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To the theory of the mechanism of mixing in turbulent flows with coherent structures
IN HEAT A~D ~%SS TRANSFER 0094-4548/78/0901-0253502.00/0 Vol. 5, pp. 253-258, 1978 © Pergamon Press Ltd. Printed in Great Bri#a~n
TO THE THEORY OF THE MECHANISM OF MIXING IN TUEBULENT PLOWS WITH COHERENT STRUCTURES B. A. Kolovandin, V. A. Sosinovich and S. V. Eravar Heat and Mass Transfer Institute, Minsk, U.S.S.R. (Ccmmmicated by O.G. Martynenko and R.I. Soloukin)
ABSTRACT A system o f linear equations has been obtained which governs the fields of velocity and of the scalar passive impurity in a turbulent incompressible liquid flow. This system is considered as a simple model characterizing the large-scale coherent structures which have been found experimentally in many turbulent flows. The means %o a c c o u n t f o r t h e s e s % r u c t u r e s when d e s c r i b i n g t h e m e c h a n i s m o~ m i x i n g a r e d i s c u s s e d .
As a c o n s e q u e n c e o f t h e e x p e r i m e n t a l s t u d i e s o f t u r b u l e n t flows, it becomes increasingly obvious that the classical pic-
ture of turbulence as a state of ~ the flow, in which the properties of different dyn=mioal variables are determined in tote statistically, lacks completeness. Recent experimental findings indicate %o the existence in turbulent flows, along with random events, of large-scale stzuotures which have the detemninistic character and which contribute much to the general characteristics of the turbulent flows [13. Of particular importance are these large-scale coherent sf~ru~tures in the mechanism of turbulent mixing up to the molecular level. I% has been shown experimentally in [2 ] %hat the process of mixing is closely related %0 the specific features of the dynamics of lazEe-soale coherent structures, i.e. with the mechanism of pairing and coalescence of vortices. An adequate theoretical description of the mech--ism for the mixing of impurities or equilibration of temperature inhomogeneities requires an equation that governs the transport 2~
254 of
B.A. Kolovandin, V.A. Sosinovich and S.V. Kravar
Vol. 5, No. 5
passive impurity by the turbulent velocity field. It is
a
clear that the large-scale coherent structures of the velocity field organize the same structures in the passive impurity [33. Therefore we shall postulate that the turbulent field of the impurity, as well as the velocity field, should be represented as the sum of the three components
f =~+~+f' Here
f
(1)
is the time-averaged value of the turbulent field
~
,
is the large-scale coherent component of the turbulent field and f' is the stochastic part of the turbulent field. By splitting the turbulent velocity field in the form of (1), the authors of [4] have obtained the equations for the wave disturbances with small amplitudes. In the present paper, the system of equations for the velocity field is supplemented with the equation for the wave part of the scalar field which is transported by the velocity field. The governing set of equations is
ui ~ul = ~p 1 t + uj axj - ~x--~ + He
a2ul ~xj~xj
(2)
xj
aT ~T a-~ +uj~xj
=-~
1
a2T axj~xj
(4)
This system has been dimensionalized by the use of the characteristic velocity
U and length ~ scales.
By splitting the fields of u i , p , T , as in (i), it is easy to obtain the system of equations which describes the behavior of the time-average and wave magnitudes ~ui =
uj ~xj aui
aP
i
~x i + Re aui
~ui=
e2u i
~xjaxj a~
~
u,u~" -
axj
i S
@2u i
(9
@
~ ~
6~j' uius ~ ~
- ,x~ rij
(5)
~ -
(6)
Vol. 5, No. 5
a xj
: o (a);
MECHANISM CF MIXING IN ~
uj_ = o
a xS
(b) ;
~xS
FIDWS
~ o (o)
255
('/)
(9) Here the following notation is used
< u1.u > -
(lo)
where the overbar denotes the time average and sharp brackets, the ensemble or phase average. Equations (5) through (9) describe distribution of the time average and coherent values and are no~ closed. W i ~ regard for the values riJ and Sj ~ e following simple & s 81~lmlD.t i o n s will be made
ri;l
= - 2~u~ij
;
~;I
=-~T
~Xj'
(12)
where
in which £u and 8T are the turbulent coefficients of viscosi1~y and diffusion, which, generally speaking, depend on the coordinate of the flow field ( for the time being, the form of these coefficients will not be specified ). We shall not require equations (5) and (8), since in what follows a change in ui and T , wi~h the form of ~i and known, will be considered. Among the approaches to the turbulent flows with coherent structures there is one based on the analogy with the stability
256
B.A. Kolovandin, V.A. Sosin(mrichand S.V. Kravar
Vol. 5, No. 5
theory of laminar flows. A linear version of this approach has been developed in [4 ~ where the long-wave regular disturbances immltate large-scale coherent structures.
