Wall Functions for the k - [Epsilon] Turbulence Model in Generali

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1/177

Iowa State University

Digital Repository @ Iowa State University

R+& a*! D&a&+*

1992

Wall functions for the k - [epsilon] turbulencemodel in generalized nonorthogonal curvilinear

coordinatesDouglas L. SondakIowa State University

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& D&a&+* & b+0$% + 3+0 #+ # a*! +* a b3 D&$&a R+&+3 @ I+a Sa U*&&3. I %a b* a! #+ &*0&+* &*

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http://lib.dr.iastate.edu/?utm_source=lib.dr.iastate.edu%2Frtd%2F9954&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://lib.dr.iastate.edu/rtd?utm_source=lib.dr.iastate.edu%2Frtd%2F9954&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://lib.dr.iastate.edu/rtd?utm_source=lib.dr.iastate.edu%2Frtd%2F9954&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://network.bepress.com/hgg/discipline/218?utm_source=lib.dr.iastate.edu%2Frtd%2F9954&utm_medium=PDF&utm_campaign=PDFCoverPagesmailto:[email protected]:[email protected]://network.bepress.com/hgg/discipline/218?utm_source=lib.dr.iastate.edu%2Frtd%2F9954&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://lib.dr.iastate.edu/rtd?utm_source=lib.dr.iastate.edu%2Frtd%2F9954&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://lib.dr.iastate.edu/rtd?utm_source=lib.dr.iastate.edu%2Frtd%2F9954&utm_medium=PDF&utm_campaign=PDFCoverPageshttp://lib.dr.iastate.edu/?utm_source=lib.dr.iastate.edu%2Frtd%2F9954&utm_medium=PDF&utm_campaign=PDFCoverPages


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4/177

Order Number 9228968

Wall

functions for

the k -

turbulence model in generalized

nonorthogonal

curvilinear

coordinates

Sondak, Douglas L.,

Ph.D.

Iowa State University, 1992

U M I

300N .ZeebRd .

Ann

Aibor,

MI

48106


5/177


6/177

Wall functions for the k

( .

turbulence model in generalized

nonorthogonal curvilinear coordinates

b y

Douglas

L . S o n d a k

A Disser ta t ion

Submitted

to

the

Graduate Faculty in

Partial

Fulfi l lment of the

Requiremen ts

for

the Degree of

DOCTOR OF PHILOSOPHY

Major : M e c hanic a l Eng ine e r ing

A pproved:

In

Charge

of Major Work

For

the

Major

Department

Graduate

College

Iowa

State Universi ty

A m e s ,

Iowa

1992


7/177

ii

TABLE OF CONTENTS

NOTATION X

PREFACE

AND

ACKNOWLEDGEMENTS

xvii

1.

INTRODUCTION

1

LI

Problem Description 1

1.2 Historical Review 2

1 .2 .1 Turbu le nc e mod e l ing 2

1.2.2

Near-wall

mode l ing 8

1.3

Scope

of

the

Pre se n t

Research 1 0

2. CONSERVATION OF MASS, MOMENTUM,

AND

ENERGY . 13

2.1 Introduction

13

2.2

Instantaneous Equations 13

2 . 3 A veraging Techn iques 1 4

2.4 M ass-A veraged Transport Equations

16

2.4.1

Cont inui ty

17

2.4.2 M o m e n t u m 17

2.4 .3

Ene rgy 17

2.5 Closure Problem 19


8/177

iii

3. k - e MODEL 21

3.1 Introduction

2 1

3 . 2 Turbulent

Kinetic

E n e r g y Transport Equation

24

3.3 M o d e l e d Turbulent

Kinetic

E n e r g y

Equation 26

3 .4 Mode le d Dissipat ion Rate Equation 28

4.

WALL FUNCTIONS 30

4.1 Bac kground

3 0

4.2

Detai led

Formulat ion

3 4

4.2.1 Introduction

3 4

4.2.2 Friction

velocity

3 4

4.2.3

B o u n d a r y

conditions

for

k

and e

3 5

4.2.4 Appl ica t ion of

tw

to

the

Navier-Stokes

equations 3 9

5. OTHER TURBULENCE MODELS 52

5.1 Introduction

52

5.2 Chie n Low-R e yno lds -N umb e r M ode l 52

5.3

Baldwin-Lom ax A lge bra ic

M odel 54

6. NUMERICAL

METHOD

56

6.1 N o n d i m e n s i o n a l

Equations 56

6.2

Vector

Form

of

Equations 57

6.3

Coordinate

Transformation 60

6.4 Navier-Stokes Solver 65

6.5 k Solver 68


9/177

iv

7. RESULTS

72

7.1

Introduction

72

7.2

Flat

plate 73

7.2 .1

Skewed

grid

79

7.2.2

A n g l e d

domain 80

7 .2 .3 Low-Re yno lds -numbe r

m o d e l test

case 83

7.3

Body

of

Re volu t ion 89

7.3.1

Introduction

89

7.3.2 Fine grid

90

7 .3 .3 Me dium

grid 96

7.3.4 Coarse grid

1 0 1

7.4 Prolate Spheroid

1 0 1

7.4.1 Introduction 1 0 1

7.4.2

Grid

108

7.4.3 Boundary conditions Ill

7.4.4 Additional

considerations

116

7.4.5 Results

119

8.

CONCLUSIONS

AND RECOMMENDATIONS 1 3 3

9. REFERENCES 138

10.

APPENDIX

A: FLUX

JACOBIANS

147

10.1

Navier-Stokes

147

1 0 . 2 fc- 150


10/177

V

11.

APPENDIX B: fc

-

SOLVER 151

1 1 . 1 B a n d e d

Solver

151

11.2

Block Solver

154


11/177

vi

LIST OF TABLES

Table

4.1:

Summary of shear stress

transformations

50

Table

7.1:

Previous

computations

of

DFVLR prola te

spheroid

1 09


12/177

vii

LIST

OF FIGURES

Figure 4.1: Typical turbulent

boundary

layer velocity profile 3 2

Figure

4.2: Defini t ion

of

7

coordinate

direct ion

40

Figure 4.3: Covariant base vectors

41

Figure

4.4:

Example contravariant base vector 42

Figure 4.5: Physical velocity

components parallel

to wall 47

Figure 7.1: Orthogonal flat plate

grid

75

Figure

7.2: Friction coefiicient, flat plate, orthogonal grid 78

Figure

7.3: Ve locity profile,

flat

plate,

orthogonal

grid

78

Figure

7.4:

S ke w e d flat plate grid 81

Figure 7.5:

Friction coefficient,

flat plate, skewed grid 82

Figure 7.6: Velocity

profile,

flat plate, skewed grid 83

Figure

7.7: Friction

coeflcient,

flat plate,

a n g l ed

d o m a i n 84

Figure

7.8: Ve locity profile, flat plate, angled d o m a i n 84

Figure

7.9:

Flat

plate grid,

low -Reynolds -number mode l

test

case

....

86

Figure 7.10: Friction coefiicient, flat plate, low -Reynolds -number mode l

test case 87

Figure

7.11: Velocity profl le, flat plate, low -Reynolds -number m o d e l test

case 88


13/177

viii

Figure

7.12: Body

of

revolution, 101

x 1 04 grid 92

Figure

7.13:

Body

of

revolut ion

coordinate system

94

Figure

7.14:

Pressure coefficient ,

b o d y

of

revolut ion, 101

X

1 0 4 grid....

9 5

Figure

7.15: Friction

coefficient , body of revolut ion, 101 x 104 grid

....

