Three-Dimensional Time Dependent Computation of Turbulent Flow · 5.6 Simulation of the...

Three-Dimensional Time Dependent

Computation of Turbulent Flow

D. KwakW. C. Reynolds

andJ. H. Ferziger

Thermosc fences DivisionDepartment of Mechanical Engineering

Stanford UniversityStanford, California

May 1975REPRODUCED BYNATIONAL TECHNICALINFORMATION SERVICE

U.S. DEPARTMENT OF COMMERCESPRING*IEID. VA. 22161 :

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THREE-DIMENSIONAL TIME DEPENDENT

COMPUTATION OF TURBULENT FLOW

by

D. Kwak, W. C. Reynolds, and J. H. Ferzlger

Prepared from work done under Grant

NASA-NgR-05-020-622

Report No. TF-5

Thermosciences DivisionDepartment of Mechanical Engineering

Stanford UniversityStanford, California

May, 1975

ABSTRACT

The three-dimensional, primitive equations of motion have been solved

numerically for the case of isotropic box turbulence and the distortion of

homogeneous turbulence by irrotational plane strain at large Reynolds

numbers. A Gaussian filter was applied to governing equations to define

the large scale field. This gives rise to additional second order computed

scale stresses (Leonard stresses). The residual stresses are simulated

through an eddy viscosity. 16x16x16 and 32x32x32 uniform grids were used,

with a fourth order differencing scheme in space and a second order

Adams-Bashforth predictor for explicit time stepping. The results were

compared to the experiments and statistical Information was extracted

from the computer generated data.

ill

ACKNOWLEDGMENTS

The authors gratefully acknowledge the useful comments of Drs.

A. Leonard, M. Rubesin, H. Lomax and Mr. S. Shaanan. We appreciate

Ms Nancy Silvis for the speedy typing of the final copy.

This work was sponsored by NASA Ames Research Center under Grant

NASA-NgR-05-020-622.

IV

TABLE OF CONTENTS

Page

Abstract iAi

Acknowledgments . *v

Table of Contents v

List of Figures 'v±±

Nomenclature ix

Chapter I. INTRODUCTION 1

1.1 Historical Background 1

1.2 Motivation and Objectives , . 2

1.3 Summary of the Results 2

Chapter II. THEORETICAL FOUNDATIONS 4

2.1 Definition of the Filtered and Residual Fields 4

2.2 The Dynamical Equations 6

2.3 Filter Selection 7

2.4 Residual Stress Models 11

2.5 Governing Equations for the Filtered Field 12

Chapter III. NUMERICAL METHOD 13

3.1 Grid Layout and Notation 13

3.2 Space Differencing 14

3.3 DG Operator 16

3.4 Transport Difference Operator 17

3.5 Time Differencing 18

3.6 Pressure Field Solution 19

3.7 Summary of Difference Equations 21

Chapter IV. DECAY OF ISOTROPIC TURBULENCE 23

4.1 Problem Description 23

4.2 The Benchmark Experiment 23

4.3 Mesh Size Selection 24

4.4 Initial Conditions 25

4.5 Boundary Conditions 28

4.6 Extraction of Statistical Information from the Compu-tation 28

7 Selection of the Averaging Scale 29

4.8 Selection of C and C 30s v4.9 Energetics of the Filtered Field 31

4.10 Other Aspects 32

4.11 Computational Details 33

Chapter V. DISTORTION OF HOMOGENEOUS TURBULENCE BY IRROTATIONALPLANE STRAIN 35

5.1 Problem Description 35

5.2 Governing Equations 36

5.3 Anisotropic Initial Condition 39

5.4 Results 41

5.5 Energetics of the Filtered Field 43

5.6 Assessment of Turbulence Closure Models 43

Chapter VI. CONCLUSIONS AND RECOMMENDATIONS 46

Figures 48

Appendix I. On the Conservative Space Differencing Scheme 84

Appendix II. Flow Charts and Program 90

References ,122

vi

LIST OF FIGURES

Figure Page

2.1 Gaussian filter 48

2.2 Filtered energy spectra, E(k,t) 49

3.1 Comparison of modified wave numbers 50

3.2 Comparison of DIV(GRAD) operator 51

4.1 Sketch of wind tunnel test section for a generation of iso-tropic flow 52

4.2 Determination of A or B vector 53

4.3 Unfiltered initial energy spectrum 54

4.4 Evolution of energy spectra: 16 x 16 x 16 mesh:Smagorinsky model: A = 0 55

4.5 Filtered energy spectra: 16 x 16 x 16 mesh: Smagorinskymodel: A. = A 56

4.6 Filtered energy spectra - a comparison of Smagorinsky andvorticity model 16 x 16 x 16 mesh: A = 2A -. 57

4.7 Filtered energy spectra - effect of Leonard term: 16 x 16x 16 mesh: vorticity model: A = 2A 58

4.8 Filtered energy spectra: 32 x 32 x 32 mesh: Smagorinskymodel: A. = 2A 59A

4.9 Sensitivity of filtered mean square velocity decay rate toSmagorinsky model constant, c : 16 x 16 x 16 mesh:A. = 2A ? 60A

4.10 Decay of mean square filtered velocity: 16 x 16 x 16 mesh:AA = 2A 61

4.11 Decay of mean square filtered velocity: 32 x 32 x 32 mesh:Smagorinsky model: A. = 2A 62

A

4.12 Skewness: 16 x 16 x 16 mesh 63

4.13 Comparison of skewness 64

4.14 Components of dissipation 65

5.la Schematic of wind tunnel producing constant rate of strainin x-y plane 66

5.1b Equivalent representation of the plane strain in a box . . 66

5.2 Tucker-Reynolds flow: turbulent intensities 67

5.3 Tucker-Reynolds flow: turbulent energy ratios 68

5.4 Tucker-Reynolds flow: change in structural parameter, K. . 69

5.5 Simulation of the Tucker-Reynolds flow: turbulent inten-sities (anisotropic Initial condition) 70

vii

Figure Page

5.6 Simulation of the Tucker-Reynolds flow: turbulent en-ergy ratios under the plane strain and the return toisotropy in parallel flow (anisotropic initial con-dition) 71

5.7 Simulation of the Tucker-Reynolds flow: change in struc-tural parameter, 1C. (anisotropic initial condition). . 72

5.8 Simulation of the Tucker-Reynolds flow: turbulent inten-sities (isotropic initial condition) 73

5.9 Simulation of the Tucker-Reynolds flow: turbulent en-ergy ratios under the plane strain and the return to iso-tropy in parallel flow (isotropic initial condition)... 74

5.10 Simulation of the Tucker-Reynolds flow: change in struc-tural parameter, K. (isotropic initial condition) ... 75

5.11a,b,c One dimensional energy spectra (isotropic initial con-dition) 76

5.12a,b,c One dimensional energy spectra 79

5.13 Filtered energy balance 82

5.14 Comparison of the pressure-strain correlations 83

viii

NOMENCLATURE

A,B • random unit vectors

A = constant in Rotta's modeloC = constant in the Smagorinsky model

C = constant in the vorticity model

D » finite difference form of divergence operator

3u 9uD . . = 2vij

E • three-dimensional energy spectrum

E- - = one-dimensional energy spectrum

E, E-, = filtered spectra

e , e , e = unit vecotrs in x-, y-, and z-directionx y z *F « i-component of forcing term due to the mean strain

F. » finite difference form of F.

f « a general field variable; f = filtered field component;f ' = residual field component

G = filter function; finite difference form of gradientoperator

g, *» RHS of Poisson equation for pressure; g- finitedifference form

H = heaviside step function_ —2 —2 —2 —2K = (< v > - )/(<v > + ) , structural parameter

k. = wave number in i-direction

k' = modified wave number in i-direction (fourth order) ;k" (second order)

2 2E = finite difference form of V operator

L = NA , computational box size.; length scale of large, eddies

M = grid size of turbulence generator in experiments

N = number of finite difference mesh points in one direction/3 —Nc = At vq /3 /A for isotropic turbulence;

= At »OJ + q /3 /A for straining flowmax

p + -r- R ; imposed pressure field

ix

• production term

- = Leonard production term

p = pressure

q2 = u±u±

R •= degree of anisotropy

RT = qL/v

R, . = u!u' + ulu. + u'u". , residual stress

T i j E ; (T1:))c= computed; (I±^ = Rotta's

model; (T .) obtained from R. . historyij K. ij

t = time

U = free stream velocity in x-direction

U. = i-components imposed mean velocity

u « i-components fluctuating velocity; u. = filteredfield component; u1 =• residual field component;u. = Fourier transformed u. ; xj = vector notation

6/6 , 6/6,. = finite difference form of ^- , -X t OX ot

e = energy dissipation; e = residual fields contribution;eT = Leonard terms contributionLI

$ = spectrum function

4>,6 = random angle

v = y/p , kinematic viscosity

V_ •. eddy viscosity

p = fluid density

F = constant strain-rate

y = constant in Gaussian filter function

ri = Kolmogorov microscale

w = filtered vorticity, curl u

A = averaging length scale

A - computational mesh width

< > - ensemble average in real space; shell average inwave-number space

Superscripts

(n) • time step

Subscripts

(k,£,m) - computational mesh number index

xi

CHAPTER I

INTRODUCTION

1.1 Historical Background

As computer capabilities grow, the three-dimensional time-dependent

computation of turbulence is becoming possible. However, the retention

of all scales of motion is not yet feasible (and probably never will be),

so the best one can hope for is the simulation of the large scale

structures. The large scale structures are strongly dependent upon the

nature of the flow, but there is considerable evidence that the structure

of the smaller scales is independent of the large scale structure. This

suggests a mixed approach in which one computes the large scale motions

and models the small scales.

To define the large scale motions, some sort of averaging operator

has to be applied to the governing equations to filter out the small

scale motions. However, in three-dimensional computations to date, the

definition of the small scale motions was not precisely related to the

filtering operation and consequently the meaning of those motions was

not very clear. In any case, the resulting equations for the filtered

field contain so-called sub-grid scale or residual scale Reynolds stresses,

which must be modeled in the computation.

Two distinctive solution methods have been used in solving the re-

sulting equations. The first is a conventional finite difference mesh

calculation. In this approach, the simplest and perhaps the most usual

way of relating the residual scale turbulence to the filtered motion is

by a local eddy viscosity model. Smagorinsky (1963, 1965) related eddy

viscosity to the local strain-rate of the filtered field. Deardorff

(1970a,b) applied this model to three-dimensional turbulent channel flow

and planetary boundary layer problems, and Schumann (1973) applied it to

plane channels and annuli. Deardorff (1973) and Schumann later introduced

more sophisticated residual Reynolds stress transport equations. Other

examples of this approach are given by Lilly (1964, 1967), Smagorinsky

et al (1965), Fox and Lilly (1972), Fox and Deardorff (1972). Previous

work mentioned above has not paid sufficient attention to the basic

aspects of this type of simulation, so as yet this approach has not

really progressed very far.

The second approach is the spectral (Fourier) method much advocated

by Orszag (1969, 1971a,b). While this method has mathematically attrac-

tive features for certain problems, it is generally more difficult to

extend to flows with interesting geometries. Moreover, work to date

ignored the residual Reynolds stresses, and it is not clear how these

could be incorporated in a Fourier calculation.

1.2 Motivation and Objectives

At the time this work was initiated there were some serious problems

with work done previously:

(1) There was a need to define the large-scale field precisely,

so that the equations can be systematically developed.

(2) There was a need to carefully evaluate the accuracy requirements

to be sure that computational errors are higher order than

the residual stresses.

(3) There was a need to carefully assess just what can really

be learned from this type of turbulence simulation.

The main objective of this study was to carefully develop a numerical

simulation method for turbulent flows away from solid or free boundaries,

and to apply this method to study decaying isotropic turbulence and

homogeneous turbulence with irrotational plane strain.

The present study is one in a systematic program investigating large

eddy simulation of turbulence, and reports the details of the initial

computations under this program.

1.3 Summary

The contribution of the present work includes

(a) a precise definition for the large-scale field (after

Leonard (1973)),

(b) a study of the optimum averaging scale, as compared to the

grid mesh scale,

(c) a study of two residual stress models, and evaluation of the

model constant for each,

(d) a demonstration that the model constant Is independent of

mesh size,

(e) a fourth-order differencing scheme that properly conserves

energy and momentum,

(f) a method for calculating the pressure so as to conserve

mass at subsequent time steps,

(g) a demonstration that a coarse mesh can be used to obtain

surprisingly good predictions for the Reynolds stresses in

a straining flow,

(h) an evaluation of certain aspects of simple turbulence closure

models.

Although many questions have been answered by this work, new ones

have been raised. Suggestions for follow-on work in this project are

made in Chapter VI.

