analisys of turbulence in shear flows using the stabilization principle

8/11/2019 Analisys of Turbulence in Shear Flows Using the Stabilization Principle

1/6

Math1 Comput. Modelling Vol. 12 No. 8 pp. 985-990 1989

Printed in Great Britain. All rights reserved

0895-7 177/89 63.00 + 0.00

Copyright 0 1989 Maxwell Pergamon Macmillan plc

ANALYSIS OF TURBULENCE IN SHEAR FLOWS

USING THE STABILIZATION PRINCIPLE

M ZAK

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, U.S.A.

Receiv ed N ovember 1987; accept ed for publ icat i on November 1988)

Communicated by X. J. R. Avula

Abstract-The analysis presented herein is based upon the concept that the velocity fluctuations, and

therefore, the Reynolds stresses, driven by the instability of the original flow grow until a new stable state

is approached. The Reynolds stresses incorporated into the Orr-Sommerfeld equation are coupled with

the main flow such that all the imaginary parts of the complex eigenvalues vanish, i.e. the original

instability is eliminated. Using this stabilization principle, it is possible to find the Reynolds stresses as

well as the mean velocity for plane Poiseuille flow with the Reynolds number slightly higher than the

critical.

INTRODUCTION

This study illustrates the application of the stabilization principle to the analysis of turbulence in

shear flows. The stabilization principle was introduced in Refs [l, 21. It shows that velocity

fluctuations driven by the instability of the original (unperturbed) motion can grow until a new

stable state is approached. Hence, the closure to the Reynolds equations can be sought in the form

of such a feedback,

f P, k , iVk, . . )= 0,

1)

between the Reynolds stresses ub = vvj and the mean velocity vk (and its gradients V,v) which

stabilizes the mean flow. Obviously, the velocity fluctuations can grow only as long as the instability

persists, and consequently, the instability of the original laminar flow must be replaced by a neutral

(marginal) stability of the new turbulent flow.

If we confine our study to flows whose instability can be found from linear analysis (plane

Poiseuille flow, boundary layers), then the closure problem can be formulated as follows: let the

original laminar flow described by the Navier-Stokes equations be unstable, i.e. some of the

eigenvalues for the corresponding Orr-Sommerfeld equation have positive imaginary parts. Then,

the closure (1) is found from the condition that all these positive imaginary parts vanish, and

therefore, the solution possesses a neutral stability. However, the closure (1) can be written in the

explicit form only if the criteria for the onset of instability are formulated explicitly. Since such

a situation is an exception rather than a rule, one can apply a step-by-step strategy proposed in

Ref. [2]. This strategy is based upon the fact that the Reynolds stress disturbances grow much faster

than the mean motion disturbances [see 21. Hence, one can assume that these stresses will be large

enough to stabilize the mean flow which is still sufficiently close to its original unperturbed state.

But the Reynolds stresses being substituted in the Reynolds equations will change the mean velocity

profile, and consequently, the conditions of instability. These new conditions, in turn, will change

the Reynolds stresses etc. By choosing the iteration steps to be sufficiently small, one can obtain

acceptable accuracy. In this article the first step approximation will be applied to a plane Poiseuille

flow.

FORMULATION OF THE PROBLEM

Let us consider a plane shear flow with a dimensionless velocity profile:

985


2/6

986

M. Z K

with boundaries

y,=o, y*= 1,

(3)

and the x coordinate being along the axis of symmetry. The stream function representing a single

oscillation of the disturbance is assumed to be of the form

$(x, y, 1) = (p(y)ei(-PJ.

(4)

The function q(y) must satisfy the Orr-Sommerfeld equations:

(U - C) D* - u*)q - Uq = icl Re)-(D* - CX*)~~,

(5)

in which o and p are constants, Re is the Reynolds number and

and

Equation (5) should be solved subject to the boundary conditions, which in the case of a symmetric

flow between rigid walls are

cp=Dq=O aty=y*,

Dcp = D3q = 0

at y = y,

8)

We will start with the velocity profile characterized by the critical Reynolds number:

Re = Ret,.

(9)

Any increase in velocity when

Re* > Ret,

(10)

leads to instability of the laminar flow and to transition to a new turbulent flow.

