Convergence of the Hybrid WENO Scheme for Steady Compressible Navier-Stokes Equations in Curved Geometries Using Cartesian Grids

Classical weighted essentially non-oscillatory (WENO) and other nonlinear schemes often face challenges in achieving steady-state convergence, although substantial progress has already been made in the simulations of compressible Euler equations, there are few contributions for the steady-state simulations of compressible Navier-Stokes (NS) equations. To address this issue, we adopt the fifth-order hybrid WENO (WENO-H) finite difference scheme, designed to achieve machine-zero residual in numerical iterations. The WENO-H scheme employs fifth-order linear reconstruction in smooth regions, guided by an effective smoothness detector, and smoothly degrades to lower-order reconstruction to prevent nonphysical oscillations. It ensures seamless transitions from smooth to discontinuous regions through a smoothing transition zone. In solving compressible NS equations in curved geometries on Cartesian grids, ghost point values outside the physical domain are determined using the fifth-order WENO extrapolation method coupled with simplified inverse Lax-Wendroff procedures. Two sets of ghost point values are utilized to handle convective and diffusive terms discretization near boundaries. Furthermore, the pressure term in primitive variables is substituted with the temperature term to facilitate the imposition of the adiabatic boundary condition. A transitional interpolation technique is proposed to enhance steady-state convergence near free-stream boundaries. Numerical experiments demonstrate that the WENO-H scheme achieves robust steady-state convergence across extensive benchmark examples of NS equations in curved geometries. The scheme exhibits good non-oscillatory property, particularly in scenarios involving strong discontinuities. Moreover, it provides high resolution for both steady and unsteady problems containing multi-scale structures.