High Order Extrapolation Techniques for WENO Finite-Difference Schemes Applied to NACA Airfoil Profiles

TL;DR

The paper addresses preserving high-order accuracy in finite-difference WENO schemes on Cartesian grids when handling complex geometries such as NACA airfoil profiles. It introduces a high-order boundary extrapolation technique for ghost cells, based on a scale- and dimension-independent weighting of interpolation stencils, with smoothness indicators $I_k$ and global weight $\omega$ to blend polynomial extrapolation with nearest-neighbor data. The method is combined with a Shu-Osher WENO spatial discretization and RK time stepping to yield high-order accuracy in smooth regions and robust behavior near discontinuities. Numerical experiments on a supersonic flow around a NACA0012 profile show results in agreement with established finite-volume methods, validating the approach.

Abstract

Finite-difference WENO schemes are capable of approximating accurately and efficiently weak solutions of hyperbolic conservation laws. In this context high order numerical boundary conditions have been proven to increase significantly the resolution of the numerical solutions. In this paper a finite-difference WENO scheme is combined with a high order boundary extrapolation technique at ghost cells to solve problems involving NACA airfoil profiles. The results obtained are comparable with those obtained through other techniques involving unstructured meshes.

Figures (2)

• Figure 1: Examples of choice of stencil ($R=3$). We use the stencil $\mathcal{S}(P)=\{N_1, N_2, N_3, N_4\}$ in case of outflow boundary and conditions and the stencil $\mathcal{S}(P)=\{P_0, P_1, P_2, P_3\}$ in case of Dirichlet boundary conditions.
• Figure 2: Density and pressure fields for the supersonic NACA0012 interaction test.