Explicit force formlulas for two dimensional potential flow with multiple bodies and multiple free vortices

TL;DR

This paper derives new force formulas for unsteady two-dimensional potential flow with multiple bodies and vortices, applicable even with vortex production, validated through classical flow problems.

Contribution

It introduces force formulas that do not require auxiliary potential functions, extending the singularity approach to flows with vortex production and multiple bodies.

Findings

Force formulas applicable to multibody, multivortex flows with vortex production.

Validation through classical flow problems like Karman vortex street and airfoil interactions.

Formulas applicable in both singularity and integral approaches without auxiliary functions.

Abstract

For problems with multiple bodies, the current integral approach needs the use of auxiliary potential functions in order to have an individual force formula for each body. While the singularity approach, based on an extension of the unsteady Lagally theorem, is restricted to multibody and multivortex flows without bound vortex and vortex production. In this paper, we consider multibody and multivortex flow and derive force formulas, in both forms of singularity approach and integral approach but without auxiliary function, that give individual forces of each body for unsteady two dimensional potential flow with vortex production on the surface of bodies. A number of problems, including Karman vortex street, Wagner problem of impulsively starting flow, interaction of two circular cylinders with circulation, and interaction of an airfoil with a bound vortex, are used to validate the force formulas.