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Closure in turbulence theory using stabilization principle
Volume 118, number 3
PHYSICS LETTERS A
6 October 1986
CLOSURE IN TURBULENCE THEORY USING STABILIZATION PRINCIPLE Michail Z A K Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Received 16 April 1986; revised manuscript received 15 July 1986; accepted for publication 22 July 1986
The approach presented herein is based upon the concept that the velocity fluctuations driven by the instability of the original flow grow until a new stable state is approached. The application of this stabilization principle is demonstrated by describing the smoothing out of a velocity discontinuity.
1. Introduction
The problem of turbulence arose almost a hundred years ago as a result of a discrepancy between theoretical fluid dynamics and applied problems. However, in spite of considerable research activity, there is no general deductive theory of high Reynolds number turbulence: the direct utilization of the Navier-Stokes equations leads to unstable and unpredictable (chaotic) solutions. The central problem of turbulence theory is to find suitable methods of converting the infinite hierarchy of equations following from the averaging process of Navier-Stokes equations into a closed set. New developments and achievements in this direction are presented in refs. [3-5]. This note is based upon a different interpretation of the closure problem. The guiding principles of this interpretation can be illustrated by the following example. Let us consider a Couette flow between two parallel flat plates which are displaced relative to each other. The flow remains laminar as long as the Reynolds number R < 1500 and the velocity distribution is then linear. For R > 1500 this flow becomes unstable, and therefore, practically it can never be observed: any small disturbances (which always exist) are amplified, and being driven by the mechanism of the instability, grow until a new (turbulent) stable state is approached. The turbulent velocity pro-
files are very flat near the centre and become very steep near the walls, while with increasing Reynolds number, the curvature of these profiles becomes more pronounced. The stability here should be understood in a sense that the flow is reproducible, while a small change of R leads to small change of the flow characteristics. Prandtl obtained the mean velocity profile analytically based upon the Reynolds equations and additional rheological (the mixing length) assumption which couples the mean velocity v k and the fluctuation velocities ~k:
However, the same eq. (1) can be interpreted as a "feedback" which stabilizes the originally unstable laminar flow by driving it to a new turbulent state. In this paper the closure in turbulence theory based upon the stabilization principle will be formulated and applied to the problem of the smoothing out of a velocity discontinuity.
2. Formulation of the closure
Thus, as follows from the above example, the closure in turbulence can be related to the stabilization effect of the fluctuation velocities. Obviously, the fluctuations can grow only as long as the instability persists, and consequently, the in-
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
139
Volume 118, number 3
PHYSICS LETI'ERS A
6 October 1986
stability of the original laminar flow must be replaced by a neutral stability of the new turbulent flow. This stabilization principle allows to formulate the closure problem as follows: Let the original laminar flow described by the Navier-Stokes (NS) equations
found from experiments. By assuming that l is proportional to the width of the mixing zone b Prandtl obtained the following solution for the mean horizontal velocity:
P(~uk//ol -[- HiVi Uk ) = --VI,-P q- PV 2Uk,
while
VtU i = O,
(2) be unstable, i.e., some of the eigenvalues of the corresponding Orr-Sommerfeld equation have negative imaginary parts. Then, the fluctuation velocity ~k in the corresponding Reynolds equation p (ark~at + uivi vk ) = - V k p -[- pV 2vk -- pal ~)i~)k
u(y,t)
=
u + = ½(u
u++u
-
+ u2),
u
V=y/b,
i-3
),
(7)
= ½(ul - u2), (g)
t,
in which ul and u2 are the horizontal velocities before the mixing, and 13 is an experimental coefficient. In this section, we will obtain the analog of eq. (7) without any experimental coefficients.
(3) 3. l. Posedness of the problem
must be coupled with the mean velocity vk by a feedback
f(v*, V,v k. . . . ~/~3,...) = O,
(4)
which is found from the condition that all these negative imaginary parts vanish, and therefore, the solution possesses a neutral stability. Some mathematical aspects of the stabilization principle are discussed in refs. [1-3].
Consider the basic flow of incompressible inviscid fluids of the same density p in two horizontal parallel streams of widths H and velocities hi, h2 ( h l > U2).