We will concentrate our attention on the situation when the increase in the Reynolds number

is sufficiently small,

Re* - Ret,

Ret,

6 1.

11)

In this case we will be able to formulate a linearized version of the closure (1) explicitly based upon

the conditions of the instability of the Orr-Sommerfeld equation written for Re = Re,.,, and to

obtain the mean velocity profile and Reynolds stress for the corresponding turbulent flow.

GENERALIZED ORR SOMMERFELD EQUATION

In order to apply the stabilization principle and formulate the closure problem we have to

incorporate the Reynolds stresses into the Orr-Sommerfeld equation. For this purpose let us start

with the Reynolds equations for a plane shear flow expressed in terms of small perturbations:

12)

(13)

and

(14)


3/6

Analysis of turbulence in shear flows

987

using the boundary layer approximation. Here U(y) is the mean velocity profile, 0, P and P are

small velocity and pressure perturbations, v is the kinematic viscosity and f is the shearing Reynolds

stress which is sought in the form

f = me - 80.

(15)

Introducing equations (4) and (15) into equations (12)-(14) we obtain, after the elimination of

pressure, the generalized Orr-Sommerfeld equation in dimensionless form:

B- c) D - ct2)q - @cp

ict

Re))(D* - CZ~)~CJJ-i (D* + cr2)r,

in which

i

r=z.

(17)

It contains an additional term on the r.h.s.: the Reynolds stress disturbance, as yet unknown.

THE CLOSURE PROBLEM

Returning to our problem, let us apply equation (16) to the case when

Re = Re*, U = U(y).

(18)

Substituting equations (18) into equation (16), one obtains

(U -

C) D- a)cp - Ucp - ict Re*)- D2 - ct*)*cp= -i D2+ a)~.

19)

With zero Reynolds stress (T = 0), equation (19) would have eigenvalues with positive imaginary

parts since Re* > Ret,. These positive imaginary parts of the eigenvalues would vanish if Re* is

replaced by Recr. Hence, according to the stabilization principle, the Reynolds stress r should be

selected such that equation (19) is converted to equation (5) at Re = Ret,, i.e.

-k D2 + ~r)r + ia Re*)-(D - cr2)cp= (iol Re,,))(D2- a2)(p

or

D2+a2)z

=( &- &) CD' - a ' +

Equation (20) relates the disturbance of the mean flow velocity and the Reynolds stress T. It

allows us to reproduce a linearized version of the closure (1):

21)

in which i and 3 are the dimensionless Reynolds stress and the stream function characterizing the

unperturbed flow (for instance, rl/ = -8UjdX). Indeed, after perturbing equation (21) and

substituting equations (4) and (15), one returns back to equation (20).

It is important to emphasize that equation (21) is not universal closure: it contains two numbers

(Re,, and LX)which characterize a particular laminar flow. Here Recr is the smallest value of the

Reynolds number below which all initially imparted disturbances decay, whereas above that value

those disturbances which are characterized by GI see equations (4) and (15)] are amplified. Both

of these numbers can be found from equation (5) as result of classical analysis of hydrodynamics

stability performed for a particular laminar flow. One should recall that the closure (21) implies

a small increment of the Reynolds number over its critical value [see equation (1 l)]. For large

increments the procedure must be performed by steps: for each new mean velocity profile (which

is sufficiently close to the previous one) the new Re:, and c( are supposed to be found from the

solution of the eigenvalue problem for the Orr-Sommerfeld equation. Substituting Ref, and u into


4/6

988

M

ZAK

the closure (21) and solving it together with the corresponding Reynolds equations, one finds the

mean velocity profile and the Reynolds stress for the next increase of the Reynolds number Re*

etc.

PLANE POISEUILLE FLOW

In this section we will apply the approach developed above to a plane Poiseuille flow with the

velocity profile

80(y)=

1 -y*

(22)

and [see 31

Ret, = 5772.2, GL 1.021.

(23)

As a new (supercritical) Reynolds number we will take

Re* = 6000.