96

Figure 7.16: Ve locity profi les, body

of

revolution, 101 x

1 04 grid

9 7

Figure

7.17: B o d y

of

revolution, 101

x

9 1

grid

99

Figure

7.18: Pressure coefficient , body of

revolution, 101

X

91

grid

....

0 0

Figure

7.19:

Friction

coefficient ,

body

of

revolution,

101 x 91

grid

1 0 0

Figure

7.20:

Ve locity profi les,

body

of

revolution,

1 0 1 x 91 grid

102

Figure

7.21: Body

of

revolution, 101

x

80

grid

1 0 4

Figure 7.22: Pressure coefficient , b o d y of revolution, 101

X

80 grid

.... 0 5

Figure

7.23:

Friction

coefficient ,

body of

revolut ion,

101 x

8 0

grid

105

Figure

7.24: Ve locity profi les, body

of

revolution,

1 01 x

80 grid

1 06

Figure

7.25:

Prolate

spheroid,

121

x

53

x

57

grid 112

Figure

7.26: Friction coefficient

map

f rom Kreplin , Vollmers,

and

M ei er

(1982)

117

Figure 7.27: U n w r a p p e d surface

grid with trip

points ,

prolate

spheroid . . 118

Figure

7.28:

Pressure coefficient , prolate spheroid 121

Figure

7.29: Friction coefficient,

prolate spheroid 1 24

Figure

7.30:

Friction

coefficient

angle ,

prolate

spheroid

128

Figure

7.31:

Surface

oil f low pattern, top view 1 3 1

Figure

7.32:

Surface oil f low pattern, side

view 132

Figure 11.1: Structure of banded matrix

152

Figure 11.2: Structure

of vector

of u n k n o w n s 152


14/177

ix

Figure 11.3: Structure

of

right hand side

Figure

11.4;

Structure

of

block

matrix


15/177

X

NOTATION

Roman Symbols

a

s p e e d of

sound

flux

Jacobian matrices

H

covariant base

vector

contravariant

base

vector

b

proportionality

constant b e t w e e n

metrics

B

constant

for

l aw

of

the

wall, 5.0

H

constant for Norris and Re yno lds ne ar -wal l

len gth scale

equa t ion

c p

specific heat at

constant pressure

C l X

constant for

A : e

mode l , 0 .09

C e p

constant for

Baldwin -Lomax

mod e l , 1 .6

'^f

friction

coefficient

^kleb

constant

for

Baldwin -Lomax mode l , 0 . 3

^wk

constant

for Baldwin -Lomax mode l , 0 .25

Cl

constant for

A : e model ,

1 .44

C 2

constant for

A : e

mode l ,

1.92

Cz

constant for

Chien

mode l ,

0 .0115


16/177

xi

C4

constant

for Chien

mode l , 0.5

D dissipation

term in

turbulent

kinet ic

e n e r g y

transport equation

D k

e source term Jacobian matrix

V

smoothing

operator

e j - unit covariant b a s e

vector

e g , / g , e n e r g y

q u a t i o n

f l u x e s

E, F , G inviscid f lux ve ctors

EviFv,

G v viscous

flux vectors

total internai

e ne rgy pe r unit

volume

/ damping

func t ion for

Chi en

mode l

^kleb Klebanoff intermittancy funct ion , for B a l d w in - L o m a x m o d e l

^wake

Klebanoff wake

function,

for

B a l d w i n - L o m a x

m o d e l

metr ic

tensor

G

metr ic

matrix

h static enthalpy per

unit

mass

H

total

enthalpy per unit mass

Hf source term vector for turbulence transport equations

J

Jacobian

of

coordinate

transformation

k

turbulent

kinet ic e ne rgy

Kc

Clauser

constant

fo r Baldwin -Lomax m ode l ,

0.02688

K geometric stretching

ratio

I length scale

le

near-wal l length

scale of Norris and R e y n o l d s

W

def ined

b y equation (6.83)


17/177

xii

n coordinate

direct ion normal

to

wall

p

static

pressure

P production

of

turbulent

kine t ic energy

V term proportional

to

product ion

of

turbulent

kinet ic

e ne rgy

P physical

shear

stress matrix

Pr Prandtl n u m b e r , 0.72

Pri

turbulent

Prandtl n u m b e r , 0.9

q

heat transfer rate

Q

dependent

variable vector

r posi t ion vector

R

gas constant

R e

R e y n o l d s n u m b e r

base d on

f re e s tre am spe e d

of

sound

R e ^

turbulent R e y n o l d s

n u m b e r

R euQ Q

R e y n o l d s

n u m b e r

based on

f reestream

velocity

and

reference

l ength

Rex R e y n o l d s

n u m b e r

based on freestream

velocity

and distance from

virtual origin

s spectral radius

of inviscid

flux Jacobian

S

constant for

Sutherland's

law

S diagona l matrix con ta in ing func t ions

of

metr ics

T

static

temperature

T matrix, c o lumns of which are right eigenvectors of inviscid flux

matrix

T shear stress matrix

Tq constant for Sutherland's

law


18/177

xiii

t time

u

velocity parallel

to

wall (2-D examples)

u,V , w

velocity

components

in

physical coordinate direct ions

w " * "

velocity normal ized b y friction

velocity

K* friction velocity

u{a) physical

velocity c o m p o n e n t

in a

direction

V,

W contravariant

velocity

c o m p o n e n t s

W

Coles '

w ake

func t ion

x , y , z physical coordinate direct ions

y

coordinate

di rect ion normal

to

wall (2-D e xample s )

distance

from

wall

in

wall

coordinates

poin t where viscous sublayer in tersects log region,

neglect ing

buffer

region

Z

func t ion

of

Cy,

def ined

b y

equa t ion

(7.9)

Greek Symbols

P func t ion to switch from fourth to secon d order

smoothing

near

shocks

7 ratio

of specific heats, 1.4

7 coordinate

di rect ion norm al

to

wall

7

friction

coefiicient angle

Kronecker delta


19/177

xiv

6

central difference operator for

inviscid

fluxes

boundary layer thickness

6

central difference

operator for viscous fluxes

A

forward difference operator

e

dissipation

rate

of

turbulent kinetic

e n e r g y

^2

se c ond difference smoothing

coefficient

H

fourth difference

smoothing

coefficient

e

total

internal

e n e r g y

per unit

mass

9

circumferent ia l angle

d

velocity

scale

K

V o n

Karman constant,

0.41

A

diagonal matrix containing

eigenvalues

of inviscid f lux Jacobian

M

molecular

viscosity

M O

constant

for

Sutherland's

law

M / i

turbulent diffusion coefficient for static

enthalpy

M (

turbulent viscosity

V

kinematic

viscosity

diffusion coefficient

for

passive

scalar

&

) 7 , C,

T

transformed coordinates

n

constant

for

Coles '

l aw

of

the

wake,

0.5

p

density

0 - e

constant

for fc

e

mode l , 1.3

constant

for A :

e

mod e l , 1 .0

T shear

stress


20/177

XV

T { a / 3 )

physical

shear

stress

components

in a, f 3 direct ions

u

vorticity

Subscripts

e x p

explicit

tensor

indices

imp

implicit

r e f

reference

t

turbulent

V

viscous

w

wall

partial

dif ferent ia t ion

in

physical

coordinate

direct ions

a,/)

tensor

indices

oo

freestream

Superscripts

n

time

level

V viscous

+

wall

variable

+ posi t ive eigenvalue

negat ive

eigenvalue


21/177

xvi

fluctuating

quantity, R e y n o l d s average

fluctuating

quantity,

Favre average

Other

Symbols

backward

difference

operator

gradient

operator

time average

t ensor in t r ansformed coordinates

modif ied b y wall functions

mass-weighted average

vector in

transformed coordinates, conservat ion

law

form


22/177

xvii

PREFACE

AND ACKNOWLEDGEMENTS

What aspect

of

human

nature

drives us

to

unshroud nature's se c re t s? Many

of history's greatest

th inkers

have grappled

with

this

quest ion

and c ome up

e m p t y -

handed,

and

I

do

not

intend to attack it here. However, whatever the motivat ions,

m a n k i n d

has

struggled to

understand his/her

sur roundings ever

since the

d a w n of

civilization.