Chapter II

Theoretical Foundations

2.1 Definition of the Filtered and Residual Fields

To resolve the smallest scales of turbulence in a grid-based cal-

culation, the mesh size has to be smaller than the dissipation length,3 1/4which is on the order of the Kolmogorov microscale, r| - (v /e)

Here v is the kinematic viscosity and e is the energy dissipation

rate per unit mass. It is known that (see Tennekes and Lumley 1972)

e - q /L where q is the R.M.S. velocity and L is the length scale

of large eddies. Thus the minimum number of mesh points that must be

used in a three-dimensional grid computation that resolves both the

large and small scales can be estimated as

" - (5)!Using this estimate, we find that an IL, of 10 , typical of turbulent

7 *•flows, would require 2 x 10 worlds of storage for four variables. This

is approximately 50 times the Large Core Memory of the CDC 7600, and

about 10 times the available memory of the ILLIAC IV disk.

It is clear that one can not do a full simulation, except at

extremely low Reynolds number. The best one can hope for is a computa-

tion that will yield the large-scale motions. Fortunately these contain

most of the turbulence energy, and are responsible for most of the tur-

bulent transport, and so a large-eddy simulation technique would be

very useful, especially if it could handle arbitrary flows.

The first problem one faces is in defining the large scale field.

Conventionally the large scale motions have been defined by volume-

averaging in a continuous manner over computational grid boxes (e.g.

Deardorff 1970a). Schumann (1973, 1974) applied a slightly different

technique involving averaging over the surface of grid boxes.A more general approach that recognizes the continuous nature of

the flow variables is the "filter function" approach of Leonard (1973).

Let f(jc) denote a field variable, for example velocity, f may contain

large and very small components. Then, we define the filtered field

f"by

I(x) - /G(x-x') f(x') dx' (2.la)

where G is a selected filter function. For C « C , where C is a

constant, the filter G must satisfy

/ G(x) dx = 1 (2.1b)

Now f can be decomposed into its filtered field (FF) component, f ,

and residual-field (RF) components, f , by

f - F + f' (2.2)

Note that f is not the conventional mean used in the classic-turbulence

literature.

In the present work we treat only flows that are homogeneous, for

which the integration in (2.1) extends over all space. Careful considera-

tion will have to be given to the domain of integration when one desires

to treat a flow near a wall.

Now note that, if G is piecewise continuously differentiable and

G(r) g'oes to zero as r -*• °° at least as fast as 1/r ,

3f_3x

CO

f G(x-x') ||, dx'

CO

- f f(x') £, G(X-X') dx'

—00 —00

0

jg f- 5J / G(x-x') f(x') dx'

—CO

- ^F (2.3a)

Also, |f - II- (2.3b)

However,

fg (2.3c)

2.2 The Dynamical Equations

Applying (2.1) to the Navier-Stokes equations, and using (2.2) and

(2.3), one obtains (for incompressible flows)

- 0

3uiIT Uiuj p 3x

2_U

(2.4)

(2.5)

The advection term is

u.u. + u!u. + u.u!

uiuj (2.6)

where

*ij - u i u j + ui"j+ uiujR.. is the residual field contribution to the advection term. ~PR.i.i *s

called the "residual stress."

To localize the first term on the right in (2.6) we carry out a Taylor

series expansion,

fc x

oo

/|=,<v.—oo

I- 0(x-x_)

(xk-xko) + 2

3u

(2.7)

For the above filtering of the dynamical equations to be useful, the

integrals in (2.7) must exist. This requires that G(r) -»• 0 exponentially

as r -»• °° .

2.3 Filter Selection

(a) Sub-Grid-Scale filter

Let us first seek a filter that makes the scales of motion in the

residual field smaller than the scales in the filtered field, in the

Fourier sense. Let

—CO

Then, °° °°

00

f - j f(k) eik-'-dk (2.8)

I - I f f(k) G(x-x') e1-"-' dx1 dk (2.9)

—CO — OO

We want to have f contain all scales larger than a cut-off scale. Thus

we want +k— f ~ <k«vf - / f(k) e-- dk (2.10)

-kcwhere k is the cut-off wave number. Hence, we can write

OO OO 00

f H(k) f (k) e1-'- dk = f f (k) f G(x-x') e1-'-' dx1 dk

where

!

0 if k. > k for any i1 c (2. lib)

1 otherwise

So we have an integral equation for G ,OO

/ G(x-x'H(k;kc) - G(x-x') e ~ d x ' (2.12)

The solution to (2.12) is

3 sin TT(X -x!)/An —,/.. ..,;:—- (2.13)

where A. = ir/k is the averaging or filtering length scale. This isA C

the proper filter if one wishes to have the residual field really be

"sub-grid scale." A grid-based computation made using this filter

would be equivalent to a Fourier computation.

The second moment of G involves integrals likev00 2x sin(TTx/A.)

— dx (2.14)irx

This integral does not exist, and hence the expansion (2.7) could not

be used. This filter is not suitable for a. grid-based numerical method;

hence one can not expect to really have the residual field be sub-grid

scale.

(b) Top-hat filter

The filter used implicitly by many workers is the top-hat,

G(x-x')

1/AA for |x-x'| < -f

(2.15)

-*A/2

Then the filtered velocity is

/

~A/2

u(x+£) dg. (2.16)

V A/2

This is equivalent to volume averaging. The Fourier transform of (2.16) isAA/2

u(k) - - u(x+i) e~ '- dxd£-ij I IAA - -AA/2

A.3J u(k) e^dC

( 3sin(V

\ i=J- i

/2) )ai7)

Here u(k) is the Fourier transform of 11 .

8

Equation (2.17) shows that the spectrum of the filtered field will

contain components of all wave-numbers. Moreover, at the wave-numbers

for which the coefficient of u_(k) in (2.17) is zero, the inverse trans-

form will be singular. This makes it impossible to predict the actual

spectrum .uQc) from the filtered spectrum ia(k) , and this very unde-

sirable feature of the top-hat filter renders it useless if we want to

compute spectral features with a grid-based method. However, the top-

hat filter could be used if spectral results were not sought.

Using (2.15) in (2.7), and carrying out the integration over 3£ »

— *! 2u.u., (x ,t) = vi.u'.U »t) + -sr V (u.u.)+ 0(A7) (2.18).i J —o 1 J —o 24 1 j A

2 2 - —The second term on the right is called the Leonard term; ~pA. V (u.u.)/24

is called the "Leonard stress." As will be shown later,2

R.. - °(AA) and A - 0(A) . So both the residual stresses and the

Leonard stresses have to be included; moreover, the computational dif-2

ference scheme must be accurate to 0(A ) to avoid Introduction of

numerical errors comparable with these stresses.

(c) Gaussain filter

A filter with much more desirable properties is

( I— 1/3 ( )G(x-xf) - jyj^p ( exp j-Y(x-x')2/A [ (2.19)

where Y is a constant. Then the filtered velocity is

e ~~ A dx1 (2.20)

The Fourier transform of this is

_oo _oo

00

-T— I I u(k) e —* AA / J ~-

-ik-x ~ Ae ~ - e A dx

-Y £2/A2

e A d£

= G(k) expl -Tk' I (2.21)

Consider now the three-dimensional energy spectra of the actual and

filtered fields:

E(k) - 27T k2 < G(k) • Q*(k)> (2.22a)

E(k) = 27T k2 < G(k) • S*(k)> (2.22b)

Here < > denotes an average over an ensemble of experiments, and

* denotes a complex conjugate. Equations (2.21) and (2.22) show that

E(k) = E(k) exp ( - •£- k2 I (2.23)

We see that the use of the Gaussian filter will result in a filtered

field that misses only a very small amount of large scale motion; most

of the small scale motions are placed in the residual field. Thus, in

many respects this filter has the desirable properties of the sub-grid-

scale filter. However, its behavior at r •*• °° makes the integrals in

(2.7) exist. Moreover, the conversion back and forth between the

spectrum of the filtered field and the spectrum of the actual field is

easily accomplished, and hence the Gaussian filter is perferable to the

top-hat filter.

10

Using (2.19) and (2.7), one obtains

T-T- _ . Al 2 - - 4UjU (x t) = Ulu (i .t) +^ VZ(uiUj) + 0(Ap (2.24)

When y ° 6 , the Leonard term in (2.24) is exactly the same as in (2.18)

Hence the Gaussian filter with y = 6 was chosen for the present study.

This filter is illustrated on Fig. 2.1 and an example of E and E

relation (2.23) is shown on Fig. 2.2.

2.4 Residual Stress Models

The following eddy viscosity model is used for R. . .

(2-25)

where

is the strain-rate tensor, and v is an effective viscosity associated

with the residual field motions.

(a) Smagorinsky model

Smagorinsky (1963) suggested a model for VT ,

(2.26)

where cc is a constant. In experiments one observes a sharp separationo

of turbulent regions, containing vorticity and non-turbulent regions which

are irrotational. A weakness of this model is that, in a non-turbulent

ir rotational region, VT will have a non-zero value. This will give

rise to residual stresses in the non-turbulent flow outside of a boundary

layer .

11

A way around this drawback is to relate VT directly to vorticity.

(b) Vorticity model

A way around this

A likely possibility is

i fTT(2.27)

where u> - curl u is the vorticity, and c is a constant.

2.5 Governing Equations for the Filtered Field

Now, neglicting the molecular viscosity term, and dropping terms of2

higher order than A , filtered momentum equations becomeA

»i 2 - - - \ SP** n .. .. o%. o 1 _ af /o oo\~~gt~ """ "^^ 1 U ^ U J "•" •>/. v U^UJ ~ ^v^jj I " ~ ^^ {t.tO)

where P * ^ + T R1:. • Thle may be written as

!i . h _JL A H9 t i 9x. i

where / .2 \„ A ._8_(G ; +£,2;s .2,1 )i 3x. \ i j 24 i j T i j /

J * '

It is in this form that we shall deal with the problem computationally.

The manner in which continuity was used to fix P is discussed in the

next chapter.

12

CHAPTER III

NUMERICAL METHOD

3.1 Grid Layout and Notation

A uniform cubic mesh is used, as sketched below. The mesh width A

need not be the same as the averaging scale A. introduced in the

previous chapter.

m+1

m

m-1

I^*

k-1/£

k+1

The i-component of the filtered velocity at the nth time step is

written as

(k.Jl.m)

where (k,A,m) is the meshpoint index for (x, y, z),

We now define the following operator notations :

6/6x. - finite difference operator corresponding to

6/6t • finite difference operator corresponding to 3/8t

G » finite difference form of gradient operator

D • finite difference form of divergence operatort- 0

y— (u. f) ** transport operator corresponding to -5 — (u. f)0 X • 1 ' T

Further details of these terms are given next .

13

3.2 Space Differencing

A fourth order differencing scheme is applied where fourth order

accuracy is needed. Since the Leonard and residual stress terms are

second order, they can be approximated by second order formula to give

the same accuracy. The central difference fourth order scheme is, for

example,

£u 16x " 12A" U(k-2)

For simplicity, the subscripts I and m are not shown.

Suppose we represent u by a discrete Fourier expansion,

e±-- (3-2)n

where . 0 TT = wavg numt,er in the x. direction1 NA 1 J.

nx = -N/2 0,1 (| - 1)

N » Number of mesh points in one direction

The sum extends over all n. , n_ , and n. . Substituting (3.2) in (3.1),

the Fourier transform of 6u/6x is identified as

£• ± -i2Ak- -iAk iA^ i2Ak;L-g . _ (e - . 8. + 8e . e ) u

= | 8 sin(Ak1) - sin(2Ak;L) \ u (3.3)

If a modified wave number, k' , is defined by

k' = -rr < 8 sin(Ak ) - sin(2Ak ) > (3.4)

then the Fourier transform of Dxi = 0 can be written as

k^ - 0 (3.5)

y\

Note that the exact transform of div 11 is k.u. . Hence, k' may be

interpreted as the wave number that allows continuity to be satisfied in

grid space.

14

If instead one were to use a second-order central difference scheme,

6u6x u

_

2A (k+l)(3.6)

The modified wave number , k'1 , would be

c« A I (3.7)

k.,, k' and k" are compared in Fig. 3.1.

The fourth-order D and G operators are therefore (again only

subscripts different from k, £, or m are explicitly shown).

