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1971-09-02

A numerical investigation of the

non-linear mechanics of wave

disturbances in plane Poiseuille flows

Clark, W. H.

Monterey, California. Naval Postgraduate School

http://hdl.handle.net/10945/29019

NAVAL POSTGf ADUATE SCHOOCMONTEREY. CALIFORNIA 93940

NPS-57Gn71091A

United StatesNaval Postgraduate School

A NUMERICAL INVESTIGATION OF THE NON-LINEAR

MECHANICS OF WAVE DISTURBANCES

POISEUILLE FLOWS

by

IN PLANE

T. H . Gawain and W. H. Clark

2 September 1971

This document has been approved for public releaseand sale; its distribution is unlimited.

FEDDOCSD 208.14/2:NPS-57GN71091A

ctcM ,09/

NAVAL POSTGRADUATE SCHOOLMonterey, California

DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SPHru-MMONTEREY CA 93wl51<£

Rear Admiral A. S. Goodfellow, Jr. M. U. ClauserSuperintendent Academic Dean

ABSTRACT:

The response of a plane Poiseuille flow to disturbances ofvarious initial wavenumbers and amplitudes is investigated bynumerically integrating the equation of motion. It is shown thatfor very low amplitude disturbances the numerical integration schemeyields results that are consistent with those predictable fromlinear theory. It is also shown that because of non-linear interactionsa growing unstable disturbance excites higher wavenumber modes whichhave the same frequency, or phase velocity, as the primary mode.For very low amplitude disturbances these spontaneously generatedhigher wavenumber modes have a strong resemblance to certain modescomputed from the linear Orr-Sommerfeld equation.

In general it is found that the disturbance is dominated for a

long time by the primary mode and that there is little alteration ofthe original parabolic mean velocity profile. There is evidence of

the existence of an energy equilibrium state which is common to allfinite -amplitude disturbances despite their initial wavenumbers.This equilibrium energy level is roughly 3-5$ of the energy in the

mean flow which is an order of magnitude higher than the equilibriumvalue predicted by existing non-linear theories.

NP3-57Gn71091A

2 September 1971

TABLE OF CONTENTS

Page

1. Introduction 1

2. Background 2

3. Basic Equations 6

h. Numerical Model 11

5. Linear Calculations l6

6. Numerical Integration of the Vorticity Equation 21

a) Small Amplitude Disturbances 21

b) Finite Amplitude Disturbances 25

7. Conclusions 36

References 39

Figures Ul

Appendix (Program Listing) 78

Distribution List 118

Form DD IU73 120

1. Introduction

In this paper we present some results from a numerical investigation

of the phenomena of stability, transition and turbulence in incompressible

channel flows. Ideally, the goal of such an investigation would be to

numerically integrate the Navier-Stokes equations and thereby be able

to follow the growth of some unstable disturbance which perturbs an

initial laminar Poiseuille flow. It would then be possible to observe

in detail how this growing disturbance eventually produces a fully

developed, stationary turbulent flow. Presumably, if one were to repeat

such a calculation for a large number of initial disturbances then the

final turbulent flow could be described by an ensemble average over all

the initial conditions

.

The results of such a computation would represent an exact (to

within the limits of the numerical model) solution to the turbulence

problem. Since turbulence is an inherently three-dimensional phenomenon,

these computations would have to be carried out on a three-dimensional

grid; unfortunately, the computational grid would have to be large enough

to contain the large "eddies" representing the initial disturbance and

still be able to resolve the smallest energy dissipation length scales

of the final turbulence . Even at low Reynolds numbers the ratio of the

larger length scale to the smallest scale is several orders of magnitude.

The number of grid points is proportional to the cube of this ratio;

hence, a complete three-dimensional solution is impractical, if not

impossible, despite the large storage capacity of modern computers.

(Emmons 1970 ) To overcome this impractically large storage requirement

we have drastically simplified the problem by treating the flow on a

two-dimensional basis. It is recognized that a two-dimensional treatment

can not adequately represent true physical turbulence. Never-the-less

the two-dimensional problem can be handled without further approximation

and, as will be pointed out in the next section, two-dimensional

solutions are of interest in their own right.

2 . Background

During the past decade considerable effort was devoted to developing

theories for the response of plane Poiseuille flows to finite -amplitude

disturbances. The most significant contribution of these theories

(in particular, those of Stuart and Watson (i960) and Reynolds and

Potter (1967)) was in showing that the square of the amplitude ( |^ | )

of an initially infinitesimal unstable disturbance is governed by an

equation of the form

dl

Ai I p U

^RT— kl K' + k

2 l

All W

Equation 1, which can be derived from the Navier-Stokes equations, is

an approximation valid in a region in the (a-Re ) -plane which is close to

the "neutral curve". Here R denotes the Reynolds number and a the

wave number of the disturbance. For disturbances with very small

amplitudes the second term on the right of equation 1 must be neglible;

hence, the square of the disturbance amplitude has an exponential

growth and the constant, le , is related to the exponential amplification

factor of the linear theory (i.e., t = -2^5 P-1-1 ) • of particular interest

is the case of k > for which the flow is unstable to small disturbances

,

If k < then it is possible that the higher order terms will

eventually balance the leading term and the amplitude growth is limited

such that

—k- -°

(lb)

Kl2

--Vk2

as

+ 00

Under such conditions a "supercritical equilibrium state" is

said to exist. On the other hand, if k > and k > then no

opositive limiting value for |A | " is possible according to equation 1.

It is possible to evaluate k based on Stuart's (i960) theory and

the necessary calculations have been carried out by Pekeris and

Shkoller (1967) and Reynolds and Potter (1967). These authors have

found that under certain conditions k < for some unstable disturbances.

Both Pekeris and Shkoller and Reynolds and Potter have found that for

regions near the lower branch of the "neutral curve" k < while

kp > near the upper branch. Hence, one can conclude that disturbances

whose wavelength corresponds to a point near the lower branch can grow

according to the non-linear theory and reach a supercritical equilibrium

state. On the other hand, for unstable disturbances whose wavelength

corresponds to a point in the (a, R ) plane near the upper branch the

non-linear theory does not seem to be applicable. At least the existing

theories give us no information about the growth of such a disturbance

beyond the linear range of amplitudes

.

Such "supercritical equilibrium states" are not observed experimentally

in channel flows; instead the initial instabilities lead immediately

to three dimensional turbulent flows. It seems, that no experimental

verification of this aspect of finite -amplitude theory is possible.

A computer simulation of a two dimensional channel flow is, perhaps,

the best way of investigating the two dimensional response of a plane

Poiseuille flow to an initial unstable disturbance. A computer

simulation offers, in addition to its two dimensionality, the advantage

of providing an "exact" solution to the equations of motion ("exact"

to within the limits of a finite difference representation) without

introducing any approximations concerning the relative magnitudes of

the various modes of which the disturbance is composed.

The present paper presents the results of a numerical investigation

of the response of a plane Poiseuille flow to various initial disturbances.

