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Analysis of turbulence in shear flows using the stabilization principle
0895-7 177/89 63.00 + 0.00 Copyright 0 1989 Maxwell Pergamon Macmillan plc
Math1 Comput. Modelling, Vol. 12, No. 8, pp. 985-990, 1989
Printed in Great Britain. All rights reserved
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.
(Received November 1987; accepted for publication 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, ck,ViVk,. . . )= 0,
(1)
between the Reynolds stresses ub = v’vj 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
986
M.
ZAK
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(“‘-P’J. The function q(y) must satisfy the Orr-Sommerfeld
(4)
equations:
(U - C)(D* - u*)q - U’q = (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
Dcp = D3q = 0
aty=y*,
We will start with the velocity profile characterized
at y = y,
(8)
by the critical Reynolds number:
Re = Ret,.
(9)
Re* > Ret,
(10)
Any increase in velocity when
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)
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
(17)
r=z. It contains an additional term on the r.h.s.: the Reynolds stress disturbance, THE
CLOSURE
as yet unknown.
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 - U”cp - (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
M.
988
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)
(0; = F at y = 1).
(29)
since U = 0 at y = 0, and c, = 1XJ 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)
*;=o.
(33)
Since at the rigid wall 0 = 0, one obtains
989
Analysis of turbulence in shear flows
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: $,=
;(l-y2)dy=$
(35)
s 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. (37)
Applying conditions (32) and (34) one finds that c2 = CA= 0.
(38)
&’ = -0.00703 C, I- 0.00712 C,,
(39)
Taking into account that
one obtains $ = 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~;
(40)
o= 2.1653 cos 0.19199~ - 51.8584 cash 0.19199y + 50.692.
(41)
and therefore,
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
(42)
SOLUTION
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: Dmax= 0.9989 < pm,, = 1.
(43)
l&b’l= 1.99132 < \@“I = 2.
(44)
\o;I = 2.00259 > I&;1= 2.
(45)
Also,
At the same time
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].
M.
990
ZAK
Turning to the Reynolds stress profile (41) one finds that the maximum is shifted toward the wall: y* = 0.58, which expresses the well-known Finally, the pressure gradient
of the stress module
1~1
(46)
wall effect
(47) for the new turbulent
flow is greater
than
for the original
2.002586
2 ‘Re*=z.
laminar
flow:
ajY (48) I
I
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. 2. 3. 4.
M. Zak, Closure in turbulence using the stabilization principle. Phys. Letf. 118(3) (1986). M. Zak, Deterministic representation of chaos with appiication to turbulence. M&z/ ModeNing 9, 599-612 (1987) P. I. G. Drazin and W. M. Reid, Hydrodynamic Sfability, p. 192. Cambridge Univ. Press, New York (1984). H. Schlichting, Boundary-Layer Theory. McGraw-Hill, New York (1963).
Math1 Comput. Modelling, Vol. 12, No. 8, pp. 985-990, 1989
Printed in Great Britain. All rights reserved
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.
(Received November 1987; accepted for publication 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, ck,ViVk,. . . )= 0,
(1)
between the Reynolds stresses ub = v’vj 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
986
M.
ZAK
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(“‘-P’J. The function q(y) must satisfy the Orr-Sommerfeld
(4)
equations:
(U - C)(D* - u*)q - U’q = (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
Dcp = D3q = 0
aty=y*,
We will start with the velocity profile characterized
at y = y,
(8)
by the critical Reynolds number:
Re = Ret,.
(9)
Re* > Ret,
(10)
Any increase in velocity when
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)
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
(17)
r=z. It contains an additional term on the r.h.s.: the Reynolds stress disturbance, THE
CLOSURE
as yet unknown.
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 - U”cp - (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
M.
988
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)
(0; = F at y = 1).
(29)
since U = 0 at y = 0, and c, = 1XJ 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)
*;=o.
(33)
Since at the rigid wall 0 = 0, one obtains
989
Analysis of turbulence in shear flows
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: $,=
;(l-y2)dy=$
(35)
s 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. (37)
Applying conditions (32) and (34) one finds that c2 = CA= 0.
(38)
&’ = -0.00703 C, I- 0.00712 C,,
(39)
Taking into account that
one obtains $ = 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~;
(40)
o= 2.1653 cos 0.19199~ - 51.8584 cash 0.19199y + 50.692.
(41)
and therefore,
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
(42)
SOLUTION
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: Dmax= 0.9989 < pm,, = 1.
(43)
l&b’l= 1.99132 < \@“I = 2.
(44)
\o;I = 2.00259 > I&;1= 2.
(45)
Also,
At the same time
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].
M.
990
ZAK
Turning to the Reynolds stress profile (41) one finds that the maximum is shifted toward the wall: y* = 0.58, which expresses the well-known Finally, the pressure gradient
of the stress module
1~1
(46)
wall effect
(47) for the new turbulent
flow is greater
than
for the original
2.002586
2 ‘Re*=z.
laminar
flow:
ajY (48) I
I
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. 2. 3. 4.
M. Zak, Closure in turbulence using the stabilization principle. Phys. Letf. 118(3) (1986). M. Zak, Deterministic representation of chaos with appiication to turbulence. M&z/ ModeNing 9, 599-612 (1987) P. I. G. Drazin and W. M. Reid, Hydrodynamic Sfability, p. 192. Cambridge Univ. Press, New York (1984). H. Schlichting, Boundary-Layer Theory. McGraw-Hill, New York (1963).