Singularities in vorticity-induced velocity field eliminated from linearized supersonic flow

Horseshoe line-vortex distributions are used to mimic downwash velocities from supersonic aircraft. They induce a velocity field within linearized supersonic flow that contains singularities. The study of supersonic vortex dynamics in various supersonic flows is a challenging task because of these singularities. They occur at the vortex line itself, as well as on the surface of Mach cones that emanate from the vertices of the flow’s horseshoe structure. Aerodynamics researchers have sought to clarify the nature of these singularities and report a means to eliminate them in Physics of Fluids.

“It’s important to eliminate mathematical singularities in linearized supersonic flow in order to faithfully capture the flow physics,” said co-author Vijay Gopal. Gopal’s team used a time-dependent approach to show that the sudden appearance of a line vortex in supersonic flow generates a ripple of singularities in the induced velocity field. This ripple convects downstream, laterally spreading to the surface of the Mach cone. Based on this finding, the group explored potential mathematical analyses for vorticity distributions and investigated a smoother vortex-source distribution that more accurately models the flow physics.

The new vorticity distributions eliminated singularities along the Mach cones and replaced them with ripples of finite magnitude in the induced velocity field. The researchers think that the final derived, induced velocity expressions will be helpful for studying the dynamics of complex vortex systems in internal and external flows. The potential of this work was demonstrated by applying the distribution to a confined supersonic flow.

“What we have developed will be useful in our studies in streamwise vortex interactions at high Mach numbers,” said Gopal, adding that understanding singularity origins will help overcome similar challenges in other domains.

Source: “Influence of vorticity distribution on singularities in linearized supersonic flow,” by Vijay Gopal and Luca Maddalena, Physics of Fluids (2018). The article can be accessed at https://doi.org/10.1063/1.5030091 .