Integral Equation Analysis of Laminar Natural Convection Problems
The coupled convective and conductive heat transfer phenomena due to the buoyancy-driven flow under heating of incompressible viscous fluids are commonly encountered in many important natural science and engineering problems. These include many practical
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Integral Equation Analysis of Laminar Natural Convection Problems
N. Tosaka, N. Fukushima Department 01 Mathematical Engineering, College 01Industriai Technology, Nihon University, Narashino-shi, Chiba 275, Japan INTRODUCTION The coupled convective and conductive heat transfer phenomena due to the buoyancy-driven flow under heating of incompressible viscous fluids are commonly encountered in many important natural science and engineering problems. These include many practical problems, which are known as the natural convection problems, such as thermal insulation of buildings, cooling of electronic equipment and general circulation of planetary atomospheres. The differential equations governing the phenomena, the Navier-Stokes equations and the equation of energy, are a highly nonlinear coupled set. Because of difficulties in obtaining analytical solutions of the problem, several numerical procedures based on the finite difference method 1 and the finite element one 2 ,3 have been developed with the advance of electronic computers. However, a limited attempt 4-8 has been only done to solve the two-dimensional natural convection problems by using the integral equation method. In this paper, we wish to ex amine possibilities on the integral equation analysis of laminar natural convection problems. The integral equation formulation of the problems is given in terms of velocity vector, pressure and temperature known as the primitive variable formulation 8 . With this formulation, we can treat easily three-dimensional problems as weIl as two-dimensional ones. But, the scope of this study is limited to steady-state two-dimensional laminar convection problems. Following the proposed methodology6, 7 we derive the nonlinear integral equation set by means of the weighted residual form and the fundamental solution tensor for linear operators of the system of governing differential equations. Fundamental
M. Tanaka et al. (eds.), Boundary Elements VIII © Springer-Verlag Berlin Heidelberg 1986
804 BOUNDARY ELEMENTS VIII CONFERENCE 1986
solutions which are indispensable for construction of our fundamental solution tensor are shown exp1icit1y concerning the problems. Numerica1 solution procedures based on a discretized scheme by using of both boundary elements and interior ce11s are presented. The derived nonlinear system of equations in terms of the unknown noda1 vectors is solved with aid of the Newton-Raphson method which is efficient in ana1yzing of problems for the higher Ray1eigh number. Samp1e resu1ts for two-dimensiona1 natural convection problems in a square, and some nonrectangu1ar enc10sures are presented in order to i11ustrate the accuracy and efficiency of the new1y proposed method. A The summation convention is used for the repeated index. comma "" is used to denote partial differentiation with respect to aspace variable (i.e.,i = D. = a/ax. ) and 6 = a2/axi denotes the Lap1acian operator. 1 1 STATEMENT OF PROBLEM Let us consider the steady problem of an incompressib1e viscous heat-conducting Newtonian fluid. Th
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