It is clear that
such an approach to describing the coherent structures cannot be considered as quite adequate, but it enables one to reveal the basic features of the interaction dynamics of the average magnitudes,
coherent structures and of the stochastic part of
the turbulent flow. This approach will be followed below when we shall consider the transport of a passive impurity. Since calculation will be made in the linear approximation, we should omit those terms in equations (6) and (9) which are quadratic with respect to the wavenumbers. Consider a parallel, on the average stationary, turbulent flow in which the time average values are the functions of nothing but the transverse coordinate y
~i = [~(y)'O'O ] ;
T = T(y)
(14)
Each of the wave variables in equations (6)~ (Tb) and (9) will be represented as
=
½[ i ( y , t ) e i(kx-
t) + o.o.J
(15)
x 3 on the RHS of (15) is The absence of the coordinate due to our assumption that the coherent structures are twodimensional. This postulation is supported by a number of experimental studies [I] . The dependence on time is accounted for dually: the factor e -i~Jt
presupposes %ime-fluctuational
character of solution~ while the dependence of the amplitude o~
t
implies that its change with time is accounted for;
k
is the complex wave vector;cO is the frequency (real number). Substituting all of the wavenumbers in the form of (15) into equations (6)~ (7b) and (9) yields the following system of coupled equations for the amplitudes of these values in the linear approximation
i~
+D~
= o
(16)
Vol. 5, No. 5
t-ff~CHANISMOF MIXING IN ~JRBUIa~%T FLOWS
( ~ - - ( - ~•) + , ~ e ) u~.
257
.. * ~S-Z2) ( ~ ) ~ = -i~# + - ~1( ~ 2-~ 2)u-(i~Sz1+ (17)
(~-~-(-i~).
i~)~
(~-(-i~).
i ~ ) ~ + (D{)~ = -~6(~2-k2)~-(ik~z
Here
D m
~
= -~
;
,
+ _~(~2_~2)~_
(i~l.
~22)
+ ~2 )
(18)
(19)
are the amplitudes of the wave
quantities riJ and Sj , respectively. With the use of equation (16) one may express ~ in terms of v , substitute this expression into (17) and then eliminate from the ensuing equation and equation (18). As a result, the following equation will be obtained which determines =
+
"k2D(~zz-S22)* i~(D2* k2) S12
(2o)
Employing formulas (12) and (13) we may express the quantities Sij and ~'i in terms of v and T . Simple manipulations give
~n- s22 = ~ ;-uD~ ; = - ik ~ ' ~
~12 ~ -~u ~(~2 + k2)~
(21)
e 2 = - ~.~
(22)
With the help of formulas (21) and (22), equations (19) and (20) may be presented as
(23)
~=~
(2~)
et
(2s}
258
B.A. K O I ~ ,
V.A. Sosinoxrichand S.V. Kravar
Vol. 5, No. 5
Here the following notation is used
L = - ik(~ - o) + ( ~ e
_- i k ( D 2 ~ )
+ Eu)~ + 2(DEu)D
+ (D2~,u)(D2 + k 2)
(26) (2?)
N = D2 - k2
(28)
P = - i k ( ~ - c) + ("~e + ET)~ + ( ~ ' T ) D
(29)
R =-
(3o)
(D~) ~u
~'u= v ~
&T
;
;'T = 9 " T
ob
;
c =-.k--
(31)
Our further aim is to solve the system of equations (23) through (25) for different expressions of E u and E T. It may be well if, using the solutions obtained, we could calculate the Reynolds stresses and the flows induced by the large-scale coherent values ui and T . This is Just that very information about the large-scale turbulent structure which is indispensible in deriving the equations for fine-scale qttsntities that determine the degree of mixing up to the molecular level. The sufficient characteristics of mixing are the scale and intensity of segregation [5 ] • The knowledge of the wave quantities ui and T is required also to determine the behavior of the boundary between the turbulent and non-turbulent parts of the fluid, which is necessary to calculate the degree of the flow intermittency [6 J .
References 1.
Roshko, A., AIAAJ, 14 , 10,p. 13~9 (1976).
2.
Dimotakis. P.E., Brown, G.L., J.Fluid Mech.,78 ,pt. 3,
p. s35 (1976). 3.
Kovasznay, L.S.G., Second Australasian Conference on Heat and Mass Transfer, The University of Sydney, p. 295 (1977).
4.
Reynolds, W.C., Hussain, A.K.M.F., J.Fluid Mech., 54 , pt. 2, p. 263 (1972J.
5.
Brodkey, R.S., Mixing in Turbulent Fields, in: Turbulence in Mixing Operations, Ed. by R.S.Brodkey , Academic Press New York (1975). Wygna.nski, l.~Fiedler, H.E.,J.Fluid Mech.,41, pt.2,p.327 (1970).