Those

of us w ho have

chosen

e ng ine e r ing as our

field

of study are fortu

nate in that, a long with the

intellectual

chal lenges i nhe re n t in all

areas

of inquiry, it

often has direct application

to

the improve me nt

of

the

h u m a n

condition.

Of

course,

the

results are not

always

so

rosy.

We therefore must always

keep

in m i n d

the

pos

sib le implicat ions

of

our work,

and

direct ourselves accordingly. W e c a n then enjoy

the e xc i t e me nt

of discovery

to the fullest

extent possible.

I

w ould

like to give

thanks to

Dr. Fletcher

for givin g

m e

the f reedom

to forge m y

o w n path, m a k e

m y

o w n errors, and

learn

m y o w n truths.

I

would

also

like

to

give spe

cial

thanks to

Dr.

L y n n e D e u t sc h

for

her

careful

proofreading (and

ruthless

edi t ing )

of

this thesis, and her extra support and understanding dur ing its complet ion.

This research was supported b y N A S A A m e s R e s e ar ch Ce nte r

contracts

NCC2-476

and NCA 2-526 .


23/177

1

1. INTRODUCTION

1.1 Problem Description

The

understanding

of

turbulence

is

of

critical

importance

for

the

predic t ion

of

flows e nc ounte re d

in

many important e ng ine e r ing applicat ions

such as flow over

flight

vehicles,

i m p i n g m e n t cool ing

in

industrial processes, and the transport

of

atmospheric

pollutants.

In principle, these

flowfields could be predic ted

by solving the full Navier-

Stokes equations.

This

approach

is

not practical, however, s ince

present

computers

do not have the spe e d and m e m o r y

required

to

resolve the wide range

of

length and

time

scales in

most

turbulent

flows.

In

practice,

the

Navier-Stokes

equations are

empl oyed

to

resolve

large scales, and

turbulence mode ls

are

rel ied u p o n

to simulate

the

eff'ects of the small-scale motion .

Turbu le nc e is

diffusive, a n d most

approaches to tu rbu le nc e mode l ing are directed

toward computing the rates

of

turbulent

diff'usion

of

m o m e n t u m

and

energy. Unfor

tunately,

a

ge ne ra l

method

for

determining these

diffusion rates has proven elusive.

Turbulence mode ls

have been d eve loped which

work

well

for

cer ta in

classes

of

flows,

but

their range of applicability is limited. S o m e

models ,

for example ,

work

well for

attached f lows,

but perform poorly

in

regions of separated flow.

A s i d e

f rom the generali ty

of

turbulence

mode ls , another concern

is

the amount

of

comput ing power required

to apply them. Computations of

complex f lows may


24/177

2

require millions

of

grid points and hundreds of hours of CPU

time,

even

on the

fastest

available

computers.

It

is

therefore

important

to

consider

both

accuracy

and

computing

re qu i re me nts

in the de ve lopme nt

and

application

of

turbulence models .

1.2 Historical

Review

1.2.1

Turbulence modeling

The

earliest attempt to analyze the turbulence prob le m is

usually attributed

to

Rey nolds (1 895) .

He was

trying

to expla in the result

of

his f amous

transition

exper

ime nt

in which he showed that

pipe f low becomes turbulent

at

a distinct

Re yno lds

n u m b e r . Bei ng familiar with the

kinet ic

theory

of

gase s , Re yno lds tried

an

analo

gous

approach for

fluid flow,

decom posing ve locit ies into m e a n and fluctuating parts.

W h e n

express ions for the de c ompose d velocit ies

were

substituted into the

Navier-

Stokes equations,

a

set of additional terms

appeared.

These

terms

are the

gremlins

which we n o w call

the R e y n o l d s

stresses,

and the subse que nt

ninety years o r so have

b e e n li t tered

with

attempts

to

f i nd

a

ge ne ra l method of

predict ing

their values.

Since viscous stress

in

a N e w t o n i a n fluid is

a l inear

func t ion

of

the

velocity

gradient,

it

was

hypothesized that

R e y n o l d s

stresses

behave

in

the same

manner.

Unfortunate ly , determining the proportionality constant, the

turbulent

viscosity,

at

first proved

to be as intractable as determining the R e y n o l d s stresses themselves .

In

the

1920s , it was

s h o w n

that

transport

equations could b e

written

for m o m e n t s

of arbitrary

order

( M o n i n

and

Yaglom

1987). However,

each equation

for a specified

m o m e n t con ta ins

the next higher

m o m e n t as

an

u n k n o w n . For example ,

the

equations

for the

R e y n o l d s stresses, which

are

s e c o n d

order m o m e n t s (the correlation be t w een

two

velocity componen ts ) , con ta in third order moments

(the

correlation be t w een


25/177

3

three velocity components) as

u n k n o w n s .

This is the "closure problem" and was a

harb inger

of

difficulties

to

come.

S o m e

headway was

achieved by

Prandtl's "mixing

length" hypothesis.

It is

interesting

to

note

that Prandtl, like R e y n o l d s

before him, turned toward

the

kine t ic

theory

of gases

for inspiration. Ac c ord ing

to the kinetic

theory,

kinematic viscosity

is

proportional

to the

product

of a velocity scale (the rms velocity of the molecules)

and a

length scale

(the

m e a n

free path of the molecules)

(Hinze

1987). Treating

"lumps

of

fluid

l ike molecules,

Prandtl

hypothesized

that

the turbulent

viscosity

is

also

proportional

to the product of a velocity scale

and

a length scale. Unfor tuna te ly ,

the ana logy with molecular

motion

is on

shaky

g r o u n d at best. Molecules retain their

ident i ty , while lumps

of fluid

do not. Also, the length scale

of

molecular

motion is

small c o m p a r e d

to

the overall system, and this is not the

case

for turbulent

fluid

f low

(Te nne ke s

and Lum le y 197 2) . Eve n with

these

weaknesses ,

the

mixing length

theory

has

proven

to

b e

useful for

the

predict ion

of

s imple

flowfields

such

as

free

jets

and

boundary layers on flat

plates.

Its m a i n drawbac k is that the

proportionality

constant

must

be

determined

empirically, and

a

given constant is useful only

for

a

very l imited class

of flows.

A n

approach

very

different

from mixing

length

theory was taken b y G.

I.

Taylor

(1935). Since the Re yn olds s tr esses are

expressed

as correla t ions b e t w e e n fluctuating

c o m p o n e n t s

of

velocity,

it

was

natural

to

apply

statistical

methods

to

attempt

to

flnd genera l express ions

for

these

correlations.

Taylor

de ve lope d this

method for

isotropic and, to a

lesser

de gre e , homoge ne ous

turbulence.

A

great deal

of insight

into

the

m e c h a n i s m s

of

turbulent

e n e r g y

transfer has b e e n gle ane d f rom this work.