For potential motion the Reynolds equation reduces to
3~i + 1 ( aePi12 Pi a-5- 5 21= - - ; lyl~H,
+ const,
i = 1 , 2,
(9)
3. Example The meaningfulness of the principle formulated above will be illustrated by the consideration of the smoothing out of a velocity discontinuity which was first treated by Prandtl [8], who closed the Reynolds equation: 3u 3u 3u a-7 + UTx + v a y
1 3T p ~y,
~- = - our,
(5)
in which x is parallel and y is perpendicular to the streams, q5i are the mean velocity potentials, Pi are the pressures which include the Reynolds stresses p:': P:'=½O(~),
pi=P:+P:',
(10)
In addition, q~i must satisfy the Laplace equation (11)
V 2~ i = 0
by using the mixing length theory:
i=1,2.
and the boundary conditions '1"= pl 2
~y,
(6)
in which u and u are the mean and fluctuation velocity projections on the coordinates x and y, respectively. ¢ is the turbulent shearing stress, and l is the mixing length which is supposed to be 140
3ep,/Oy= 0
at
]y[ = H.
(12)
3.2. The closure problem
Linear stability analysis of this system leads to
Volume 118, number 3
PHYSICS LETTERSA ak2 2_ s i n h [ k ( y - H)I
the following characteristic equation: ()k -/~/l) 2 - ()k - 22) 2 = ( p * / p a k ) t a n h ( k H ) , (13) in which a and 2,¢r/k are the amplitude and the wavelength of a small perturbation, respectively (a << H), 2t is the speed of wave propagation, and p* is the Reynolds stress discontinuity: p * - - p , " - p~' .
(14)
Assuming that the basic flow is laminar (p* = 0) one arrives at the well-known solution of eq. (13): X = 2+ _+ ih_,
(15)
which shows that this flow is unstable because of negative imaginary part of )~. Due to this instability small disturbances (which always exist) will grow until their contribution to the Reynolds stress p* stabilizes the system by eliminating the imaginary parts in eq. (15). Hence, p* must be selected such that ), = 2+
(16)
P*
22_,
(17)
tanh(kH)
i.e., the Reynolds stress discontinuity is proportional to the square of the velocity discontinuity. Eq. (17) can be considered as a feedback (4) which makes the system (9)-(12), (17) closed. The solution to this system is
•1 = 2IX
a2 sinh(kH) c o s h [ k ( y + H)]
× s i n [ k ( x - ~t+t)], 0 > y > - H , a2_ ~2 = U2X sinh(kH) c o s h [ k ( y - U)] ×sin[k(x-~+t)],
sinh(kH)tan(kH)
x (cos[
(x- 2+t)] - rcos[k(x- .+t)]
0 ~
O<~y<~H,
(18)
(21)
3.2. The smoothing out of the velocity discontinuity
Let us consider the case when the system is conservative, and therefore, due to fluctuations the velocity discontinuity is smoothing out. In order to obtain a closed form solution we will assume that ak << 1,
(22)
i.e., that the disturbances are given i~ the form of shallow waves with amplitude much smaller than the wavelength. As will be shown below, in this case the smoothing out process is much slower than the oscillations with the frequency k2+. Differentiating with respect to x the momentum equations (9), then substituting them into the solution (18), (19) and averaging the resulting expressions over the period 2¢r/k2+, one obtains at y = 0:
+ Icos[
b/I It;0 ~--"~/1' in which
-
-
u2) 2,
du2/dt = A(u 1
U2 [,=0 = 22,
A = (2/~r) ak 2 coth(kH).
-
u2) 2, (23) (24)
Now the averaging procedure can be justified. Indeed, the time derivative characterizing the smoothing out process has the order: (25)
d u / d q - ak222,
(26)
d u / d t 2 - k2 2,
and therefore (19)
d u / d t 2>> d u / d t I
if
ak << 1.