An attempt has been made to present the results in a manner that will

facilitate comparison with the existing non-linear theory. The

difficulty in making such comparisons is due to the fact that the

theories mentioned concern the growth of an unstable disturbance with

zero initial amplitude (at time t = - » ) which grows at first according

to linear theory and finally reaches an amplitude of such magnitude

that the non-linear equations are appropriate. In a numerical simulation,

with limited computation time available, the initial disturbance must

have some finite amplitude. The growth rates predicted by the finite-

amplitude theories are very small and the time required to follow

the growth of a very small disturbance to the non-linear range consumes

enormous amounts of computer time. For this reason the numerical

investigator is tempted to start his calculations by using a disturbance

whose amplitude is of such magnitude that the entire linear range can

be bypassed. It will be pointed out in Section 6 that there seem to be

significant qualitative differences in the flow when large initial

amplitudes are used as compared with those for which very small initial

amplitudes are used in the calculations

.

Another aspect of the present investigation was to determine under

what circumstances, if any, a two dimensional representation of the

flow could be made to resemble in some respects three dimensional

turbulent flows

.

It is clear that the "supercritical equilibrium states" in which

the unstable disturbance mode remains dominant do not resemble

turbulence in which the disturbance energy is distributed over a

wide spectral range. Also in a "supercritical equilibrium state"

the resultant mean velocity profile differs only slightly from the

parabolic laminar profile while for turbulent flows the mean velocity

profiles bear little resemblance to the original laminar one . It

is recognized that there are significant differences between two

dimensional and true three dimensional turbulence. This is particularly

evident in the energy spectrum and mechanism for the transfer of

energy and vorticity between the various spectral components (Lilly (1968)

and Kraichnan (1967)).

However, could it be possible that for some choice of initial

disturbance mode shapes and amplitudes the structure of the resultant

two dimensional flow could lie somewhere between the (relative)

simplicity of a supercritical equilibrium state and the "complete

chaos" characterizing a true turbulent flow? This aspect of the

present paper will be discussed in Section 6.

3- Basic Equations

Consider the two dimensional flow of an incompressible fluid

between two parallel planes. The governing equations of motion are

Bt 9y 5x+

3x By_Re

V Q {d&)

and

C = V2Y (2b)

where Q and Y are the vorticity and stream function respectively with

C and Y defined by

--S' -i (3a)

The variables are assumed to be normalized on suitable characteristic

lengths and velocities which are shown in Figure 1 along with other

parameters of interest. The velocity components can be expressed as

u = U(y) +u; V = V ; Q = - — + Q

Y = 3/2 (|y3

- y) + Y'

where the primed quantities which depend on x, y, t represent the

instantaneous departure of the flow from the original laminar flow ,

Substituting equations k into equations 2 results in

(*)

dtu

3x J 3x dy c»x dx dy RV C ^a;

C = V2

Y* (5b)

The conditions of zero slip at the wall and constant mass flow rate

imply that the stream function, Y' , satisfy the boundary conditions

Y* = 3Y'/Sy = at y = ± 1. (6)

By considering periodic solutions to equation 2 the boundary conditions

in the streamwise direction are fixed by

r (x,y,t) = Y* ( x ± 2nL, y, t) (7a)

C" (x,y,t) = C' (x ± 2nL, y, t) (7b)

where the basic period is 2L. Equations 5a and 5b with the boundary

conditions (6) and (7) are to be solved for Y'(x,y,t) for a given

initial value of Y'(x,y,0).

At this point it is convenient to define some parameters that are

useful in describing the flow structure. The assumed periodicity

implies homogeneity in the streamwise direction; hence, all averaging

is done with respect to the streamwise coordinate, x. The average

value of any quantity, q(x,y,t), is denoted by an overbar and obtained

by integrating over one period of the basic wavelength.

L

r

-Lq (y»t) =

2L J_q(x,y,t) dx (8)

It should again be emphasized that in our representation the flow is

arbitrarily divided into a laminar and a disturbance part. Hence,

the total streamwise velocity component is given by

u(x,y,t) = U(y) + u'(x,y,t) (9)

and the mean velocity is

u(y,t) = U(y) + u^" (y,t) (10)

The difference between the mean and local values of any quantity is

referred to as the turbulent part of that quantity and is denoted by

a circumflex. Thus we have

u (x,y,t) = u(x,y,t) - u(y,t) = u' - u' (lla)

v (x,y,t) = v' (x,y,t) (lib)

Y (x,y,t) = V (x,y,t) - Y7 (y,t) (lie)

C (x,y,t) = V (x,y,t) - V (y,t) (lid)

and

a by

By

— BY'u' = - x—

By

ABY

V =Bx"

dx

The turbulent kinetic energy is

(He)

(11*)

(llg)

V = !^r

= (llh)

S (x^.t) = i [(- f) (g)2

](12a)

The mean value is

g (y?t) =£l I * (x >y^) **

(12t )

—

L

and the normalized sum of turbulent kinetic energy over a domain extending

from the upper to the lower wall and for one period of the basic

disturbance is

a i r , l A 7E(t) = \ J dy \ ±-

IE (x,y,t) dx i (12c)

-1 -L

Similarly, the kinetic energy in the mean flow is

_ _p

E (y,t) =i[u(y) -|^_

(13a)

and the normalized sum over the domain is

1

E (t) = i[ E (y,t) dy (13b)

' -1

The normalized energy in the initial laminar flow is

E^|J i U

2dy = .600 . (13c)

-1

We can represent the periodic disturbance by means of a Fourier

series as follows:

r (x,y,t) = E f (y,t) e"1 nax

(lUa)n=-co

V (x,y,t) = s gn(y,t) e-

inaK(lift,)

n=-oo

with

fn

(y ' t} = h I T (x'y ' t}

elnQK ^ (l5a)

f = f *-n n

gn(y,t) = A- / C (x,y,t)e

inaKdx (15b)

where a - rr/L, and superscript * denotes a complex conjugate. From 5b

g = f _(n af f (15c)n n n

Notice that the mean value of the disturbance streamfunction and vorticity

is simply

Y1" (y,t) = fQ

(y,t) (16)

V (y,t) = gQ(y,t) (17)

The mean value of the turbulent kinetic energy is given by

2 (y,t) = 2 En(y,t) - E

Q(y,t) (l8)

where each of the components of the energy spectrum function, E (y,t),

is given by

En(y,t) - i ( \*'J (ff \tj) (19)

10

The normalized total turbulent kinetic energy over the domain is

E(t) = Z E (t) - E (t) (19a)

where

1

En(t) 4 J E

n(y,t)dy (19b)

k. The Numerical Model

A rectangular grid of dimensions 2 x 2L is used with M x N grid

points located as shown in Figure 2. For the calculations presented

herein 6k x 201 grid points were used. The indices k, j, and i denote

stations along x, y, and t respectively and 6x, 6y, 6t are the

corresponding intervals . The value of the disturbance streamfunction

Y' (x,y,t) is then denoted by Y . The use of the primes will be

discarded from this point unless necessary for clarity. Equations

5a and 5b are expressed in finite difference form as follows

i+1 £-1 I I - i - I

^K,J " ^K,J=

*K+1,JTK-1,J _ y

bK+l,J " bK-l,J

2 6t 26xJ

2 6x

(20a)

t l 1 A2 , *>

+ \j +R-

A £KJ7 e

r * = a2 Yl =-±-h Ji

- 2Y * + Y£

^KjJ %J6x

2 V K+1,J *K,JTK-1,J

+ _L_ (yZ

_ 2Y + Y. 2 VK.J+1 T

K,JTK,J-1

6y ' » '

(20b)

11

2where A represents the modified form of the Laplacian differential

operator of DuFort and Frankel (1958) defined as

A2 rl ~ 1 v r

i (ri+1 + r1 -1

^ + rl

1' J 6x

2 L K+1,J " V K,J %J7 ^K-ljJj

_1_ YJL ( 1+1 i-l\ 4 "I

.2 LbK,J+l " \>K,J WK,d7 bK,J-lJ

(21)

+

and J is the total energy and mean square vorticity conservation

form of the non-linear advection terms introduced by Arakawa (1966).

i.e.

