Euler and Navier-Stokes solutions for flow over a conical delta wing

Abstract

The simulation of vortical flow about a conical delta wing (14:1 elliptic cone) by the Euler equations is studied. Navier-Stokes solutions are generated for comparison with the inviscid results. At the flight condition considered, the viscous solutions reveal a large primary leading-edge separation vortex and a smaller secondary vortex. The viscous solutions are shown to be grid-resolved. On a coarse grid, the Euler solutions result in a primary separation vortex but no secondary vortex. A comparison of pressure coefficient with the viscous solution shows surprising agreement. The agreement is, unfortunately, fortuitous. When the Euler equations are solved on the fine, viscous grid, the large separation vortex is eliminated. Instead, a cross-flow shock appears. Shock curvature introduces sufficient vorticity to produce a small vortex downstream of the shock. This inviscid result, fundamentally different from the viscous case, is a valid solution to the Euler equations. The coarse-grid Euler solution is characterized by the production of spurious vorticity and entropy due to numerical error at the leading edge. It is neither a valid solution to the Euler equations nor a reliable approximation to the real viscous flow. The use of a Kutta condition is investigated for both the fineand coarse-grid Euler solutions. It is shown that the Kutta condition fails to produce a grid-independent simulation of the actual viscous flow.