which contains NS values of the streamX(JX) wise mesh coordinate.

If AP is input .GT.0 only X(1)and X(2) need be input.

If AP is input .LT.0 NS values of X must be input.

If IAXI=I, X is a physical but dimensionless distance.

If IAXI=2 and AD_ code geometry is used, X represents

the dimensionless computational distance from the

initial plane.

The maximum allowable value of X is given as

(j STEPMA X _ JSTEPXENTR)*DS/U SCALE where USCA LE=YS (2,I) ]

(YS(2,1)-YS(I,I))- The maximum value of X should be

slightly less than this value to avoid marching out of

the ADD code geometry which will result in an error

termination- See Fig. BI.

IPERIO =0, normal calculation

=I, periodic option activated. Input must be set up

for periodic flow field. Check NE(2) specification.

=0, nonswirling flow. ISWIRL

=i, swirling flow. Requires IPERIO =I- Also requires

values for RSWI, RSW2, VSWRL, and DELSWL.

SEND

NAMELIST IN3

Restart and plot file information is specified in namelist $IN3.

._melist Description

$IN3

The number of the axial station to be read in for a

IRSTIN

restart. No restart if IRSTIN =0 (default).

Increment for saving restart information. Restart files will

IRSTOT be written every IRSTOT steps from starting value of JX, the

axial station counter. Default assumes no restarts written.

On a restart, the starting value of JX = IRSTIN.

JRSTIN File number from which restart information is to be read. Default file number is Ii.

File number onto which restart information is to be

JRSTOT written. Default file number is Ii.

5O

w

. .
Namelist

Sequence number in file JRSTIN cf the desired restart

NFILE informationThe number of restart data blocks saved on file

NSAVED JRSTOT.

If IPL OT>I a plot file must be assigned in the run

IPLOT stream and information will be written in this file

for subsequent use in plotting. IPLOT=0 gives no

p]ot file- Default is IPLOT =0- A plot file will be

written eyery IPLOT th station.

SEND

NAMELIST IN4

Velocity and temperature ratios between the two streams are specified in

namelist $IN4, as are turbulence levels and information used to construct the

initial secondary flows.

Namelist Description

$IN4

USTRM2 Ratio of secondary stream (fan) velocity to primary

stream (turbine) velocity. For matched average inlet

Mach numbers and total pressures USTRM2=_ -_RM2"

Default value is USTRM2 =-707-

TSTRM2 Ratio of secondary stream temperature to primary stream

temperature. Default is TST_M2=-5 -

TUSTRI Primary stream turbulence intensityTul= u--_/u_ where uI=UZEROTUSTR2 Secondary stream turbulence intensity.

Tu2= _/u_ where u2=UZEROxUST_M_-

If TUSTR2 is input _ 0.0 the free stream turbulence

in both streams (k=3Tu2/2) is assumed equal to 3Tu_/2-

If IDIS =I is input (as recommended) only TUSTRI is used.

Dimensionless free stream turbulent length scale,

ALEN £ref/YZERO. To be estimated by the user based on

geometric constraints of the mixing duct and boundary

layers. ALEN should be on the order of .i of

estimated free shear layer or boundary layer thickness.

AMEXIT AMEXIT is an assumed exit planeHach number which is

used to specify the isentropic exit plane pressure for

the thrust calculation. Defmult value is AMEXIT=0-9"

51

l •
Namelist Description

DEI/dlX DELMIX is input with IDATA .GT. 0 as a dimensionless

boundary layer thickness presumed to be representative

of the boundary layers on the lobe surfaces. It is used

in computing the initial turbulence length scale.

DELHUB DELHUB is input with IDATA .GT. 0 as a dimensionless boum£mr_-

layer thickness, 6/YZERO, respresentative of the hub bou_r!_

layer. It is used in computing the initial turbulence lezgt_m

scale. See IHUB in $IN5.

DELS_P similar to DELHUB but pertaining to the shroud boundary

layer. See ISHR in $IN5.

RHUB RHUB is a dimensionaless secondary flow hub boundarY

layer thickness used to construct initial secondary

flow profiles required for IDATA=0 or 2.

RTIP Similar to RBIIB but pertaining to the shroud secondary

flow boundary layer.

"" DELSEC DELSEC is a dimensionless secondary flow boundary layer

thickness used to construct the initial vorticlty Pr ofil_-

A value on the order of (RTIP+RHUB)/2 should be used.

VFAN Used with IDATA=0- VFAN is the ratio of the local radial

velocity to the local streamwise velocity (positive

outward) in the fan stream.