,

+3 &<* \<j -V V>U (23)

where A and A denote ordinary first central differences,x y

Z+lEquation 20a is solved explicity for £ at each time step from

K, d

a a x-ithe known values of Y„ T , Q„ T , and £v T • The corresponding values

Jijd IS-jd ft,d

4+1of the Y„ _ are obtained from equation 20b. Equation 20b represents

Kj J

a set of simultaneous algebraic equations to be solved for the YT, TK,J

K = 1,2,... M; J = 1,2,... N at each time step. Because of the large

number of mesh points used in this investigation, conventional

techniques are not suitable and an alternative method is used. We

a %express the variables £„ _ and Y~ T in terms of their discrete Fourier

IV, d rL,d

transforms as follows

:

12

\l

,J - \ f»'j e_1 9n > K <^>5 n=l '

M

' n=l '

with

M

>A = h S *A e1 n,K (2Uc)

k=ln,J - M ^ K

5J

M

3n,J M . \ &K,Je ^^

' k=l '

where

s _ 2TT(n-l)(k-l)_

n,K M

Substituting equations 2U into equation 20b yields the following

6

equation for the Fourier components of Yy T :

f T-Ll " a f T+ f TT

=62S T (25)n,J+l n n,J n,J-l y &n,J

n = 1,2, ...M

J = 1,2, ...N

where

or* = 2[l - (6y/6x)2

(cos Q - l)]

The transforms and inverse transforms defined by equations 2U are

computed by the fast Fourier algorithm of Cooley and Tukey (1965) while

equations 25 are tri-diagonal and are easily solved for the f ' s

by Gauss elimination.

13

The assumed periodicity in x is automatically satisfied through

equations 2k. The boundary conditions at the walls (equation 6) are

solved by setting Y„ . = YT, ,, = for all times and the no slip

condition is satisfied in the following way. We express Yv T in

terms of Y„ -, through a Taylor series expansion about a point on theK.,1

wall.

yk!2 = v!i +a

i6y + A

26y2+A

36y3 + --

y ^K 3

= YKV ^1 6y + kA

26y + 8A

36y +

(26)

YA = YKV ^h 6y + ^2 6y2 + 27A

36y3 +

'*

'

etc.

The no slip condition (dY'/^y = 0) is satisfied if we require that

n

A., = which implies that (since Y„ n= 0)

1 IV, l

\% - i \% " | *A (27a)

and

\,N-1=

2 YK,N-2 "

9YK,N-3

(27b ^

for all k and i . The Poisson equation (5b) is satisfied (to second

order) at the walls by

C^i " 2*2 " ^2 I3/2 *k'3 - V9 YkM (28a)

^ - 72 fr2 ^K-2" V9 ^N. 3}

(28b)

1U

The length of the time increment, 6t, is fixed by the semi -empirical

stability limit

6t = f 6x 6y/ (U + u) 6y + v 6x1 (29)L o o J

where the velocity components (U+u') to v ' are the maximum absoluteo o

magnitudes of velocity selected from a large sample of grid points

.

Numerical experimentation has shown that for values of f £ .6 consistent

results are obtained. Values of f near unity may lead to catastrophic

instability in this explicit technique.

Since equation 20a is not "self starting" (it requires three time

levels), the computation procedure starts with the forward difference

representation

1+1 i 1+1 1+1 - 1+1 1+1bK,J " bK,J i f~ K+1,J " *K-1,J bK+l,J " bK-l,J

6t 2 r 2 6x " J 26x

Y ^ _ ¥X r l

- CZ

1 f~ K+1,J *K-1,J bK+l,J ^K-1,J+ 2 V 26x *

UJ 26x

(30)

T 1 1 A2 1 \

Equation 30 is also used periodically during the calculations in order

to suppress any instabilities associated with the central time differencing

used in equation 20a.

The computation procedure can now be outlined step-by-step.

1. An initial distribution of Y^ T is chosen and the correspondingis., J

oQv _ are computed from equation 20b.K, J

15

2. The time increment, 6t, is computed from equation 29.

3. The values of Q are computed for all the interior points

(J = 3jU,...N-2) by iteratively solving equation 30.

h. The Fourier coefficients, g are computed from equation1

n' 1

24d, equation 25 is solved for the f and the corresponding2.

n ? <J

values for the YR

- found from equation 2k& for the interior

points (J = 3,U,...N-2).

5. The remaining grid points for Y , Q • J = 1,2,N-1,N)

are computed from equations 27a, 27b, 28a ,b and 20b.

6. For each time step the procedure in steps 3, k, and 5 is

repeated except the explicit central difference equation 20a

is used in step 3 rather than equation 30.

7. Periodically, say every 50 time steps, a new time increment,

6t, is computed (step 2) and a single time step using the

implicit forward difference equation 30 is used.

For all of the computations presented in this paper a grid with

dimensions M = 6k, N = 201 was used which required a computation time

of approximately 17 seconds per time step.

5. Linear Calculations

In an investigation such as this it is desirable to make some

comparison between the results of our numerical computations and some

known solution to the Navier Stokes equations. The imposed restraint

of two dimensionality precludes any comparison with experimental data

and, of course, there are no exact solutions to the non-linear equations

of motion. The only alternative is to consider the limiting case of

infinitesimal disturbances for which the results from linear theory

16

are available. For this reason, a rather thorough investigation of

the linear solutions to equations 5& and 5b was undertaken. For small

amplitude disturbances the non-linear terms in the vorticity transport

equation are neglected and the equations of motion become

f-i'-3f=^V2C- (31a)

V = v2 r (3ib)

The general solution to equations 31 can be expressed in the form

oo oo -i(n ax - p t)

r(x,y,t) =2 I $nm (y)

e^

(32a)n=-oo m=l

oo oo -i(n ox - P t)

C'(x,y,t) = S S 6nm(y)

e^

(32b)n=-co m=l

with

,29 = § - (n a)nm nm v nm

a = 2tt/\

(32c)

where \ is the wavelength of the disturbance.

Substituting equation 32 into 31 results in the well known Orr-

Sommerfeld equation for $ (y); i.e.,nm

IV 2 " h !~Y$ - 2(na) $ + (no) $ + i naR ('

nm nm nm e |_\

- (no)2

$ ) + U $ 1 =nm/ nmj

na /\ nm

(33a)

17

with boundary conditions

t

$ = $ =o at y = I i . (33b)nm nm J \~>-> >

Solutions of 33a for $ (y) for given a, n, and R can be made to satisfynm ' ' e

the homogeneous boundary conditions (33b) only for the discrete eigen-

values, (3 , 1 £ m £ oo. For the given a, n, and R we are usually' nm' D ' ' e

interested in determining if there exist corresponding 8 's which* D nm

have negative imaginary parts. The corresponding disturbance, $ (y),

then grows exponentially with time and the flow is unstable to disturbances

at the specified a, n, and R .

