A generalised and low-dissipative multi-directional characteristics-based scheme with inclusion of the local Riemann problem investigating incompressible flows without free-surfaces

Abstract

In the present study, we develop a generalised Godunov-type multi-directional characteristics-based (MCB) scheme which is applicable to any hyperbolic system for modelling incompressible flows. We further extend the MCB scheme to include the solution of the local Riemann problem which leads to a hybrid mathematical treatment of the system of equations. We employ the proposed scheme to hyperbolic-type incompressible flow solvers and apply it to the Artificial Compressibility (AC) and Fractional-Step, Artificial Compressibility with Pressure Projection (FSAC-PP) method. In this work, we show that the MCB scheme may improve the accuracy and convergence properties over the classical single-directional characteristics-based (SCB) and non-characteristic treatments. The inclusion of a Riemann solver in conjunction with the MCB scheme is capable of reducing the number of iterations up to a factor of 4.7 times compared to a solution when a Riemann solver is not included. Furthermore, we found that both the AC and FSAC-PP method showed similar levels of accuracy while the FSAC-PP method converged up to 5.8 times faster than the AC method for steady state flows. Independent of the characteristics- and Riemann solver-based treatment of all primitive variables, we found that the FSAC-PP method is 7–200 times faster than the AC method per pseudo-time step for unsteady flows. We investigate low- and high-Reynolds number problems for well-established validation benchmark test cases focusing on a flow inside of a lid driven cavity, evolution of the Taylor–Green vortex and forced separated flow over a backward-facing step. In addition to this, comparisons between a central difference scheme with artificial dissipation and a low-dissipative interpolation scheme have been performed. The results show that the latter approach may not provide enough numerical dissipation to develop the flow at high-Reynolds numbers. We found that the inclusion of a Riemann solver is able to overcome this shortcoming. Overall, the proposed generalised Godunov-type MCB scheme provides an accurate numerical treatment with improved convergence properties for hyperbolic-type incompressible flow solvers. Program summaryProgram Title: unified2D-CProgram Files doi:http://dx.doi.org/10.17632/8m3dw6zkgc.1Licensing provisions: CC BY NC 3.0Programming language: C++Nature of problem: Incompressible flow solver have generally a non-hyperbolic type and thus the method of characteristics and Riemann solvers cannot be used without modifications for low speed flows. In the framework of compressible flows, the Riemann problem – and the method of characteristics which is closely related to it – is an essential part of the solution procedure. The Riemann solver is able to preserve the conservativeness and, through the evaluation of the eigenstructure of the system, introduces transportiveness into the spatial reconstruction schemes. The characteristics-based scheme allows to couple the pressure and velocity in a physical manner which, together with the Riemann solver, presents a new multi-directional Godunov framework for incompressible flows.Solution method: We show a generalised description of a multi-directional characteristics-based schemes which may be used with any incompressible and hyperbolic system of equations. The Finite Volume approach is used where inter-cell fluxes are reconstructed through a simple but higher-order polynomial interpolation scheme which only adds numerical dissipation proportional to its Taylor-series truncation error. We use the Rusanov Riemann solver which provides the needed conservativeness and transportiveness. It also adds just enough numerical dissipation for cases where the dissipation of the numerical scheme is not sufficient while retaining a high level of accuracy.