Some Nonlinear Vortex Solutions

We consider the steady-state two-dimensional motion of an inviscid incompressible fluid which obeys a nonlinear Poisson equation. By seeking solutions of a specific form, we arrive at some interesting new nonlinear vortex solutions. 1. Introduction The study of wakes behind bodies has generated significant interest due to the problem’s important applications. An excellent review of current theories on the evolution of such wakes is given by Chomaz [1]. Numerical models of these flows are typically based on the full three-dimensional Navier-Stokes equations. Still, two-dimensional exact solutions of the Euler equation add significantly to our understanding of such flows [2]. Recently, vortex solutions of the steady two-dimensional inviscid problem have been used as initial conditions for solvers applied to the full three-dimensional time-dependent problem. This approach was adopted by Faddy and Pullin [3] to model the three-dimensional wake behind an aircraft wing. Flows which bear strong resemblance to exact vortex solutions occur in many other applications; a recent experimental study [4] involving cavity flows provides some good examples of this. In the present contribution we examine a class of exact solutions of the two-dimensional inviscid problem which are related to vortex flows. Similar theoretical efforts are well documented; in particular, we note two recent contributions [5, 6]. In plane two-dimensional hydrodynamics, the equation governing the motion of an inviscid incompressible fluid can be written in terms of a stream function as where is a Jacobian and is the two-dimensional Laplacian. The stream function equation (1.1) admits steady-state solutions of the form for any function . Equation (1.2) is a nonlinear Poisson or an elliptic Klein-Gordon equation, and a number of solutions are known for two-dimensional hydrodynamics. In particular, the solution corresponding to a row of corotating Stuart vortices [7] is , while the solution corresponding to the counter-rotating Mallier-Maslowe (M&M) vortices [8] is Stuart vortices satisfy Liouville’s equation [9], , and a number of additional solutions of Liouville’s equation are known, with some of these given in [10–12]. M&M vortices satisfy the sinh-Poisson, or elliptic sinh-Gordon, equation, . Again, a number of additional solutions are known [5, 13, 14], some of which involve Jacobian elliptic functions [15] such as and , which are doubly periodic functions of . The goal of this study is to find solutions to (1.2), that are nonlinear vortex solutions of the form with , , constants and and .