For the present analysis we are interested in solutions to equation

20a; hence, we must consider the finite difference representation of

the linearized equation. For small disturbances equation 20a becomes

4t,J " ^K,J _ K+1,J TK-1,J bK+l,J bK-l,J

26t 5 2 6x "J 2 6x , ^

Again we will consider solutions of the form (32); hence, we have

-i(nc*(k-l)6x - B £ 6t)

Y = ($ ) eYK,J ^ Vnm e

CfT .T " ( 9j)r,m e

(35)-i(na(k-l)6x - 8 i 6t)

nmK,J vwJ'nm

'

Substituting into 3*+ and 20b results in

p,>jU " in4W + 3(§j ) ]

+ 7T2 [«W +"(*P * (9J-i^ ]u nm nm R 6y nm nm nm

(36a)

18

where

nm 6y nm nm nm

uj = 2^(6y/fix) cos na 6x) (37a)

p

«> = -2[l +(||) (1 - cos no 6x)] (37b)

o 1 = sin na 6x/n6x (37c)

and

_p_ sin 3 6t

B* =21 + l/R_6x cos B 6t + i ..nm

(37d)nm L e J nm 6t w

'

i = V-l

When expression 36b for (0 ) is substituted into equation 36a thereJ nm

results a homogeneous set of linear algebraic equations for the ($ )J nmFor given values of a, n, R , 6x, 6y, and 6t we wish to determine the

eigenvalues and eigenvectors of the matrix of coefficients of the (§ )J nmIn particular, we are interested in those eigenvalues B for which the

nm

imaginary part of B is negative and in the associated eigenvectors

($ ) . The effects of 6x, 6y, and 6t on the linear solution are especiallyJ nm

important for the present application.

The eigenvalues and eigenvectors have been computed for a number of

Reynolds numbers and for a wide range of a1

s using the QR algorithm

(Wilkinson 19&5 or Fair 1971) • For more details on these calculations

the reader is referred to O'Brien (1970) or to the report by Gawain and

Clark (1971). In the latter report some of the eigenvalues (B ) and

19

eigenfunctions ($ (y)) are tabulated for the case a = 1.0, R = 6667.

In the limit as 6x, 6t -* 0, the problem is the same as that solved by-

Thomas (1953) although the methods used are quite different from those

employed by Thomas. For the case a = 1.0, R = 6667, and 6y = .01 (in

Thomas' notation this corresponds to a = 1.0, R = 10,000, 6y = .01) we

have solved equations 36 for n = 1, 2, 3? and k. Our solutions show

that there is only one unstable eigenvalue and this unstable solution

occurs for n = 1. We shall order the eigenvalues, p , so that for a' nnr

given n, the m's are arranged according to the stability of the computed

eigenvalues. Hence, for the case a - 1.0, Re = 6667, P,-. is the unstable

eigenvalue with the corresponding eigenfunction $, ,(y). Thomas gives a

value of p.. = .3653 - i .0055 and the corresponding mode shape, $-.n(y)

is tabulated in his Table V. (Using our non-dimensionalization, P is

exactly 1.5 times the value given by Thomas.) Our calculations yield

the value p = .3593 - i .00Ul and the corresponding mode shape, $-.-,(y),

is shown in Figure 3 for comparison with that of Thomas

.

Now consider the influence of the finite difference parameters

6x, 6y, and 6t on the solutions to 36. The effect of 6y on the growth

rate (-^ P ) is illustrated in Figure h. As 6y increases the flow

becomes more and more stable and when 5y = .05, the phenomenon of linear

instability has been completely obscured by the crude finite difference

representation of the equations of motion. Indeed, calculations with

iy = .05 for a wide range of Reynolds numbers and a's have shown no

indication of linear instability.

The effect of varying 6x was investigated at R = 6667, ot = 1.0,

6y = .01 and 5t < .02. Figure 5 shows the variation with 6x of the growth

20

rate (-y P-,-,) and phase velocity (Q(& ) for the unstable mode. For

6x < .8 the changes in the disturbance profile ($, ,(y)) where insignificant.

Perhaps the most surprising discovery in this investigation of the

linear behavior of the discrete representation of the equations of motion

was the significance of the time interval, 6t, on the disturbance growth

rate (-^ P-, -, ) • The effects of 6t on this parameter are shown in Figure 6.

There are significant departures from the "exact" value ofO 3-,-, for

6t > .Ok while the phase velocity and mode shape are not significantly

changed. Hence, in addition to the usual stability constraint (29) on

the time interval, 6t, it is necessary that 6t < .Ok in the numerical

integration scheme described in section h.

We can now summarize the results presented in the preceeding discussion.

Increasing the y increment, 6y, tends to make the flow more stable and

for coase grids ( 6y > .05) the phenomenon of linear instability is

completely obscured. The linear solution is comparatively insensitive

to changes in 6x, at least, until the x-increment is approximately the

same length as the channel half-width. The growth rate of the disturbance

is strongly influenced by the 6t increment (due to the DuFort-Frankel

representation of the diffusion term) and for large 6t the growth rate

is much higher than that predicted by the "exact" linear solution.

6. Numerical Integration of the Vorticity Equation

(a) Small Amplitude Disturbances

We now return to the numerical integration of the vorticity

equation using the techniques described in Section h. As a check on

the validity of the solution we first consider the limiting case of

21

small amplitude disturbances and compare the results obtained with the

known results from the linear solution as described in the preceeding

section. To be more explicit, for a given Reynolds number and initial

disturbance streamfunction, Y (x,y,0), are the results from the numerical

integration consistent with the values of p and $ (y) predicted bynm nm

linear theory?

A series of calculations were made at R = 6667 and the initiale

disturbance given by

Y*(x,y,0) = • {^(7,0) e"iQK

+ f/(y,0) e+i "*} (38)

where e is an amplitude factor. For e sufficiently small the non-linear

terms in equation 5a should be neglible and the numerical integration

should give results which are in agreement with the linear theory. As

a first test we choose a = 1.0, R = 6667? e = .005 and the initial mode

shape identical to the unstable mode computed from linear theory; i.e.,

f (y,0) = $i;L

(y), where &n (y) was snown in Figure 3. The growth

Arate of the disturbance kinetic energy, E (t), as defined by equation 12c

is shown in Figure 7. The computed growth rate gives a value ofjB = .0050

which is in good agreement with the value given by Thomas . From equation 1*+

we can determine the phase, cp (y>t), of any of the spectral componentsT n

of Y (x,y,t) by

% (y,t) = tan"1^ f

n/(Kf) (39)

The phase velocity, c , or rate of propagation, of any of the spectral

components is then

c = i BcVat ^°)n na '

22

For these computations the grid length, 21, was set equal to the wavelength,

\, of the disturbance; so for the disturbance (primary) mode, n = 1, and

in the linear range we can identify c with (R. 3 from the linear theory.

Figure 10 shows that c is, indeed, a constant and the slope of this

phase versus time plot shows that c, = .363 which is in excellent

agreement with Thomas' value of (K P = .3653 and with our value of

$ pi;l

= 3593-

Other short computations were carried out with, again, a = 1.0,

R = 6667, e = .005, but with the initial mode shape, f (y,0), chosen to

correspond to one of the stable modes computed from linear theory. These

calculations were carried out by arbitrarily setting f, (y,0) equal to

the mode $ , $? n,

and $__ which are shown in Figure 9« The energy

decay rates for these three cases are shown in Figure 10. From the energy

decay rates and from the computed phase velocities (not shown here) the

numerical integration technique yields the following values for the

appropriate eigenvalues

= 1.351 + i .138, P2g

= 1.U25 + i .207

P22= I.I+56 + i .076U.

