Novel, Robust and Accurate Central Solvers for Real, Dense and Multicomponent Gases

Abstract

The nonlinear convection terms in the governing equations of inviscid compressible fluid flows are nontrivial for modelling and numerical simulation. The traditional Riemann solvers, which are strongly dependent on the underlying eigen-structure of the governing equations. Extension of the existing methods for generalized Equation of State (EOS), to incorporate real gas effects and for multicomponent fluids, is not straight forward as the eigen-structure can become complicated. Objective of the present work is to develop simple algorithms which are not dependent on the eigen-structure and can tackle easily hyperbolic systems with various equations of state. Central schemes with smart diffusion mechanisms are apt for this purpose. For fi xing the numerical diffusion, the basic ideas of satisfying the Rankine-Hugoniot con- ditions along with generalized Riemann invariants are proposed. Two such interesting algorithms are presented, which capture steady contact discontinuity exactly and have minimum numerical diffusion in smooth regions to avoid numerical instabilities. First, an interesting modi cation of a recently developed central solver (Method of Op- timal Viscosity for Enhanced Resolution of Shocks (MOVERS)), based on enforcing Rankine-Hugoniot jump conditions at the discrete level, is presented. The proposed modi cation avoids the wave-speed correction mechanism required for MOVERS and the modi ed algorithm is termed as MOVERS+. Further, a shock sensor is introduced to choose appropriate numerical diffusion in different regions. The second novel algorithm introduced in this thesis is based on selecting the numeri- cal diffusion by utilizing the generalized Riemann invariants. In this Riemann Invariant based Contact-discontinuty Capturing Algorithm (RICCA), additional numerical diffusion is also introduced based on the scaled speed of sound for robustness. Both the algorithms presented are robust in avoiding shock instabilities, are accurate in capturing grid aligned steady contact discontinuities, do not need wave speed corrections and are independent of eigen-strutures of the underlying hyperbolic systems. These algorithms have been tested with perfect gas EOS, stiffened gas, van der Waals and 5th order (5O) Virial EOS and also multicomponent gases. For multicomponent gases, both mass fraction and -based models have been used and are tested for condi- tions which are known to generate pressure oscillations. The proposed algorithms have also been utilized in simulating dense gas flows, in which the non-classical mixed shock- expansion waves and expansion shocks are physical features, due to a change in the sign of the fundamental derivative. These algorithms perform very well, without needing any modi cations for such dense gas flows. Further these numerical methods are also used to simulate viscous flows.