(24)

The closure (21) should be considered together with the governing equation for the unidirectional

mean flow:

voll +

5 = C = const

(25)

or

vB +i=C,y+C2

(26)

The constants c, and Q can be found from the condition

i = 0

at y = 1 and y = 0,

(27)

expressing the fact that the Reynolds stress vanishes at the rigid wall and in the middle of the flow.

Hence,

G =O,

(28)

since U = 0 at y = 0, and

c, = 1XJ (0; = F at y = 1).

(29)

Thus,

i=v(U;y - U )

or, in dimensionless form,

Substituting equation (30) into the closure (21) oie obtains the governing equation for the mean

velocity profile in terms of the stream function rl/, while u = all/ ay :

(31)

in which qj = F at y =y,.

Without loss of generality it can be set

$&J=o.

(32)

Since at the rigid wall 0 = 0, one obtains

*;=o.

(33)


5/6

Analysis of turbulence in shear flows

989

In the middle of the flow due to symmetry:

VA= 0, i.e. *i = 0.

(34)

Finally, the flux of the turbulent flow should be the same as the flux of the original (unstable)

laminar flow:

$,= ;(l-y2)dy=$

s

(35)

These four (non-homogeneous) boundary conditions (32)-(35) allow one to find four arbitrary

constants appearing as a result of integration of equation (31). After substituting the numbers [see

equations (23) and (24)] one arrives at the following differential equation:

F - 1.08202$ - 0.04124@ = l.O44$y,

(36)

whence

7 = C, sin 0.19199~ + C, cos 0.19199~ + C, sinh 0.19199~ + C,cosh 0.191994, - 25.3152+yy.

Applying conditions (32) and (34) one finds that

c2 = CA= 0.

Taking into account that

one obtains

& = -0.00703 C, I- 0.00712 C,,

$ = C, sin 0.19199~ + C, sinh 0.19199~ + (0.17797 C, - 0.18024 C,)y.

Now applying conditions (33) and (35) one arrives at the following solution:

q = 11.278 sin 0.19199~ - 270.11 sinh 0.19199~ + 50.692~;

and therefore,

o= 2.1653 cos 0.19199~ - 51.8584 cash 0.19199y + 50.692.

Substituting solution (40) into equation (30) one obtains the Reynolds stress profile:

Re* r = 0.41572 sin 0.19199~ + 9.9563 sinh 0.19199~ - 2.00259~.

ANALYSIS OF THE SOLUTION

(37)

(38)

(39)

(40)

(41)

(42)

We will start with the comparison of the original laminar velocity profile (22) and the mean

velocity profile (40). Both of them envelop the same area, i.e. the fluxes of the original laminar and

post-instability turbulent flows are the same. However, the maximum turbulent mean velocity is

smaller than the maximum of the original laminar flow:

D

max 0.9989 < pm,, = 1.

(43)

Also,

l&bl 1.99132 < \@I = 2. (44)

At the same time

\o;I = 2.00259 > I&;1= 2.

(45)

Hence, the turbulent mean velocity profile is more flat at the centre and more steep at the walls

in comparison with the corresponding laminar flow. This property is typical for turbulent flows [4].


6/6

990

M. ZAK

Turning to the Reynolds stress profile (41) one finds that the maximum of the stress module 1~1

is shifted toward the wall:

y* = 0.58,

(46)

which expresses the well-known wall effect

Finally, the pressure gradient

(47)

for the new turbulent flow is greater than for the original laminar flow:

2.002586 2 ajY

Re*=z.

I I

(48)

Thus, despite the fact that the Reynolds number Re* slightly exceeds the critical value Re,,, all the

typical features of turbulent flows are clearly pronounced in the solution obtained above.

REFERENCES

I. M. Zak, Closure in turbulence using the stabilization principle. Phys. Letf. 118(3) (1986).

2. M. Zak, Deterministic representation of chaos with appiication to turbulence. M&z/

ModeNing 9, 599-612 1987)

3.

P. I. G. Drazin and W. M. Reid,

Hydrodynamic Sfability,

p. 192. Cambridge Univ. Press, New York (1984).

4. H. Schlichting, Boundary-Layer Theory. McGraw-Hill, New York (1963).

analisys of turbulence in shear flows using the stabilization principle

Documents