Validation of a numerical algorithm based on transformed equations

Generally, a typical equation governing a physical process, such as fluid flow or heat transfer, has three types of terms that involve partial derivatives, namely, the transient term, the convective terms and the diffusion terms. The major difficulty in obtaining numerical solutions of these partial differential equations is the discretization of the convective terms. The transient term is usually discretized using the first-order forward or backward differencing scheme. The diffusion terms are usually discretized using the central differencing scheme and no difficulty arises since these terms involve second-order spatial derivatives of the flow variables. The convective terms are non-linear and contain first-order spatial derivatives. The main difference between various numerical algorithms is the discretization of the convective terms. In the present study, an alternative approach to discretizing the governing equations is presented. In this algorithm, the governing equations are first transformed by introducing an exponential function to eliminate the convective terms in the equations. The proposed algorithm is applied to simulate some fluid flows with exact solutions to validate the proposed algorithm. The fluid flows used in this study are a self-designed quasi-fluid flow problem, stagnation in plane flow (Hiemenz flow), and flow between two concentric cylinders. The comparisons with the power-law scheme indicate that the proposed scheme exhibits better performance. (author)