The linear matrix solution described in the preceeding section gave the

values

p = 1.363 + i .136; P2g= 1.1+03 + i .192;

p22= 1.U62 + i .071+2.

* Since equation 33a is symmetrical in y it is possible to divide the

solutions for $nm(y) into even and odd solutions. The boundary conditions

are $nm(+l) = $Am( +1 ) = 0, with $nm(o) - $*" (0) = for even solutions and

$nm (0) = $^m(0) - for odd solutions. For our numerical model with N

lateral stations there are N-2 unknown (Jj^'s; hence, N-2 eigenvalues

are computed for each n, a, Rp . Of these N-2 eigenvalues, N-l correspond

to even solutions and N-3 correspond to odd solutions

.

23

A more critical test of the numerical integration is the ability to

predict the correct mode shape, $ (y), for a disturbance of known wave

—

length and at a fixed R . Suppose in equation 38 we choose a = 1.0

but arbitrarily select the initial disturbance mode shape f (y,0). The

arbitrary function f (y,0) can be expressed in terms of the eigenfunctions

,

$, (y)j °f the Orr-Sommerfeld equation (33a) for the chosen a and R

(Schensted, 1961). i.e.,

f,(y,0) = S A $ (y) (Ula)1 m=1 m lm

with

1

A =J

f (y,0) yjy) dy (Ulb)

and

v(y) =(72 - *

2

)§

dy

where §, (y) is the solution to the adjoint of equation 33a. However,

the linear solution tells us that only one of the $ in equation Ula is

unstable; all other modes will decay with time. Hence, in the linear

range only one mode, $,,(y)j grows and will eventually dominate all the

other terms in the expansion (Ula). Therefore, the mode shape

^(yjt) - $i:L

(y) as t - 00

where $n ,(y) is as shown in Figure 3 for the case a = 1.0, R = 6667.

The initial shape chosen for f (y,0) is shown in Figure 11. The

disturbance kinetic energy, E(t), is given in Figure 12 and the mode

shapes, f (y,t), are shown for various times during the calculations,

2k

As expected, the disturbance energy at first rapidly decreases then

grows according to linear theory while the computed mode shape is

clearly tending towards the predicted shape of $ (y)

.

From the results discussed in this section and from calculations

at other wavenumbers, a, we have found that the numerical integration

scheme consistently gives results that are in excellent agreement with

the linear theory. The agreement between the results of the numerical

model and the predicted results for small amplitude disturbances is

gratifying and gives some confidence in the non-linear calculations

now to be discussed.

b) Finite Amplitude Disturbances

Before discussing the results of the non-linear aspects of this

investigation it is appropriate to give a very brief outline of the

features of the existing non-linear theories which predict equations

of the form of equation 1. The following discussion is from the

paper by Pekeris and Shkoller (1967 ) which in turn outlines Stuart's

(i960) theory for finite amplitude disturbances. If the Fourier

representations for the disturbance streamfunction, Y' , and vorticity,

£'; (equations Ik and 15) are substituted into the equation of motion

(5a) the following equation for the modes results:

R~ (gn_ (na) gn ) + i n ° |_(U " f )gn " (U -f )

fnj 0*2)at

where

- iaH n = 0,1,2,. .

.

n 9 9 9

2

ln= f

n- (na) fn (15c)

25

and

HB = l {<»-«>(V V S

"SS

' fn-s }

(tea)

If the parameter, e =|\J P , , |, is small (implying proximity to the

neutral curve) then Stuart showed that in an approximation in which

terms of order e' are retained, only the f , f. , and f in (ll+a)

are significant. Now expand f (y,t) in terms of the eigenfunctions,

$ , of the Orr-Sommerfeld equation as was done in equation (1+la)

,

so that

f (y,t) = eo

z A (t) V (U3)

m=l

Stuart's asymptotic analysis for small e shows that only the first

term in (1+3) is important, i.e.,

fx(y,t) m e

QA1(t) $

i:L(y) + (e

Q3

) (1+1+)

and that the terms f and f„ are of smaller order so thato 2

fQ(y,t) - «

2|A1(t)| G

Q(y) + 0(e

Q

k) (1+5)

f2(y,t) = Sq

2Ax

2(t) G

2(y) 0(^)

k(1+6)

The functions G and G„ are determined from Pekeris and Shkoller '

s

o 2

equations (22), (23), and (2k). By substituting equations (1+1+) , (1+5),

and (1+6) into (1+2) one can obtain an expression of the form (l) for

2the amplitude parameter, (A., (t)

|, i.e.,

26

2d

l

All 2 k

"dT-= \ l

AJ +k2 l

Ail CD

The important point from our viewpoint is that according to the above

analysis, the disturbance energy should be contained primarily in the

fl(y>t) mode and that the growth of this energy should behave according

to equation (l). Assuming that the disturbance streamfunction can be

adequately represented by f then we have from (Uh)

Y'(x,y,t) = eQ(A1(t) $

11(y)e"

i ax+ j£(t) $^(y)e

+i ax)

(U7 )

The disturbance kinetic energy as defined by equation (12c) (since we

are neglecting the contribution from f the "disturbance" and "turbulent"

kinetic energies are the same) becomes

E (t) = eQ

2I ^(t)!

2(US)

where

1 =I {

I

$ii

I

2+ * l

$nl2

}dy ( u9)

It was pointed out in part 2 of this paper that the constant, k^ , in

(l) is related to the amplification factor of linear theory (k =-2J p )

and if kp < then the growth of an unstable disturbance is limited

so that

\\\ -\/*2

(50)

Pekeris and Shkoller found that these conditions are satisfied for a

and R inside the lower portion of the neutral curve. For example,

at R = 6667 and a = .90 their calculations give

27

I*! \- 1/9

Our solutions to the linear eigenvalue problem at R^ = 6667 and

a = .875 gives Pnn = -.OO39U and I = 1.8U8. Hence,

E(t) - (l.8U8)(.0039U)/9.0 = 8.08 x 10"4

Figure 13 shows the energy growth for modes E , E , E , and E for

a run with the initial disturbance given by (38) with a = .875,

Rg

= 6667, e = .005, and f^yjO) = $i:L

(y) where I in the unstable

eigenfunction for a - .875. The disturbance is, indeed dominated by

E.. (t) and the mode shape, f (y,t), normalized so that f (0,t) = 1.0,

never departed from the initial shape given by $-|1

(y). A longer

calculation was made using the same initial conditions except in this

case a = 1.0. Again the energy remained predominately in the n = 1

mode and again the mode shape for f (y?t) never departed from the initial

$ (y) shape indicating that (hk) and (hj) do indeed adequately represent

the flow as far as the overall disturbance energy is concerned.

Although this latter calculation required 3^«5 hours of computer

time, we can see that the energy growth is still within the linear

range. This demonstrates that, as was pointed out in part 2, it is

impractical using the present techniques to follow the disturbance

through the linear and into the non-linear range of amplitudes. For

this reason we have been unable to establish whether or not the .

