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Staggered grids discretization in three-dimensional Darcy convection

888 B. Karasözen et al. / Computer Physics Communications 178 (2008) 885–893 d 3 θ i,j,k+1/2 = θ i,j,k+1 − θ i,j,k z k+1 − z k ≈ (θ z ) i,j,k+1/2 , and weighted averag**in**g operators on the coord**in**ates (18) δ 1 θ i+1/2,j,k = (x i+1 − x i+1/2 )θ i+1,j,k + (x i+1/2 − x i )θ i,j,k x i+1 − x i ≈ (θ) i+1/2,j,k , δ 2 θ i,j+1/2,k = (y j+1 − y j+1/2 )θ i,j+1,k + (y j+1/2 − y j )θ i,j,k y j+1 − y j ≈ (θ) i,j+1/2,k , δ 3 θ i,j,k+1/2 = (z k+1 − z k+1/2 )θ i,j,k+1 + (z k+1/2 − z k )θ i,j,k z k+1 − z k ≈ (θ) i,j,k+1/2 . (19) The formulas (18)–(19) are valid both for **in**teger and half**in**teger values of i, j and k. Then the discrete analog of the Laplacian on the seven-nodes stencil can be written as △ h = d 1 d 1 + d 2 d 2 + d 3 d 3 ≈△, (20) and the averag**in**g operator on **three**-**dimensional** cell is given as δ 0 = δ 1 δ 2 δ 3 . (21) The nonl**in**ear term approximation is constructed us**in**g a l**in**ear comb**in**ation of two terms (v ·∇θ) i,j,k ≈ J(θ,v) i,j,k = 1 3[ d1 δ 1 (θδ 2 δ 3 v 1 ) + d 2 δ 2 (θδ 1 δ 3 v 2 ) + d 3 δ 3 (θδ 1 δ 2 v 3 ) ] i,j,k + 2 3[ d1 δ 2 δ 3 (δ 0 θδ 1 v 1 ) + d 2 δ 1 δ 3 (δ 0 θδ 2 v 2 ) + d 3 δ 1 δ 2 (δ 0 θδ 3 v 3 ) ] i,j,k . (22) This provides second order accuracy for the uniform grid and an asymptotically second order accuracy for the quasi-uniform mesh. It allows conservation of energy and constitute a mimetic **discretization** of the underly**in**g problem. 2.2. Semi-**discretization** To reach some steady state it is useful to apply an approach based on artificial compressibility [16,17]. Thus **in**stead of Eq. (2) we consider the follow**in**g equation with coefficient η ṗ + 1 ∇·v = 0. (23) η Us**in**g the operators (18)–(22) we discretize system (1), (3) and (23) **in** the follow**in**g form [ ˙θ −△ h θ − λδ 1 δ 2 v 3 + J(θ,v) ] (24) i,j,k = 0, [ ˙v 1 − ε(−d 1 p − v 1 ) ] (25) i,j+1/2,k+1/2 = 0, [ ˙v 2 − ε(−d 2 p − v 2 ) ] (26) i+1/2,j,k+1/2 = 0, [ ˙v 3 − ε(−d 3 p − v 3 + δ 1 δ 2 θ) ] (27) i+1/2,j+1/2,k [ = 0, ṗ + 1 ] (28) η (d 1v 1 + d 2 v 2 + d 3 v 3 ) = 0. i+1/2,j+1/2,k+1/2 For the problem with Dirichlet boundary conditions (1)–(5) the **grids** are **in**troduced **in** such a way that the nodes for the transversal velocity are located on the wall. It allows us to fulfill the boundary conditions on a rigid wall (5) **in** a very simple way for each pair of planes: for x = 0(i = 0) and x = L x (i = N x + 1) we have vi,j+1/2,k+1/2 1 = 0, j = 0,...,N y,k= 0,...,N z , θ i,j,k = 0, j = 0,...,N y + 1, k= 0,...,N z + 1, for y = 0(j = 0) and y = L y (j = N y + 1) we have vi+1/2,j,k+1/2 2 = 0, i = 0,...,N x,k= 0,...,N z , θ i,j,k = 0, i = 0,...,N x + 1, k= 0,...,N z + 1, and for z = 0(k = 0) and z = L z (k = N z + 1) we have (29) (30) vi+1/2,j+1/2,k 3 = 0, i = 0,...,N x,j= 0,...,N y , θ i,j,k = 0, i = 0,...,N x + 1, j= 0,...,N y + 1. (31) The problem with mixed boundary conditions (1)–(4) and (6) is discretized us**in**g fictitious nodes to satisfy the boundary conditions on the planes y = 0 and y = L y . Thus **in**stead of (30) we have vi+1/2,0,k+1/2 2 =−v2 i+1/2,1,k+1/2 , i = 0,...,N x ,k= 0,...,N z , vi+1/2,N 2 y +1,k+1/2 =−v2 i+1/2,N y ,k+1/2 , i = 0,...,N x ,k= 0,...,N z , θ i,0,k = θ i,1,k , θ i,Ny +1,k = θ i,Ny ,k, i = 0,...,N x + 1, k= 0,...,N z + 1. 2.3. Discrete equations for the planar problem (32) The system of ord**in**ary differential equations (24)–(28), (29), (31) and (32) has a solution such that v 2 i+1/2,j,k+1/2 = 0 and the other variables are **in**variant with respect to **in**dex j. Then, we can exclude Eq. (26) and consider the two**dimensional** problem. Moreover, we can elim**in**ate pressure and velocities and reformulate the system with the discrete stream function and temperature. Lets’ def**in**e the stream function ψ at the nodes ω 0 us**in**g difference operators (18) (d 3 ψ) i,k+1/2 =−v 1 i,k+1/2 , (d 1ψ) i+1/2,k = v 3 i+1/2,k . (33) The discrete analog of the cont**in**uity equation (2) is automatically fulfilled by (33). After the substitution of (33) **in** (25) and (27) and the comb**in**ation of the result**in**g formulas we can deduce the analog of (13) (**in**ertia terms are omitted) [ 2,h ψ − D x θ] i,k = 0. Similarly we f**in**d from (24) [ ˙θ i,k = 2,h θ + λD x ψ + 1 3 J D + 2 ] 3 J d . i,k (34) (35) Here D x = d 1 δ 1 , D z = d 3 δ 3 are the first order differenc**in**g operators on **three**-nodes stencils. The Laplacian and Jacobian are approximated us**in**g

