MODELING OF TL'RBI'LENT TRANSPORT
1x1
(30) O U ; ~ ~
= 6 ; , i ( ~ s , . ~ q ~ P g i ~ o
+ A ~ ~ C I ' . ' ~ ( B U ; ) , , )
+ A , , . T ~ ( ~ U , U ~ / ~ ~
- bii)u2g/r,, + A ~ , , @ - [ ( ~ ~ U J ~
+ (OU,).,]
Only the terms in A,, and A,, will remain if gravitational effects are rclalively weak. The term in A,, simply corrects the term in .q5, for anisotropy.
We may give a simple physical interpretation to the terms in A,, and A S , :
Og/To is the buoyant acceleration, and .Tq the turbulent length scale: hence,
.Y@g/To is that part of the turbulent energy that is correlated with the
temperature 'fluctuation. The terms in A,, and AS6 are simply gradient
transport .
The purely mechanical transport terms are also relatively simple. For the
mechanical dissipation we have
_.
.- ~
(31)
EU; = & { t l i U I , U l l i ,T, g/T,)
leading t o
I n an exactl? siiiiilar way. we find
( 3 2 )
CU; = - d421(Y2/3)(q2/2).; - Azz3(y2/3b.;
.- . .. . __ .- .
. -
(33)
(34)
..
-. - . . ..~
( I l i U , + 2siKp/3p,)uj = *FiKj;U;uj, uju, E, g/T0}
(u,/I,, -t i ~ i ~ p p / 3 p o ) ~ i
= - A , I . ~ ( ( ~ ~ j 3 ) [ 6 i ~ f i , ,
+ a ( S ; j S , , + b ; , t ~ ~ , ) ] q <
which gives (to first order)
- A 1 2~F'(q2/3)[6;, 6,; + b ( S ; j d , , + ij;,b,j)F,,
where a and h are constants. In each case we have tried to choose signs and
numerical coefficients which will result in the constants A, being positive
and of order one.
Just as we found for the transport of thermal flux, there is a situation in
which the first order transport of Reynolds stress vanishes, and second order
terms are consequently necessary. Specifically, in a simple shear U I (.xZ), in
which only two gradients are nonzero, ul u2 u2 = 0 from (33). Since this is a
dynamically important quantity, we will need second order terms (in the
wake calculations to be described later, the omission of this transport results
in undesirable, and unrealistic, hyperbolic behavior, with steepening lronts,
etc.). The second order terms are
(35)
*T('?/3)[aIai~.l
QZ(aij.K
4~j.i) " ~ 6 i ~ a j p . p
+ a , ( S i j a K p , p s ~ j Q i p , p ) ]
+ , S [ P I i j i K u j / + B 2 u i K S j / + B 3 ( 6 i j ~ l + 6 , j a i l ) + 84(6i/a,j + b , l a i j ) ] q :
+ ; y 2 [ ) J ~
(S;,,(Ij, + ')'2U;,hjl + . r ' j ( h ; j U , , + b,jCJ;,) + ')*,+8ilU,j
-t ('x/aij)li;.,
1x1
(30) O U ; ~ ~
= 6 ; , i ( ~ s , . ~ q ~ P g i ~ o
+ A ~ ~ C I ' . ' ~ ( B U ; ) , , )
+ A , , . T ~ ( ~ U , U ~ / ~ ~
- bii)u2g/r,, + A ~ , , @ - [ ( ~ ~ U J ~
+ (OU,).,]
Only the terms in A,, and A,, will remain if gravitational effects are rclalively weak. The term in A,, simply corrects the term in .q5, for anisotropy.
We may give a simple physical interpretation to the terms in A,, and A S , :
Og/To is the buoyant acceleration, and .Tq the turbulent length scale: hence,
.Y@g/To is that part of the turbulent energy that is correlated with the
temperature 'fluctuation. The terms in A,, and AS6 are simply gradient
transport .
The purely mechanical transport terms are also relatively simple. For the
mechanical dissipation we have
_.
.- ~
(31)
EU; = & { t l i U I , U l l i ,T, g/T,)
leading t o
I n an exactl? siiiiilar way. we find
( 3 2 )
CU; = - d421(Y2/3)(q2/2).; - Azz3(y2/3b.;
.- . .. . __ .- .
. -
(33)
(34)
..
-. - . . ..~
( I l i U , + 2siKp/3p,)uj = *FiKj;U;uj, uju, E, g/T0}
(u,/I,, -t i ~ i ~ p p / 3 p o ) ~ i
= - A , I . ~ ( ( ~ ~ j 3 ) [ 6 i ~ f i , ,
+ a ( S ; j S , , + b ; , t ~ ~ , ) ] q <
which gives (to first order)
- A 1 2~F'(q2/3)[6;, 6,; + b ( S ; j d , , + ij;,b,j)F,,
where a and h are constants. In each case we have tried to choose signs and
numerical coefficients which will result in the constants A, being positive
and of order one.
Just as we found for the transport of thermal flux, there is a situation in
which the first order transport of Reynolds stress vanishes, and second order
terms are consequently necessary. Specifically, in a simple shear U I (.xZ), in
which only two gradients are nonzero, ul u2 u2 = 0 from (33). Since this is a
dynamically important quantity, we will need second order terms (in the
wake calculations to be described later, the omission of this transport results
in undesirable, and unrealistic, hyperbolic behavior, with steepening lronts,
etc.). The second order terms are
(35)
*T('?/3)[aIai~.l
QZ(aij.K
4~j.i) " ~ 6 i ~ a j p . p
+ a , ( S i j a K p , p s ~ j Q i p , p ) ]
+ , S [ P I i j i K u j / + B 2 u i K S j / + B 3 ( 6 i j ~ l + 6 , j a i l ) + 84(6i/a,j + b , l a i j ) ] q :
+ ; y 2 [ ) J ~
(S;,,(Ij, + ')'2U;,hjl + . r ' j ( h ; j U , , + b,jCJ;,) + ')*,+8ilU,j
-t ('x/aij)li;.,