Its application to useful turbulencemodels has

b e e n limited,

though, s ince turbulen ce


26/177

4

is not actually isotropic, and o n l y approximates homoge ne i ty

for

certain

very

simple

flows,

such

as

w i n d

tunnel turbulence

behind

a

grid .

Whi le

statistical

methods w ere being de ve lope d , other approaches

to

improving

upon mixing

length theory w ere investigated. One

of

the disadvantages

of

mixing

length models is that

they do not account for "history"

(transport) effects

on the

turbulence. To alleviate this shortcoming,

o n e

o r

more

transport equations

can be

employed . It is possible to derive an exact

equation

for the transport of

turbulent

kinetic

energy,

although

additional

u n k n o w n s

are

introduced in

the

process.

The

n e w

u n k n o w n s

can

b e m o d e l e d , and

the

resulting equation

can be use d

to de duc e a ve

locity scale distribution of the

turbulence.

Specification of a

length

scale

distribution

then closes the problem. If

the length

scale is calculated algebraically,

the

result ing

model is k n o w n as

a "one-equa t ion

model," s ince o n e partial differential equa t ion

is

employed .

One-equa t ion

mode ls

yield

better

results

than

mixing

length

mode ls

for flows in

which convect ion

and difl 'usion of

turbulent

kinet ic e ne rgy are

important

( L a u n d e r

et

al.

1972). For many complex flows,

however , algebraic specification of

the

length

scale c a n be difficult. The next logical step

w ould

therefore be to develop a transport

equation for

length scale, or a

quantity

which can be easi ly

related to

a l ength scale.

This equation, a long

with

the turbulent

kinetic

e ne rgy

transport equation,

yields a

two-equat ion mod e l .

The

second

equation

is

usually

written

for

the

rate

of

dissipat ion

of turbulent kinet ic energy,

e,

although other quantities are

sometimes

used, such as

the length scale, L, (Rodi

and Spalding

1970) ,

the

rate of dissipation per uni t energy,

w , (W ilcox 1988) ,

and

the time scale, r

(Abid, Speziale, and

Thangam

1991) .

T w o -

equation

m o d e l s came to

the

forefront upon publ icat ion of

a

series of papers from

Los


27/177

5

A l a m o s Scientific Laboratory (Harlow

and

N akayama 1967;

Harlow

a n d

N a k a y a m a

1968;

Daly

and

Harlow

1970) . Derivat ion

of

the

second

equation

is

not

as

rigorous

as that

of

the turbulent kinetic energy equation, a n d this

is

o f t e n cited as a poin t

of

w eakness

of two-equation

models.

Even so, calculation of

the length

scale as part of

the

mode l has

proven to b e advantageous for

m a n y

flowfields.

Daunted

b y

the prospect

of

so lving

the

comple te se c ond-mome nt equations

and searching for a method

to

improve the pe rformanc e

of

two-equat ion

models ,

Rodi

(1972)

invest igated

the

possibi l i ty

of

simpli fy ing

the

se c ond-mome nt

equations.

He

deve loped

an algebraic

expression for the

R e y n o l d s

stresses as a

func t ion of the

dependen t va r iab les in

his

two-equa tion mod e l ,

and the mode l

is

therefore referred

to as an

algebraic

R e y n o l d s

stress

model. Since the n e w equation is algebraic,

little

computational

effort is required a b o v e that for the two-equat ion model . Although

alge braic Re yn o lds stress

m o d e l s

show promise,

they

have no t exhibi ted the

expec ted

improvemen ts

over

two-equat ion m ode ls

(Ferziger

1987).

Other

variations

of two-equation mode ls have

also

b e e n i nves t iga ted . On e weak

ne ss

of

two-equa tion m ode ls is that a single velocity

scale

and a

single

length

scale

are assume d

to

be

sufficient to

describe the turbulence . This implies that the e ne rgy

spectrum is

similar

in different regions

of the

fiowfield, which is not general ly

true.

In

"mult iscale"

two-equat ion models, the energy spectrum is

divided

i n to

two

parts

(Launder

1979).

The

first

is

the

production

range ,

which

is

the

region

of

highest

energy. The se c ond

is

the transfer range , w here the e ne rgy

is t ransferred

from large

scales

to small

scales . Separate k and e

transport

equations

are written for each

range .

Mult iscale

m o d e l s have

s h o w n

improvemen ts over standard two-equat ion models for

flowfields

such as flow over a backward-facing

step (Kim and C h e n

1989)

and

swirl ing


28/177

6

jets

(Ko and

Rhode

1990). The results are not consis tent ly better,

however,

and a

significant

increase

in

computer

power

is

required

due

to

the

addition

of

two

transport

equations.

Another variation of two-e quat ion mode ls is the

"non l inear"

m o d e l . I n s o m e

flowfields, anisotropy

of the normal

turbulent

stresses

is

important.

A n example

of

this is the secondary

f low observed

to occur

in

turbulent f low through straight

rect

angular channe ls . Since the Boussinesq

approximation

d o e s not admit anisotropy of

the

normal

turbulent

stresses,

it

is

impossible

to

predict

the se se c ondary

f lows

with

the standard model. In

n o n l i n e a r

m o d e l s

(Speziale

1987; Yoshizawa 1988; Barton,

Rubinstein, and

Kirtley 1991), the

Boussinesq

approximation

i s replaced

b y a

n o n l i n

ear func t ion of the m e a n strain

rate.

This method is not restr icted to two-equat ion

mode ls , but can be applied to

other

mode ls which

utilize

the Boussinesq

approxima

tion

(e.g. ,

algebraic models) . Initial results from these models look promising, but

m o r e

applications

n e e d

to

be

invest igated

before

their

value

c a n

b e

fully

evaluated.

A s m e n t i o n e d

earlier,

the closure problem precludes the solut ion of the trans

port equations for

correlations

b e t w e e n fluctuating

velocity componen ts . Also,

these

equations contain

terms

such as pressure-velocity correlations,

which

are

general ly

u n k n o w n . C h o u (1945) made various assumptions about the u n k n o w n quantities

in

the se c ond and third

m o m e n t

transport equations

in

order to

close

them, creating

what

is

n o w

referred

to

as

a

R e y n o l d s

stress

transport

mode l .

A n

advan tage

of

this

type of m o d e l is that the Boussinesq approximation

is

not employed . Although the

Boussinesq approximation is

effective

fo r many types of flows, it is

k n o w n

to be in

accurate

for s o m e f lowfields such as wall

jets.

Chou's model laid fairly dormant for

many years , because means for solvin g the equations

for

ge ne ra l

cases

were not avail


29/177

7

able . A s computer s came

into

prominence and

i mproved in

capability,

greater

efforts

were

put

into

the

de ve lopme nt

of

R e y n o l d s

stress

models .

These

mo dels require

a

great dea l

of computational effort,

and they do not present ly yie ld results which are

general ly

better

than

two-equat ion

models . A s they are further ref ined, it is

expected

that

they

will c ome into

greater

use in the future.

The goal

of

all techniques discussed so far is

to compute the

R e y n o l d s

stresses.

The R e y n o l d s stresses represent momentum transfer averaged

over

a

w i de range of

scales.

If

a

flowfield

is computed

us ing

a

very

fine

grid,

large-scale

structures

c a n

b e

resolved, a n d only the

momentum transfer occuring

at smaller

scales n e e d s to b e

mode le d . Since the

required

m o d e l

represents

a

subset of

the full

range of

scales,

it

c a n be

simpler

in

fo rm

than models

which

represent

the full

R e y n o l d s

stresses.

