On the computation of transonic leading-edge vortices using the Euler equations

Abstract

Separation from the leading edge of a delta wing with the subsequent roll-up into a vortex has been simulated in numerical solutions to the Euler equations. Such simulations raise a number of questions that are still outstanding, including the process of inviscid separation from a smooth edge, the role of artificial viscosity in the creation and capturing of vortex sheets, the roll-up mechanism and core features, losses in total pressure, and the stability of the vortical flow structures to three-dimensional disturbances. These matters are discussed in the context of two numerical experiments, both carried out in a sequence of three simulations that starts with a coarse-mesh discretization and ends with a fine mesh of over one million cells. The first experiment is for transonic flow, M∞ = 0.7, α = 10°, around a pure delta wing. The sequence converges to the expected classical steady vortex flow. In the second experiment, transonic flow, M∞ = 0.9, α = 8°, past a twisted cranked-and-cropped delta wing, the sequence does not converge. Instead the crank is observed in the fine-mesh solution to set off an instability in the vortex sheet that causes the vortex to burst into a thin chaotic vortical layer embedded in laminar flow. The mesh sequence suggests that it is the shortest waves resolved that are most unstable, but the energy contained in them comes from the large-scale motion and seems to be small.

Comparison of Euler and Navier-Stokes solutions for vortex flow over a delta wing

“…Numerical solution of Euler equations is currently being proposed as a model to solve the problem of leading-edge separation from a delta wing and the consequent formation of a vortex over the wing. The approach seems reasonable enough, and indeed convincing results have been produced when the separation occurs from a sharp edge' [1][2][3] '. One concern about a finite difference solution on a grid is that the vorticity diffuses because of the numerical method.…”

Large-scale viscous simulation of laminar vortex flow over a delta wing

“…(1) over the boundary of cell P is approximated by assuming the mean-value of the flux tensor A. RIZZI AND B. MULLER AIAA JOURNAL on each side to be equal to the arithmetic average of the flux tensor in the adjacent cells: (3) where and d V Pk denotes the common part of the boundaries of P and its neighboring cell k.…”

“…The approach seems reasonable enough and, indeed, convincing results have been produced when the separation occurs from a sharp edge. 1 ' 3 One concern with a solution to finite differences taken on a grid is that the vorticity diffuses because of the numerical method. A mesh with a large number of grid points usually is needed in order to limit the diffusion to a low level.…”

Computation of 3D vortex flows past a flat plate at incidence through a variational approach of the full steady euler equations

“…Thus we have to take fl as small as possible; in practice we set fl=O. 1 and w = 1.5 for our problem. As solving (14) is embedded into the iterations of the Newton linearization , it is not Table I1 Mesh necessary to compute X very accurately, so a few iterations (5-10) of SBSOR is enough.…”