6.
TO THE THEORY OF THE MECHANISM OF MIXING IN TUEBULENT PLOWS WITH COHERENT STRUCTURES B. A. Kolovandin, V. A. Sosinovich and S. V. Eravar Heat and Mass Transfer Institute, Minsk, U.S.S.R. (Ccmmmicated by O.G. Martynenko and R.I. Soloukin)
ABSTRACT A system o f linear equations has been obtained which governs the fields of velocity and of the scalar passive impurity in a turbulent incompressible liquid flow. This system is considered as a simple model characterizing the large-scale coherent structures which have been found experimentally in many turbulent flows. The means %o a c c o u n t f o r t h e s e s % r u c t u r e s when d e s c r i b i n g t h e m e c h a n i s m o~ m i x i n g a r e d i s c u s s e d .
As a c o n s e q u e n c e o f t h e e x p e r i m e n t a l s t u d i e s o f t u r b u l e n t flows, it becomes increasingly obvious that the classical pic-
ture of turbulence as a state of ~ the flow, in which the properties of different dyn=mioal variables are determined in tote statistically, lacks completeness. Recent experimental findings indicate %o the existence in turbulent flows, along with random events, of large-scale stzuotures which have the detemninistic character and which contribute much to the general characteristics of the turbulent flows [13. Of particular importance are these large-scale coherent sf~ru~tures in the mechanism of turbulent mixing up to the molecular level. I% has been shown experimentally in [2 ] %hat the process of mixing is closely related %0 the specific features of the dynamics of lazEe-soale coherent structures, i.e. with the mechanism of pairing and coalescence of vortices. An adequate theoretical description of the mech--ism for the mixing of impurities or equilibration of temperature inhomogeneities requires an equation that governs the transport 2~
254 of
B.A. Kolovandin, V.A. Sosinovich and S.V. Kravar
Vol. 5, No. 5
passive impurity by the turbulent velocity field. It is
a
clear that the large-scale coherent structures of the velocity field organize the same structures in the passive impurity [33. Therefore we shall postulate that the turbulent field of the impurity, as well as the velocity field, should be represented as the sum of the three components
f =~+~+f' Here
f
(1)
is the time-averaged value of the turbulent field
~
,
is the large-scale coherent component of the turbulent field and f' is the stochastic part of the turbulent field. By splitting the turbulent velocity field in the form of (1), the authors of [4] have obtained the equations for the wave disturbances with small amplitudes. In the present paper, the system of equations for the velocity field is supplemented with the equation for the wave part of the scalar field which is transported by the velocity field. The governing set of equations is
ui ~ul = ~p 1 t + uj axj - ~x--~ + He
a2ul ~xj~xj
(2)
xj
aT ~T a-~ +uj~xj
=-~
1
a2T axj~xj
(4)
This system has been dimensionalized by the use of the characteristic velocity
U and length ~ scales.
By splitting the fields of u i , p , T , as in (i), it is easy to obtain the system of equations which describes the behavior of the time-average and wave magnitudes ~ui =
uj ~xj aui
aP
i
~x i + Re aui
~ui=
e2u i
~xjaxj a~
~
u,u~" -
axj
i S
@2u i
(9
@
~ ~
6~j' uius ~ ~
- ,x~ rij
(5)
~ -
(6)
Vol. 5, No. 5
a xj
: o (a);
MECHANISM CF MIXING IN ~
uj_ = o
a xS
(b) ;
~xS
FIDWS
~ o (o)
255
('/)
(9) Here the following notation is used
< u1.u > -
(lo)
where the overbar denotes the time average and sharp brackets, the ensemble or phase average. Equations (5) through (9) describe distribution of the time average and coherent values and are no~ closed. W i ~ regard for the values riJ and Sj ~ e following simple & s 81~lmlD.t i o n s will be made
ri;l
= - 2~u~ij
;
~;I
=-~T
~Xj'
(12)
where
in which £u and 8T are the turbulent coefficients of viscosi1~y and diffusion, which, generally speaking, depend on the coordinate of the flow field ( for the time being, the form of these coefficients will not be specified ). We shall not require equations (5) and (8), since in what follows a change in ui and T , wi~h the form of ~i and known, will be considered. Among the approaches to the turbulent flows with coherent structures there is one based on the analogy with the stability
256
B.A. Kolovandin, V.A. Sosin(mrichand S.V. Kravar
Vol. 5, No. 5
theory of laminar flows. A linear version of this approach has been developed in [4 ~ where the long-wave regular disturbances immltate large-scale coherent structures.