This

approach is

cal led

"large eddy simulation." The disadvan tage of

large

e d d y

simulation

is

the great amount

of

computer power required to

run

with such

a

fine

grid .

This

method is

therefore

presen t ly cons t rained

to

relatively

simple

fiowfields.

In theory, a

grid

could be constructed which is fine e nough to resolve the full

spectrum

of

scales

e nc ounte re d

in

turbulent

motion, obviat ing

the

n e e d

for

a n y turbu

lence m o d e l at all.

This

approach, "d irect numerical simulation,"

has

b e e n applied to

very s imple geometries at low

turbulent

Re y no lds n umb e rs (e .g . , Rai and M o i n 1989) .

Since

a

doub l ing

of the turbulent R e y n o l d s

n u m b e r

requires

an

orde r-o f -magni tude

increase in

computer capability

(Yakhot

and

Orszag

1986),

it

will

not

b e

possible

to use direct simulation to solve "real world" prob le ms

in

the

near

term future.

It

has b e e n estimated

that if

a

terraflop

(1 0^^ floating point operations per

se c ond)

mac hine

were available, several hundred

thousand

years of

CPU time

would still

b e

required

to

compute a

direct simulation

of f low over an entire aircraft (Peterson

e t


30/177

8

al. 1989) . This would prove

to

be a major a n n o y a n c e to typical computer system

managers,

and

is

therefore

untenable.

E v e n

so,

present

direct simulation results

are

valuable for studying the

detailed structure

of turbulence.

Quantities

which are

not

me asurab le

c a n

b e

extracted

f rom the simulation

results,

and this

is

an excel lent

wa y

to

check

details of

turbulence models.

1.2.2 Near-wall modeling

A s solid walls

are

approached,

the

structure

of

turbulent

flow

c hange s

due

to the

increasing

importance

(and

eventual dominance) of

viscous effects. M a n y turbulence

mode ls

have

b e e n de ve lope d

with

the

assumption

that the flow

is

fully turbulent

(i.e.,

far from walls), and they require additional attention in order to m o d e l wall regions

correctly.

A n

early

near-wal l

m o d e l

which

has proven quite useful , and of ten appears today

i n m a n y

guises, is

that

of

V a n

Driest

(1956). V a n

Driest

was

looking

for

a

wa y

to

modi fy

the

Prandtl mixing length to account for damping of turbulent e dd ie s near

walls.

He n o t e d

that in

Stokes' solution

for

flow

over

an

oscillating

flat plate, the

amplitude of

m otion fa l ls

off exponentially

with distance from the

plate. This

funct ion

m a y b e interpreted as quantifying the region

of

viscous

in f luence .

V a n Driest used a

similar funct ion to damp the mixing length

near

walls, since turbulent effects decrease

as

viscous effects

increase .

A s

more

complex

turbulence

models came

into

use , new approaches

to m o d e l i n g

near-wal l

behavior were

required.

M o s t

of these near-wal l mode ls

attempt

to approx

imate the

effects

of

anisotropy,

which are neglec t ed

elsewhere in

the

f lowfleld.

These

models

are somet imes referred to as " low-Re yn o lds -num be r m ode ls ," s i nc e they c ome


31/177

9

into play in regions of low turbulent R e y n o l d s number. Harlow and N a k a y a m a (1967)

presented

a

tentative

anisotropy correction

to the

turbulent

kinetic

e ne rgy i n

their

two-equation model, but

they

showe d

no

results. Daly and

Harlow (1970)

used a

"wall-effect

tensor"

to modify the

fluctuating pressure/strain

rate

correlation term in

their R e y n o l d s stress model . They

show ed

that this

term drove the peak turbulent

kinetic en ergy c loser

to

the wall as the R e y n o l d s number increased, which is in accord

with experimental

data.

A di fferent

approach,

"wall

functions,"

was

applied

b y

Patankar

and

Spa ld ing

(1970). They

r easoned that

equations descr ibing

the

structure

of

turbulent boundary

layers, e.g.,

the

law of the

wall,

could be

coupled

with numerical

solution

schemes,

thereby eliminating the n e e d

to

resolve the turbulent boundary layer in

the

region

where

anisotropy is

important.

This

technique is limited b y the accuracy

and

range

of

applicability

of the

equations

employed .

A n

early

two-equation near-wall

model

which

has

b e e n

quite

inf luent ial is

that

of

Jones and Launder (1972). They interpreted

the e transport

equation as mode l ing

only the isotropic part of the dissipation rate. Using asymptotic analysis , a

term

was

added

to the

turbulent

kinetic

e ne rgy transport equation

to account,

for anisotropy

of the dissipation rate. Damping

functions

were employed for

several

terms in the

e

equation, and an ad-hoc term was

added to

bring the maximum

level of

turbulent

kinetic

e ne rgy

into

line

with

experimental

data.

De ve lopme nt of both

wall

functions and low-Re yn o lds -num be r mode ls has

con

tinued

in parallel. Chieng and

Launder

(1980)

refined

the computation of the

wall

shear stress, and their approach has b e e n i m p le m e n te d b y many investigators.

Their

method was

further

general ized

for

compressible , separated

flow

b y

Viegas,

Rubesin ,


32/177

10

and Horstman (1985). Chien (1982) took an approach similar

to that

of Jones and

Launder

(1972)

to

create

a

low-Reynolds-number

model

which

has

ga ined

wide accep

tance. There has b e e n a great

deal

of

activity

in recent y ears in

the

de ve lopme nt of

improve d low-Re y no lds -numbe r models, usually base d on asymptoticanalysis .

A use

ful comparison of eight of

these

models is

given

b y

Patel,

Rodi, and Sheuerer (1985),

whe re it

is

conc luded that even the best per forming m ode ls n e e d more development

if they are

to

b e used with confidence . Avva, Smith,

and

Singhal

(1990)

directly

compared

results

of

wall

functions

and

a

c o m m o n

low-Re yno lds -numbe r mode l

for

three two-dimensional flowfields, and f o u n d

that wall

functions gave comparable or

better results in all three cases.

The best choice be twe e n the two techniques

has yet

to

be conclusively deter

m i n e d . Wall

functions yield

g o o d

results for many

problems,

and they require

less

computer power

than low-Reynolds-number mode ls .

Low-R e yno lds -numbe r mode ls

have

the potential to

b e

more

general and

to

give

better

results

for

s o m e

flowfields,

but that potential

has yet to

be

demonstrated.

Both approaches will most

likely

con t inue

to be used

in

the future.

1.3 Scope

of

the Present Research

O n e disadvantage

of

wall functions is

that

they are

difficult

to apply to complex

geometries .

Early applications

general ly

involved

two-dimensional

flows

over

flat

surfaces such as duct flows, backward-facing steps, and

compression corners .

C o m

putation

of

flow

over complex

three-dimensional geom etries is

n ow

c ommonplac e , but

a

method

for

applying

wall

functions

to

these

geometr ies

has

not

b e e n

available .

In

the present work,

a

method has b e e n

de ve lope d

for

the application of wall

funct ions


33/177

11

to three-dimensional

general ized curvilinear coordinates

with

nonorthogonal grids.

A

high-Reynolds-number

k

-

turbulence

mode l with

the

n e w wall

function

for

mulation has b e e n added to F3D,

a

Reynolds-averaged compressible Navier-Stokes

solver.

F3D utilizes an implicit, partially flux-split, two-factor approximate factor

ization

algorithm, and the ke model utilizes an implicit, fully flux-split, three-factor

a p p r o x i m a t e f a c t o r i z a t i o n

a l g o r i t h m .

