Finite-element analysis of compressible and incompressible viscous flow and heat transfer problems

There are numerous problems of two-dimensional viscous, compressible or incompressible steady-state flows where the governing hydrodynamic equations are difficult to solve even numerically due to their elliptic-hyperbolic character and the complex geometry of the flow configuration. For such problems, a finite-element numerical technique has been developed whereby the steady-state hydrodynamic equations and the associated boundary conditions are solved taking into full account the nonlinear convective terms, viscous terms, heat conduction terms, and variable fluid properties. The numerical technique is based upon a general formulation for the system of hydrodynamic equations making use of the method of weighted residuals, applied over discrete, distorted finite elements of the flow domain where the unknown fluid variables are expressed continuously in terms of polynomial approximating functions and nodal parameters. This process results in a set of nonlinear algebraic equations for the nodal parameters which are solved iteratively by using a multi-dimensional Newton–Raphson scheme. To assess its accuracy, the method is applied to problems of compressible and incompressible viscous flow and heat transfer in diverging channels with plane walls, and also to a problem of normal shock wave in one-dimensional flow. The results in all cases are in satisfactory agreement with existing analytical solutions and experimental data. Furthermore, the numerical scheme appears to be stable.