A generalised multi-directional characteristic-based Godunov-type framework for elliptic, parabolic and hyperbolic pressure-based incompressible methods

Abstract

The objective of the current research is to construct numerical methods based on physical principles to reduce modelling errors in the field of computational fluid dynamics. In order to investigate the non-linearities of the convective flux term, a multi-directional characteristic-based scheme has been developed in this work to capture the anisotropic behaviour of the incompressible Navier{Stokes equations. To avoid the pressure-velocity decoupling and to promote stability at high Reynolds numbers, the Riemann problem has been incorporated into the scheme which creates a multi-directional Godunov-type framework. In order to capture the pressure correctly, which through its coupling to the velocity field is depending on the velocity's non-linear effects, it is postulated that the pressure should have its own transport equation which should have a parabolic type. This is necessary to align the pressure with the mathematical properties of the Navier{Stokes equations. Thus, a novel incompressible method has been developed which features a pressure transport equation which is referred to as the Fractional-Step with Velocity Projection (or FSVP) method. It is further extended through a perturbed continuity equation of the Arti cial Compressibility (AC) method to hyperbolise the first Fractional-Step of the system of equations, while the second Fractional-Step retains the required parabolic behaviour, which is called the FSAC-VP method in turn. Through the hyperbolic Fractional-Step, the multi-directional Godunov-type framework is directly applicable to the newly developed method. Parametric simulations for the lid driven cavity, backward facing step, sudden expan- sion and Taylor{Green vortex problem have been performed using the AC, FSVP, FSAC-VP and the Fractional-Step, Arti cial Compressibility with Pressure Projection, or FSAC-PP, method. The FSVP and FSAC-VP method showed superior convergence properties compared to the AC method for unsteady flows, where a speed up of a factor up to 193.0 times has been observed. Since the parabolic pressure transport equation has a memory of the time history of the flow, smooth error curves have been produced over time while the other methods showed oscillatory profi les. Generally speaking, the most accurate results have been obtained with the FSAC-PP method, closely followed by the FSAC-VP and FSVP method. The inclusion of the multi-directional Godunov-type framework showed generally better or equally well resolved results compared to the benchmark numerical scheme for the FSAC-PP and FSAC-VP / FSVP method. Furthermore, the multi-directional scheme by itself showed its capabilities to predict vortical flows better than a simple numerical reconstruction scheme. The FSAC-VP method has shown a higher degree of scheme independence where velocity and pressure curves showed little variations compared to reference data. This was particularly pronounced for the sudden expansion which had consequences on the prediction of the correct bifurcation behaviour. Finally, it has been argued that what the numerical scheme development is to the non-linear term of the Navier{Stokes equations should be similarly done with incompressible flow method development to capture the correct pressure behaviour. This work shows that differences between elliptic, parabolic and hyperbolic pressure treatments do exist which can have a significant effect on the overall prediction of the flow features.