(27)
The solution to eq. (23) is written as 1 u l=~t+ 1/2 +4At'
sinh(kH) t a n h ( k H )
× {cos[k(x0 >/y > / - u,
[},
while the time derivative characterizing oscillations with the frequency ku+ is evaluated as
ak2 2_ s i n h [ k ( y + H)] P~'=
P~'= -
du,/dt = -A(u,
and therefore 2pak
6 October 1986
(x- .+tl I}, (20)
u2=2++ 1/2
1 +4At"
(28) 141
Volume 118, number 3
PHYSICS LETTERSA
tion velocities can be found by the substitution of the discontinuity (ul - u2) from eq. (28) into eqs. (20) and (21) After averaging these equations over the period 2~r/kh+ one gets:
It is easy to conclude from this solution that ul + u2 = hl + h2 = c o n s t
and ul, u 2 ~ h +
at
t--*oo.
6 October 1986
(29)
( ~ ) =
2ak sinh[k(y+H)]
v s i n h ( k H ) tanh( k H )
For the mean horizontal velocity profiles one obtains: sinh[k(y-
h1= h1-
×
o
y
+ [(Sak2/'rr)
coth(kH)]t
(30)
1/u
o y H.
+ [(8ak2/Ir) coth(kH)]t
The evolution of the mean horizontal velocity fields in the course of the smoothing out of the velocity discontinuity described by eqs. (30) and (31) has the same character as those obtained from the Prandtl solution (7), fig. 1. However, it is worth mentioning again, that eqs. (30) and (31) do not depend upon any experimental coefficients as eq. (7) does. The decay of the kinetic energy of the fluctua-
~ U2
u2
I
U1
O
Fig. 1. 142
1/h
+ [(8ak2/~r)coth(kH)]t
(33)
'
(31)
tll
9
×
sinh(kH)
O<~y~H.
t-O
(32)
2ak sinh[k(y - H)] ,~ sinh(kH ) t a n h ( k H )
'
sinh[k(y+ H)]
h_-
)2
+ [(8ak2/~r)coth(kH)]t
0>~y>/ - H ,
-.,
h2= h2+
×
- 1/h
1 1/h
sinh(kH)
h
x(
H)]
1
4. Discussion
Thus, it has been demonstrated that the closure in turbulence theory is based upon the principle of stabilization of the original laminar flow by fluctuation velocities. What is the mathematical meaning of this procedure? It is well known that the concept of stability is related to a certain class of functions: a solution which is unstable in a class of smooth functions can be stable in an enlarged (non-smooth) class of functions. Reynolds enlarged the class of smooth functions by introducing the field of fluctuation velocities which generated additional (Reynolds) stresses in the Navier-Stokes equations. Now it is reasonable to extend this procedure by choosing these Reynolds stresses such that they eliminate the original instability, i.e., by applying the stabilization principle. It should be emphasized that one cannot except that the solution of the type (30)-(33) will describe all the peculiarities of turbulent motion; it will rather extract the most essential properties of the motion, i.e., such properties which are reproducible, and therefore, have certain physical meaning. Description of finer details of turbulent motions will require further enlarging the class of
Volume 118, number 3
PHYSICS LETTERS A
f u n c t i o n s (which p r o b a b l y can be p e r f o r m e d b y u s i n g m e t h o d s of fractal geometry). However, all new closures must be based u p o n the stabilization principle which will provide the reproducibility of the t u r b u l e n c e structure.
Acknowledgement This research was carried out b y the Jet Propulsion Laboratory, C a l i f o r n i a Institute of Technology, u n d e r C o n t r a c t NAS7-918.
6 October 1986
References [1] M. Zak, Physica D, Kruskal Festschrift Volume (1985). [2] M. Zak, Phys. Lett. A 107 (1985) 125. [3] M. Zak, Deterministic representation of chaos with application to turbulence, Int. J. Math. Modell. (1986). [4] J.L. Lumley, J. Appl. Mech. 50 (1983) 1097. [5] D.C. Leslie, Developments in the theory of turbulence (Clarendon, Oxford, 1973). [6] R.G. Dessler, Rev. Mod. Phys. (1984) No. 2, part I. [7] P.G. Drazin and W.H. Reid, Hydrodynamic stability (Cambridge Univ. Press, London, 1984). [8] L. Prandtl, The mechanics of viscous fluids, in: Aerodynamic theory, Vol. 3, ed. W.F. Durand (1935) p. 166.
