XMa N.Zabares/Joumal f Computational Physics()44-471 8453 where the expectation value(can be evaluated by any numerical integration rule. Next.we consider a general non-linear function g(x.).where e is the random porosity.We need to express this function (27) 28) Here.))))ornd)oMThis is beca heterm Hermite poly- nomials are justo()By writing the k-Le jection onto each basis element)we can obtain from Eq.(27)the following: (2）-明 29 Thus,we obtain 哥-gK62 (30) ( This expression may be evaluated usng discusd-quraturebased approach suchas the ased on the Latin-Hypercubestatethiscase wer be The ide. subdivide the space of then calculate the value of the integrand in the numerator of Eq.(30).Summing all the values,the expectation value ()is just the arithmetic mean of these realizations.Also.the value of(can be pre-computed using quadrature rule. 4.2.GPCE-based formulation 院 By no nedia.n the sto cussed in the pre 议可. 1 31 --含 32) e(Xω) (33) (X.0 where Pis the number of expansion terms determined by the stochastic dimension and expansion order as in Eq.(23) hus.the stochastic problem is to fir the stochas ch tha e loe aand (150Da)e(x,t,) +PrV2v(x.t.@)-c(x.t.w)Vp(x.t.@)-c(x.t.@)PrRa0(x.t.w)eg. (34 7vx,t,=0. 65 .vx.t.)-V..)-vo.t.) (36) where the expectation value h; i can be evaluated by any numerical integration rule. Next, we consider a general non-linear function gðx;Þ, where is the random porosity. We need to express this function as gðx;Þ ¼ XP i¼0 giWi; ð27Þ where gi is the expansion coefficient onto the polynomial basis and the porosity ðx;xÞ is written here based on the K–L expansion as follows: ðx;xÞ ¼ ðxÞ þXM i¼1 ffiffiffiffi ki p fiðxÞniðxÞ ¼ XP i¼0 iðxÞWiðxÞ: ð28Þ Here, 0ðxÞ ¼ ðxÞ, iðxÞ ¼ ffiffiffiffi ki p fiðxÞ, for i ¼ 1; ... ; M and iðxÞ ¼ 0, for i > M. This is because the first M þ 1 term Hermite poly￾nomials are just W0 ¼ 1; W1 ¼ n1ðxÞ; ... ; WM ¼ nMðxÞ. By writing the K–L expansion as Eq. (28) instead of Eq. (16), it is easy to formulate and perform the polynomial chaos calculations. Using the same method as before (i.e., performing a Galerkin pro￾jection onto each basis element), we can obtain from Eq. (27) the following: g x; XP i¼0 iWi !; Wj * + ¼ gjhW2 j i: ð29Þ Thus, we obtain gj ¼ hgðx; PP i¼0iWiÞ; Wji hW2 j i : ð30Þ This expression may be evaluated using quadrature rule as discussed in [43] or a non-quadrature-based approach such as the integration, Taylor series and sampling approach discussed in [42]. Here, we employ a Monte-Carlo based sampling approach based on the Latin-Hypercube sampling (LHS) strategy [44]. In this case, we first generate samples of uncorrelated standard normal variables n using LHS. The idea of LHS is to subdivide the stochastic support space of the joint PDF of n into N sub￾intervals along each stochastic dimension and to ensure that one sample of n lies in each subinterval. For each sample, we then calculate the value of the integrand in the numerator of Eq. (30). Summing all the values, the expectation value h; i is just the arithmetic mean of these realizations. Also, the value of hW2 j i can be pre-computed using quadrature rule. 4.2. GPCE-based formulation By now, we have developed all the tools we need to formulate natural convection in random porous media. In the sto￾chastic natural convection problem, the input uncertainties are due to the Gaussian random field of porosity. Note that since there are non-linear functions ofðx; xÞ in the governing equation (1), we need to first express them in the polynomial basis using the method discussed in the previous section: 1 ðx;xÞ ¼ XP i¼0 ^iWi; ð31Þ ð1  ðx;xÞÞ2 ðx;xÞ 2 ¼ XP i¼0 iWi; ð32Þ 1  ðx;xÞ ðx;xÞ 2 ¼ XP i¼0 ~iWi; ð33Þ where P is the number of expansion terms determined by the stochastic dimension and expansion order as in Eq. (23). Thus, the stochastic problem is to find the stochastic functions that describe the velocity field vðx;t;xÞ : D  ½0; T  X ! Rd , the pressure field pðx;t;xÞ : D  ½0; T  X ! R and the temperature field hðx;t;xÞ : D  ½0; T  X ! R, such that the following equations are satisfied: ovðx;t;xÞ ot þ vðx;t;xÞ ðx;t;xÞ rvðx;t;xÞ¼ Pr Da ð1  ðx;t;xÞÞ2 ðx;t;xÞ 2 vðx;t;xÞ  1:75kvðx;t;xÞkð1  ðx;t;xÞÞ ð150DaÞ 1=2 ðx;t;xÞ 2 vðx;t;xÞ þ Prr2 vðx;t;xÞ  ðx;t;xÞrpðx;t;xÞ  ðx;t;xÞPrRahðx;t;xÞeg; ð34Þ r vðx;t;xÞ ¼ 0; ð35Þ ohðx;t;xÞ ot þ vðx;t;xÞ rhðx;t;xÞ ¼ r2 hðx;t;xÞ: ð36Þ X. Ma, N. Zabaras / Journal of Computational Physics 227 (2008) 8448–8471 8453