6u 6v . 5w6x * 6y 6z

12A

12A i

u(k-2)

U-2)

- 8U/

- 8v,

•f 8u - u

/ o\ ~ w, 1N(m-2) (m-1) /_. 1v(m+1) - w

div u + 0(AH)

G(P) - [S5 ^ 6 _,. .6 \T~ + e -T— -f e T— I6x y oy z oz /

12A ?(k-2) " 8P(k-l) + 8P

(3.8)

(k+1) " r(k+2)i

y 12A

ez 12A ]P(m-2)

grad P + 0(A4)

I" 8P(m-l) + 8P(m+l) ~ P(m+2)|

(3.9)

15

where e , e , e are unit vectors in the x-, y-, and z-directions.x y z * '

3.3 DG Operator

DG(P) - •£- (•£-*) (3<10)

Expanding one term of (3.10), using the fourth-order difference scheme,

ov + 64P,£-(±T>\ o 16x \6x / (k-2)

+ 16P(k-l) - 130P(k) + 16P(k+l)

-16P(k+3)+P(k+4)j

If P is expanded in a discrete Fourier series, similar to (3.2), the

Fourier Transform of (3.11) is identified as,

(f ($*I6x\6x

- 16ei3Akl + 64ei2Akl + 16eiAkl(12A)2 I

- 130 + 16e-lAkl 4- 64e-12Akl -

- — j -65 + 16 cos(Ak,) + 64 cos(2Ak1)72AZ I

- 16 cosCSA^) + cos(4Ak1) ! P

= ~42p ' (3.12)

(3.12) may be obtained directly from (3.4). Therefore, in Fourier space DG

operator becomes^DG - -k^ kj (3.13)

Compare (3.12) to the following fourth order central differencing scheme,2 2

which is a commonly used approximation to 3 /8x operator;

16

62

P — n 1 ~t /, ,,x i J.WJ. /,_ n x JWJ. ,,_x T j-Ui /•!_., \ i /I.J.2^

(3.14)

15 - 16 cos(Ak1) + cos(2Ak1)j p

- - k p (3.15)

~ 2 2 ~ 2where SL is defined by (3.15). k-, V.1 , and E. are compared in Fig.

3.2 for N - 16 .

3.4 Transport Difference Operator

Differencing the transport terms in the form of (2.28) will auto-

matically conserve momentum in an inviscid flow. But the

computation becomes unstable and the kinetic energy increases. This

happens in real flows in spite of the dissipative nature of R . and

the Leonard term. This non-linear instability, first reported by Phillips

(1959), arises because the momentum conservative form does not necessarily

guarantee energy conservation, and truncation errors in the energy equa-

tion are not negligible.

Arakawa (1966) devised a differencing scheme that conserves both

mean square vorticity and energy in two-dimensional calculations that use

the vorticity and stream function as dependent variables. This is not

useful in a three-dimensional flow.

A fourth-order transport differencing scheme that does conserve

energy and momentum was developed for the present work:

17

- f(k-D} + f(

1 ( — _24A (uf)(k+2) * (uf)(k-2) + u(f(k+2) - f(k-2)}

Again we only show subscripts that differs from k, JL, or m . For details

see Appendix I. In the present work this was used for the terms

3(u u.)/3x. ; for the Leonard term a second-order version of (3.16) was

used,

(uf) - (uf)(k+1) - (3)

+ G(f(k+l) - !(k-l))+ 1(5(fcfl)

The familiar second-order central difference approximation was used for

v in the Leonard term,

2 = ~~!) '6x tT

For the residual stress terms, (3.17) was used. The strain-rates and

vorticity were computed from the second-order central difference (3.6).

3.5 Time Differencing

A second order Adams-Bashforth method was used for the time inte-

gration. As shown by Lilly (1965), this method is very weakly unstable,

but the total spurious computational production of kinetic energy is

small.

The Adams-Bashf orth formula for u. at time step n-fl is

18

-(n+1) _ -(n) + At / 3 R(n) _ 1 R(n-l

where H is defined by (2.29). Note that this is an explicit scheme.

3.6 Pressure Field Solution

To study this problem in detail, let us rewrite (2.29) as

ItT ~ hi ** ~ "dT" (3.20a)

Again, continuity is

-~- = 0 (3.20b)dx±

Taking the divergence of (3.20a)

-- -3x. i 3t 3x

(3.21)

The usual computational procedure involves choosing the pressure

field at the current time step such that continuity is satisfied at the

next time step, i.e. so that the new flow field will be divergence free.

This must be done very carefully. Let's look at three possibilities.

These take advantage of Fourier transformation, for it is known that

fast Fourier transforms provide an excellent way to solve the Poisson

equation, at least in a rectangular domain.

(a) Method 1

The Fourier transform of (3.21) is

-k2P = ^ (3.22)

~ 2 2 2 2where g.. is the difference approximation to g.. and k = k +k + kj. x x y zk f k , and k are wave numbers in x-, y-, and z-directions. By inversex y z

19

transformation, P can be obtained. However, as will be shown shortly,

this method does not give a next velocity field that is divergence-

free in grid space. Hence this approach is unsatisfactory.

(b) Method 2

A second approach is to use the difference form of (3.21),

*2 *2 K2^T + + T P - ~h (3.23)6x <5y 6z

Then P can be obtained by Fourier transforming (3.23),

-k±k± P - x (3.24)

As will be shown shortly, the pressure from this does not give a divergence-

free field in grid space at the next time step. Hence, this method, which

was used by Jain (1967) , is also unsatisfactory.

(c) Method 3

The finite difference forms of (3.20a) and (3.20b) are

<Su. _ .

Du± - 0 (3.25b)

where h. is difference approximation to h. . If we apply the D

operator to (3.25a),

DG P - ^ (Du±) + Dh± -' gj (3.26)

Then, taking the Fourier transform of (3.26), and using (3.13),

-kjkj P = §[ (3.27)

Now let's compare the three methods. The Fourier transform of

(3.25a) is

20

6t = -i (3.28)

To satisfy continuity in grid space, we want to have

_ _fit

6xifor a flow that has Du. = 0 to start. From (3.5), this is equivalent

to k'u, =0 at the first and next time steps. Using (3.21), and sub-1 x />» Astituting P in (3.28) by P from (3.22), (3.24) or (3.27) and operating

with D on the resulting equations, one obtains,

for method 1:_fi_fit ( k'lr' \•*v ^ • *x . xv %

• /

(3.29)

for method 2:

( k'k1 \/\ R-. N, ^ \

^i - £J J M

(3.30)

for method 3:

It (ki"l) ( k!k! f. \fii-7^vk;s (3.31)

The error introduced by method 1 and method 2 can be seen clearly by22 2 ~2observing the magnitude of the ratio k1 /k or k' /k as illustrated

in Fig. 3.2. Method 3 satisfies continuity at the next time step in

grid space, and hence is chosen for pressure field solution here.

3.7 Summary of Difference Equations

Momentum equation:

fit

5Pfix.

2

24"

_5x

(3.32)

21

Poisson equation for P:

DG(P) - 0(1 ) (3.33)

where

P - 4 + 7*!!^ 1

P *

2 1/2VT = <

CSV ^ J j' : Sma8°rinsky

2 - - 1/2= (cA) (w) ' : Vorticity model

Wi = £ijk

Co.c = constant

22

CHAPTER IV

DECAY OF ISOTROPIC TURBULENCE

4.1 Problem Description

Perhaps the most basic problem in turbulence is the decay of incom-

pressible homogeneous isotropic turbulence. It is with this primitive

turbulent flow that our study began. This flow was used to determine the

value of the residual stress model constant (C or C ), for use in sub-

sequent calculations of other flows. It also provided a basic testing

ground for the computational methods being developed in this program.

The experimental grid turbulence data of Comte-Bellot and Corrsin

(1971) were used as the "target" for these predictions. Such experiments

closely approximate homogeneous isotropic turbulence, when viewed in a

coordinate frame translating at the mean flow velocity.

4.2 The Benchmark Experiment

The pertinent information from Comte-Bellot and Corrsin's (1971)

experiments will be reviewed now. The wind tunnel test section, which

has a slight secondary contraction to isotropize the turbulence, is

sketched in Fig. 4.1 . The turbulence was generated by a biplane square

rod grid with mesh size, M , of 5.08 cm. The free-stream air speed, U ,

was 10 m/sec, giving grid mesh Reynolds number U M/v of- 34,000 . The2 2 2 °

streamwise () and transverse (<v > , <w >) turbulent energy components

remained nearly equal to each other during the decay along the test sec-

tion. These were closely fit by

J2 ,Dt -1'25o

<u2:

u2 u2 1'25

_°_ „ _2_ e 20<V

2> <w2>

/u t X(-£--3.5)V M '

Correlations, energy spectra and other quantities were measured using

23

hot wire anemometry at U t/M » 42, 98 and 171. The Reynolds number

based on the Taylor microscale, IU = \(u )A/\> , was 71.6, 65.3 and

60.7 at these points.

4.3 Mesh Size Selection

The choice of the computational mesh size requires consideration of

the turbulence spectrum (Fig. 4.3). The smallest scales that can be re-

solved have wave number TT/A , where A is the mesh width. If there

are N points in one direction, the largest scales that will be repre-

sented in the computation have wave number 27T/(NA). N and A must be

chosen such that the computation captures as much of the turbulence

energy as possible. It is also desirable that the computation extend

to the so-called inertial subrange (Tennekes and Lumley 1972).

The mesh systems used were as follows:3

16 mesh

A = 1.5 cm, N = 16 , At = 6.25 x 10~3sec

323 mesh

1.0 cm, N = 32 , At = 6.25 x 10~3sec

The model constants were first evaluated using the 16 mesh; the 32

calculation then verified that the constants are independent of mesh size.

The corresponding Courant numbers were:

Nc = \<l/3 At/Ax , q2 = <u2> + <v2> + <w2>

163 mesh

Nc < 0.06

323 mesh

Nc < 0.1

24

4.4 Initial Conditions

We want to prescribe an initial profile that has the proper energy

content and spectrum, and is isotropic. A technique for generating a

random field that meets these conditions, and also satisfies continuity,

was developed for this purpose. Since the computation will treat the

filtered field, we matched the initial filtered spectrum and not the

initial measured spectrum.

We generated initial filtered field, u. (x) , by first establishing/N 1 ~~

its discrete Fourier transform, u.(k ) ,

*- 12 *, ik • x- EEL u(k )e~n (4.2)

N

Here k is the wave number vector defined by (k ) = TJT- n , where n

is an integer ranging from -1/2N to 1/2N-1 for an N mesh system. Note

that the maximum wave number is k = IT/A . If u is discretized atmax —N points, then the Fourier transform u can only be evaluated at N

discrete wave numbers, and that is why the summation must have non-sym-

metric limits.

The commonly used fast Fourier transform requires N to be 2 ,

where m is an integer (see Cochran et al. 1967). Physically N has

to be large enough so that wave-number spectra can be treated as smooth3

functions. As will be shown later, 16

smooth three-dimensional energy spectra.

Now, we can approximate the spectru

field (see Tennekes and Lumley 1972) as,

(4.3)

3 3functions. As will be shown later, 16 or 32 mesh systems gave fairly

Now, we can approximate the spectrum function, $.. of the filtered

where < > denotes an average over an esemble of experiments or, alter-

natively, over a spherical shell in k-space with radius k (see Section

4.6).

25

The filtered 3-D energy spectrum, E , is given by

~E(k) = 2Trk2$i:L(k) (A. 4)

E(k)dk is the energy content of a differentially thick spherical shell

in wave-number space. Using (4.3), E(k) is approximated by

E(k ) ~ 2irk2< u. (k )u,*(k )> (4.5)n n i n i n

To establish the initial field we need to fix the Fourier amplitudes^ /\ /v

u.(k ) . Equation (4.5) was used to fix u,(k )u.*(k ) for each k .in _^ i n i nThe vector components u. were chosen by a technique described below.

/s

To get u. to satisfy continuity in grid space, the real and imag-1 ^

inary parts of the transformed velocity vector, \± , must be perpendicular

to the modified wave-number vector, ' (see Equation (3.5)). In the

actual computation, we have N points in k^-space, and, for any k^ ,A ™—

_k' can be obtained by (3.4). Then, ^i has to be selected on a perpen-

dicular plane to k' in k-space.^_ *_• ^

To ensure statistical isotropy, the real and imaginary parts of _u

must be chosen randomly. First we picked a unit vector A , perpendicular

to ^' , by turning a random angle, <}> , from a reference frame (Fig.

4.2). Here <{> was selected with uniform probability over the interval

0 to 2ir . We then repeated to get a second random unit vector JB ,/N "™"

also perpendicular to k.' . The real part of u was made proportional

to the vector A^ and the imaginary to ]i , and hence continuity was sat-

isfied. We still needed to fix the relative magnitudes of the real and/\imaginary parts of ii(k) , which we did by a random choice of an angle,

*

9 . Then we defined a and b by

a = cos6 , b = sin6 (4.6)

Finally, we set

26

Now, by Inverse transforming u. , we obtained u. , which must be

real and will be real if the Fourier transform satisfies

u(-k ) - u (k ) (4.8)— —n — ~~n

In essence, the imaginary contribution for each negative k exactly

cancels the imaginary contribution of the same positive k . Hence, we/\ nonly needed to generate u. by (4.7) for the upper half of the k-space,

N •— Ni.e. for 0 < n < -T - 1 . However, this won't fix u for n. = - -r- .—" •"" £. ~— 1 Z

/\

Moreover, if ju is not zero, the velocity field will have an imaginary

part (see (4.2)). If instead we wrote (4.2) as

N/2 ^ ik • x

u±(x) = E E E "iQ e ""-N/2

then u. would be real. However, then we could not take advantage ofl ^the FFT routine to invert u. . As a practical solution to this dilemma

— iwe set u equal to zero for the wave numbers corresponding to n =

-1/2N . Then (4.2) is essentially the same as

( - 1)^2 ; ik • x

u(x) = EEE u

-(--4

and u. will be real, and an FFT routine may be used. The resulting

energy spectrum was therefore slightly low at the highest wave number.

