Gradient Based Reconstruction: Inviscid and viscous flux discretizations, shock capturing, and its application to single and multicomponent flows

TL;DR

This paper introduces a gradient-based reconstruction method for compressible Navier-Stokes equations that enhances accuracy and efficiency in shock capturing and viscous flow simulations by sharing derivatives between schemes.

Contribution

The novel algorithm efficiently reuses derivatives for both inviscid and viscous flux calculations, improving accuracy and stability in complex flow simulations.

Findings

Fourth-order accuracy for viscous schemes

Effective shock capturing with monotonicity-preserving scheme

Demonstrated robustness in complex viscous flow simulations

Abstract

This paper presents a gradient-based reconstruction approach for simulations of compressible single and multi-species Navier-Stokes equations. The novel feature of the proposed algorithm is the efficient reconstruction via derivative sharing between the inviscid and viscous schemes: highly accurate explicit and implicit gradients are used for the solution reconstruction expressed in terms of derivatives. The higher-order accurate gradients of the velocity components are reused to compute the viscous fluxes for efficiency and significantly improve the solution and gradient quality, as demonstrated by several viscous-flow test cases. The viscous schemes are fourth-order accurate and carefully designed with a high-frequency damping property, which has been identified as a critically important property for stable compressible-flow simulations with shock waves [Chamarthi et al., JCP, 2022]. Shocks and material discontinuities are captured using a monotonicity-preserving (MP) scheme, which is also improved by reusing the gradients. For inviscid test cases, The proposed scheme is fourth-order for linear and second-order accurate for non-linear problems. Several numerical results obtained for simulations of complex viscous flows are presented to demonstrate the accuracy and robustness of the proposed methodology.