VCORE Used with IDATA=0- VCORE is the ratio of the local

radial velucity to the local streamwise velocity

(positive ouv;ard) in the turbine stream.

VSWRL VSWRL is used only with ISWI RL=I to specify the

dimensionless swirl velocity magnitude.

RSWI, RSW2 RSWI and RSW2 are the radial limits over which swirl

velocity is to be specified. They must be input such

that YS(I,I) < RSWI < RSW2<-YS(2,1)-

DELSWL DELS_% is a dimensionless length scale over which the

swirl velocity in increased from zero to its' maximum

_ value. DELS_ must be less than (RSW2 - RSWI)/2.

SEND

NAMELIST IN5

The lobe shape and primary flow boundary layer parameters are

/

specified in namelist $IN5. Spezification of this information should be

performed using Fig. B2 or B3 as a guide.

Namelist Description

$1N5

.... ILOBE ILOBE is a flag used in the automated starting routine.

"_ ILOBE=0 indlcates that no boundary layers are to be

i-"._ set up on the lobe surfaces.

- , ILOBE=2 indicates that lobe boundary layers are to be

, set up.
Description

Namelist

IHUB=O indicates that hub boundary layers will not be

IHUB present in the initial profile, regardless of other

starting options. If IDATA is input .GT.0 and IHUB is

input .GT.0 DELI[UB must be input. If the IDATA option

is not used, and hub boundary layers are desired, IHUB

must be input equal to the number of _cints input to _

describe the hub boundary layer in the array BLHUB.

ISHR is similar to IHUB except it pertains to the shroud

IS}D% boundary layer.

BLIIUB(I,J) If hub and/or shroud boundary layers are to be construc- ted using the automated starting routine (IDATA=O) the

boundary layer displacement thickness and shape factor

must be input as functions of 8 along the hub and/or

shroud surfaces. This is accomplished using the array

BLHUB. See Fig. B2.

BLHUB(I,I) contains the values of 8 (in radians) where

hub boundary layer parameters are specified. The 8

values are arbitrary but they must increase monotonicallyFor calculation usinB the IPERIO=O option the first

value must be slightly less than 8min and the last value

must be slightly greater than 8max- For IPERIO=I, the

first 8 value must be sliBhtly less than that of

(Stain+ flmax)/2- A maximum of 20 values maY be specified.

I_ The number of 0 values specified must be equal to the

.... input value of IHUB.

BLHUB(I,2)=e /YZERO, the dimensionless displacement

thickness of the hub boundary layer at each specified

O location.

BLHUB(I,3)=H, the shape facto" at each e location.

j_ BLHUB(I,4) is the shroud equivalent of BLHUB(I,I)-

_ ISHR values of 8 must be loaded in BI/_JB(I,4)-

BLHUB(I,5) and BLHUB(I,6) are similar to BLHUB(I,2)

and BLHUB(I,3) except they pertain to the shroud

.. boundary layer parameters.

NLOBE is the number of data points input to describe

i N'LOBE the lobe sPmpeand lobe boundary,layers.

__ YLOBE(I,J) YLOBE is an array which contains the data needed to

specify the lobe shape and boundary layers which may

be on each side of the lobe surface- NLOBK values of

each parameter in YLOBE must be specified. See Fig. B2

for ordering of information- Specification of the lobe

shape must start where the lobe crosses the 8max boundary.

53
YLOBE(I,I) contains NLOBE dimensionless radial coordinates,

$LOBE(I,J) R[YZERO, of the lobe. The first point must be the radial

coordinate where the lobe crosses the 0max boundary. The

last point must be the radial coordinate value where the

lobe crosses the Omln boundary for IPE RIO=0" For IPER IO=I

only half the lobe shape is specified, thus the last point

/ is theradialcoordinatewhereO = (Omin+Omax)12.

See Fig. B2.

YLOBE(I,2) contains the O values covalue rresp(oI=I) ndingmustto bethe sliR ghvalues tly

loaded in YLOBE(I,I)- The first the last value (I=NLOBE)

-J greater than Omax- For IPE RIO=O For IPE RIO=I the last value

must be slightly less than 0rainmust be slightly less than (0rain+ 0max)/2"

YLOBE(I,3) contains N-LOBE values of the dimensionless disi_ placement thickness, 6*/YZERO, for the boundary layer on the

turbine stream side of the lobe surface at the lobe coordinate

locations specified in YLOBE(I,I) and YLOBE(I,2)-

YLOBE(I,4) is the same as YLOBE(I,3) except it pertains to

the fan stream side of the lobe.

