A Transformation for Imposing the Rigid Bodies on the Solution of the Vorticity-Stream Function Formulation of Incompressible Navier–Stokes Equations

Abstract

A new penalization method is proposed for implementing the rigid bodies on the solution of the vorticity-stream function formulation of the incompressible Navier–Stokes equations. The method is based upon an active transformation of dependent variables. The transformation may be interpreted as time dilation. In this interpretation, the rigid body is considered as a region where the time is dilated infinitely, that is, time is stopped. The transformation is introduced in the vorticity and stream function equations to achieve a set of modified equations. The, in the modified equations, the time dilation of the solid region is approached to infinity. The mathematical and physical properties of the modified equations are investigated and implementation of the no-slip and no-penetration conditions are justified. Moreover, a suitable numerical method is presented for the solution of the modified equations. In the proposed numerical method, time integration is performed via the Crank–Nicolson method, and the semi-discrete equations are spatially discretized via second-order finite differencing on a uniform Cartesian grid. The method is applied to the fluid flow around a square obstacle placed in a channel, a sudden flow perpendicular to a thin flat plate, and the flow around a circular cylinder. The accuracy of the numerical solutions is evaluated.