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# Staggered grids discretization in three-dimensional Darcy convection

Staggered grids discretization in three-dimensional Darcy convection

## 886 B. Karasözen et al.

886 B. Karasözen et al. / Computer Physics Communications 178 (2008) 885–893 In this work we consider the three-dimensional problem of natural convection in a porous medium and develop a finitedifference scheme for a system in primitive variables (velocities, pressure and temperature). The discretization is based on nonuniform staggered grids [14] with five types of nodes by using the differencing and averaging operators on two-nodes stencil. The present finite difference scheme is constructed in the spirit of the fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids [5]. An algorithm for the computation of the family of steady states is described, and is used in numerical computations. This paper is organized as follows. The equations for the three-dimensional Darcy convection problem in primitive variables are formulated in Section 1. In Section 2, the grids, discrete operators and discretization in space are described. The computation of the continuous family of steady states with varying stability spectra (cosymmetric family) is described in Section 3. Numerical results on the branching off of the family of steady states and isolated regimes from the state of rest are presented in Section 4. 1. Darcy convection 1.1. Darcy equations in primitive variables We consider an enclosure filled by a porous medium saturated by an incompressible fluid which is heated from below. We assume that the fluid is incompressible according to the Boussinesq approximation [6]. The system of equations consists of the momentum equation based on the Darcy law 1 ˙v =−∇p + θγ − v, (1) ε and the continuity and energy equations ∇·v = 0, ˙θ = θ − λv · γ − (v ·∇)θ. (2) (3) Here v = (v 1 ,v 2 ,v 3 ) T is the velocity vector, γ = (0, 0, −1) T is the direction of gravity, p(x,y,z,t) is the pressure, θ(x,y,z,t) is the deviation of the temperature from a linear (in z) profile, ɛ is the porosity of the medium, x, y, z are the space variables and the dot accent denotes differentiation with respect to time t. The Rayleigh number is defined as λ = αgT Kl/χμ, where α is the thermal expansion coefficient, g is the gravity acceleration, μ is the kinematic viscosity, χ is the thermal diffusivity of the fluid, K is the permeability coefficient, T is the characteristic temperature difference and l is the length parameter. We suppose that the temperature at the boundary is given by a linear function on the vertical coordinate z. The parallelepiped D =[0,L x ]×[0,L y ]×[0,L z ], with length L x , depth L y and height L z is filled with the fluid. The normal component of the velocity is equal to zero at the boundary V · n = 0, (x,y,z) ∈ ∂D. We consider two problems with different boundary conditions for the temperature. The first problem is characterized by (4) a temperature deviation θ equal to zero at the boundary ∂D (Dirichlet boundary condition): θ = 0, (x,y,z) ∈ ∂D. The second problem has mixed boundary conditions: the heat flux equals zero on two lateral faces ∂ 1 D ={y = 0}∪{y = L y } and the temperature deviation θ is equal to zero on the remaining faces ∂ 2 D = D \ ∂ 1 D θ y = 0, (x,y,z) ∈ ∂ 1 D, θ = 0, (x,y,z) ∈ ∂ 2 D. The initial condition is given as follows θ(x,y,z,0) = θ 0 (x,y,z), v(x,y,z,0) = v 0 (x,y,z). It is simple to check that Eqs. (1)–(5) are invariant with respect to the discrete symmetries R x : {x,y,z,v 1 ,v 2 ,v 3 ,p,θ} ↦→{L x − x,y,z,−v 1 ,v 2 ,v 3 ,p,θ}, (8) R y : {x,y,z,v 1 ,v 2 ,v 3 ,p,θ} ↦→{x,L y − y,z,v 1 , −v 2 ,v 3 ,p,θ}, (9) R z : {x,y,z,v 1 ,v 2 ,v 3 ,p,θ} ↦→{x,y,L z − z, v 1 ,v 2 , −v 3 ,p,−θ}. (10) This implies that with appropriate transformations of velocity, pressure and temperature deviation, a given set of solutions produces a new set. 1.2. Darcy equations for the temperature and stream function When the initial velocity v 0 and the initial temperature distribution θ 0 do not depend on y the system (1)–(3), (5) has a two-dimensional solution v 1 = v 1 (x,z,t), v 2 = 0, v 3 = v 3 (x,z,t), p = p(x,z,t), θ = θ(x,z,t). (11) In this case we can write our system as a system containing temperature and stream function. We follow the usual assumption in porous media flow and neglect inertia in the momentum equation. The continuity equation (2) is fulfilled automatically when the stream function ψ is given by v 1 =−ψ z , v 3 = ψ x . (5) (6) (7) (12) Then the underlying system can be transformed to a new form. After application of the curl-operator to (1) we deduce 0 = 2 ψ − θ x ≡ G, 2 () = () xx + () zz , and using (12) we obtain from (3) ˙θ = 2 θ + λψ x + J(θ,ψ)≡ F, J(θ,ψ)= θ x ψ z − θ z ψ x . (13) (14) The boundary conditions for the system (13), (14) follow from (6):

B. Karasözen et al. / Computer Physics Communications 178 (2008) 885–893 887 ψ = θ = 0, (x,z)∈ ∂ ̂D, where ̂D =[0,L x ]×[0,L z ] (15) and the initial condition is formulated only for the temperature θ(x,z,0) = θ 0 (x, z), (16) where θ 0 denotes the initial temperature distribution. For a given θ 0 , the stream function ψ can be obtained from (13) and (6) as the solution of the Dirichlet problem via Green’s operator ψ = Gθ x . Eqs. (13)–(15) require that the equilibrium θ = ψ = 0 (state of rest), be stable if λ

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