"supercritical" equilibrium states as predicted by (l) and (50) do,

in fact, exist for our exact treatment. Our calculations, as shown

in Figures 13 and lU, do indicate, however, that there is no significant

28

difference between cases for which a = .875 or a = 1.0 in equation (38).

The calculation of Pekeris and Shkoller, however, show that "super-

critical" equilibrium states are possible for a = .875 but not for

a - 1.0. All our calculations for both low and high amplitude disturbances

have shown no strong dependence on a as far as the final result is

concerned.

Although this last run was entirely within the linear energy

growth range we will see that there were very interesting non-linear

features to the calculations. The appearance of energy in modes

En , E , and E as shown in Figure ik is, of course, due to the

presence of the non-linear terms in equation (5a). In identical

calculations for which the non-linear terms in (5a) were deliberately

omitted the primary (n=l) mode grew according to linear theory while

the energy in the other modes remained constant or decreased and the

energy levels (due to numerical "noise") were twelve to fifteen orders

of magnitude lower than that of the primary mode . Figure 1*+ shows

that the energy in the other modes (n = 0,2,3) grows very rapidly at

first and then seems to follow an exponential growth rate, i.e.,

2p t

En(t) ~ e

nn = 0,1,2,3

for large t. From the data shown in Figure lU the values for (3

are PQ

= .OllU, p = .0050, P2= .0080, and = .0117.

29

The most interesting feature is the shape of the modes, f , f,

with respect to y . These shapes are shown in Figure 15 where the

modes are normalized so that f (0,t) = 1.0 for even modes and forn

odd modes the maximum real and imaginary parts of f (y) are unity.

The phase velocities for these modes averaged between t = 65 and

t = 71 were cp= .362 and c = . 363 • These phase velocities are the

same as the phase velocity of the primary mode predictable from linear

theory (i.e^ Pn = -363).

The striking feature is the strong resemblence between these modes

and the modes calculated from the Orr-Sommerfeld equation. Indeed, it

appears that the non-linear solution has spontaneously generated modes

that can be predicted from the linear theory even though the linear

solution indicates that these are stable modes and would decay with

time. This means that if we represent each mode by an expression of

the form of (k3) i.e.,

f (y,t) = L A (t) $ (y) (51)n ' m=l m nm

•*From equation (15c) we see that g has the same symmetry or anti-

symmetry as does fn *, i.e. if fn (y) is an even function of y then gnis also even. Then from equation (U2) we can establish that if

Y'(x,y,0) is given by (38) and f-|_(y,0) is even in y then f (y,t) is

odd, f2(y>t) is odd, fo(y,t) is even, fl|.(y,t) is odd, etc. likewise

if fi(y,0) is odd then fo(y,t) is odd, f2 (y,t) even, f3(y,t) odd,

fU(y^) is ev"en ? etc For example, let fi(y,0) be even. Then since

f (y,0) = n f 1n

Bg2W t "iaH2l

=-±0l[fl gl" g

l'f

t=0 t=0

= odd function of y.

At the next time step, t - 6t

g2(y>6t) =Pg2/Bt)6t = odd function of

30

y

then one mode seems to dominate the expansion. It is interesting to

notice that in each case the dominant mode is one for which there exist

a step gradient near the wall. MDst of the stable modes that we have

computed from linear theory are similar to the ones shown in Figure 9

which have a detailed structure near the center of the channel rather

than near the wall.

A series of higher energy runs with various initial disturbance

amplitudes was made at R = 6667 using as initial conditions a

disturbance with wave number a = 1.0 and that mode shape given by Thomas

(see Figure 3)' The total energy growth (or decay) is shown in Figure l6

for several different initial energy levels . A further energy breakdown

is given in Figure 17 for the cases e = (.05)v5- For this case the

total energy remains almost constant while there is a continual

exchange of energy between the mean flow and the turbulence. The

turbulent kinetic energy oscillates about a value of E(t) = .03 which

is roughly ten times the equilibrium value predicted from equation 50.

From Figure lUa it seems that there is a tendency towards some common

energy equilibrium state for the different initial conditions . A more

detailed inspection of the flow structure shows that there are significant

differences between the three flows . Starting with the high energy run

we summarize the structure of the flow by presenting plots of the mean

velocity profile, Reynolds stresses, and energy spectrum at various

times throughout the run (Figures 18, 19, and 20). Also, in Figure 21

the shape of the primary (n = l) mode at the end of the calculations

is shown for each of the three energy levels. For the high energy run

(although the time parameter is comparatively small) there are drastic

31

alterations of the mean velocity profile and primary mode shape and

considerable energy transfer from the primary mode to the higher

harmonics. For the two lower energy runs there is little alteration

of the parabolic velocity profile and only small amounts of energy-

transfer to higher wave numbers . For the lowest initial energy

level (e = .05 / >f2) the primary mode shape remains essentially unchanged

throughout the entire calculation. More details of this low energy

run are shown in Figure 22 where the growth of the total turbulent

energy in the first three spectral components (equation 19b) is

shown. The phase velocities (equation 1+0 ) were computed near the

end of the run and are summarized below:

c±

= .1+92, c2

= .1+8U, c3

= .1+73

The measured C were independent of y and all are roughly k0% higher

than the wave velocity (#t(3 ) of the primary unstable mode predicted

by linear theory. This low energy run has some qualitative similarities

to the predictions of the non-linear theory which was outlined at the

beginning of this section. That is, there appears to be an asymptotic

energy level, the primary mode shape is only slightly altered and the

energy in the primary mode remains dominant.

In order to investigate the influence of initial wavenumber on

the flow two runs were made at R = 6667 with a - .875 (near the

lower branch of the neutral curve) and with a = 1.05 (near the upper

branch). For both cases the initial mode shape was determined from

the eigenvalue solution to the linear equations and the initial

energy was chosen to be slightly lower than that of the run (e = .05/J5,

32

Figure l6) which seemed to remain in an energy equilibrium state. The

energy variation for these cases is shown in Figure 23. There is no

obvious qualitative difference between the three wavenumbers; the initial

high energy level, however, apparently causes oscillations of the

disturbance kinetic energy and makes any direct comparison with non-

linear theory rather difficult.

Another important question which arises in the consideration of

non-linear aspects in the stability of plane Poiseuille flows is whether

or not a flow which is stable to infinitesimal disturbances might be

unstable to disturbances of some finite amplitude. Such instabilities

are generally referred to as "subcritical" instabilities . To investigate

this aspect of the problem several runs were made using as initial

conditions a disturbance that is known to be stable in the linear

range. The case chosen was for a = .78, R = 6667 for which £ = .2^9

+ i .0012^ for the least stable eigenfunction determined from the linear

solution described in Section 5« Again the initial disturbance was

given by equation 38 with the function f (y,0) = § (y) determined

from the linear solution. For a small amplitude disturbance (e = .005)

the disturbance energy decreased with time according to linear theory

while for a larger amplitude (e = .0635). The energy variation was

as shewn in Figure 2k. The familiar oscillation of the turbulent

kinetic energy which is associated with an initially high disturbance

energy is present in this case; while the total kinetic energy appears

to remain approximately constant after an initial decrease.

