FINITE ELEMENT METHOD FOR NATURAL CONVECTION PROBLEM INSIDE PARABOLIC CROSS-SECTION ENCLSOURE
Keywords:
Finite element method, natural convection, parabolic enclosure, coordinate transformationAbstract
Numerical solution procedure for natural convection in parabolic cross-section enclosure is proposed in this study. The governing equations in term of finite element equations are solved by using iterative method. According to solution procedure is performed on computational domain, transformations X = +( / 4)(1-2) and Y = ( / 4)+(1 - )( - 1/2)2 are defined to transform solutions back to physical domain which is original problem. Numerical tests involving natural convection heat transfer are carried out to display the results in forms of flow circulation and temperature distribution. Two cases of temperature boundary conditions are considered. First, the top of the enclosure is cooled and the parabolic curve is uniformly heated. Second, the top of the enclosure is adiabatic and the parabolic curve is nonuniformly heated. The interested parameter is Rayleigh number (5104 Ra 1106) with constant Darcy number (Da = 10-2) and Prandtl number (Pr = 0.71). The solution for natural convection problem shows that the increase in Ra results in stronger flow intensity and affects the characteristic of temperature distribution indicating the change from conduction mode to convection mode. The variation of heat transfer along heated curve for case 1 is maximum at the left and the right sides and becomes constant at the middle portion for Ra = 5104 and Ra = 105 while it is slightly increased near the center for higher Ra. For case 2, heat transfer is increased along the curve and symmetric with respect to vertical center line. The maximum heat transfer occurs for two points when Ra = 5105 and Ra = 106. Graphs of relative error values are also plotted to show the convergence of numerical solution obtained from the proposed method. From the numerical results, it is demonstrated that the proposed method is efficient and suitable for natural convection problem in the enclosure with parabolic curve.
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