However, the effect of this discrepancy was insignificant and became

invisible after a few time steps in the computation.

We remark that the field generated by this procedure is quite iso-

tropic. However, as will be shown it has zero skewness, whereas real

turbulence has a non-zero skewness. As will be seen, this condition

corrected .itself in only a few time steps.

27

4.5 Boundary Conditions

The computational problem can only extend over a part of the experi-

mental region. To get around this difficulty we have used "periodic"

boundary conditions, which are of course not really correct. However,

if the computational grid system extends over a distance large compared

to the scale of the energy- containing motions, the periodic boundary con-

ditions should not introduce appreciable error. The periodic boundary

conditions have a great advantage in dealing with the Poisson equation

for the pressure by fast Fourier transform (FFT) methods.

4.6 Extraction of Statistical Information from the Computation

The statistical quantities of interest are averages over ensembles

of experiments. Since we made only one computational realization in

each case, the statistical quantities had to be inferred from appropriate

ergodic hypotheses.

In physical space the ensemble average < > was replaced by an

average over the flow field. This was done by taking a mean value over

N mesh points, i.e.

N

C4'9)

The differencing schemes described in Chapter III were used to calculate

these quantities.

In wave number space, the ensemble average was replaced by an aver-3

age over a shell in k-space ("shell average"). Since we have only N

discrete points in k-space, the < > average was made by taking a mean

value over the points between the two shells with radius (k-l/2Ak) and

(k+l/2Ak).^

To get the filtered spectrum, E(k) , the transformed velocity, u ,— — * ~~

was obtained by FFT (see 4.2). Then, was calculated

by shell-averaging. The choice of the band width, Ak , is somewhat

arbitrary and was set to be 0.1 cm here. Then, from (4.5),

28

k =k-Akn

where N, is the number of points between the two shells with radius-1(k-l/2Ak) and (k+l/2Ak) . The resulting spectra, evaluated at 0.1 cm

wave-number intervals, were smooth enough to be represented by continuous

curves as shown by solid lines in Fig. 4.4, 4.5, and 4.7 . The filtered

spectrum, E(k) , was then compared to the filtered experimental spectrum,

which we obtained using (2.23).

4. 7 Selection of the Averaging Scale

Considerable thought was given to the choice of the averaging scale

A. . Our failures are as important as our successes, and both will now

be discussed.

Consider first the computation with A = 0 . This zero averaging

length is equivalent to the unfiltered calculations used in laminar flow,

and implies that we are trying to resolve the complete spectrum by a fin-

ite difference method. The Leonard term in this case is equal to zero,

i.e.

The unfiltered initial energy spectrum is plotted in Fig. 4.3 . The3 3amounts of unfiltered energy for the 16 mesh and 32 mesh systems are

also shown. Figure 4.4 shows the computation for a value of C thatsgives the proper rate of energy decay. Note that for k > l/2k the

spectrum is distorted considerably at tU /M - 86.5 and become worse as

time increases.

In an instantaneously fluctuating field, higher derivatives are not

small and the convergence of the Taylor series is expected to be slow.

Use of A = 0 and the consequent exclusion of the Leonard term causedf\

much distortion of the spectrum, i.e. aliasing error. Indeed, the fin-

ite difference method with N mesh points in one direction can only re-

solve the unfiltered field up to k = ir/(2A) = k /2 , which is a halfmaxthe maximum wave number in one direction (see Orszag 1969, Orszag and

29

Israeli! 1974). Judging from Fig. 4.4, in a computation without filter-

ing the non-linear interactions transfer too much energy from large to

small scales. This excess up-scale energy transfer could be somewhat re-

duced by using a larger coefficient in the residual stress model. How-

ever, this brings an unreasonably high energy decay rate. We conclude

that one should never use A. = 0 in a turbulence simulation; filter-A

ing is essential.

Let's look next at what happens when the averaging scale A is

equal to the mesh scale A . Figure 4.5 shows a computation with a value

of C that gives the proper energy decay. Note that the errors in thespredictions of the filtered spectrum are significant at high wave numbers.

Consider now the computations with A. = 2A , shown in Fig. 4.6, runA

for values of C and C that give the proper energy decay. The pre-

dictions of the filtered spectrum for both residual stress models are3

remarkably accurate, even in the coarse 16 calculation!

To investigate the effect of the Leonard term separately from the

filtering, an additional calculation with A. = 2A was run with the

vorticity model, excluding the Leonard terms (Fig. 4.7). The prediction

is poor on high wave number side. Evidently the Leonard terms assists

in removal of energy from high wave numbers. We conclude that good re-

sults will be obtained with A = 2A , and that the Leonard terms mustA

be included.

4.8 Selection of C and Cs v

An analytical way of determining the residual stress model constants,

C or C , is not known. Lilly (1966) estimated C =0.2 using sev-S v 5

eral ad-hoc assumptions. Later workers (Deardorff 1971, Fox and Deardorff

1972) calibrated this constant to get the best computational results. In

these cases, the required C was between 0.10 and 0.22 .8 3In the present study a series of 16 mesh calculations were run with

different values of each constant, and values selected that gave the best

prediction for the filtered rate of energy decay as judged by consideration

of the slope of the curve (Fig. 4.10). The constants obtained were as

30

follows;

C - 0.206 , C - 0.254

Figure 4.9 shows the sensitivity of the predicted filtered energy decay

to C . Figure 4.10 shows the excellent agreement of the energy historys

with the data for the final constants.

Figure 4.11 shows the energy decay rate from the 32 calculation

with these same constants. The spectral results are shown in Fig. 4.8 .

The excellent agreement with data confirms that the model constants do

not vary with the mesh size, at least in the range covered.

In comparing Fig. 4.10 and 4.11, it must be remembered that these

are the filtered energies, which are different in these two mesh systems.

Because of the discrete Fourier approximation, not all the turbulent

energy is captured (see Fig. 4.3). Filtering improves the situation, be-

cause less energy is omitted from the filtered field at high wave number.

However, the energy in the discrete approximation to the filtered field

was still less than that in the filtered experimental field. To facili-

tate selection of the constants, the filtered experimental history was

shifted as shown in Fig. 4.10 and 4.11 .

4.9 Energetics of the Filtered Field

Multiplying (2.28) by u. , and taking an ensemble average, and as-

suming homogeneity, one finds

2 — —where q = <u. u > .

The dissipation e is seen to have two parts, a part representing

transfer to the residual field,

The three digits are not meant to imply accuracy. We actually ranwith

(2Cg)2 =0.17 , (2Cv)

2 =0.26 .

31

and a Leonard term part,

To see the relative contributions of the Leonard term and the re-

sidual scale motions to the energy decay rate, e, /£ and £_/£ areL O K O

shown in Fig. A.1A. Here e is the sum of eT and en at (U t/M -o L R o

3.5) = 42 where the computation started, e is much smaller thanL

Leonard estimated (1973). As shown by Leonard (1975), e takes energyL

mostly from the large wave number side thus preventing the damming up of

energy in the smaller eddies.

4.10 Other Aspects

No significant difference is observed between Smagorinsky and vor-

ticity models. However, some differences are expected in future applica-

tions to unbounded flow problems with turbulent and non-turbulent regions.

The skewness, which is a measure of vorticity production in the

energy cascade process, is shown in Fig. 4.12 and 4.13. Since the initial

field is randomly generated, the skewness is zero initially, but quickly3

adjusts to essentially a constant value. For the 16 calculation the

value is clearly too low (the experimental skewness is about -0.4). For3

the 32 calculation the skewness seems slightly high.

We have emphasized the need for a fourth order differencing scheme,

and wonder, why others have been able to do so well with second order

schemes. The reason may be that the second order difference form of the

advection term implicitly includes Leonard-like second order truncation

terms and thus the Leonard term is partially taken care of by the trunca-

tion. If a fourth order scheme is to be used, the Leonard terms should

be included explicitly. We have seen that they are important, particularly

at the high wave numbers. We conclude that, for a grid calculation of

the type run here, the best results will be given by the fourth-order

difference scheme that incorporates the Leonard terms.

32

A question arose as to the behavior of the vorticity under the dif-

ference scheme used here. A two-dimensional irrotational flow was input,

v was set equal to zero, and two time steps were taken. The vorticity

remained exactly zero, indicating that, at least in a two-dimensional

flow, the differencing scheme will not produce unwanted vorticity. This

aspect of the computation should receive further study in the future.

4.11 Computational Details

The calculations described above were executed on the CDC 7600 at

Lawrence Berkeley Laboratory, using programs written in FORTRAN (Appendix

II). The total storage requirements (octal) for 60 bit words were as

follows:3

16 calculation

Large Core Memory: 230,360

Field Length (Small Core) required to load: 121,200

332 calculation

Large Core Memory: 1,100,234

Field Length (Small Core) required to load: 121,200

The computer time per computational step was approximately as follows:

16 calculation; CPU time = 3 sec3

32 calculation: CPU time = 20 sec

The calculation program was carefully checked before these production

runs. To check each term in the difference equation (3.32) and (3.33),

we imposed systematically artificial flow fields. For the terms involving

first derivatives of velocities such as S , 10 , v_ , -=— (u. u.) and

Du. , the following linear velocity field was used:

u = x + 2y + 3z

v = 4x + 5y + 6z

w = 7x + 8y + 9z

33

Then the computed results were compared to the exact values. For the

terms with second derivatives, the following quadratic expressions were

used:

u =

V =

w =

X2 H

4x2 ^

7x 2 <

^ 2 y 2

^y2

^8y 2

2+ 3z

+ 6z2

+ 9z2

Then, at randomly picked mesh points, the computer results for the ad-

vection terms, the Leonard terms, fi , and D(H ) were compared to the

exact values obtained analytically.

For the Poisson solver, a sinusoidal pressure field was used to

generate DG(p) , then the computer results were checked against the im-

posed pressure field. The initial field was generated as described in

Section 4.4 and two time steps were advanced. The subsequent results

provided a testing ground for time stepping, the maintenance of a di-

vergence-free velocity field, the overall sequence of computing, and

input, output, tape handling, and data reduction routines.

The computer program is given in Appendix II.

34

CHAPTER V

DISTORTION OF HOMOGENEOUS TURBULENCE BY

IRROTATIONAL PLANE STRAIN

5.1 Problem Description

Shear may be viewed as a combination of pure strain and rotation.

Therefore, a basic problem is that of homogeneous turbulence acted upon

by an imposed uniform homogeneous irrotational strain. Tucker and Reynolds

(1968) approximated such a flow experimentally by passing grid-generated

turbulence through a passage designed to produce uniform strain in a

coordinate system translating with the mean flow velocity. This experi-

mental flow approximates the problem of box turbulence with a constant

rate of strain, shown in Fig. 5.1b.

In this chapter we discuss the computation of an idealized homo-

geneous flow with irrotational pure strain, comparable to the Tucker-

Reynolds laterally strained flow. In addition, we treat the return to

isotropy following the removal of strain, which roughly corresponds to

the experiment in the uniform channel downstream of the straining section.

Tucker and Reynolds did not measure the energy spectrum and hence we

cannot make an exact comparison with their data. However, for a qualita-

tive comparison, the initial turbulent intensities in the computation

were set to be equal to the experimental values at the beginning of the

strained section. Two cases were run. The first case was run with

approximately the same initial anisotropy as the experiments. However,

there are problems in that the anisotropic field so generated had improper

shearing stresses. Therefore, a second calculation was made with an

initially isotropic field, and this flow has been used to study the effects

of pure strain on homogeneous turbulence.

The initial field for the computation was based on an energy spectrum

similar to that used in Chapter IV. However, the Tucker-Reynolds initial

energy level was much higher than the energy in the grid flow studied in

Chapter IV. To adjust the energy, the amplitude of the Fourier coefficients

were multiplied by a constant. The initial one-dimensional energy spectra

are shown in Figs. 5.11 and.5.12 by solid curves.

35

The strain-rate used in the calculation was different from that of

the Tucker-Reynolds experiments. In an initial calculation their strain

rate was used, but the energy-decay rate in the relaxation section did

not match their experiments. The difference was attributed to differences

between their (unmeasured) spectrum and that used to start the calculation.

We first computed the energy-decay rate in the absence of strain as shown

by the solid lines in Fig. 5.5 and 5.8. Then, the strain-rate was deter-

mined to get the same total strain as in the Tucker-Reynolds experiments

at a point in the flow that would have the same energy in the absence of

strain. The final calculations were performed using this strain rate.

Therefore, the calculation should be regarded as a "Tucker-Reynolds-like"

flow, and not as a simulation of their flow.

5.2 Governing Equations

To handle the imposed mean strain, we express the local velocity

and pressure field as*

u±(x,t) - U x) + u '(x,t) (5. la)

p(x,t) - P(x) + p"(x,t) (5.1b)

where

U± - (U.V.O) - (Fx.-ry.O) (5.2a)

P - - jpF2 (x2 + y2) (5.2b)

F is the constant strain-rate. With this decomposition, the Navier-

Stokes equations for incompressible flow become

Note that we place the strain in the ^^ plane, while Tucker andReynolds placed it in the Xi~x3 plane.

