Spectral collocation method for natural convection in a square porous cavity with local thermal equilibrium and non-equilibrium models

In this paper, detailed numerical calculations of the spectral collocation method (SCM) to simulate natural convection in a square porous cavity using two different local thermal models, i.e., the local thermal equilibrium model and the local thermal non-equilibrium model, are presented and formulated. In this approach, stream function and temperature are approximated by the Chebyshev polynomials and the dimensionless governing equations are discretized by the Chebyshev-Gauss-Lobatto collocation points. The two step method, i.e., the matrix diagonalization method, is used to solve the discretized equations. The present results are compared with those reported in literature. It is found that these results are in excellent agreement. Furthermore, the exact solutions for local thermal equilibrium model have been constructed to test the accuracy of the method, and it can be observed from the test that the Chebyshev spectral collocation method can produce high accuracy. Meanwhile, the isotherms and the streamlines are analyzed to reveal the fluid flow and heat transfer for various parameters, i.e., the Rayleigh number, the inter-phase heat transfer coefficient, and the thermal conductivity ratio.