A new flux-based scheme for compressible flows

• A new flux-based scheme (PVU-M+ scheme) for compressible flows is presented.
• Convective flux estimation via novel combination of upwind/central molecules.
• The new scheme performs better than Van Leer and AUSMPW+ schemes for 1D Test cases.
• Multi-dimensional low (Mach = 0.1) and high Mach number (Mach = 10) test cases.
• Shock/slip line discontinuities and flow instabilities are resolved quite well.

A new algorithm for the computing of compressible flows governed by Euler/Navier–Stokes equations is presented in this paper. The inter-cell numerical convective flux is estimated through a weighted combination of fourth order central/third order upwind biased/first order upwind interpolations of inter-cell numerical fluid velocity and convective transport vector. The higher order/lower order interpolations are carefully combined via two types of local solution sensitive weight functions. One of the weight functions is designed to control the balance of upwind/central contributions via flow speeds while the other one performs the dual purpose of detecting non-smooth or discontinuous features in the solution and regulating the balance between the higher order and first order upwind interpolations. The present work, through several one-dimensional (scalar and vector hyperbolic conservation laws) and multi-dimensional (Euler/Navier–Stokes) test cases, demonstrates that a carefully designed flux-based scheme can deliver a comparable performance in terms of robustness, accuracy and efficiency and is much simple to implement in comparison to some of the popular wave based TVD schemes like Van Leer and AUSMPW+ of Flux-Vector Splitting type and HLL scheme of Reconstruction-Evolution (Godunov) type. Employing multi-dimensional test cases, it is shown that, the new scheme is very robust and can be utilized for computing flows over a very wide range of flow speeds, ranging from incompressible limit (Mach No. ∼0.1) to very high speed compressible flows (Mach No. ∼10).