A Novel Velocity–Vorticity Formulation of the Navier–Stokes Equations with Applications to Boundary Layer Disturbance Evolution

Abstract

A novel velocity–vorticity formulation of the unsteady, three-dimensional, Navier–Stokes equations is presented. The formulation is particularly suitable for simulating the evolution of three-dimensional disturbances in boundary layers. A key advantage is that there are only three governing equations for three primary dependent variables. Another advantage is that no wall boundary conditions are needed for the vorticity. Instead the conditions placed on the velocity are linked to the vorticity field through integral constraints based on the definition of vorticity.Numerical methods are presented in the context of application to the three-dimensional boundary layer over a rotating disc. The discretization scheme uses spectral expansions in the wall-normal and azimuthal (or spanwise) directions and compact finite differences in the radial (or streamwise) direction. The scheme is implemented so that the advantages of spectral convergence can be combined with the use of an efficient line-iteration solution procedure. The linearized form of the new velocity–vorticity method is validated for the case of convective instabilities evolving over both rigid and compliant discs.