A METHOD TO SOLVE A PROBLEM OF FLOW IN A CHANNEL WITH AN AXISYMMETRIC RECTANGULAR EXPANSION

Abstract

A numerical method is developed to solve a problem of fluid motion in a straight flat rigid channel with a local axisymmetric rigid-walled expansion of rectangular shape. The method has a second order of accuracy both in spatial coordinates and time. In the developed method, the Navier-Stokes and discontinuity equations are solved in the variables velocity-pressure by integrating over the elementary volumes (in which the integration domain is divided), spatial and temporal discretization of the obtained integral equations, and subsequent solving the non-linear algebraic equations. In making the noted discretization, its temporal part is carried out on the basis of the implicit three-point asymmetric backward differencing scheme, whereas the spatial one is based on the TVD-scheme and the appropriate scheme of discretization of the spatial derivatives. Solving of the noted non-linear algebraic equations is divided in three stages. Initially, the discrete momentum equation is rewritten in the form of equation for the velocity. Then, based on the discrete discontinuity equation, an equation for the pressure is derived. After that the obtained coupled nonlinear algebraic equations are solved by finding gradual approximations of the velocity and the pressure and their corresponding agreeing with one another. Herewith the number of the approximations is determined from the desired accuracy of the solution. In finding the first approximations of the noted magnitudes, the equation for the velocity is modified by replacing in it the unknown pressure and velocity (only in the flow term that is a part of the equation) with their known values, obtained at the previous time moment. In finding the next approximations, the unknown velocity (again in the flow term only) and pressure are replaced in the equation for the velocity by their known previous approximations. These replacements result in solving uncoupled systems of linear algebraic equations for the velocity and pressure instead of the above-noted coupled non-linear ones. An iterative method, which uses the deferred correction implementation method and the method of conjugate gradients, as well as the solvers ICCG (for symmetric matrices) and Bi-CGSTAB (for asymmetric matrices), is applied to solve the noted systems of linear algebraic equations.