Numerical Simulation of the Inter-Equation Couplings in All-Speed Flows via the Method of Proper Closure Equations

Abstract

In the classical finite-volume method, the governing partial differential equations are first integrated on a control volume to obtain balance equations containing integration point flux quantities. Profile assumptions are made afterwards to relate the integration point flux terms to nodal values, and to finally obtain the set of algebraic governing equations. To numerically model the pressure-velocity coupling on a colocated grid arrangement, two integration point velocity definitions have been traditionally employed to avoid non-physical oscillatory solutions. The method of proper closure equations (MPCE), already used to solve steady incompressible flows, advocates a procedure in which a single velocity concept is needed in the numerical solution of flow equations on colocated grids. In this article, the implementation of the MPCE in the numerical solution of unsteady and compressible flows is discussed. Numerical tests, including quasi one-dimensional flows at all speeds and the flow in a shock tube, show that the method successfully models the transient behavior as well as the compressibility effects and discontinuities in fluid flow problems.