B. Karasözen et al. / Computer Physics Communications 178 (2008) 885–893 889 2,h = d 1 d 1 + d 3 d 3 , J D = D x (θD y ψ)− D y (θD x ψ), J d = ̂d x (̂d 0 θ ̂d z ψ)− ̂d z (̂d 0 θ ̂d x ψ), (36) where ̂d 0 = δ 1 δ 3 , ̂d x = d 1 δ 3 , ̂d z = d 3 δ 1 . The result**in**g scheme (34)–(36) ensures the fulfillment of a discrete analog of the cosymmetry property (17) as well as the nullification of the gyroscopic terms [12]. Equations **in** [11] follow from (34)–(36) for the case of the uniform grid x i = ih, z k = kg, h = L x /(N x + 1), g = L z /(N z + 1). Then the Jacobian approximation gives the famous Arakawa formula [15] on uniform **grids** h = g. 3. Computation of the family of steady states We rewrite the result**in**g system of equations **in** vector form. Let us **in**troduce vectors which conta**in** only unknowns at **in**ternal nodes Θ = (θ 1,1,1 ,...,θ Nx ,1,1,θ 1,2,1 ,...,θ Nx ,N y ,N z ), V 1 = ( v1,1/2,1/2 1 ,...,v1 N x ,1/2,1/2 ,v1 1,3/2,1/2 ,..., vN 1 x ,N y +1/2,N z +1/2) , V 2 = ( v1/2,1,1/2 2 ,...,v2 N x +1/2,1,1/2 ,v2 1/2,2,1/2 ,..., vN 2 x +1/2,N y ,N z +1/2) , V 3 = ( v1/2,1/2,1 3 ,...,v3 N x +1/2,1/2,1 ,v3 1/2,3/2,1 ,..., vN 3 ) x +1/2,N y +1/2,N z , P = ( p 1/2,1/2,1/2 ,...,p Nx +1/2,1/2,1/2,p 1/2,3/2,1/2 ,..., p Nx +1/2,N y +1/2,N z +1/2) , and obta**in** the system which corresponds (24)–(28) ˙Θ = A 1 Θ + λC 1 V 3 − F(Θ,V), ˙V 1 =−B 4 P − C 2 V 1 , ˙V 2 =−B 5 P − C 3 V 2 , ˙V 3 =−B 6 P − C 4 V 3 + C 5 Θ, P ˙ =−B 1 V 1 − B 2 V 2 − B 3 V 3 . (37) Here the matrices B k , k = 1,...,6, are constructed by the application of first order difference operators, and the matrices C k , k = 1,...,5, by the averag**in**g operators. The matrix A 1 presents the discrete form of the Laplacian. The nonl**in**ear term is given by F(Θ,V).Eqs.(37) form a system of 5N x N y N z + 3(N x N y + N x N z + N y + N z ) + 2(N x + N y + N z ) + 1 unknowns. From (37) at J = 0 we can derive the perturbation equations (σ is a decrement of l**in**ear growth) to analyze the stability of the state of rest σΘ = A 1 Θ + λC 1 V 3 , σV 1 =−B 4 P − C 2 V 1 , (38) (39) σV 2 =−B 5 P − C 3 V 2 , σV 3 =−B 6 P − C 4 V 3 + C 5 Θ, σP =−B 1 V 1 − B 2 V 2 − B 3 V 3 . (40) (41) (42) For the decrement σ = 0 we obta**in** the system from which we can determ**in**e the threshold value of the Rayleigh number correspond**in**g to the monotonic loss of stability. We can express P , V 1 , V 2 è V 3 via Θ from (38)–(42) and obta**in** a system of N x N y N z equations for the unknown vector Θ. Substitut**in**g (39), (40) and (41) **in**to (42) we deduce A 1 Θ = λC 1 (C 5 − B 6 S)Θ. (43) Here we f**in**d the vector P = SΘ from the system of l**in**ear algebraic equations with rank deficiency 1 ( B1 C −1 2 B 4 + B 2 C −1 3 B 5 + B 3 C −1 4 B ) 6 P = B3 C −1 4 C 5Θ. S**in**ce for an **in**compressible flow the pressure may differ by a constant we can exclude one component of P and respectively one equation. To f**in**d an isolated convective pattern we apply the direct approach and **in**tegrate the system of ord**in**ary differential equations (37) by the classical fourth order Runge–Kutta method up to convergence. To compute a family of steady states we apply the technique based on the cosymmetric version of the implicit function theorem [18] and the algorithm developed **in** [10,11,19]. The zero equilibrium V 1 = V 2 = V 3 = P = Θ = 0 is globally stable for λ

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