YLOBE(I,5) contains the distribution of the shape factors, H,

at each coordinate point for the turbine stream side of the

lobe.

Finally, YLOBE(I,6) contains the shape factor distribution

for the fan stream side.

YLOBE(I,3)throughYLOBE(I,6)need not be specifiedif the

IDATA option is specified,or if ILOBE=0" YLOBE(I,I)and

YLOBE(I,2),the r-@ coordinatesof the lobe,must be

specified for all starting optionsIf IWAKE=I the slots for shape factor in BI/{UBand YLOBE i

IWAKE need not be input as a value of the wake function

: - in the boundarylayer profile is assumed-

,.

of

SEND

When settingup a lobe shape it is advantageousto considerspecification

BLHUB and YLOBE for a _ calculation(IPEKIO=0)" Under these conditions

. the computation segment appears as in Fig. B2. If 8min is specified as zero,

and 0me x is specified as positive, and the lobe shape is properly specified,

then a periodic calculation can be performed = without adjusting changingthe tnumber he YLOBofE ii

or BLHUB arrays simply by setting 8mi n -8max'

54
O grid points, and by specifyingIPERIO=I. In general,if the arrays YLOBE

and BI/{UBare properlyspecifiedfor a nonperiodiccalculationof limits 0min

to 8max, the same lobe shape specificationwill be correctfor periodiccalculations in which 8max remainsunchanged,and 8mln is reset as Omin=2emin-Smax"

NAMELIST IN6

Boundaryconditionsare specifiedin namellstIN6. For periodicflows 8

direction boundary conditions are ignored. The boundary conditions are set to

defaults appropriate for mixer calculations performed with wall functions.

Only XCBC must be specified.

Namellst Description

$IN6

NBCON(I,J,K) NBCON is a three-dimensionalarray which allows the user

to select the desired boundary conditions for theprlmary

flow momentumequation,the energy equation,the secondary

flow streamfunction-vorticitysystem,and the turbulence

model equations. The subscript"J" determinesthe

boundarysurface. J=l for the surfaceat Rmln, J=2 for

Rmax, J=3 for 8min, J=4 for emax"

NBCON(I,J,I) NBCON(I,J,I)is used to specifyboundaryconditionfor the

primaryflow momentumequation.

= 0 - no slip

= 1 - zero normalgradient

= 2 - zero secondderivative

= 4 - wall function

At the hub (J=l)and shroud (J=2)eitherno sllp or wall

functionsshouldbe specified. (Wallfunctionsare preferred,

and requiredwhen LOWRE=O in $IN2).

At emin (J=3) and 8max (J=4)symmetry(zerogradient)

conditionsare requiredwhen IPERIO=0.

-: NBCON(2,J,I) NBCON (2,J,l)is used for the energyequation.

= 1 - zero normal gradient (adiabatic)

= 2 - zero secondderivative.

Adiabaticconditionsshould be specifiedat all surfaces

. (J=l to 4).

NBCON(4,J,I) NBCON(4,J,I)is used to specifyboundaryconditionsfor

tha vortlcity-vectorpotentialsystem,and results in

eitherno sllp or slip conditionsappliedto the local

tangentialcomponentof the secondaryflow velocity. I

= 0 - no sllp

= 1 - sllp with zerovortlclty

55

/
At r (J=l) and r (J=2) no slip should be specified.

_rain (J=3) and Amax (J=4) symmetry implies slip with

At Omin -max

zero vorticity should be specified. The no sllp condition

should be used at a centerllneNBCON(I,J,2) NBCON(I,J,2) is used to specify boundary conditions for the turbulence kinetic energy equation which is solved if

ILAM=O or -3.

= 0 - zero kinetic energy

= i - zero normal gradient

= 2 - zero second derivative

At r__ n (J=l) and rma x (J=2) the turbulence kinetic energy

shou_ be set to zero if LOWRE =I, or the normal gradient

should be set to zero if LOWRE =0- The latter is consistent

with the wall functions used for the primary flow momentum,

and is recommended. At 8min (J=3) and Omax (J=4) zero normal

._° gradient should be specified.

NBCON(2,J,2) NBCON(2,J,2) is used to specify boundary conditions for

the turbulence dissipation equation (ILAM=0 only).

