Vortex sheet roll-up revisited

Abstract

The classical problem of roll-up of a two-dimensional free inviscid vortex sheet is reconsidered. The singular governing equation brings with it considerable difficulty in terms of actual calculation of the sheet dynamics. Here, the sheet is discretized into segments that maintain it as a continuous object with curvature. A model for the self-induced velocity of a finite segment is derived based on the physical consideration that the velocity remain bounded. This allows direct integration through the singularity of the Birkhoff–Rott equation. The self-induced velocity of the segments represents the explicit inclusion of stretching of the sheet and thus vorticity transport. The method is applied to two benchmark cases. The first is a finite vortex sheet with an elliptical circulation distribution. It is found that the self-induced velocity is most relevant in regions where the curvature and the sheet strength or its gradient are large. The second is the Kelvin–Helmholtz instability of an infinite vortex sheet. The critical time at which the sheet forms a singularity in curvature is accurately predicted. As observed by others, the vortex sheet strength forms a finite-valued cusp at this time. Here, it is shown that the cusp value rapidly increases after the critical time and is the impetus that initiates the roll-up process.

Discrete shedding of secondary vortices along a modified Kaden spiral

“…The distortions can be investigated by modelling the inner portion of the spiral as a single point vortex located at the centre (Moore 1974). The entire shear layer roll-up can also be predicted by a point-vortex representation of an initially straight vortex sheet (Krasny 1987;DeVoria & Mohseni 2018). The degree of the elliptical distortions depends on the shape of the object.…”

Wingtip vortex structure in the near-field of swept-tapered wings

“…Substantial efforts have been made to develop theoretical and numerical models for the roll-up process of trailing wingtip vortices [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] . The bulk of experimental investigations, largely required for the investigation of vortex dynamics and numerical validation, have been either qualitative in nature or directed toward tracking the mean tangential/axial velocities of the trailing vortex as a function of downstream range.…”

Finite rotating and translating vortex sheets

“…Thus, this problem is ill-posed in a similar way to vortex sheets exhibiting the classical Kelvin-Helmholtz instability. As suggested by the form of the generalized eigenvectors shown in figure 3, this analysis captures the general form of the instability actually observed in numerical computations (Krasny 1987;DeVoria & Mohseni 2018), although direct comparisons are made difficult by the fact that the computations relied on various regularized forms of the Birkhoff-Rott equation (1.3), while no such regularization was used in our stability analysis. The Prandtl-Munk vortex is thus the only known equilibrium involving a vortex sheet which does not have exponentially unstable modes.…”

“…The conserved quantity is defined as , while is determined by the parameterization. Specific forms of parameterization which have been considered in the literature include parameterization in terms of the arclength (DeVoria & Mohseni 2018) and in terms of the graph of a function with (Marchioro & Pulvirenti 1993). However, for the particular parameterization in terms of the circulation contained between the sheet endpoint and the point with the arclength coordinate , we have and .…”

“…For these and other related reasons, this equation has been at the centre of much mathematical research, and many important results are summarized in the collection edited by Caflisch (1989) and in the monograph by Majda & Bertozzi (2002). Because of its compact form, the Birkhoff–Rott equation (1.3) has been used in many numerical studies of the evolution of vortex sheets typically involving some form of regularization (Krasny 1986 a , b ; Sakajo & Okamoto 1996; Krasny & Nitsche 2002; DeVoria & Mohseni 2018). Similarly, we will also use it, albeit without any regularization, as the point of departure for the different analyses in the present study.…”