143
PHYSICS LETTERS A
6 October 1986
CLOSURE IN TURBULENCE THEORY USING STABILIZATION PRINCIPLE Michail Z A K Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA Received 16 April 1986; revised manuscript received 15 July 1986; accepted for publication 22 July 1986
The approach presented herein is based upon the concept that the velocity fluctuations driven by the instability of the original flow grow until a new stable state is approached. The application of this stabilization principle is demonstrated by describing the smoothing out of a velocity discontinuity.
1. Introduction
The problem of turbulence arose almost a hundred years ago as a result of a discrepancy between theoretical fluid dynamics and applied problems. However, in spite of considerable research activity, there is no general deductive theory of high Reynolds number turbulence: the direct utilization of the Navier-Stokes equations leads to unstable and unpredictable (chaotic) solutions. The central problem of turbulence theory is to find suitable methods of converting the infinite hierarchy of equations following from the averaging process of Navier-Stokes equations into a closed set. New developments and achievements in this direction are presented in refs. [3-5]. This note is based upon a different interpretation of the closure problem. The guiding principles of this interpretation can be illustrated by the following example. Let us consider a Couette flow between two parallel flat plates which are displaced relative to each other. The flow remains laminar as long as the Reynolds number R < 1500 and the velocity distribution is then linear. For R > 1500 this flow becomes unstable, and therefore, practically it can never be observed: any small disturbances (which always exist) are amplified, and being driven by the mechanism of the instability, grow until a new (turbulent) stable state is approached. The turbulent velocity pro-
files are very flat near the centre and become very steep near the walls, while with increasing Reynolds number, the curvature of these profiles becomes more pronounced. The stability here should be understood in a sense that the flow is reproducible, while a small change of R leads to small change of the flow characteristics. Prandtl obtained the mean velocity profile analytically based upon the Reynolds equations and additional rheological (the mixing length) assumption which couples the mean velocity v k and the fluctuation velocities ~k:
However, the same eq. (1) can be interpreted as a "feedback" which stabilizes the originally unstable laminar flow by driving it to a new turbulent state. In this paper the closure in turbulence theory based upon the stabilization principle will be formulated and applied to the problem of the smoothing out of a velocity discontinuity.
2. Formulation of the closure
Thus, as follows from the above example, the closure in turbulence can be related to the stabilization effect of the fluctuation velocities. Obviously, the fluctuations can grow only as long as the instability persists, and consequently, the in-
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
139
Volume 118, number 3
PHYSICS LETI'ERS A
6 October 1986
stability of the original laminar flow must be replaced by a neutral stability of the new turbulent flow. This stabilization principle allows to formulate the closure problem as follows: Let the original laminar flow described by the Navier-Stokes (NS) equations
found from experiments. By assuming that l is proportional to the width of the mixing zone b Prandtl obtained the following solution for the mean horizontal velocity:
P(~uk//ol -[- HiVi Uk ) = --VI,-P q- PV 2Uk,
while
VtU i = O,
(2) be unstable, i.e., some of the eigenvalues of the corresponding Orr-Sommerfeld equation have negative imaginary parts. Then, the fluctuation velocity ~k in the corresponding Reynolds equation p (ark~at + uivi vk ) = - V k p -[- pV 2vk -- pal ~)i~)k
u(y,t)
=
u + = ½(u
u++u
-
+ u2),
u
V=y/b,
i-3
),
(7)
= ½(ul - u2), (g)
t,
in which ul and u2 are the horizontal velocities before the mixing, and 13 is an experimental coefficient. In this section, we will obtain the analog of eq. (7) without any experimental coefficients.
(3) 3. l. Posedness of the problem
must be coupled with the mean velocity vk by a feedback
f(v*, V,v k. . . . ~/~3,...) = O,
(4)
which is found from the condition that all these negative imaginary parts vanish, and therefore, the solution possesses a neutral stability. Some mathematical aspects of the stabilization principle are discussed in refs. [1-3].
Consider the basic flow of incompressible inviscid fluids of the same density p in two horizontal parallel streams of widths H and velocities hi, h2 ( h l > U2).