= 0 - zero dissipation

I = i - zero normal gradient

....... 2 - zero second derivative

= 4 - wall function type boundary condition

At r . (J=l) and rma (J=2) the dissipation should be

set _nzero if LOWRE =_, or the wall function should be

used if LOWRE =0- The later is recommended. At Bmin and

0 zero gradients should be applied.

max

XCBC XCBC is used in MIXER calculations to change boundary

conditions as the flow moves downstream off the hub,

where wall function boundary conditions are applied,

to a center line where sy_netry conditions are applied.

The user must specify the value of XCBC by examination

of the geometry from tileADD code. XCBC Is ci_mputed

as XCBC=(JSTEPxcBc - JSTEPxENTR)aDSw_? re JSTEPxcBc is

the number of the ADD code potentia_ ilne where the

plug degenerates to a center line and JSTEPxENT R is the

number of the potential line representing the initial

MIXER calculation plane. See Fig. BI. The default

value causes no changes.

56
%

NBCONC(I,J,K) NBCONChas the same function as NBCONbut is used to

specify which boundary conditions are to be changed

for X>XCBC and what they are to be. Default value

will cause no change in boundary conditions regardless

of the value of XCBC. For a calculation with hub

_I boundary layer, NBCONC is used to change from wall

functions to symmetry conditions as the hub degenerates

to a center line.

SEND

P_._EL_ISTIH7

SIN7

IPRN(1) IPRN is an array which selects variables to be printed

at each axial station. If IPRN(1)=I variable "I" will

be printed. If IPRN(1)=0 variable "I" will not be

printed. The variables are numbered as follows:

I= Variable

I u

2 v

3 w

4 p

5 E

6 PT

7 k

8 c

9

I0

11 H

12 static temperature ratio T/T r

13 total temperature T°/T °

r

14 pressure coefficient

15 static pressure ratio P/Pr

16 total pressure ratio pO/pO

r

17 swirl angle, degrees

18 u'v'

19 u'w'

20 v'w'

ICON ICON controls printer contour plots of the printed

flow field variable. "

ICON=I produces printer contour plots.

ICON=O, no contour plots.

57

Qo
.L/

IPRINT iPRINT must be input _I. Printout will be made

every IPRINT'th station.

SEND

Experimental Flow Field Input

The following cards are needed only if starting profiles are to be read

in (IDATA=2)- Each card contains velocity components u, v, w in feet

per second and static temperature, T, in degrees Rankine at each computational

grid point. One card per grid point is required, thus a total of NE(1)_NE(2)

" data cards are needed. Format 4FI0.0. The data is read for the grid point

numbering shown in Fig. B4.

CARDS _ SEND+I to $END+NE(1)*NE(2)

Col. I-i0 u - velocity component in the computational streamwise

direction, ft/sec.

11-20 v - velocity component in the computational radial direction,

ft/sec.

21-30 w - velocity component in the computational azimuthal direction,

ft/sec.

31-40 T - static temperature, OR58
q

FigureBI.- Typicalcoordinatesystemand definition0£ inputparameters.

[

i

...... • i •
OR;GINAL PAGE IS

OF POOR QUALITY

Casing boundary layer parameters

specified along this surface as

a function of 9 Boundary 2

e

e max

rain

NLOBE NLOBE-1

y4

Boundary

: Specify lobe data points

in the numbered order

3

2 1

Hub boundary layer parameters

specified along this surface i

as a function of e

\\

r

Figure B2. - Specification of the lobe shape and hub and casing

_ boundary layers.

6O
ORIGINALPAGEIit

OFPOORQUALITY

:nin maz

9

"+nit; /"-"- _ max

\

"\

!:' " /

\ /

"-" \ !

: /' Specify lobe data point_

in the numbvred order.

"\ _2 [.,_bc shape need only t_ / :q,_-cificd fro= -' t_

# 1 ) ,o ma:-: / " (" + :;rain '-"

it

'\.v_ _¢........... _--..] ,-" o . ,';

"¥- ; .- . /

V - , !!

!

Figure B3. - Specification of lobe shape for

periodic flow domain.

61

/ __
DR!C-{_tAPAL (_,2m

OF POORQUALITY

2(NE(1))

NE(2) * (NE(1)

NE(1)

O O

O * (NE(2)-I)

NE(1)-] + (NE(1)-l)

0 ,0 0

1_E(1)+2

O O O NE(I) * (NE(2)-I) + 2

2

NE(1)+I

NE(1)* (NE(2)-I)+ I

Figure B4. - Ordering of grid points for IDATA option.
•

"m

NASA

3 1176 01419 3032