36

o 0 (5.3b)

Now, using the definition (5.2), and noting that 3

this reduces to

0 ,

u -I'J

3uV•r-i3x

(5.4b)

These are the equations that will solve by the methods presented pre-

viously. The only modification (compare (2.5)) comes from the third

term on the left; this term may be regarded as a forcing function due

to the mean strain.

Now, we express each variable quantity, f" , as

f" = ?+ f

where

f"(x) /G(x-x'

(5.5a)

(5.5b)

Note that ? is now the filtered f" field. The l u" term in (5.4a)

is filtered using the method described in chapter II, giving

+ . .u! djc

J

"2T Vj + °(AA) (5>6)

37

-The model for the residual stress, R.. , Is taken as

(5'8)

where now S.. is the total strain-rate,

1 \ a - a - /S = i j_£_ (ui+Ui) +^£_ (n +u )j (5.9a)

Since we expect the strain-rate to be dominated by small scales, the

eddy viscosity, v , was evaluated using the vorticity model,

VT - (CVAA)2 GYV" (5.9b)

a) • curl _u (5.9c)

Since the imposed flow is irrotational, v_ is based on the total RMS

vorticity.

Then, filtering (5.4), and again neglecting the viscous term, the

following equations are obtained (compare (2.28)).

,23u,

3t 3Xj{vj+lKoyj ) -

(5-10a)

9uT-i - 0 (5.10b)9xi

where _

P - + R (S.lla)

38

F. are the terns through which the major effects of the mean strain come

in. These are

Note the appearance of a Leonard correction term.

For the computation, the difference form of (5.10) is used,

6u

IF ~

- G(P) + F± (5.12)

where F is the difference form of F. and

(5.13)

The exact expressions for were used. As before, the Poisson

equation for P is obtained by operating with D on (5.10a),

DG(P) - D(h±) + D(F±) - -£ D(u±) (5.14)

where . f ( AT g2

' 6x7 uiuj + itJ \ K. K.

The space differencing and the time advancing schemes are the same as

those explained in Chapter III. Periodic boundary conditions in all

three directions were imposed, and the same solution procedure as

described in Chapter IV was applied.

5.3 Anisotropic Initial Condition

Anisotropy in grid generated turbulence is not negligible in many

experiments (e.g. Grant and Nisbet 0-957), Uberoi (1963), Tucker and

Reynolds (1968)). Therefore, to make the initial condition reasonably

39

close to the experiments, it was felt desirable to generate an initial

field with anisotropy. A method for this will now be described.

Suppose that the u and v components of turbulence energy are

equal while the w component is different,

<u2> - <v2> (5.16a)

<w2> - (1+R) <u2> (5.16b)

where < > denotes an average over an ensemble of experiments. For

the Tucker-Reynolds experiments , w is in the mean flow direction and

R = 0.45. Now let us decompose w into its isotropic part, w_ , and

anisotropic part, w , such that

and

 - < v 2 > - < v 2 > (5.17a)

w = WT + w (5.17b)

If we assume that the isotropic part can be generated by the method in

Chapter IV, then, for continuity to be satisfied,

D (WA) = 0 (5.18)

This is a crude assumption. However, unless we know more about the

initial turbulence structure, this is perhaps the best we can do. Now

(5.18) in Fourier transformed space can be written as,

(5.19)

where "" denotes a Fourier transform and kl is defined by (3. A).st J

Therefore, w can have a non-zero values only when kj «= 0 . Then,

following the same procedure discussed in Section 4.4, we get

40

2. / p ?\1^2wA(krk2,0) - (3 $ ) (a+ib) (5.20)

wheref» A»/\ /\/\ /*. /V ^ ^

q = uu* + w* + WTW* 4- w.w*I I A A

and a • cos 9 , and b a sin 6 are obtained from a random angle, 6 ,

with uniform probability from 0 to 2ir .

The initial condition for the first run was generated by this pro-

cedure and, for an input R of 0.45 , the generated field emerged with

R = 0.43 . This field had shearing stresses not present in the actual

flow, and so the second run was made using an isotropic initial field

generated by the method described in Chapter IV. Further studies in

Section 5.5 and 5.6 are based on the second run. Both runs are reported

for completeness.

;

5.4 Results

The results of the following two cases are presented.

(1) Anisotropic initial field

<w2> <w > _ - ..- 2" - ~^r - l-43 <v >

(2) Isotropic initial field

<^2> = <v > = <w >

For both cases, the mean stream speed was taken as w - 240 in/sec,

T *• 1.457/sec., A = 0.59 in and At = 5.36xlO~ sec. This corresponds

to a Courant number wq /3 + IT At/A £ 0.15 . The 16 mesh system

was run using the vorticity i

the isotropic decay studies.

was run using the vorticity model, with C = 0.206 as obtained from

41

For convenience of reader, three of the Tucker-Reynolds measurements

for the turbulent intensities, <u, > /w ,. the turbulent energy ratios,—2 2 . o 2 —9 2<ui >/q , and the structural parameter, 1^ = {<v > - }/{<v > + } ,

are replotted in Figs. 5.2, 5.3 and 5.4 For the case 1 , the same quanti-

ties are plotted in Figs. 5.5, 5.6 and 5.7. Here, for comparison, the

time in computed results, t , are converted into the downstream distance,

Z , by Z « W t . The behavior of these are comparable to Tucker and

Reynolds results. However, when the strain is turned off, the rate of

return to isotropy is much slower than in the experiments. It is inter-

esting that this is consistent with the feelings of some workers that the

return to isotropy .in the Tucker-Reynolds flow is too rapid (Reynolds

1975) , perhaps because of defects in the experimental simulation of

homogeneity. The same quantities for case 2 are shown in Figs. 5.8,

5.9 and 5.10.

One-dimensional energy spectra were obtained in a similar manner

to that described in Chapter IV (see Tennekes and Lumley (1972)). The

only difference is that the shell average in Chapter IV is replaced by

a plane average in wave-number space, i.e.

E(k,) = -% EE u(k ) u*(k ) (5.21)11 -1 IT k2 k3 . "* "*

Here, the notations are the same as before. These spectra were computed

at three different times or downstream locations (Figs. 5.11 and 5.12).

At zeroth time step, ¥,, (kj) , E.2(k2) and ¥,3(k3) are almost identical,

as they should be. By the end of the straining period (75th time step),

Ej, and E22 have become quite different. E... is flatter on the large

eddy side while both small-scale spectra are nearly the same. Over the

last period, in which there is no strain, the spectra approached one

another very slowly, as seen by the spectra at the 125th time step.

The calculations were run on a CDC 7600, and required approximately

7 minutes for each case. Storage requirements were similar to those for

isotropic decay.

42

5.5 Energetics of the Filtered Field

Multiplying (5.10) by u. , taking the ensemble average, and

assuming homogeneity, one finds (compare (4.11))

3q2/23t

where

q *> <u.u> (5.23a)

(5'23b)

£ and e. are the dissipation terms discussed in Section 4.9. (p and

/p are the production terms. Note the appearance of a Leonard production

term, QX , that comes from the non-linear interaction between the mean

strain and the filtered field.

The computed behavior of these terms is shown in Fig. 5.13. Note

that Q} contributes significantly, particularly where the anisotropyLIis large near the end of straining period.

5.6 Assessment of Turbulence Closure Models

Turbulence computation of the conventionally averaged (ensemble or

space) quantities has been based on some ad hoc closure models with a

number of adjustable constants. As pointed out by Reynolds (1974a,b),

more systematic approaches are desirable for generalized turbulence

models. Even though laboratory experiments provide actual quantities,

experiments are limited because important properties like the pressure-

strain correlations are difficult to measure directly. On the other

hand, computer-generated experiments provide a vast amount of data on the

flow field, and hence the numerical experiments can be used to study

the closure models. We have attempted to study the pressure-strain terms

and other statistical quantities using the present computation. Even

43

though the 16x16x16 mesh calculation gives good results for the energy

components, as will be shown shortly we cannot use the computer generated

field to compute the pressure-strain term directly, at least not in the

present calculation.

The exact Reynolds stress equations for homogeneous flow without

mean deformation are

dR

IT - Tij * Dij (5'24)

where T . «• </ 3u. 3u.

3u. 3u

Here u. denotes the turbulent components of velocity and p is the

turbulent pressure divided by the fluid density.

The pressure-strain term T.. is responsible for the return to

isotropy following removal of strain. The modeling of T. . has been

the subject of much discussion. Since no direct measurement of this

term is known, we tried to estimate this term using the present compu-

tation in the return-to-isotropy portion of the computation.

The computed pressure-strain term, (T .) , was obtained from

<Vc •<p^ + ;> (5-25)

Here < > denotes an average over the flow field. The fourth order

central differencing scheme was used for the 6/6x. terms.

For comparison T.. was obtained a second way. Using D . = 2/3 e

(see Reynolds (1974b), (1975)), T was obtained from the computed R..

history as

(T*4>» " TT <"J 4> + TEfi^ (5.26)

e was estimated from the energy equation, which in the absence of strain

is

d <u.ue o -- |-i-_ (5.27)

e agreed well with £_ + e , computed directly. The time derivatives

were approximated by a second order central differencing formula.

Finally, we predicted T . by Rotta's (1951) model for T . in the

absence of mean strain,

where A is a constant ando

We used A = 2.5 , as suggested by Reynolds (1975).

These three results are shown in Fig. 5.14. It appears that (T..)

is quantitatively poor. This is attributed to the coarseness of the3

16 mesh . We conclude that T . undoubtedly contains some Leonard-like

terms, and this must account for the difference between (T. . ) andij c

(T. .)„ . While T. . can be estimated from the R. . equation a laij K ij ij(5.26), it cannot be computed directly from the calculated field with

3such a coarse grid. A repeat of this work using a 32 grid is recommended.

This should be accompanied by a careful analysis of the Leonard terms

arising in the R . equations.

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

In this thesis we have developed the basic approach to computation

of three-dimensional time-dependent turbulent flows. We have seen that,

with a modest 16x16x16 mesh and a residual scale stress model, many

interesting features of experimental flows can be computed. Work remains

to be done in the development of better approaches for using this type

of computation to assess turbulence model equations, and to extend the

procedure to other flows.

It would be informative to study the effect of the initial spectrum

on the rate of return to isotropy. This might be done by removing the

strain at different points along the straining portion of the Tucker-

Reynolds flow, allowing isotropy to be restored from points with different

spectra.

In extending the method to other interesting flows, problems to be

resolved include the handling of non-periodic boundary conditions, solid

boundaries, and free boundaries connecting the region of computation to

irrotational flow outside.

One useful problem to study would be the case of homogeneous tur-

bulence near a wall, without mean shear, for which some experimental data

exist (Uzkan and Reynolds (1967)), and would be the diffusion of tur-

bulence into a non-turbulent region, again without shear. It is recommended

that experience with these simple problems be gained before a more com-

plex flow is attempted.

When one moves on to handle flows like jets, wakes, and mixing regions,

it should be possible to take advantage of the fact that the flow outside

of the superlayer is irrotational, and to use the exact solution for

unsteady irrotational flow to extend the calculation to infinity out

beyond the mesh. Care must be taken that the numerical scheme does

not produce vorticity in an irrotational flow; the diffusion of vorticity

by VT will also have to be handled in a way that prevents its diffusion

into the irrotational external field.

46

Eventually it may be possible to treat practical flows, such as

boundary layers, wakes, combustion, etc.; by these methods. But much

more effort should first be devoted to fully understanding the nuances

of the residual scale models, grid-schemes, differencing schemes, filters,

etc. that are the bases for this type of numerical simulation.

47

G.

•1.0 A -.5 A 0

xrx',

Fig. 2.1. Gaussian Filter

.5A 1.0 A

48

10-

10'

CMU

if)Eu

UJ

10

1 I I I I l I I | I I I I I I I

\- EXPERIMENTALSPECTRA,E

FILTERED SPECTRA, E

I I I I I I I I I I * I I I I I \ II 1.0

k (cm'1)

Fig. 2.2. Filtered Energy Spectra, ¥(k,t)

10.0

AA - 2A (see Eqn. 2.23)

4

n

8

Fig. 3.1. Comparison of Modified Wave Numbers

(see Eqn. 3.4 and 3.7)

50

(KA)'

Fig. 3.2. Comparison of Div(Grad) Operator2

k£ = exact

k1 » fourth order (see Eqn. 3.15)

kj2 - fourth order (see Eqn. 3.12)

N 16

51

GRID(meshM)

Uo='0m/sec = I.27U0

<^\\\\\\\\\\\\\\\\\\\\\\\\\N

Fig. 4.1. Sketch of Wind Tunnel Test Section for aGeneration of Isotropic Flow.