For potential motion the Reynolds equation reduces to
3~i + 1 ( aePi12 Pi a-5- 5 21= - - ; lyl~H,
+ const,
i = 1 , 2,
(9)
3. Example The meaningfulness of the principle formulated above will be illustrated by the consideration of the smoothing out of a velocity discontinuity which was first treated by Prandtl [8], who closed the Reynolds equation: 3u 3u 3u a-7 + UTx + v a y
1 3T p ~y,
~- = - our,
(5)
in which x is parallel and y is perpendicular to the streams, q5i are the mean velocity potentials, Pi are the pressures which include the Reynolds stresses p:': P:'=½O(~),
pi=P:+P:',
(10)
In addition, q~i must satisfy the Laplace equation (11)
V 2~ i = 0
by using the mixing length theory:
i=1,2.
and the boundary conditions '1"= pl 2
~y,
(6)
in which u and u are the mean and fluctuation velocity projections on the coordinates x and y, respectively. ¢ is the turbulent shearing stress, and l is the mixing length which is supposed to be 140
3ep,/Oy= 0
at
]y[ = H.
(12)
3.2. The closure problem
Linear stability analysis of this system leads to
Volume 118, number 3
PHYSICS LETTERSA ak2 2_ s i n h [ k ( y - H)I
the following characteristic equation: ()k -/~/l) 2 - ()k - 22) 2 = ( p * / p a k ) t a n h ( k H ) , (13) in which a and 2,¢r/k are the amplitude and the wavelength of a small perturbation, respectively (a << H), 2t is the speed of wave propagation, and p* is the Reynolds stress discontinuity: p * - - p , " - p~' .
(14)
Assuming that the basic flow is laminar (p* = 0) one arrives at the well-known solution of eq. (13): X = 2+ _+ ih_,
(15)
which shows that this flow is unstable because of negative imaginary part of )~. Due to this instability small disturbances (which always exist) will grow until their contribution to the Reynolds stress p* stabilizes the system by eliminating the imaginary parts in eq. (15). Hence, p* must be selected such that ), = 2+
(16)
P*
22_,
(17)
tanh(kH)
i.e., the Reynolds stress discontinuity is proportional to the square of the velocity discontinuity. Eq. (17) can be considered as a feedback (4) which makes the system (9)-(12), (17) closed. The solution to this system is
•1 = 2IX
a2 sinh(kH) c o s h [ k ( y + H)]
× s i n [ k ( x - ~t+t)], 0 > y > - H , a2_ ~2 = U2X sinh(kH) c o s h [ k ( y - U)] ×sin[k(x-~+t)],
sinh(kH)tan(kH)
x (cos[
(x- 2+t)] - rcos[k(x- .+t)]
0 ~
O<~y<~H,
(18)
(21)
3.2. The smoothing out of the velocity discontinuity
Let us consider the case when the system is conservative, and therefore, due to fluctuations the velocity discontinuity is smoothing out. In order to obtain a closed form solution we will assume that ak << 1,
(22)
i.e., that the disturbances are given i~ the form of shallow waves with amplitude much smaller than the wavelength. As will be shown below, in this case the smoothing out process is much slower than the oscillations with the frequency k2+. Differentiating with respect to x the momentum equations (9), then substituting them into the solution (18), (19) and averaging the resulting expressions over the period 2¢r/k2+, one obtains at y = 0:
+ Icos[
b/I It;0 ~--"~/1' in which
-
-
u2) 2,
du2/dt = A(u 1
U2 [,=0 = 22,
A = (2/~r) ak 2 coth(kH).
-
u2) 2, (23) (24)
Now the averaging procedure can be justified. Indeed, the time derivative characterizing the smoothing out process has the order: (25)
d u / d q - ak222,
(26)
d u / d t 2 - k2 2,
and therefore (19)
d u / d t 2>> d u / d t I
if
ak << 1.
