Finite element solutions to fluid flow problems using hybrid functions for numerical integration

Abstract

A new attempt has been made to obtain the finite element solutions of fluids flowing within closed cavities using hybrid functions. The temperature distribution and the fluid profiles are analysed for inclined square and right-angled triangle cavities for various thermal boundary conditions. The governing equations, consisting of mass, momentum and energy equations representing a nonlinear coupled system of partial differential equations, were solved using Galerkin finite element method with penalty parameter. The Finite Element (FE) equations obtained for each elemental domain represent a nonlinear system of algebraic equations. Instead of the traditional Gauss-Legendre quadrature method being used in the literature, an integration scheme based on Hybrid functions of Block-pulse function and Lagrange polynomial (HBL2) is employed for the evaluation of definite integrals appearing in the FE equations. Hybrid functions provide flexibility of choosing the appropriate number of nodes in the respective directions of the integrals. Among hybrid functions, the present HBL2 gives better accuracy with less nodes. Numerical simulation results for the streamlines and isotherms in terms of non-dimensional numbers, Rayleigh number (and#12310;10and#12311;^3 and#8804; Ra and#8804; and#12310;10and#12311;^6)and Prandtl number (0.71 and#8804; P r and#8804; 1000) are presented for various fluids of importance to industry. The convergence achieved by the streamlines and isotherms are matching well with those of the existing results. The increase in Rayleigh Number (Ra) and angle of inclination and#951; in the square cavity and varying the apex angles and#958; in the triangular cavity, impacts the spreading of the temperature and the circulation of the fluid. It is noted in both the cavities considered, when Ra = 10^3 the heat transfer is conduction dominant; whereas as Ra increases, it shifts to being convection dominant; and when Ra = 10^6, turbulence is very much evident. The finite element source code for the study of the temperature profile and fluid flow were developed with Wolfram Mathematica.