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Staggered grids discretization in three-dimensional Darcy convection

Staggered grids discretization in three-dimensional Darcy convection

888 B. Karasözen et al.

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 averaging operators on the coordinates (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 integer and halfinteger 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 averaging operator on three-dimensional cell is given as δ 0 = δ 1 δ 2 δ 3 . (21) The nonlinear term approximation is constructed using a linear combination 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 underlying problem. 2.2. Semi-discretization To reach some steady state it is useful to apply an approach based on artificial compressibility [16,17]. Thus instead of Eq. (2) we consider the following equation with coefficient η ṗ + 1 ∇·v = 0. (23) η Using the operators (18)–(22) we discretize system (1), (3) and (23) in the following 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 introduced 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 using fictitious nodes to satisfy the boundary conditions on the planes y = 0 and y = L y . Thus instead 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 ordinary 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 invariant with respect to index j. Then, we can exclude Eq. (26) and consider the twodimensional problem. Moreover, we can eliminate pressure and velocities and reformulate the system with the discrete stream function and temperature. Lets’ define the stream function ψ at the nodes ω 0 using 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 continuity equation (2) is automatically fulfilled by (33). After the substitution of (33) in (25) and (27) and the combination of the resulting formulas we can deduce the analog of (13) (inertia terms are omitted) [ 2,h ψ − D x θ] i,k = 0. Similarly we find 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 differencing operators on three-nodes stencils. The Laplacian and Jacobian are approximated using

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 resulting 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 resulting system of equations in vector form. Let us introduce vectors which contain only unknowns at internal 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 obtain 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 averaging operators. The matrix A 1 presents the discrete form of the Laplacian. The nonlinear 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 linear 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 obtain the system from which we can determine the threshold value of the Rayleigh number corresponding to the monotonic loss of stability. We can express P , V 1 , V 2 è V 3 via Θ from (38)–(42) and obtain a system of N x N y N z equations for the unknown vector Θ. Substituting (39), (40) and (41) into (42) we deduce A 1 Θ = λC 1 (C 5 − B 6 S)Θ. (43) Here we find the vector P = SΘ from the system of linear 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. Since for an incompressible flow the pressure may differ by a constant we can exclude one component of P and respectively one equation. To find an isolated convective pattern we apply the direct approach and integrate the system of ordinary 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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