(27)
The solution to eq. (23) is written as 1 u l=~t+ 1/2 +4At'
sinh(kH) t a n h ( k H )
× {cos[k(x0 >/y > / - u,
[},
while the time derivative characterizing oscillations with the frequency ku+ is evaluated as
ak2 2_ s i n h [ k ( y + H)] P~'=
P~'= -
du,/dt = -A(u,
and therefore 2pak
6 October 1986
(x- .+tl I}, (20)
u2=2++ 1/2
1 +4At"
(28) 141
Volume 118, number 3
PHYSICS LETTERSA
tion velocities can be found by the substitution of the discontinuity (ul - u2) from eq. (28) into eqs. (20) and (21) After averaging these equations over the period 2~r/kh+ one gets:
It is easy to conclude from this solution that ul + u2 = hl + h2 = c o n s t
and ul, u 2 ~ h +
at
t--*oo.
6 October 1986
(29)
( ~ ) =
2ak sinh[k(y+H)]
v s i n h ( k H ) tanh( k H )
For the mean horizontal velocity profiles one obtains: sinh[k(y-
h1= h1-
×
o
y
+ [(Sak2/'rr)
coth(kH)]t
(30)
1/u
o y H.
+ [(8ak2/Ir) coth(kH)]t
The evolution of the mean horizontal velocity fields in the course of the smoothing out of the velocity discontinuity described by eqs. (30) and (31) has the same character as those obtained from the Prandtl solution (7), fig. 1. However, it is worth mentioning again, that eqs. (30) and (31) do not depend upon any experimental coefficients as eq. (7) does. The decay of the kinetic energy of the fluctua-
~ U2
u2
I
U1
O
Fig. 1. 142
1/h
+ [(8ak2/~r)coth(kH)]t
(33)
'
(31)
tll
9
×
sinh(kH)
O<~y~H.
t-O
(32)
2ak sinh[k(y - H)] ,~ sinh(kH ) t a n h ( k H )
'
sinh[k(y+ H)]
h_-
)2
+ [(8ak2/~r)coth(kH)]t
0>~y>/ - H ,
-.,
h2= h2+
×
- 1/h
1 1/h
sinh(kH)
h
x(
H)]
1
4. Discussion
Thus, it has been demonstrated that the closure in turbulence theory is based upon the principle of stabilization of the original laminar flow by fluctuation velocities. What is the mathematical meaning of this procedure? It is well known that the concept of stability is related to a certain class of functions: a solution which is unstable in a class of smooth functions can be stable in an enlarged (non-smooth) class of functions. Reynolds enlarged the class of smooth functions by introducing the field of fluctuation velocities which generated additional (Reynolds) stresses in the Navier-Stokes equations. Now it is reasonable to extend this procedure by choosing these Reynolds stresses such that they eliminate the original instability, i.e., by applying the stabilization principle. It should be emphasized that one cannot except that the solution of the type (30)-(33) will describe all the peculiarities of turbulent motion; it will rather extract the most essential properties of the motion, i.e., such properties which are reproducible, and therefore, have certain physical meaning. Description of finer details of turbulent motions will require further enlarging the class of
Volume 118, number 3
PHYSICS LETTERS A
f u n c t i o n s (which p r o b a b l y can be p e r f o r m e d b y u s i n g m e t h o d s of fractal geometry). However, all new closures must be based u p o n the stabilization principle which will provide the reproducibility of the t u r b u l e n c e structure.
Acknowledgement This research was carried out b y the Jet Propulsion Laboratory, C a l i f o r n i a Institute of Technology, u n d e r C o n t r a c t NAS7-918.
6 October 1986
References [1] M. Zak, Physica D, Kruskal Festschrift Volume (1985). [2] M. Zak, Phys. Lett. A 107 (1985) 125. [3] M. Zak, Deterministic representation of chaos with application to turbulence, Int. J. Math. Modell. (1986). [4] J.L. Lumley, J. Appl. Mech. 50 (1983) 1097. [5] D.C. Leslie, Developments in the theory of turbulence (Clarendon, Oxford, 1973). [6] R.G. Dessler, Rev. Mod. Phys. (1984) No. 2, part I. [7] P.G. Drazin and W.H. Reid, Hydrodynamic stability (Cambridge Univ. Press, London, 1984). [8] L. Prandtl, The mechanics of viscous fluids, in: Aerodynamic theory, Vol. 3, ed. W.F. Durand (1935) p. 166.
143