Singularities in the spatial complex plane for vortex sheets and thin vortex layers
Golubeva, Natalia Yurievna
Baker, Gregory R.
An important special case of shear layers is the narrowly confined vortex layer whose mathematical model is a vortex sheet with the motion described by the Birkhoff equation. Consistent discretizations of the Birkhoff equation fail to yield reliable results; in particular, a curvature singularity forms in finite time. This failure motivated the development of alternative approaches. One of them is to replace the vortex sheet by a layer of finite thickness and uniform vorticity. That prevents singularity formation. The limiting behaviour of a thin layer may be addressed by understanding the formation and subsequent motion of singularities formed in the spatial complex plane. Next, the motion of these singularities can be compared to that of the equivalent vortex sheet. The difficulty is that the parametrizations for vortex sheet and the boundaries of the thin layer do not coincide. Therefore, a new and easy way is found to compare the motion of a thin layer with the motion of a vortex sheet. The motion of the fluid particles on one side of the sheet is used, and this choice is referred to as a one-sided vortex sheet. As a result, the fluid motion of the boundary of the thin layer becomes the motion of the fluid particles of a one-sided vortex sheet in the limit of vanishing thickness. The equations of motion for a one-sided vortex sheet are derived and analytically continued into the complex plane of the new parametrization. Asymptotic methods applied to these equations reveal the presence of a 3/2-power singularity moving towards the real axis. This singularity is related to the one for the classical vortex sheet by studying the mapping between the parametrizations. The equations of motion for the thin layer are also analytically continued into the complex plane of the parametrization variable. By studying the limit of vanishing thickness, confirmation is made that the motion of the boundary of the thin layer does become the motion of the one-sided vortex sheet. Finally, asymptotic methods are applied to the analytically continued equations to understand the initial formation of singularities. Numerical calculations of the boundary motion confirm the presence of 3/2-power singularities. The trajectory of these singularities approach the one for the vortex sheet in the limit of vanishing thickness before the singularity time.