52

REF

Fig. 4.2. Determination of A or B Vector

53

I05

CMO

(OEu

UJ

10

I I I I I I I I | I I I I I I I '.

-5/3

(71%OF TOTAL ENERGY)

(80% OF TOTAL ENERGY)

I I I I I I I 1 I I I I I I I I I1.0

k (cm'1)

10.0

Fig. 4.3. Unfiltered Initial Energy Spectrum(Data by G. Comte-Bellot and S. Corrsin 1971)

54

10s

o0>in

10EoLJ

10

1 I 1 I I I 1 I 1 I I I 1 I I I L

= 42

= 86.5

-5/3 -

EXPERIMENTAL DATA

COMPUTED

I I I I I I I I I I I I I I I III 1.0

Mem'1)

Fig. 4.4. Evolution of Energy Spectra16x16x16 Mesh: Smagorinsky Model:

A. = 0 ; A = 1.5 cm

10.0

55

IOJ

10*

ego

wEu

UJ

10

I I I I I I I I I I I I I I I i .

X V ° - A ?\ s —rr- = 42

- EXPERIMENTAL DATA(FILTERED)

COMPUTED

I I I I I I I I I I I I I 1 I I I.1 1.0

k (cm'1)

10.0

Fig. 4.5. Filtered Energy Spectra16x16x16 Mesh: Smagorinsky Model:

A. = A ; A » 1.5 cm

56

I03

I02

CJuo>m\

10Eu

10

\

\°\\\\\

s>COMPUTED POINTS

*\\\ -O SMOGORINSKY MODEL » \

j\ \

A VORTICITY MODEL j? \

EXPERIMENTAL DATA *p I(FILTERED) V '

I I I I I I I I I | I I I I I I I.1 1.0 10.0

k(cm" ' )

Fig. A.6. Filtered Energy Spectra — A Comparison ofSmagorinsky and Vorticity Model16x16x16 Mesh: A = 2A ; A = 1.5 cm

57

1 I I I i i r i i i i i i i i L

I0k

u0)

inEu

UJ

10

= 42

EXPERIMENTAL DATA(FILTERED)

COMPUTED (WITHOUTLEONARD TERM)

I I 1 1 I I I I I I I I I

Fig. 4.7.

1.0

k(cm~ ' )

Filtered Energy Spectra-Effect of Leonard Term16x16x16 Mesh: Vorticity Model:

10.0

= 2A 1.5 cm

58

I03

I02

CVJu«in

toEuUJ

10

1 1 1 1 1 1 1 1 1

/ \/ \

/w V

^^ — ^^

/px ND \!/ \Q \d ^ \

^ \

\\

\ \\ \

\ \\ \

A \EXPERIMENTAL DATA H .(FILTERED) A \

\ \D COMPUTED POINTS

\I I I I I I I I I I I I I I I I I

1.0 10.0

Mem'1)

Fig. 4.8. Filtered Energy Spectra32x32x32 Mesh: Smagorinsky Model:

A = 2A ; A = 1.0 cm

59

3U§

10'

.9

.7

.5

.4

30 40I

I I I I

I I50

tU

M

60

3.5

I I70 80 90 100

Fig. 4.9. Sensitivity of Filtered Mean Square VelocityDecay Rate to Smagorinsky Model Constant, C

B

16x16x16 Mesh: A. 2A A - 1.5 cm

60

3U§

1 1

10*

.9

.8

.6

.5

.4

T T

30

COMPUTED POINTS

O SMOGORINSKY MODEL

VORTICITY MODEL

EXPERIMENTAL DATA

ACTUAL FILTERDDATA

I I l I

40 50

tU0

M

60

- 3.5

7Q 80 90 100

Fig. 4.10. Decay of Mean Square Filtered Velocity16x16x16 Mesh: A. 2A 1.5 cm

< > : Average Over all Space

61

I0«

.9

,2 .8

.6

.5

I I

EXPERIMENTAL DATA

D COMPUTED POINTS

30

Fig. 4.11.

SHIFTED DATA

• ACTUALFILTERED DATA

UNFILTEREDDATA

I 140 50

tUpM

60

- 3.5

70 80 90 100

Decay of Mean Square Filtered Velocity.32x32x32 Mesh: Smagorinsky Model:

A. = 2A ; A • 1.0 cm

62

inrd

CO

ACN

'5S^x

V•-»A

U

0)CO(U

9)

X\D

NO

60•H

63

-.5

-.4

-.3

-.2

-. I

30 40 50 60

tU0M

70

3.5

80 90

Fig. 4.13 Comparison of Skewness, S . (see Fig.4.12 forDefinition of S ): Smagorinsky Model:

2A

100

64

Fig. 4.14. Components of Dissipation.

16x16x16 Mesh: AA = 2A ; A = 1.5 cm

Vorticity Model.

e - e0 + e. at (U t/M - 3.5) = 42o R L o

65

PARALLELFLOW

W0 = 240,in/sec

CONST RATE" OF STRAIN

PARALLELFLOW

Fig. 5.la. Schematic of Wind Tunnel Producing ConstantRate of Strain in x-y Plane

Fig. 5.1b. Equivalent Representation of the Plane Strainin a Box T « 1.457

66

CONST RATE OF STRAIN

200

(Z-HO) (in)

Tucker-Reynolds Flow.Turbulent Intensities. The solid lines are fordecaying turbulence in the absence of strain.Z = Downstream distance from the beginning ofstrained section.

67

.8


A A

OOQ

°°°0°oo0

PARALLEL FLOW

D D

I I

<v2>/q2

D D D

A

<w2>/q2

O O

<u2>/q2

40 80 120DOWNSTREAM DISTANCE (in.)

160 200

Fig. 5.3. Tucker-Reynolds FlowTurbulent Energy Ratios

68

1.0


.8

.4

0 -

PARALLEL FLOW

-.2 I-40 40 80 120

DOWNSTREAM DISTANCE (in.)160 200

Fig. 5.4. Tucker-Reynolds FlowChange in Structural Parameter, TL

Kl<v > - 

<v2> + <u2>

69


200

Fig. 5.5. Simulation of the Tucker-Reynolds FlowTurbulent Intensities. The solid linesare for decaying turbulence in the absenceof strain. (Anisotropic Initial Condition)

Z = W to

70

.8


D

8

.2O

I

A AA

PARALLEL FLOW

<w2>/q2

0 o 0 oooo

<U2>/q2

000

I I I40 80 120 160

DOWNSTREAM DISTANCE (in.)200

Fig. 5.6. Simulation of the Tucker-Reynolds FlowTurbulent Energy Ratios under the PlaneStrain and the Return to Isotropy inParallel Flow. (Anisotropic InitialCondition)

71

1.0

.8

CONST RATEOF STRAIN "

I

PARALLELFLOW

.6

.2

I I I _L40 80 120 160

DOWNSTREAM DISTANCE (in.)

2OO

Fig. 5.7. Simulation of the Tucker-Reynolds FlowChange in Structural Parameter.(Anisotropic Initial Condition)

72

I I I I I I I \ ICONST RATE OF STRAIN '

IPARALLEL

FLOW

/W0 —

100 200(Z+IO) (in.)

Fig. 5.8. Simulation of the Tucker-Reynolds FlowTurbulent Intensities. The solid linesare for decaying turbulence in the absenceof strain. (Isotropic Initial Condition)

V

73

.6 I I


D

D

A A A A *

O

I

I 1

PARALLEL FLOW

<w2>/q21A A A A A A A A

o o o.2

I I40 80 120 160


Fig. 5.9. Simulation of the Tucker-Reynolds FlowTurbulent Energy Ratios Under the Plane Strainand the Return to Isotropy in Parallel Flow.(Isotropic Initial Condition)

7A

1.0

8

.6

.2

I

CONST RATEOF STRAIN

I

PARALLELFLOW

0°0

o

I I I I40 80 120 160


Fig. 5.10. Simulation of the Tucker-Reynolds FlowChange in Structural Parameter, IL(Isotropic Initial Condition)

75

\

I.O

E,,<0;0)

.1

.01

Oth TIME STEP

75th TIME STEP (ENDOF STRAINED SECTION)

125th TIME STEP (ENDOF PARALLEL SECTION)

1 1 I I I I I 1 I I1.0

Fig. S.lla. One-Dimensional Energy Spectra(Isotropic Initial Condition)

76

1.0

£22(1*2; t }

E22(0;0)

TT| I I I I I I I I I I I

Oth TIME STEP

75th TIME STEP (ENDOF STRAINED SECTION) \ \

\— 125th TIME STEP (ENDOF PARALLEL SECTION) \

I I I I I I I I * I I.1 1.0

Fig. 5.lib. One-Dimensional Energy Spectra

77

1.0

33E33(0;0)

.01

I I _I I I I I I 1 I

I I I

Oth TIME STEP

75th TIME STEP (ENDOF STRAINED SECTION)

125th TIME STEP (ENDOF PARALLEL SECTION)

I I I I I I I I i \ V..1 1.0

k, (in"1)

Fig. 5.lie. One-Dimensional Energy Spectra

78

1.0

EM(0;t)

.1

.01

AT 0 IN DOWNSTREAM(Oth TIME STEP)AT 96.5 IN DOWNSTREAM(75th TIME STEP)AT 160.8 IN DOWNSTREAM(125th TIME STEP)

I I I I 1 1 11.0

k, (in'1)

Fig. 5.12a. One-Dimensional Energy Spectra atThree Different Downstream Locations(Isotropic Initial Condition).

79

1.0

E2 2 (0; t )

.1

.01

i i i i i i i i

AT 0 IN DOWNSTREAM(Oth TIME STEP)

AT 96.5 IN DOWNSTREAM \(75th TIME STEP) \AT 160.8 IN DOWNSTREAM(125th TIME STEP)

I l I j l I \1.0

k2 (in"1)

I I

Fig. 5.12b. One-Dimensional Energy Spectra

80

1.0

E3 3(0; t)

.01

l l I l l l l l

AT 0 IN DOWNSTREAM(Oth TIME STEP)AT 96.5 IN DOWNSTREAM(75th TIME STEP)AT 160.8 IN DOWNSTREAM(125th TIME STEP)

I l I I 1 I I l I.1 1.0

k3(in~')

Fig. 5.12c. One-Dimensional Energy Spectra

81

4.0XICT5

2.0

-2.0

-4.0

-6 O100 120 140 160

DOWNSTREAM DISTANCE (in.)180 200

Fig. 5.14. Comparison of the Pressure-StrainCorrelations Normalized by Wand Computational Box Size,L = NA(see Eqns. 5.27, 5.28 and 5.29)

83

APPENDIX I

ON THE FOURTH ORDER CONSERVATIVE SPACE DIFFERENCING SCHEME

To explain the difference formula used here in detail, consider the

following equations.

!T+a^(ttiV ' ° (A'1}

3u

3x

(A.I)

3uui IT + ui

" ° (A>

Integrating (A.4) over all space

dTv v -

If div u = 0 ,

(A.6)

Now look at difference form of (A.3)

6 Vi J_6t 2 "Ui 6x uiuj (A.7)

Summing over all mesh points

84

The RHS of (A.8) has to be zero as in (A.6). Therefore 6u u /6x must

be devised such that RHS summation of (A.8) goes to zero thus conserving

the total kinetic energy. The difference formula for 6UJU./6X. depends

on the type of mesh and the number of neighboring points used.

The following is a fourth order energy conserving scheme using five

points in one direction. Following usual convention

—*f* - i[f(x+f)+f(x-f)]

(A.9)

Note that

I

(fi-2 - 8fi-l + 8fi+l -

Now fourth order momentum and energy conserving form of (A.I) and (A.2)

are

-^ + |(^Ju^) - i. (U2xju2xj)2 = 0 (A.ll)

0*> -* •*• J **, J J. J £.J±t

x 2x±

' * "i 3 i 2xi

To show that these are indeed energy conserving, work with the expanded

form of (A.ll).

85

l,-2x-2xx

+ <-r< l-2z-2z.(A.13)

where u« is either u, v or w . u. x (A.13) gives the energy equation.

Then sum over all mesh points.

I m _y^ /^/~x~x) _ l/-2x-2xv )V. f\^ l£^^ I 0 I ^ P TF ^ P x V I

+ s v and w - component ? I

-2x . , — x

.

+ v, w, - component

^U£UA ) f4 -x 1 -2x1-£—j L 3 U x - 3 U 2 x J

+ fl ^ . I-^2z 1 }+ l 3 W z 3 W 2 z J (

1 -2yl- 3 V 2 y J

(A. 14)

86

The first summation on RHS of (A.14) is zero for periodic boundary

condition. Therefore (A.14) becomes

^ *"() "2.ufi. ( I£u£ It " -£~^)div u in 4th order; (A.12)J (A.15)

So if the numerical divergence of velocity is zero, the RHS of (A.15)

is zero and (A.11) is indeed energy conserving.

