TRANSPORT MODEL I.IMTCA I IONS IN TURBULENCE
57
whcre dI quitIltltles arc ctgaluatcd at (z, t).
Substitilting Eq. (A.6) and the corresponding expressions for the other
vclocity and the two mean concentration terms into Eq. (A.I), we amve at
(A.7)
(2/n)&. t )
0
= - ~vr, - r v z +
+ mtT, + w,r, + w,r, + vT1,r,
+ r7i, r/, + i7vrZf + iTrKt.
The symmcttry in r and V is a reflection of that in Eq. (A.1).
After some algebra,
With the convention T = I/V, thm can be written as an expansion in 1.
For a steady state, homogeneous transport mechanism (I, = = l, =
rj = 0). to O(IZ),
M.9)
( 2 / n ) ~ -wg + w,, ,
which is Eq. (19).
For steady state field and mechanism (r, = 1, = V, = 0). to O(P),
which is Eq. (24).
It should be remarked again that any specific physiCrtl or mathematical
case may have material conservation constraints which will have to bc taken
into account.
ACKNOWLEDGMENTS
I should like to thank Michael Kanveit for critical discussion and computation. James Riley
and Lyle Horn for constructive d i n g of the manurcript, nnd the National Scitna
Foundaiion (Engineering Mechanics Section) a d Otfjce of Naval Rssearch (mid Dynamics
Program) for financial support.