We now address ourselves to the question posed at the end of

Section 2 concerning the character of a two dimensional "turbulence"

33

in channel flows . For all the runs discussed above the computational

grid length (2l) was equal to the wavelength of the initial unstable

disturbance (i.e. 2L = 2rr/a) . In any finite difference representation

of the equations of motion there are only discrete wavelengths

available to be excited by the non-linear interactions . Recall from

equation 2^a that the streamfunction is expressed as

M -i9 .v^v (24a)

and since Y^ T is real

» fn,J

= afM-n+2,J

v fn,J

= "^ fm-n+2,J

i.e.,0t f Tis even and ^ f

T is odd over the interval n = l,2,...M/2n j J n j J

. . .M. Therefore, there are M/2 distinct Fourier modes "available" to

the solution of the equations of motion. When the grid length equals

the wavelength of the initial disturbance there is only one lower mode

available to be excited; namely, the zero mode, f . It is known

that two-dimensionality constrains the transfer of energy between

the modes of which the disturbance is composed. For an isotropic

two-dimensional turbulence there can be no systematic transfer of

energy from lower to higher wave numbers without a corresponding

transfer to still lower wavenumbers (Lilly (1968), Kraichnan (1967)).

Hence, one may suspect that our discrete representation in which

there are no available modes at lower wave numbers could force the

disturbance energy to remain in the primary initial mode. The first

3*+

alternative to this dilemma would seem to be to increase the grid

length, 2L, thereby allowing modes with both higher and lower wave

numbers to fit into the available grid length. However, if the

initial disturbance energy is contained in, say, wave number n then

the non-linear interaction of this mode with itself will initially

excite wave numbers and 2n; later modes 0, 2n, 3n, hn, etc. The

modes that can be excited will be separated by a spacing of n in

wave number space, resulting in a spectrum with "holes" or regions

in which no energy can appear. Clearly, this is not a realistic

case. The alternative to this situation is to initially perturb the

flow with a disturbance in which at least two contiguous (in the

discrete wave number space) modes are present. For example, if the

initial disturbance contains modes n and n+1 then the non-linear

interaction will, at first, excite modes 0, 1, 2n, 2n+l, 2n+2; these

in turn will excite modes 0, 1, n-1, 2n-l, Un, etc. until all possible

wave numbers have been excited.

It seems reasonable to require that the wavelengths of these two

initial modes should both fall within the unstable region in the

(a-R ) plane. If we set the grid length so that 2L=i6tt (or a = l/8)

then modes with wavenumbers 7a and 8a will both fall within the

unstable region. Hence

x l\, -i7ax „ -i8ax * +i7ax ,* +i8ax\

Y'(x,y,0) = e |f?e

<

* + fQe + f

?e + fge j

35

A lengthy numerical integration of the vorticity equations using

the initial conditions just described is summarized in Figures 25

and 26 which show the turbulent kinetic energy time variation and

the energy spectrum at various times during the run. For this run

the turbulent kinetic energy initially grows and then appears to

Avary randomly about a value of E(t) = .019. The energy spectrum is

quite different from the other cases presented in that there is a

gradual transfer of significant amounts of energy to other wavenumbers.

Although it is not shown here, the mean velocity profile again differed

only slightly from the parabolic distribution. Although, there

are significant differences between this run and the others presented,

all the examples that have been discussed are qualitatively similar

in that most of the energy remains in modes whose wavelength

approximately equals that of the primary (unstable) mode.

7. Conclusion

Our numerical technique for integrating the vorticity equation

has been shown to be stable and predicts results that are consistent

with linear solutions to the vorticity equation.

The influence of the finite difference representation on the

linear eigenvalue solution to the equations of motion was investigated

and some interesting conclusions resulted. The size of the 6y

increment significantly influences the stability of the equations

and for grids with 6y > .05 (approximately) the equations are stable

to all disturbance. Because of the DuFort-Frankel representation

of the diffusion term, the growth rate of an unstable disturbance is

strongly influenced by the 6t increment. The growth rate increases

36

rapidly with 6t even though the time step increment may still be

small enough to satisfy ordinary numerical stability requirements.

The 6x increment has the smallest effect on the linear solution

and seems to be significant only when 6x is approximately equal to

the channel half-width.

Because of the long computing time required we were unable to

follow the growth of an unstable disturbance through the linear

and into the non-linear range of disturbance amplitudes . This

precluded any direct comparison of our results with those predicted

by the non-linear theories of Stuart and Watson and Reynolds and Potter.

We can, however, make some general comparisons. We find qualitative

agreement in the sense that the unstable primary mode dominates the

disturbance and there is little distortion of the mean flow. The

calculations presented indicate the existence of a disturbance energy

equilibrium state which seems to be common (for a given Reynolds

number) to all unstable disturbances despite their initial energy

or wavenumber. 'Indeed, we have found no outstanding differences

in the behaviour of disturbances with varying initial wave numbers

(at a constant R ) which is not in agreement with the non-linearv e

theories. The distortion of the primary mode shape is significant

when the initial energy is high enough to be well beyond the linear

range but for lower initial energies the distortion of the mode

shape is less important.

The computed flows which have been discussed in this paper

show little resemblance to actual turbulent shear flows . In a true

three dimensional flow one would expect an initial unstable disturbance

37

with a length scale approximately 2rr times the channel half width to

grow in amplitude and through non-linear interactions the energy of

the disturbance would be transferrred to higher wavenumbers with length

scales comparable to the channel half width. The corresponding

situation for a two dimensional turbulent channel flow is not clear.

However, for two dimensional homogeneous turbulence the energy

would cascade to the lower wavenumbers while mean square vorticity

is transferred to higher wavenumbers when the turbulent energy is

continually fed into the flow in a narrow band of spectral components.

Neither of these processes was observed for the present calculations.

One can speculate that our calculated "quasi-equilibrium" states

in which most of the energy remains in one or two spectral components

with little alteration of the parabolic velocity profile represent

two dimensional solutions to the equations of motion which are, in

some sense, highly unstable flows and, for this reason, are not

observed experimentally.

38

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1. Arakawa, A., 1966 "A Computational Design for Long Term NumericalIntegration of the Equations of Fluid Motion: Two DimensionalIncompressible Flow, Bart 1." J. Comp. Physics , 1, 1, pp.119-1^3.

2. Cooley, J. W. and Tukey, J. W., 1965. "An Algorithm for the MachineCalculation of Complex Fourier Series," Math Comp . , 19, 90,

pp. 297-301.

3. DuFort, E. G. and Frankel, S . P. , 1953- "Stability Conditionsin the Numerical Treatment of Parabolic Differential Equations,"Math Tables and Other Aids to Computation , 7, pp. 135-152.

k. Emmons, H. W., 1970. "Critique of Numerical Modeling," in AnnualReview of Fluid Mechanics, Vol 2, Palo Alto, California:Annual Reviews , Inc

.

5. Fair, G. , 1971. "Allmat: A TSS/360 Fortran IV Subroutine forEigenvalues and Eigenvectors of a General Complex Matrix,"NASA TN D-7032.

6. Gawain, T. H. and Clark, W. H., 1971. "Tables of Eigenvaluesand Eigenfunctions of the Orr-Sommerfeld Equation," to bepublished as Naval Postgraduate School Report, Monterey,California.