The accuracy of the LHS depends on the time advancing scheme and,

as mentioned earlier, the Adams-Bashforth predictor method introduces

very weak instability on computational mode. I'o show that (A. 11) and

(A.12) are really fourth order accurate, work with a typical term:

-x. 1,-2x-2xxu ) - v u )

V(k) 24ZT (vu)(k+2) - (vu)(k-2)

+ U(k) (v(k+2) ~V(k-2)) +V(k) (u(k+2) -"(k-2)

Substitute RHS by Taylor expansion as usual. After some algebra, it can

be shown that

4 —Y—Y 1 "7 Y—9Y/ « "\ •*- / **A 4»A\•r(v u ) - -r(v u )_J A O ^X

ft A4 ( /• ^T <vu) - |TT (vu)v -f vv + uv + ... (A.16)

Therefore it is indeed a fourth order scheme. An extension to higher

order differencing scheme can be done on the same basic idea. Overall

fourth order accuracy is obtained for advective term by using second

order energy conserving scheme for the Leonard term.

For convenience, fourth and second order advection term is

recapitulated below.

87

energyfourth order energy

~ N . J- (u., i- (u.v) + 3z i

„ 3- (u u) + 3y i' « 1

4- v Cu(uiV) Cl-1^4

w ( <.»i

N vi (**

(A. 17)

88

Second order scheme:

3 , v 3 3 3•r — (u,U. ) » -r— u. u + T— u.v •+• -T— u.w3x . 1 j 3xi 3y 1 3z 1

is (uiu) - ( u u ) + u

ui

ui

w

The subscript is shown whenever it is different from (k,£,m), i.e.

9 U(k+l,£,m)

89

APPENDIX II

FLOW CHARTS AND PROGRAM

Overall Flow Chart

INITIATION(SUBROUTINES INICON & START)

CALCULATE EDDY VISCOSITY(SUBROUTINE VISCS & VISCV)

CALCULATE fii (EQUATION 3.32)(SUBROUTINE VELOC)

CALCULATE RHS OF POISSON EQUATIONFOR PRESSURE

(MAIN, SUBROUTINE DIVGCE)

IPRESSURE FIELD SOLUTION

(SUBROUTINE PRESS, FFTX, FFTY AND FFTZ)

ADVANCE ONE TIME STEP BYADAMS-BASHFORTH METHOD

(MAIN)

DATA REDUCTION AND PRINT OUT

WRITE DATA ON TAPE FOR CONTINUATION

90

Flow Chart for INICON

READ INPUT

INITIAL PROBLEM ?

YES

READ IN FILTERED EXPERIMENTAL SPECTRUM

DETERMINE AMPLITUDE OF THE VELOCITYFLUCTUATION (EQUATION 4.5)

DETERMINE A, B, a AND b (EQUATION 4.7)

DETERMINE VELOCITY AT CONJUGATE POINTS(EQUATION 4.8)

INVERSE TRANSFORM THE VELOCITY FIELD(SUBROUTINES FFTX. FFTY, AND FFTZ)

READ IN DATA FROM TAPEFOR CONTINUATION

PRINT OUT THE VELOCITY FIELD AND ENERGY

91

• DEC* *MNPROf. t fA* MAIN ( INPUT, niiTPHT.

c » * * * * * * * » » * » « * « * * * * « » * * * » * » » * * * * « + * *C T H T S cn*T9r>L«i THF nvE^ALLC TV-F cO*p!.'i Al T0*< IS PF.ecris^C O A T A FRP* ThF I &PCF CP°e "

C . S T A J T 5 T K « | P ' i iN 'T ITJESC Tine, v F | C - C 1 T T f : S flNO THF.C

CORF PYr>F COMPnTI* Tnf S

IT THE ENf5 DF F A C H TI«t S T E P THEniT. AT THE FNn nF THg CO^PUT

Hf tK 'O SIDE OF Mp-iEMUM FQUiTTnMg ARE

CCCCcCCCCCCCCCC

./(1«U.*OELT»*«2)

,/(U.*rFLTft**3)a. /DELTA

cl ,/(12.«rF.LTA)

COfcf »*?./( 3, *PT)

/f Ufl.*r>ELTA)./fue.*PEI.TA)-

CPFB).n F0» THE FJPST TIME

'-'F.AK- COMVECTTVF

INITIATION

*1.S

VELOCITY

f All IMcow , rPFF;>,r.nFF3,cnEFo,CnEe5,cnEFfe,CnEF7.COEFi»,COEF<)

ORIGINAL PAGE ISOP POOR QUALITY

92

COFr

pf T f t s r

C

C00 300 T P < E « T S T A R T , T E W r »

C..-.-COMPUTE2*1

V I S C O S I T Y " F "

T R S N S F E K D A T A FBOM

S*.A|. LT ' - j r t ! ( l 1,1)Sf-ALl . P' ( V ( l ! ,1)SMALLS f t'( 1 1,11S *• A11- P (" (i 1,2)S ' l A l L T M f V ( l 1,2)

S H & I L I K (*( l 1,2)co roo

TO SCM FOW THE PLANFS L«L.MAX

1 .I " ' - V ) , ? S 6 )1,1 ).?56)

v ^ t i*"'()Mf (1

IF fl. '.F.o. IVAJO LD1«1T U 4 K - S F E W V E L C C T T I P S F O P

t. JN f U f l . 1 ,?P l ) . ' J» i r i . 1.1.011. 256)M rVf 1 . t ,Zp! ) . V K (

T^FM ^F TAK r r"- 'PUTF EPPV V I S C O S I T Y AT

r.n TO (2n,25) Mr'noFL?o C A L L V T S C S ( Z O , 7 > ' t , 7 f 1 .7.COEF?)

GL TO ?0?« : T A L I V I S C V C Z O , 7 M 1 , Z P 1 , 7 , C O F F ? )jn r.CNTlwi. iE

THf F.HPV VISCOSITY OK; THE PL-*NE*L ISsf ALL PUT (Kd ,1 ,Z),Ef i,i,D,?56)

TO C IN LCM

700M(I). THEM ST09E

ZOr?

• I.'-'S A L I.'

fUClfv

I L 1 '• Pi (V(

s M A i L11. n.(S'MA| LI'-1 (" <** A| LI K (»" {

1 , 1 11,1 )1,1),3)

!3).1 )

TN GU, GV

V(1,1 ,L"iMn,t,LI' 'd ,1,1V^'f 1 .1,1"•'•"(l.t ,1 1.76P)F- (1 , 1 .F(l,l,t),Sl2)

12^121

PAGE93

HO 700 PI.AN'E«J,lC A L L VFLOC ( 7 O,

» i.3i, (••* M.I ,fSK.f \, J.I) , 01 (I, t, PLANE)* 25*)<S1GM,1 ,?),*?n ,1 ,Pl»HF.>,2*><O

PI tPHESSHRF. f r iHlKJRl lTtO*1 HV y f A M MOTION

t , 1 ,U) ,r,H1,1 ,*)

, j , /S) . r , f

(L ,r,T.

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7t>o COM IKI.'E

C Ct.mpUTI? n i V f G l ' ) . THEM fiFT GOtV U IS STORED IN R.

.1 fvn

s* ALL IN

j .D,r ,vf i

1,31, Pi ' f 1

f * f 1 . 1 . 3 ) , P " ( 1 , ).76fl)

T A L LS^AL I CUT

71

7 P S 7 P J77t

¥ ( 1 , 1

(L .GT. L " « X ) L«l.-LMAXLLT' . fun , 1.ZP?1.e<'M,

JV f V ( J , I , /. P? ) , P.V f l , l , t > ,

770 C<c..-«-n(r,u) is I M

, 256)

ORIGINAL PAGE ISOF POOR QUALITY

94

no 760 LsJ,LHA)f0000PHl,J.t>aCnEF»»*G(I.J.l)*P*(T,J.L)

790 CONTINUEC.....WO»» C TS TFMPOPAHI I -Y STO»EP |N PI

F M

^ ( ' ( 1DO t 77 f t PI »N'CALL nivr,cfl

i. T

1 , L « A * - 1 ^

I ,l*At-1)I , 1 >,76«)l,1).76>i)1,1).788)

TF f t .GT.

fu ( i ,1 , .n n.i .1v;i (i ,1 ,i«i n, J.i

1770

TALL P«E9S .C.---->.-r>i'< STOwf PI TO PI F<?0" P« AND SWIFT G FRO* Pi TO

no J P M O L » I , L ^ A Xj, j , i ).PI r i ,1 ,L).i , i ,2), PM( i, t.i.(i ,1 , 1 ) . R ( 1 .1 ,L).>56)

PPlvT 90300 UOO Lel,l«»PRT.MT <70!J, L

?, (E(T,lO,U»I«1,Ul000

'401

PRINT 917f-0 iiOj ,is t

P » E S S ( C O f c P 3 » C O E P n ,roEF71>"»v,h,P AT FTvF STFP

(>'STO»F .ff). 5) GO TO 110

r.n TO M2o\ 10 MTJ^E nTTHE-1

W H I T E (9) NTIMF, ,U" .VM,WM.PM,PI

FND FILE 9

95

j?o C

TO «OU»''. TMfN &.OV4NCE 1 STEPno poo L«I,IHAXJ

. 7*5, 7«5.?«5,7«5.7flS.7»5,7»S. 785. 7*5.785,

I. MI «L VAXr.O TO 7«S

r.O TO 7fl?

CO TO 7957*0 I Plel

PO «00 Jm\

J°1eJ»l

r.O T" (7P6,797,7«)0,790.7«>0,7«>0,790,790.7PO,790.7<»0,790.790,7<»0.J

%Tf'. JsJvixr n T i i 7 o «

GO TO 790« JP2sl60 TO 790

7?9 JPlal

790

JPlalttTP?eIt?C,0 TD f 79 1,79?, 79?. 795, 795. 795. 795, 795, 705, 795, 795, 795, 795, 795.

791

r,o TO 705792

GO TO 70S

r.o TO 705lI

79S tr i"-(i, n«Gnri.j,i1-P"(TP?,J,L))

- )-Pf (T ,J ,LP2M

96

V * ( T » J i U » V M ( T , J » U t O T « ( r n F * « J ( ) U M f I . P > - ^ .S*°V ( I , J .L) )» J . U « i » M ( l , J .» J ,U*"C> '""U»1

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rO ?3o

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VT K V B T K V * T f t l V

vSUM«o'.

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V 5 '.' * S \, S I' - + R 0 1 1 " f L f 2 )

»"r j l1M(Lf 31

P W I V T Q 1 000*

T ! u E , T K M ,EaT 1"E*DT

T K D ] V s l , / T K S U M

T«iJlT H V 1

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K » a T K u * iv o S

DOT Ml 9 J f>, I'k, VW, Wf,'J'«lr,....nl«iSlP4TTO'>' lERMSj LFOM4BD, *GS. TOT4C

97

ORIGINAL PAGE ISOF POOR QUAUIY

DO0000

2 U 1

P H U T o i \ , o L f « . - , o s r , s , o T n TC- — •Si'.Ei--Nfc?s C H E C K

S"3eO.

no

TF (I . U T . ?) r ,0 TOr,0 Tl.' (2«i 1,2521 I

?5t

GO TOTN?sL.GC 10I l s T ^IP f t .LT . I I ) 00 TOT 2 s T « A V - T * tr ,n TO (25I"?slr-0 TC 25*

25»> C O ^ T U ' U Er«0 2*"0 J*l ,.Jno ?^n Lei ,L

?t>0

P R I N T O J ? , S KPO TO (270, 2*0)

?7f t ^ F I L T * 2RO TO 261

C O N T I N U EP W J v T QO/>

Q05

90?, (""n.fM.). Ie l ,MH4LF)"T 00?, fV»n , in ,L ) .wT QO?, ( « V f I , IP, I . ) .

