Velocity boundary conditions for vorticity formulations of the incompressible Navier-Stokes equations

Abstract

Velocity boundary conditions for the vorticity form of the incompressible, viscous fluid momentum equations are presented. Vorticity is created on boundaries to simultaneously satisfy the tangential and normal components of the velocity boundary condition. The newly created vorticity is specified by a kinematical formulation which is a generalization of Helmholtz decomposition of a vector field. Related forms of the decomposition were developed by Bykhovskiy and Smirnov in 1983, and Wu and Thompson in 1973. Though it has not been generally recognized as such, these formulations resolve the over-specification issues associated with determining a velocity field from velocity boundary conditions and a vorticity field. The generalized decomposition has not been widely used, however, apparently due to a general lack of a useful physical interpretation. An analysis is presented which shows that the generalized decomposition has a relatively simple physical interpretation which facilitates its numerical implementation. The implementation of the generalized decomposition for the normal and tangential velocity boundary conditions is discussed in detail. As an example of the use of this boundary condition, the flow in a lid-driven cavity is simulated. The solution technique is based on a Lagrangian transport algorithm in the hydrocode ALEGRE. ALEGRE`s Lagrangian transport algorithm has beenmore » modified to solve the vorticity transport equation, thus providing a new, accurate method to simulate incompressible flows. This numerical implementation and the new boundary condition formulation allow vorticity-based formulations to be used in a wider range of engineering problems.« less