Computation of vortex sheet roll-up in the Trefftz plane

Abstract

Two vortex-sheet evolution problems arising in aerodynamics are studied numerically. The approach is based on desingularizing the Cauchy principal value integral which defines the sheet's velocity. Numerical evidence is presented which indicates that the approach converges with respect to refinement in the mesh-size and the smoothing parameter. For elliptic loading, the computed roll-up is in good agreement with Kaden's asymptotic spiral at early times. Some aspects of the solution's instability to short-wavelength perturbations, for a small value of the smoothing parameter, are inferred by comparing calculations performed with different levels of computer round-off error. The tip vortices' deformation, due to their mutual interaction, is shown in a long-time calculation. Computations for a simulated fuselage-flap configuration show a complicated process of roll-up, deformation and interaction involving the tip vortex and the inboard neighbouring vortices.

Convergence of the point vortex method for 2-D vortex sheet

“…Direct discretization of the Birkhoff-Rott equation gives exactly same approximation as the one in (10) by discretizing the Euler equations except that the index j in (6) and (10) are reduced to one-dimension and hence the computation is extremely efficient [15]. It is noted that the validity of the Birkhoff-Rott equation after the formation of a singularity is not well understood.…”

“…The necessary estimates on the singular kernel K defined in (15) are provided in the following lemma, and its proof can be found in [16], see also [9]. (19) where C depends only on θ.…”

“…A vortex blob was introduced by Chorin in 1973 [7] to replace the point vortex, known as vortex blob method, and was successfully used in a computation of the flow past cylinder along with two other seminal ideas: creation of vorticity on the boundary to enforce the no-slip boundary condition and using random walk to approximate the viscosity. Krasny performed careful studies for the vortex sheet problem with the vortex blob methods [15].…”

Runway Wake Vortex, Crosswind, and Visibility Detection with a Scintillometer at Schiphol Airport

“…Therefore, small spiraling motions develop at the wing tips and landing flap. Through a phenomenon known as roll-up, these small motions develop into the wake vortex with single-and double-branched spirals (Krasny 1987). The strength of the circulation of a wake vortex is proportional to the weight of the aircraft and the order of the wing span (Gerz et al 2002).…”

Comparison of regularizations of vortex sheet motion

“…The vortex blob method convolves the singular kernel in the Biot-Savart law with a smooth function. Here we use a regularization which introduces a smoothing parameter δ into the denominator of the integrand [4][5][6][7]. This is numerically convenient, but the true physical regularization is by viscosity.…”