A viscous-term subcell limiting approach for high-order FR/CPR method in solving compressible Navier-Stokes equations on curvilinear grids

High-order flux reconstruction/correction procedure via reconstruction (FR/CPR) method has become an attractive method for simulating multi-scale flows because of its high accuracy and low dissipation. However, simulating flows with strong shocks is still challenging for FR/CPR. Recently, a priori subcell limiting based on compact nonlinear nonuniform weighted (CNNW) schemes was proposed for the CPR method to solve the Euler equations, which robustly simulates strong shocks and has a good balance between high resolution and shock-capturing robustness. The CPR method with subcell CNNW limiting is a hybrid scheme also called the hybrid CPR-CNNW scheme (abbr. HCCS). This paper extends HCCS to solve compressible Navier-Stokes equations on curvilinear grids. To improve the capability of HCCS in simulating hypersonic flows, a viscous-term subcell limiting (VSL) approach is proposed, which emphasizes the importance of introducing nonlinear mechanisms to the viscous term for capturing shocks. The main idea of the VSL is to replace the unified high-order polynomial distribution of physical variables or fluxes by a limited piecewise polynomial distribution for the viscous-term discretization in troubled cells. In this paper, the VSL is implimented by decomposing the troubled cells into subcells and limiting the distribution of physical variables for each subcell. To makes the approach efficient, the VSL reuses the limited results used in the inviscid-term discretization. Moreover, HCCS with the VSL is extended to curvilinear grids. Discrete grid metrics and discrete Jacobian are designed to make HCCS with the VSL satisfy discrete conservation laws and discrete geometric conservation laws and fulfill free-stream preservation, which are validated by two test cases on curvilinear grids. It is shown that HCCS without the VSL blows up in simulating some viscous hypersonic flows. Numerical results on several typical cases show that HCCS with the VSL has obvious superiority in high resolution, shock-capturing robustness,and accurate prediction of surface pressure, skin friction, and heat transfer.