Velocity-scalar filtered density function for large eddy simulation of ...

Phys. Fluids, Vol. 15, No. 8, August 2003Velocity-scalar filtered density function2325u i , t , t u ku i , 1x k 2 u k , 1x k 2 2 u i , x k x ku k ,u i ux k , u ikx k 1 1 3 1 S2 3 S2 u ix k Gx ik u k , C u i , ku i ,S u k ,u i , x k, 22b 2 , x k x k 1 2 1 S2 S2 x ku k , ux k , kx k 2Cx , ,S k ,S u k , , x k. 22cA term-by-term comparison of the exact moment transport equations Eqs. 4 and 6, with the modeled equations Eqs.21 and 22, suggests 1 2 3 S1 S2 2. However, this violates the realizability of the scalar field. A set ofcoefficients yielding a realizable stochastic model requires: 1 2 3 2 and S1 S2 0. That is,dX i U i dt2 dW i X ,23adU i p 2 2 u i Gx i x k x ij U j u j dt2 u idW Xk x k C 0 dW U i , 23bkd C S dt.23cThe Fokker–Planck equation for this system is ft vx k f pk x i f Gv i v ij v j u j f i2 u jx i 2 f u i u j x i v j x k x kC f S f 2 f x k x k 2 f 1 v i v j 2 C 2 f0v k v k24and the corresponding equations for the moments areu k x k0, 25au i t tu i ,u j tu i , t u ku i x k p 2 u i u k ,u i , 25bx i x k x k x k u k 2 Sx k x k x u k , , 25ck x k u ku i ,u j x k u ku i , x k 2 u i ,u j ux k x k ,u i u jkux k ,u j u iGk x ik u k ,u j G jk u k ,u i C 0 ij k u k ,u i ,u j x k, 26a 2 u i , x k x ku k ,u i x ku k , u iGx ik u k , C u i , ku i ,S u k ,u i , x k, 26bDownloaded 22 Sep 2004 to 140.121.120.39. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp