Computational Technology Resources - CCP

Keywords: numerical simulation, higher-order interpolation polynomial, non-staggerd finite difference scheme, three-dimensional problem, natural convection phenomena, high Rayleigh number.

Summary

The object of this paper is to show numerical simulations of three-dimensional natural connection problems based on a higher-order interpolation scheme in solving coupled system of incompressible viscous fluid flow with thermal convection.

The scheme to be developed in our paper is composed of the seventh-order interpolation polynomial for the non-convective term and the fourth-order polynomial for the fourth-order polynomial for the convective term in the convection dominated coupled equations. Time integration of our field unknown functions, which are the velocity vector and temperature, can be easily perform with only the above-mentioned interpolation polynomial in terms of space variables without any up-minding procedure.

Effectiveness and applicability of our scheme are shown with numerical simulations of natural convection phenomena in a cubic cavity for through a comparison with the existing numerical results on velocity vector fields, temperature field and the average .

Figure 1 shows the geometry and boundary conditions of the problem. The condition of numerical computations on three cases in which the used values of , the number of grid points, (time difference), and (convergence criterion) are shown in Table 1. All results are obtained by numerical computation with each time increment and the fixed convergence estimation . The relation of the average Nusselt number-Rayleigh number is shown in Table 2 through the comparison with other results obtained by different methods.

**Figure 1:** Computational model.

Table 2: Condition for the numerical computation.

	Case 1	Case 2	Case 3

Number of Grid
	0.1	0.4	0.625

Table 2: Comparisons of numerical results.

			Average
Present	38.67	63.47	4.345
Kakuta et al [1]	38.96	65.36	4.398
Present	75.68	217.2	8.755
Kakuta et al [1]	71.7	215.9	8.833
Haldenwang et al [2]	-	234.4	8.61

References

1: K. Kakuda, Y. Kawahara, N. Tosaka: Petrov - Galerkin finite element approach using exponential test functions for three-dimensional natural convection in cubic enclosure (in Japanese), Trans. of JSME, 60, 112-119, 1994.
2: P. Haldenwang, G. Labrosse : 2-D and 3-D spectral Chebyshev solutions for free convection at high Rayleigh number, Proc. 6th Int. Symp. Finite Elements Methods Flow Problems, 261, 1986.

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