Three-Dimensional Solutions of Trailing-Vortex Flows Using Parabolized Equations

Abstract

Flows of practical significance exist, like systems of trailing vortices, which are inhomogeneous in two directions and weakly dependent along the third spatial direction. Exploiting these characteristics, an integration of the Navier–Stokes equations using the parabolic Navier–Stokes concept is proposed for the recovery of steady solutions that might be used subsequently in the scope of primary instability analyses. The parabolic Navier–Stokes equations are first formulated in a cylindrical coordinate frame and used to calculate the solution of an isolated, axisymmetric nonparallel (axially developing) vortex. Then, the fully three-dimensional flow corresponding to a counter-rotating pair of nonparallel vortices is obtained by parabolic Navier–Stokes formulated in Cartesian coordinates. Stable high-order finite-difference-based numerical schemes are used for the spatial discretization to exploit the benefits of using a sparse direct solver for the inversion of the large matrices resulting from the discretization of the partial-derivative-based parabolic Navier–Stokes equations. Solutions recovered converge to the analytically obtained evolution of an isolated vortex in the limit of large distance between vortices and depart from this idealized situation when the distance becomes of the same order as the radii of the vortices.