T h e

C h i e n ( 19 82 ) l o w - R e y n o l d s - n u m b e r k ~ e

m o d e l

has

also b e e n

added

for comparison with

the

wall function formulation. F

3 D

contains

the

Baldwin -Lomax

(1978)

algebraic

turbulence

model ,

which was

also

run

for comparison with the present method.

The n e w wall function technique was applied to a series

of

test cases.

First,

flow

over

a flat plate was computed

using

two

difl^erent

grids, one which is orthogonal

and o n e which is

skewed at

the

wall,

to

test the

non or thogonal g r id capabilities of

the present formulation.

For

these cases,

the computed

friction

coefficients were

compared

with those

from

a

semi-empirical

equation.

Velocity

profiles were

also

compared

with experimental

data.

Flow

over

a

body of revolut ion at zero ang le

of

attack was then computed

to

show the method's effectiveness for f low

over

a

curved surface. Friction

coefficients

and

velocity profiles w ere compared with test data.

The same

case was

also

computed

using

the

Chien (1982) low-Re yno lds -numbe r k

model and the Baldwin-Lomax

(1978)

algebraic

m o d e l

for

comparison.

Each

of

the

cases

was

run on three

grids

with different

wall

spacings, demonstrating the advan tage

of wall

functions for coarse

grids.

Finally ,

f low over

a prolate spheroid at angle of attack was computed using the

wall function

formulation,

and

results

were compared

with test

data.

This

d e m o n -


34/177

12

strated the

effectiveness

of the

wall

funct ion formulat ion for a complex flowfield

with

regions

of

separated

flow.


35/177

13

2.

CONSERVATION

OF MASS,

MOMENTUM,

AND

ENERGY

2.1

Introduction

For

turbulent

flows,

it

is not

possible

to

solve

the

equations

of

motion

numerical ly

due

to the i m m e n s e

computer power which

would b e required to resolve the wide range

of l ength scales . In order to make the problem tractable, the

equations

are averaged

in time, introducing additional

u n k n o w n s .

The

additional unknowns ,

which

represent

turbulent transport of momentum and energy, are then mode le d us ing a combination

of analysis and empiricism. In

this

chapter, the

technique

for

averaging fluctuating

quantities

is

presented,

and

it

is

then

applied

to

the

equations

of

motion.

2.2 Instantaneous Equations

The

working

fluid is assumed to b e a homoge ne ous continuum,

and therefore

may not contain voids or particulates. It

is

also assumed that

the

fluid is Newton ian ,

i.e.

that the stress is proportional to the rate of strain. Stokes ' hypo thesis that the

bulk

and

molecular viscosities

(A

and

i x

respectively)

are

re la ted

by

the

equation

A = (2/3))U is employed. Finally , buoyancy and other body forces are

neglected.

Given

these

assumptions, the equations of conservat ion of mass per

unit volume.


36/177

14

m o m e n t u m p e r

unit volume, and

total

enthalpy

per unit

volume are given

by

It +

(2.1)

d

^(m)

+ + i j p

-

j )

=

o

(2.2)

-p)+ -^{pujH + q j-i T i j ) = 0 (2 .3)

where

'y

=

(

~

This

is a system of five equations with

seven

unknowns , and must

be closed with

the

aid

of an

equation

of state and an

expression for

molecular

viscosity.

The

fluid

is

assumed

to b e a perfect gas,

p

=

p R T (2.5)

where temperature is related

to total

enthalpy b y

H

=

h+^{u{ui) (2.6)

= CpT+i(jtii)

(2.7)

The molecular v iscosity

will

be calculated from Sutherland's La w (White 1974),

S) frl

where for

air,

f j

, Q

=

0.1716mP,

Tg

=

491.6i2,

and S

=

199i.

2.3 Averaging Techniques

Following

Reynolds' approach

toward dealing

with

turbulent

f low, values of

ve

locity and fluid

properties

are

de c ompose d

into m e a n and fluctuating parts. There are


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16

1

/'(+A(

1

rt+d

= TtJt '' Sk

= ( j ) - < j )

= 0

(2.13)

For mass-weighted averages,

1 _ 1

/ " i+A^

=

p ( l ) - p 4 >

rt+At 1 rt+At ~

it

At

J t

From

equation

(2.10),

p c p

=

so

p(()"

=

0

(2.14)

It

is important to note that < / > 7 ^

0.

2.4 Mass-Averaged Transport Equations

The

variables

in the

continuity,

momentum,

and

energy

equations

will

no w

b e

d e

c ompose d

into

average and

fluctuating

components, and

the results

averaged.

Equa

tion

(2.11) will b e used

to

decompose p, p, and r,

and equation

(2.12) will be used

for

U j and H.


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20

R e y n o l d s

stresses. Unfortunately,

these

equations for se c ond-orde r moments (aver

ages

of

products

of

two

fluctuating

quantities)

contain third-order moments,

which

are

also u n k n o w n . Indeed, transport equations for any moments will contain terms

with higher-order moments. This is

the i n famous "closure problem."

In addition to the R e y n o l d s stresses, there is an additional u n k n o w n quantity in

the e ne rgy equation

(2 .34),

pu'jh". This term represents the transport of e ne rgy by

turbulent

velocity

fluctuations.

Both

of

the

u n k n o w n s

will

be

computed from

a

turbulence model, thereby

re

establishing a closed

set of equations.


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24

3.2 Turbulent

Kinetic

Energy

Transport

Equation

The

momentum

equation (2.2)

may be rewritten, replacing the

subscript i with k-.

+hp-

%)

=

0 (3-14)

Now,

multiplying equation

(2.2)

b y equation (3.14) b y u ^ - , adding

the

results,

applying the continuity equation

(2.1),

and

simplifying, yie lds the m o m e n t of

m o

mentum

equation:

d d d p 7

k

d p n

j r

gilm'-k)

+ +"'T^ - ilij=

(^.w)

This equation

wil l now be averaged.

A f (>+" i ) (%+4) (%+ j)

J

+k+40+(.+


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28

This is the

mean density times

the

dissipation

rate,

ctM.

=

-a

^

d X n

pe

(3 .30)

Finally,

applying equations (3.1), (3.25), and (3 .30)

to

equation (3 .22) y ie lds the

mode le d turbulent

kinetic

energy

equation:

IW + A

d u i duj

J

3 ^

dxj^

2r

fijpk

m

d X r i

- p

(3.31)

The terms on the left hand side represent the

total

rate

of

change and rate

of convec

tion

of

turbulent kinetic e ne rgy per unit volume. On

the

right hand side

are

the rate

of

diffusion,

rate of

production, and

rate of

dissipation

of

turbulent kinetic e ne rgy

per unit volume.

3.4 Modeled Dissipation Rate Equation

A n exact equation for the dissipation rate

of

turbulent kinetic

energy may

be

derived f rom

the momentum equations. This is most

c o m m o n l y

carried out b y assum

ing

incompressible flow (e .g. Harlow and N akayama 1968) ,

although

it

has

also

b e e n

d o n e for

compressib le flow

(El Tahry

1983).

A n order-of-magnitude analysis of the

incompressib le equation reveals that

two

terms are

of much

greater order

of magni

tude than

the others, even though the

difference

be t w een

these

two terms

is expec ted

to

b e small (Launder 1984). This situation make s it extremely difficult

to

solve the

equation numerical ly . To m a k e

matters worse,

both terms consist

of

unmeasurable

quantities, so e v e n

if

they could be computed

accurately, it

w ould be impossible

to

compare the results with test data. Rather than try

to

mode l the exact equation.


