8456 X.Ma,N.Zabaras/Joumal of Computational Physics 227 (2008)8448-8471 Soourterm of interest can be reduced to).This form is now easier to calculate,since it ony evolves third-order product terms. Remark3.In the on-linerdrate -5∑o∑o(2-1e we assume that the magnitude of the velocity is determined by the mea 60010010006 Fig.3.Contour results obtained with the second-order projection method.Top row:u and v velocity component contours.Bottom row:pressure contours 00=0.u-v-0 01 u=v-0 E(w) 77777777 777777777 Fg4.Schematic of stchastic natural vectin heterogeneous porous medium.So our term of interest can be reduced to 2 hW2 k i PP j¼0 PP m¼0dm rvn j hWjWmWki. This form is now easier to calculate, since it only evolves third-order product terms. Remark 3. In the non-linear drag term  1:75kvnk ð150DaÞ 1=2 PP i¼0 PP j¼0 ~ið2vn j  vn1 j Þeijk, we assume that the magnitude of the velocity kvnk is determined by the mean velocity vn 0. Fig. 3. Contour results obtained with the second-order projection method. Top row: u and v velocity component contours. Bottom row: pressure contours. = 1 u = v =0 = 0 u = v =0 θ θ = 0, u = v =0 n ∂ θ ∂ = 0, u = v =0 n ∂ θ ∂ ε ( ) w Fig. 4. Schematic of stochastic natural convection in a heterogeneous porous medium. 8456 X. Ma, N. Zabaras / Journal of Computational Physics 227 (2008) 8448–8471