It is clear that
such an approach to describing the coherent structures cannot be considered as quite adequate, but it enables one to reveal the basic features of the interaction dynamics of the average magnitudes,
coherent structures and of the stochastic part of
the turbulent flow. This approach will be followed below when we shall consider the transport of a passive impurity. Since calculation will be made in the linear approximation, we should omit those terms in equations (6) and (9) which are quadratic with respect to the wavenumbers. Consider a parallel, on the average stationary, turbulent flow in which the time average values are the functions of nothing but the transverse coordinate y
~i = [~(y)'O'O ] ;
T = T(y)
(14)
Each of the wave variables in equations (6)~ (Tb) and (9) will be represented as
=
½[ i ( y , t ) e i(kx-
t) + o.o.J
(15)
x 3 on the RHS of (15) is The absence of the coordinate due to our assumption that the coherent structures are twodimensional. This postulation is supported by a number of experimental studies [I] . The dependence on time is accounted for dually: the factor e -i~Jt
presupposes %ime-fluctuational
character of solution~ while the dependence of the amplitude o~
t
implies that its change with time is accounted for;
k
is the complex wave vector;cO is the frequency (real number). Substituting all of the wavenumbers in the form of (15) into equations (6)~ (7b) and (9) yields the following system of coupled equations for the amplitudes of these values in the linear approximation
i~
+D~
= o
(16)
Vol. 5, No. 5
t-ff~CHANISMOF MIXING IN ~JRBUIa~%T FLOWS
( ~ - - ( - ~•) + , ~ e ) u~.
257
.. * ~S-Z2) ( ~ ) ~ = -i~# + - ~1( ~ 2-~ 2)u-(i~Sz1+ (17)
(~-~-(-i~).
i~)~
(~-(-i~).
i ~ ) ~ + (D{)~ = -~6(~2-k2)~-(ik~z
Here
D m
~
= -~
;
,
+ _~(~2_~2)~_
(i~l.
~22)
+ ~2 )
(18)
(19)
are the amplitudes of the wave
quantities riJ and Sj , respectively. With the use of equation (16) one may express ~ in terms of v , substitute this expression into (17) and then eliminate from the ensuing equation and equation (18). As a result, the following equation will be obtained which determines =
+
"k2D(~zz-S22)* i~(D2* k2) S12
(2o)
Employing formulas (12) and (13) we may express the quantities Sij and ~'i in terms of v and T . Simple manipulations give
~n- s22 = ~ ;-uD~ ; = - ik ~ ' ~
~12 ~ -~u ~(~2 + k2)~
(21)
e 2 = - ~.~
(22)
With the help of formulas (21) and (22), equations (19) and (20) may be presented as
(23)
~=~
(2~)
et
(2s}
258
B.A. K O I ~ ,
V.A. Sosinoxrichand S.V. Kravar
Vol. 5, No. 5
Here the following notation is used
L = - ik(~ - o) + ( ~ e
_- i k ( D 2 ~ )
+ Eu)~ + 2(DEu)D
+ (D2~,u)(D2 + k 2)
(26) (2?)
N = D2 - k2
(28)
P = - i k ( ~ - c) + ("~e + ET)~ + ( ~ ' T ) D
(29)
R =-
(3o)
(D~) ~u
~'u= v ~
&T
;
;'T = 9 " T
ob
;
c =-.k--
(31)
Our further aim is to solve the system of equations (23) through (25) for different expressions of E u and E T. It may be well if, using the solutions obtained, we could calculate the Reynolds stresses and the flows induced by the large-scale coherent values ui and T . This is Just that very information about the large-scale turbulent structure which is indispensible in deriving the equations for fine-scale qttsntities that determine the degree of mixing up to the molecular level. The sufficient characteristics of mixing are the scale and intensity of segregation [5 ] • The knowledge of the wave quantities ui and T is required also to determine the behavior of the boundary between the turbulent and non-turbulent parts of the fluid, which is necessary to calculate the degree of the flow intermittency [6 J .
References 1.
Roshko, A., AIAAJ, 14 , 10,p. 13~9 (1976).
2.
Dimotakis. P.E., Brown, G.L., J.Fluid Mech.,78 ,pt. 3,
p. s35 (1976). 3.
Kovasznay, L.S.G., Second Australasian Conference on Heat and Mass Transfer, The University of Sydney, p. 295 (1977).
4.
Reynolds, W.C., Hussain, A.K.M.F., J.Fluid Mech., 54 , pt. 2, p. 263 (1972J.
5.
Brodkey, R.S., Mixing in Turbulent Fields, in: Turbulence in Mixing Operations, Ed. by R.S.Brodkey , Academic Press New York (1975). Wygna.nski, l.~Fiedler, H.E.,J.Fluid Mech.,41, pt.2,p.327 (1970).
6.