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29

a

more

heuristic

equation

is normally used, which mimics

the form of

the turbulent

kinetic

e ne rgy

transport

equation.

This

equation,

including

the

compressible

terms,

is (Coakley

1983)

I

^ =

-]

+Ci

k

du

(3 ,32)

Ci and C2 are empirically determined constants.


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31

the stiffness problem encountered with

the

low-Re yno lds -numbe r

models ,

and for

some

flows

yie lds

good results.

However,

since

simpler

models

are

use d

near

the

wall,

the c onc ominant disadvantages

of

these models, such as the n e e d

to

specify a length

scale, are encountered. They also require relatively f ine grid resolution.

A

third

approach,

the one to be

further deve loped

here, is the use of

wall func

tions. Wall functions are base d

on the idea

that the basic structure

of

turbulent

boundary layers has b e e n

well

established. Before

discussing

this structure, some

defini t ions

are

required.

In

the

equations

throughout

the

remainder

of

the

present

work, all ti ldes and overbars are dropped except for those indicating correlations b e

tween fluctuating quantities. A n appropriate

velocity

scale

for f low

in the near-wall

region

is the

friction

velocity,

de f ine d b y

where

tw

is

the

wall

shear stress

and

p w

is

the

dens i ty

at

the

wall.

Using

this

velocity

scale,

a nondimensiona l velocity and

a

nondimensiona l

length

are

def i ned by

where

u

is

the

velocity

component parallel

to

the

wall,

y

is

the

distance

normal

to

the

wall, and v is the kinematic viscosity.

A

typical turbulent boundary layer velocity

profile,

similar to

that s h o w n in Anderson,

Tannehill,

and Fletcher

(1984), i s shown

in Figure

4.1. The equation which

descr ibes

the velocity

profile in

the

log region is

(4.1)

(4.2)

and

,+

=

i

(4.3)

^ln{y'^) - f - B

(4.4)


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33

adjacent

to

the wall may be

placed

well aw ay

f rom

the

wall,

and the shear stress

inferred from

the

velocity

at

that

point.

The use

of

wall functions has several advantages. The other techniques m e n

tioned above

require that

the entire

boundary

layer be resolved.

The

grid point

adjacent

to

the

wall

must therefore b e located in the

viscous

sublayer , typically at

a

of

less than five. For wall functions, the first grid point is normally located in

the lower part

of

the

log

region, at

a of

approximately 40

to

100. Gi ven

that

the

rate at

which

the

grid

may

stretch

aw ay

from

the

wall

is

limited

by

most

numerical

solution schemes, wall functions result in a

large saving

in the

n u m b e r of

grid points

and the

amount

of computer memory required. Viegas and R u b e s i n (1983) found

that approximately half as many grid points w ere required when using wall

funct ions

as compared

to low-Re yno lds -numbe r models. Since the minimum grid

spacing is

muc h larger for the

wall

function

case, a

larger time step

m a y

be used for a

given

Courant

number

(for

steady

state

computations),

resulting

in

further

saving

in

CPU

time. The re duc e d me mory and CPU required w h e n using wall functions can b e

extremely

important for

the computation

of complex three-dimensional

flowfields.

One disadvantage

of

wall functions

is

that

the log

equation is not accurate

for

some

flowfields,

such as those with regions of separated f low. Also, the standard

wall

function

formulation requires the

assumption

that

the turbulence

is in

equilibrium

at

the

first

grid point

aw ay

f rom

the

wall, which

is

not

always

the

case.

E v e n

with

these

limitations, wall functions have

b e e n

shown

to yield

results comparable, and

often

superior,

to those obtained with low-Re yno lds -numbe r m ode ls , inc lud ing com

putations

of

some complex

flowfields

(Chieng and Launder 1980;

Viegas

and

Rubesin

1983; Viegas, Rubesin, and

Horstman

1985; C h e n

and Patel

1987; Awa, Smith, and


56/177

34

Singhal

1990).

Gi ven

the

advantages

and

disadvantages

of

each

method

described above ,

wall

functions

will

b e

pursued in greater

detail.

4.2 Detailed Formulation

4.2.1

Introduction

The first step in applying wall functions is

to

compute the

friction

velocity and

the wall shear stress. The friction velocity is then used to set the boundary conditions

for

k

and

e

at the grid

point

adjacent to the wall .

Finally ,

the wall shear

stress

is

used in the

computation of

the diffusion term in the

Navier-Stokes

equations at the

grid point

adjacent

to the wall . Development of

a

genera l

method for applying the

wall

shear

stress to the Navier-Stokes equations

in

general ized

curvilinear

coordinates

with

nonorthogonal

grids is the primary contribution

of the present work.

4.2.2

Friction velocity

Substituting

equations (4.2) and (4.3)

into

(4.4) and (4.5) gives

= (4.6)

k

\J

for

the log

region and

2. = 2 (4.7)

u

for the

viscous

sublayer .

Using

u

and v from the previous

time

step, the friction velocity can b e

calculated from equation (4.6) or (4.7). If the grid point adjacent

to

the wall falls in

the log region,

equation

(4.6) is

solved

using an iterative scheme such as

N e wton ' s


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60

Before writing

the

source term Hf in vector form, it is useful to

define

a symbol

for

a

term

which

is

proportional

to the

production

of

turbulent

kinetic

energy,

P,

po1

=

R e

dxi 3 dx^

xr ,

d u j

d x o

(6.27)

Now

Hf may be

written

as

H f

=

V e

(6.28)

6.3 Coordinate Transformation

The

standard transformation

to generalized

curvilinear coordinates

(e.g. An

derson, Tannehill and Fletcher 1984)

will be

employed here. Transformed

time

is

identical with physical time, and transformed spatial coordinates are general func

tions

of

physical

spatial

coordinates and time,

T t

^

=

((r,y,

V = nix,y,z,t)

C

= C i x , y , z , t )

The Jacobian

of

the transformation is

r_d{^,V,0

d{x,y, z )

(6.29)

(6.30)

(6.31)

(6.32)

(6.33)

Partial

derivatives

of quantities

with respect to physical

coordinates may

beexpressed

in

terms of

transformed coordinates through the chain rule. Letting ( j ) represent a


83/177

61

variable to be differentiated,

< t > t =

x = + ^7x^77

+

Cx4> ( ^

^2/ = +

% < ^ 7 7

+ C y ( f > (

< l > z = +

Vz< l> r ) +

Cz (

(6.34)

(6.35)

(6.36)

(6.37)

Applying the chain rule

to

equation

(6.32), dividing

by the Jacobian, and rearranging

and cancelling terms results in the transformed set of equations.

d Q

d

,

dF

, d G _i fdv

d F v

d G v

+ +

T

=0

where

Q

=

= J ~ ' ^

P

p u

p v

p w

S

p U

p u U

+ ^ x P

p v U

+

^ y p

p w U + ^ z P

{ + p ) U

-t P

(6.38)

(6.39)

(6.40)


84/177

62

f

=

G

= J-^

^ j-l

E v

=

Fv =

pV

puV +

pvV + rjyp

pwV + -qzP

{ +p)u-r]tp

pW

puW + CxP

p vW

+

C y P

pwW + CzP

{ +p)U-Ctp

0

^ x T x x

+ y ' ^ x y +

^ x T x y

+

y ' ^ y y

+

^ z ^ y z

xTxz

+ y'^yz +

M+W5+W

0

V x T x x +Vy'^ 'xy +V z ^ x z

T j x T x y +

V y ' ^ y y

+V z ' ^ y z

V x ^ x z

+

H y ' ^ y z

+

j z ^ z z

V x e ^ + yf^ + z g ^

(6.41)

