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3, we denote f ∗ as the dimensionless density distribution function and g ∗ as the dimensionless temperature distribution √ function. Using the computational length scale = x, the velocity scale U = gβ T L, the time 2 3 q0 L , and the density scale ρf , scale t = U , the temperature scale T = gβ (ρcp )f we can obtain the following dimensionless equations for the momentum equation and heat transfer equation respectively. The D2Q9 scheme is chosen in the present simulation because direct simulations of the 240 tubes by the NDLBM is still time consuming, although it is much quicker than conventional finite difference and finite volume method.

17), if we take the divergence of the Eq. 14), we get the Poisson equation for the pressure p n+1/2 as Eq. 18), ∇ · v∗ n+1 = 0, ∇ 2 p n+1/2 = 1 ∇ · v∗ . 18) At a wall boundary, the pressure can be expressed, ∂p ∂N n+1/2 = 0. 19) Γ The numerical algorithm is the following. First, Eq. 13) is solved explicitly for v∗ and then the coupled Eqs. 18) are coupled and are solved by Gauss–Seidel iteration. Temperature is calculated at the same point as the pressure, while velocity is not. The size of the matrices used to calculate the pressure and velocity field are different.

Vafai K, Amiri A (1998) Non-Darcian effects in confined forced convective flows. In: Ingham D, Pop I (eds) Transport phenomena in porous media. Elsevier, Oxford, pp. 313–329 81. Vafai K, Kim SJ (1995) On the limitations of the Brinkman-Forchheimer-extended Darcy equation. Int J Heat Fluid Flow 16:11–15 82. Vafai K, Tien CL (1981) Boundary and inertia effects on flow and heat transfer in porous media. Int J Heat Mass Transf 108:195–203 83. Wang BX, Du JH, Peng XF (1998) Internal natural, forced and mixed convection in fluidsaturated porous medium.

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