7. Kraichnan, R. H., 19&7- "Inertial Ranges in Two-DimensionalTurbulence," Phys Fluids . 10, 7.

8. Lilly, D. K., 1970. "The Numerical Simulation of Three-DimensionalTurbulence with Two Dimensions." V. VII, SIAM-AMS Proceedings,American Mathematical Society, pp. Ul-53.

9. O'Brien, G. D., 1970. "A Numerical Investigation of Finite

-

Amplitude Disturbances in a Plane Poiseuille Flow," NavalPostgraduate School, Ph.D. thesis.

10. Pekeris, C. L. and Shkoller, B., 1967. "Stability of PlanePoiseuille Flow to Periodic Disturbances of Finite Amplitude

in the Vicinity of the Neutral Curve," JFM 29, 1.

11. Reynolds, W. C. and Potter, M. C, 1967. "Finite AmplitudeInstability of Parallel Shear Flows," JFM 27, 3-

12. Schensted, I. V., 1961. "Contributions to the Theory ofRydrodynamic Stability," University of Michigan, Ph.D.

Thesis.

39

13. Stuart, J. T., i960. "On the Non-linear Mechanics of WaveDisturbances in Stable and Unstable Parallel Flows," part1. JFM 9, 2.

lU. Thomas, L. H., 1953- "The stability of Plane Poiseuille Flow,"Phys Rev . , 91, h, pp. 780.

15. Watson, J., i960. "On the Non-linear Mechanics of Wave Disturbancesin Stable and Unstable Parallel Flows," part 2. JTM 9, 2.

16. Wilkinson, J. A., 1965. The Algebraic Eigenvalue Problem ,

pp. U85-569j Clarendon Press.

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Figure 7. Energy Growth for a Low Amplitude Run.R = 6667, a = 1.0

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Figure 8. Phase, cfx. ? Variation for a Low Amplitude

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(c) $pp . Dashed line represents decay rate

predicted from linear theory

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Figure 13. Energy Variation for the Modes E^.Ej,*^

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Figure 14. Energy Variation for the Modes E , Ej , Ejand E-,.R#

t 6667, ^ : 1.0

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Figure 15a. Mode Shape for fQ (odd) at t = 85. a = 1.0,

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.

The Laminar Energy (.600) has been Subtractedfor Convenience.

Top Curve

,

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61

= .5.

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Figure l8b. Reynolds Stresses for the Run e

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Energy Spectrum for the Run e

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Figure 20b. Reynold's Stresses for the Run e = .05/^/2"

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Figure 23. Energy Variation at R = 6667 for ThreeDifferent Wavenumbers

.

-0- Turbulent Kinetic Energy-x- Total Kinetic Energy

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DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 20Cameron StationAlexandria, Virginia 2231^

2. Library, Code 0212 2

Naval Postgraduate SchoolMonterey, California 939^+0

3. Chairman, Department of Aeronautics 1Naval Postgraduate SchoolMonterey, California 939^-0

k. Professor T. H. Gawain, Code 57Gn 5Department of AeronauticsNaval Postgraduate SchoolMonterey, California 939UO

5. Assistant Professor W. H. Clark, Code 57Cv 5

Department of AeronauticsNaval Postgraduate SchoolMonterey, California 939^-0

6. Lieutenant Commander George D. O'Brien, USN 3U8l7 Bayard BoulevardWashington, D. C. 20016

7. Mr. Ronald Bailey 1Theoretical BranchNASA Ames Research CenterMoffett Field, California 9U035

8. Information Research Associates 2

21^0 Shattuck AvenueBerkeley, California 9^720

9. Los Alamos Scientific Laboratory 1

Los Alamos, New Mexico 875UUAttn: Dr. F. Harlow

10. Commander 2

Naval Air Systems CommandAttn : Dr . E . S . Lamar - 1

Dr. H. J. Mueller - 1

Navy DepartmentWashington, D. C. 20360

118

11. Commanding OfficerNaval Ship Research and Development CenterCarderoc , MarylandAttn: Dr. F. Frenkiel - 1

Dr. J. Schott - 1

12. Office of Naval ResearchNavy DepartmentWashington, D. C. 20360Attn: Morton Cooper ; Code U38 - 1

Ralph Cooper - 1

13. Dr. W. C. ReynoldsDept. of Mechanical EngineeringStanford UniversityStanford, California 91+03 1+

Ik. Professor C. E. Menneken, Code 023Dean of Research AdministrationNaval Postgraduate SchoolMonterey, California 939^-0

15. Dr. Jacob FrommInternational Business Machines, Corp.Research LaboratoriesSan Jose, California 9511^

119

UNCLASSIFIEDSecurity Classification

DOCUMENT CONTROL DATA -R&D(Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified)

i originating ACTIVITY (Corpormte author)

Naval Postgraduate SchoolMonterey, California 939^0

2«. REPORT SECURITY C L A SSI F I C A T I 0^

Unclassified26. GROUP

3 REPOR T TITLE

A Numerical Investigation of the Non-Linear Mechanics of Wave Disturbancesin Plane Poiseuille Flows

4 DESCRIPTIVE NOTES (Type of report and.inctusive dates)

Technical Report5 authORISI (First nmme, middle initial, last name)

T. H. Gawain and W. H. Clark

S REPOR T DATE

2 September 1971

7a. TOTAL NO. OF PAGES

121

7fc. NO. OF REFS

168a. CONTRACT OR GRANT NO

b. PROJEC T NO

9a. ORIGINATORS REPORT NUMBER(S)

NPS-57Gn71091A

9b. OTHER REPORT n o I s i (Any other numbera that may be as signedthis report)

10 DISTRIBUTION STATEMENT

This document has been approved for public release and sale; its distributionis unlimited.

11. SUPPLEMENTARY NOTES 12. SPONSORING MILI TAR Y ACTIVITY

Naval Postgraduate SchoolMonterey, California 939UO

13. ABSTR AC T

The response of a plane Poiseuille flow to disturbances of various initialwavenumbers and amplitudes is investigated by numerically integrating theequation of motion. It is shown that for very low amplitude disturbances thenumerical integration scheme yields results that are consistent with thosepredictable from linear theory. It is also shown that because of non-linearinteractions a growing unstable disturbance excites higher wavenumber modeswhich have the same frequency, or phase velocity, as the primary mode. Forvery low amplitude disturbances these spontaneously generated higher wavenumbermodes have a strong resemblance to certain modes computed from the linearOrr-Sommerfeld equation.

In general it is found that the disturbance is dominated for a longtime by the primary mode and that there is little alteration of the originalparabolic mean velocity profile. There is evidence of the existence of anenergy equilibrium state which is common to all finite -amplitude disturbancesdespite their initial wavenumbers. This equilibrium energy level is roughly3-5$ of the energy in the mean flow which is an order of magnitude higherthan the equilibrium value predicted by existing non-linear theories.

DD "*" 1473I NOV • I ™T # ««J

S/N 0101 -§07-«t1 1

(PAGE 1)UNCLASSIFIED

120 Security Classification4- 3 1408

UNCLASSIFIEDSecurity Classif ication

KEY WORDS

Hydrodynamic stability

Finite-differences

Turbulence

Plane Poiseuille Flow

DD F° RvM691473 back

S/N 0101-807-682 1

LINK A

ROLE W T

LINK B

ROLE W T ROLE w T

UNCLASSIFIED121 Security Classification

J DUDLEY KNOX LIBRARY

3 2768 00391418 5