(6.42)

(6.43)

(6.44)


85/177


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64

/ g =

U T x y

+V T y y

+

W T z y

+_L_

(JL

+

ILI

(7-1)

\ P r ^

P n ,

5 5 = U T x z +V T y z +

V l T z z

+ jl BI

(7-1)\Pt Pvt.

and

the contravariant

velocities are

j

(53)

+

(6.54)

U

=

+

+yu + ^ z v }

V = } f ; + r i x u + } y v +

j z W f

W , i +

(xu

+C y v +

C z W f

The

transformed turbulence

transport equations are

dQ^ d^ dFi d^ n_

1

( tv .

where the

dependent

variable vector is

Qt

= J ^

The flux

vectors

are

f

=

J-1

F t

=J-'

7-1

t = J

pk

pe

U p k

U p e

V p k

V p e

Wpk

W p e

( 6 . 5 5 )

( 6 . 5 6 )

(6.57)

=

H f

( 6 .58 )

( 6 . 5 9 )

(6.60)

(6.61)

(6.62)


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67

were neglected ,

and

viscous

terms

were retained

only

in the direction normal

to

the

wall

(thin-layer

assumption).

Here , viscous

terms are

retained in all three

directions,

with all

cross-derivative terms neglected .

In the flux-split direction (^), the viscous

terras

cannot be l inearized if

a tri-diagonal solver is to

be employed,

so

they are

i nc lude d on ly

on

the

right hand side.

The

flux vector splitting method of Steger and Warming (1981) is used for con

vect ion

terms

in the

^

direction. The ^

direct ion inviscid Jacobian is

therefore split

as

follows,

ii++i- ( 6 . 7 4 )

where

+ =T+f (6.75)

and

-=f-f-l

( 6 . 7 6 )

"^

is

a

diagonal

matrix

containing the

positive eigenvalues of

A j ^ ,

and ""

contains

t h e

n e g a t i v e e i g e n v a l u e s .

T h e

c o l u m n s ofT a r e

t h e

r i g h t

e i g e n v e c t o rs o f ^ .

Substituting

difference

operators,

adding

smoothing, and applying

approximate

factorization to equation (6.73),

{/

+

At [V(i+)

+St-C"

-

X {/+ Ai [A(i-) + ji" Re-'-

}

AQ

= At [V

j

(b+)" + Afl-)"+ S , , F +

-Re-'-(jJ

+

-

exp>,

+^expc) ]


90/177


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70

computationally

efficient

(Shih and Chyu 1991). The

resulting

equation is

{/ [ l+At

( V ^ C / + +

-

tRe-^

X {/ [ l +

At

{Vr,V++ ArfV~y] -

tRe'^

[ S r j (j-'^ATp^Srjj)]}

X {/[l

+

At -

tRe-^

[3^

^AtD^ Qi

= -A({[V ( +) + +V;;(f+) +

C , { ^ t Y

^ C , { ^ t ) + 7 ? % + - r }

( 6 . 8 7 )

The ^ and r j

direction

factors require

solution

of uncoupled

equations, since

there

i s n o source term Jacobian present.

A

specialized solver was therefore written for

b a n d e d tridiagonal

matrices to

enhance computation speed. In the (

direction,

a

block solver is required due

to

the source

terms.

Since two-by-two blocks m a y

b e

inverted algebraically, a second specia l ized solver

was written for

the

C direction.

Details of both

of

these solvers are given in Appendix B .

A n

additional consideration in the

solution

of these

equations

is the possibility

of

obtaining negative

values

for k

and e .

Negat ive

values

are physically

impossible ,

but

are

admitted by

the mode le d

transport

equations.

Lower

limiters

are therefore

used

for

both

equations. Af te r

each

step, a n y values of p k

or

pe that are below

the

specified limit are bumped up to that limit.

The

f reestream values of

p k

and pe are

used for

the

limiters, since the physical (as opposed

to

numerical) values are

not

expected

to

fall be low those in the freestream. O n e run was started f rom scratch (all

dependent

variables set

to

f reestream

values) with the limiters

turned off,

and

it

was

unstable. Addition

of the

limiters solved the

problem. Af te r the

solution

had partly

converged,

the limiters were

again turned

off, and the solution remained stable.

The


93/177

71

starting transient had caused

the unphysical

values in

the

first

case,

and onc e the

solution

settled

d o w n ,

the

limiters

were unnecessary.


94/177

72

7.

RESULTS

7.1

Introduction

A formulation designed for general ge ome t r ie s an d

skewed

grids must also work

on

simple geometr ies and orthogonal grids. The ubiquitous flat plate

has

therefore

b e e n chosen

for

the first set of

test cases.

First, turbulent flow over a semi- inf in i te

flate plate

was

computed on

an

orthogonal grid. The

next step

was

to verify that the

formulat ion is effective

w h e n

the

grid is

skewed at the wall, so

the same flat

plate

was

solved

with

a skewed

grid.

The

tensor

transformations

in

the

present

formulat ion

contain functions

of

all

of the metrics. When any of the general ized coordinate directions are orthogonal to

physical coordinate

directions, some

of

the metrics

will

b e ident ical ly equal

to

zero.

One

f inal f lat

plate test c ase was run with the entire

d o m a i n

at a thirty

degree angle

with the physical coordinate system (at zero angle of

attack)

to br ing

additional

metrics

into play

and further

test

the method.

O n e

additional

flat

plate test

case

was

run

to

verify

the

coding

of

the

Chie n

(1982) low-Re yno lds -numbe r model.

After having verified the present formulat ion for flat surfaces,

the ne x t

step

was

to solve a

well-behaved (non-separated)

f low over a curved surface, Ramaprian,

Patel, and Choi

(1978, 1980) took

measurements of the turbulent fiowfleld

around a


95/177

73

b o d y

of

revolut ion

at

both zero and fifteen degree angles

of

attack. The zero degree

(axisymmetric)

flowfield

was

computed

using

the

present

wall

function

formulat ion

as

well

as the Chien

(1982)

low-Reynolds-num ber mode l

and

the Baldwin -Lomax (1978)

algebraic

model ,

and the

results compared

to

the

measurements.

Each

computation

was

carried

out

using grids

with three different wall

spacings

to determine the grid

sensitivity of each model .

Finally, the present

wall function

formulat ion was tested on a complex three-

dimensiona l flowfield. Separated flow

over

a

prolate

spheriod at

ten

degrees angle

of

attack was computed, and the results

compared

to the measurements of

Krepl in ,

Vollmers, and Meier

(1982).

7.2

Flat plate

The first

test

case

is

the

computation of

incompressible , turbulent flow over a

semi- inf in i te

flat

plate

with

zero

pressure

gradient.

The

freestream

Mac h

number

was set to

0.2

to

minimize

compressibility effects

without getting

too close to the

incompressible limit

of

the solver. The Reynolds

number

base d on

freestream

velocity

was

1

X

10 at the

upstream

boundary , and 8 x 1 0 at the outflow boundary . The

reference length

was def ined such

that the distance f rom the virtual

origin of

the

boundary layer to

the

upstream boundary

was

o n e unit of length. All

lengths

were

normal ized

b y

this

distance.

The

streamwise

and

normal

coordinate

directions are x

and

z in

physical coor

dinates

and

j and ( in

transformed coordinates. Wall functions

are expected to

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Wall Functions for the k - [Epsilon] Turbulence Model in Generali

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