P R I - v T gn? , (Pm. ' . i (L ,n , I« ! ,3 )«io r ON T i MI IE

P«I '»T 00?,300 ro^TiMne

W R I T E f O )F I L E «)

TO 98ORIGINAL "OF POOR

320(JO? FU»"»T (X, MF.1«t»».xnO Q 3 FC"? fAT f / X / , X , * t ' P O Y V I S C O S I T Y A T J«10M

'

005 F O C - 4 T ( / / / , X , * v E " L n C T . T Y AT .trlfl I U« *)906 F P K - w & T (X, 1 H« , SX , 0.-ii»« J/JSM-I, , F 1 'J . 7 , 5V , 9HV**? /9 ' JM« , £ 1 U , 7 ,

1 " j j y , q w w » « ? / s : i k ! s , E1-'1.7, Sy. • K 1 » ( V ? » 1 l ? i " / ( V ? t i i ? ^ x « , F1" .7 , < »X,1H*)

908 F ''"•' " * T f y • M'"i-*****«*** + * *«** * * * * * * * *«*»** * * * * *»** * * * * * * * * * * * * * * *

• * * « » * * » • * * * * >

UU.7, 5"» •••? =•» H 1 U . 7 . * Sl'"3«, E lU .7 , !«»«, \H«1910 F T P - i T ( 1 H 1 )

. * T O T 4 L s « , ?!?.«?,(x, J^ i , ^x , *BC4l . T T « f a « . Fs'.a. ex. * l) I ST

*M'J.7, u>. I ?H*»<?/i . . , ,«*? a ,E1a .7 . '4X. 12H^*»?/-.- iO*«? e , F 1 a , 7 ,

17 ei'P"AT ( ///, x, •H»pv vlSC nSTTV AT

1 fl f ••'!•> 4 r (X, *Js», 13)STOP

C * * * * « * * * + * * * * » * * * + » » + * * * « * * * * * * * * » * * * * * * * * * » * + » * » * * * * * * * * * « * * * * * * * * * * * *C T H T S t- 'Ji-^nijT IN? T ' T T I i T E * THF. f3np"AM. pn1* S T a ' J T t M r : P»nwLt M » ™f-: IMl-*C T I» I . F rein is t ? F . K i t C A T F ^ . pr» r;f1k 'T 's" | 4TI r t- 'Vj p"nHFH, THE f > 4 T * S T O R E D *C ON T»Pi : AT Twp F.Mf( r>F TKF pftF.vlnnS cil^ AOE «r»0 IM'. *

C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

, 16, 1 h) .'"S ( 1 6. 16.. v I f 1 <•, 1 ^ , 1 6) . • I (

O . o ' (1 *>, 11>. 1 *)) , M 1 ( 1 ft , 1 (S , 1

i- 1 Mf; :-.S T i'.(. W ( 1 *., 1 f,. J hi , r,, 1 fe , • •.. 1 t*) , f R ( 1 6 . 1 O J , F T ! \ •-., \ <-.) , T°e ( * • 3 J

r o v " (. > n / r A T A «> /1. o, r, j. T P ^, T <? r

If ,'- 'FFT

C.....f>.ST4t.'T*1 S l A U T I N G FOOM T!'«F.FI»:IM O M E V O H . s -JUKI

IN V-

C . . . . - T S T A R T s S T A f ' T I M r . T J H F STEPT I^E ST i -P

99OKIG1NAL PAGE it>OF POOR QUALITY

C.-C««C.-C«-C.«

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•Cs f 'Onfr'L CONS« - ' 4 v r , s . ~ > f c L T A ( A v F O A i ; i 1 > J G l / n F . L T A r K ' F.SHI--4MSO» » I* F. f;iJATink: (5.16P)

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C C s i .

r 1 , /f 1 u'J.«OEI. T * * * ? J

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" v " r j 1 1 . 3 . i u ] 5 -i >j 5 J S n -i f1 / f J H A j. )•C O t f t 1

T 4 )S T * C O F . F 1 0 / H E L T A

1 2 * F 4 C » C O N S T S

C A L L S T 4 C T (C"PF3 .COEPl 1r.r TO f t , ) non, 50001 ,

1 C l ' ^c l .OnCOh 'TsJin a v»l ,25

TC 1C-- --- !.iC..»«»F> TS THFC A M S O T Q o p i r P A R T .

no S » -B l .2« ,M

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THFN '.s TMTEPV4LL UP Tp 6.0Fni> TWF TSCTQUOJC PAPT. EMJ j.s THE

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ORIGINAL PAGE ffi

OF POOR

10°

no 5 I«I,I*AX

I.' If I.J.DBO.VI(|,J,L)»0.'•ni,J,UBO.»U(T. J.L isfl.tV(l,J,UaO.»*( J»J/L)B0.

«;DO UOU*L

o 50 .let, MM 1

**?lS*

TF (Nsr« ttT. 0.1) GP TO 20

i'-l? .L*. C- . i ) «

GO TO (310,315, 315. 315. 315, SIS. ?15)

J-: w ( <• ) *(• P * v M* t o .("1 )-t'M (•

PO TC 32A

("I )-F*:

1 (•")*FCA*V*'*?.

"AVE l,.03

101

C.....NET Ff" i» t fe» A- 'PLiTur-e PK THP I N T T I » L PTELP AD r«Esc"iREO T f< SEC a.a7

4GG1 «P1 1 *».'?

no TO (33?.3ao) K.'

C ...-r.ET ft * « VECTORC *I°ST CHOOSE

.V Y a tf G P K •( x * )

fii1 TC f /?, I 1) ' CO^T

r-o TO 12

A3*0 .i»cG"nE Xsf Y ! +0 .?

12C DETF"»iU iF A AHO B TV F.CI.iftTlnn; (u'.bl

IK ( T H f T 41

I ' l r j T , JJ,LL)«f!W*Cli i«H1

102

ORIGINAL PAGE ISOF POOE QUALITY,

If fKJ .».£. 0) CO TO 20

I SO"J

1 1,.'J. LI )•*«»( U ,.!.!. LLII,JJ.Ll)3' KIT, JJ.LL1 *»!»'"* VSIGN

3060

C *.o* TWC \\poft) W A L P rf T*E K.*p*rf HAS «Ep'J piC RFT THF TPiMSFft j f .FO VELDCI T ^ AT Tri£ COvJ :H»ATF. P O T ^ T SC..«--CO^JUGATf FORMC * -3s j TO 7, Ml K * '?s-7 TO 7C

00 til L=?,9

00 til Jst.lJ»15)/17

ro 16

CC

Ki3sn» w l a l TO 7,00 62 Is?, 8

DO Jsi,lb

TO 7

IF (J .KU" ( 1 M , .I'M

VB f 1^,J*

I I I ( I " , J«V1 ( I , J''

^O^TI^JIIE

M l00

91 GO

)* l'»

)B Vt5

)B i-P

)*-UI

) a - V I

TO( I . J( I . J( T . J( I . J( I . J( T . J

12,n.1).i),n*n

M!(tVI (j

*Soo 50

i)» v R n , j,1 )« ""(I ,.J.

. 1 ) « - l ' J f l . J ., 1 )*-vi f.l , J,

TRANSFOOMr TR4MSFORM

ORIGINAL PAGE ISPOOR QUALITY

103

.1" fMd ,n ,UJ ( t , 1 ,L ) ,2S6)CALL F F T v f S l R M )

( f ° ( \ , \ ) ,V(( F I d , 1 ) , v T (

C A L L F F T Y ( S I r ' * . C C )

, 1,1

, 1 ,L ) ,?<?<>)(* t ( 1 , ! ) , > ' ! (

C A L LC A L LS ^ A L L O H T f F f f (1 , 1 ) ,.»»(! , 1 , L)•?.•>« A L L O l - T f F 1(1 ,1 ),••.•! M , 1 ,L) .?Sbl

50 CONTINUE2 TpAMSPf )P M

(Md, 1 . I ),"T( 1 ,1CALL FIT? tSjr,*!, «*:,»)

M(I , i , , ).o«rd , \, \L nut ("( t . 1 , 1 J ,uld , J , i )

d , 1 , t ),W( » . i ,n ,'IPQ«Os^ ALL i 'i r HCI , i . n ,VT d . » . n.un«»fc)CALL FFT7tSTG".'^.w)SHALL CUT f H M f i . i, n ,wi"ri , t »n.«rt<»6>S H 4 L L O « ' T (Hf 1 , ( , i ) . v 1 ( i , 1 ,S M A L L JS f H M ( 1 , 1 , t ] ,*Ud .1 .

SÂI .L IM r^f i , i . I ).•••! f i , i , i) ,/C A L L cp i7 (S i r , \ ,H^ ,M)S*A lLC1 i 'T ("f'd , !• 1) .'-'d t 1s^ ALL OUT (H ( I , , , n.-i ( i .1 , i ) ,'

C-.-.-ThF. IMTTJ.4L HFLH wiS «F.EiJ r,F-!Ew«TEO. TH£ FOLLOÎVG IS TOC ruT TN'»-npM4Tl ON n\ THFC V t L f l C l T T F S AB£ STOWE!1* T^ I.'B. VP 4Nf>

C-.-.-TUf BlJLE^TTKUsO.TKVsO.TKwsO.TKIsO.rÔ 95 L = l ,00 95 Jsl , JM4X00 05 1st, J W A X

I , J . L 1I , J , L ) * * 2

T K U s T K " J * T n i V»TUIV

T K V B S

pS

iT-' 70*

70?, T M « , T f V , T f - v , T K S l j M70?, TKUK, TK V,706

104

115 CONTINUEPRINT 601UTOT30.

no 120T 710,

00 116 J*1,J**Xno ttfr 1st ,T*AX

, J,L)

PRINT 7(1?, (L'B(l,JO,U,Tsl.8)00 I is. T 70^, (Vttfl, 10,1. 1,1*1 ,«1Pkl«.iT 70?,

GO TO 131COFsl.S

707

t\\ ,i.)l ,UtL I « ,L( 1 in f- T I-'.F.HR, vi,-,

( S O I

&I^T 70?, (IJPfKTv l 70?,

7o? ,

700 F Q P ^ A T (v f * i * - i T i * L coi»ntTtnN, PT«* ( FI<) ,U,» O f i .TA*» ,F to ,u») * Cs * i F7 • " • * • * A Vf- H AGl NC r.K Tfs* ,F'J , 1 § * OEl.TA*)

701 FC'SnAT (V,*Le*,I3)

;C1 r"'-'- AT (1013)700 F09vAT (UE20.1U)7C5 FORu4T fV, •CO^TtMliFj) AT TT VE STF»« *, T«,/,/)70h FJRi'-T (x, 1 Ho^-«*«»•**«***»»****•**»*»***« + ******«»**»*********•**

i •»**»*» )

707 FOP* AT MM1)710711

• ,» (js*tF7,uf * lies*,

C" TO5000

60^0

105

RETURN

*OFC<

n PI STOHFD iv r,

oo"MsL/9

6*1

00 200MHsJ/9

M* I ftf

ARG?=A«G1*2.*3,

SC ARGU)*1 . *(CO?fARr, n-00

« « * » + * * * * * * * » » * * * * « * « * « * * * * * * * * * * * * * » * * » * * * * * * * ' •LARpE »»MM6, l fc f |« , ) . t ,nA,1 to , lM» l? :M* .> '» .J* ' )»En i6 .1 '> . tM,

1 01 (1*,16, i*o,r?(ih. i<>, i<o.Ff>i(6'M,e' ' i f *0) .EM r*.o).P!M!*<i*, j).2 Pnn?)

LARGE lirn6,l6,lM,v««(1*,.1*, tn l . -» t f ( l f t . 16,1*) »U I M h , > * , 1 6 )• ,VIM6, th. 1M.-T ( lh,16. 1h)

LARGE 9 U ( 1 6 , 1 h , 1 M , ! J V ( l « , , 1 f r , i 6 ) , R - A n f c , i B , i * l , P l ( i h , t 6 » ^ ). , Gl.'J (16. J*f I f t^ . r .v . do. 16 .1S1 .<V«1 (1*>, U, 16)

nlHE^STn*- H«f ) h, !»•.DIMEN3IOM F e f | f , l » «

. f M v A V P ( 1 < S l , M F F TCO^»OK: /D*T45 / r , c ,G iC O ^ » « O M / 0 4 T A 7 / F O , P tC O ^ O N i / O t T A H / v « A V

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REFERENCES

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Cochran, W. T., et al, 1967, "What is the Fast Fourier Transform?,"Proc. IEEE. Vol. 55, 10, p. 1664.

Comte-Bellot, G. and S. Corrsin, 1971, "Simple Eulerian time correlationof full- and narrow-band velocity signals in grid-generated 'isotropic1

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Deardorff, J. W., 1970b, "A three-dimensional numerical investigation ofthe idealized planetary boundary layer,"Geophys. Fluid Dyn. , Vol. ^,p. 377.

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Deardorff, J. W., 1973, "The use of subgrid transport equations in athree-dimensional model of atmospheric turbulence," J. FluidEngineering. Sept. 1973, p. 429.

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Jain, P., 1967, "Numerical study of the Navier-Stokes equations in threedimensions," MRC Technical Summary Rep. No. 751, University ofWisconsin, Madison, Wisconsin.

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Lilly, D. K. , 1967, "The representation of small-scale turbulence innumerical simulation experiments," Proc. of the IBM Scientific Com-puting Symposium on Environmental Sciences, IBM Form No. 320-1951,p. 195.

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Orszag, S. A., 1971a, "Numerical simulation of incompressible flows withinsimple boundaries. I. Galerkin (spectral) representation," Studiesin Applied Math., Vol. _!, 4, p. 293.

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Phillips, N. A., 1959, The Atmosphere and Sea in Motion, RockefellerInstitute Press, New York, 501-504.

Reynolds, W. C., 1974a, "Recent advances in the computation of turbulentflows," Advances in Chem. Engrg., Vol. 9, (Reprint of Report MD-27,Mechanical Engineering Department, Stanford University.)

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Schumann, U., 1973, "Bin Verfarhren zur direckten numerischen simulationturbulenter stromungen in platten- und ringspaltkanalen and tiberseine anwendung zur untersuchung von turbulenzmodellen," UnivesitatKarlsruhe. (NASA Technical Translation, NASA TT F 15,391.)

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