Finite volume solution of the Navier-Stokes equations in velocity-vorticity formulation

Abstract

SUMMARYFor the incompressible Navier-Stokes equations, vorticity-based formulations have many attractive features over primitive-variable velocity-pressure formulations. However, some features interfere with the use of the numerical methods based on the vorticity formulations, one of them being the lack of a boundary conditions on vorticity. In this paper, a novel approach is presented to solve the velocityvorticity integro-di erential formulations. The general numerical method is based on standard ÿnite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a socalled generalized Biot-Savart formula combined with a fast summation algorithm, which makes the velocity boundary conditions implicitly satisÿed by maintaining the kinematic compatibility of the velocity and vorticity ÿelds. The well-known fractional step approaches are used to solve the vorticity transport equation. The paper describes in detail how we accurately impose no normal-ow and no tangential-ow boundary conditions. We impose a no-ux boundary condition on solid objects by the introduction of a proper amount of vorticity at wall. The di usion term in the transport equation is treated implicitly using a conservative ÿnite update. The di usive uxes of vorticity into ow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential-ow boundary condition. As application examples, the impulsively started ows through a at plate and a circular cylinder are computed using the method. The present results are compared with the analytical solution and other numerical results and show good agreement.

BEM simulation of compressible fluid flow in an enclosure induced by thermoacoustic waves

“…The solution of the transport equation only carries the vorticity, generated at the boundary, into the flow with diffusion and advection. Other authors proposed several different schemes for the calculation of boundary vorticity [13,14]. We propose to use BEM [15] because of its unique advantage for solving the boundary vorticity values directly.…”

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

“…e.g. [1][2][3][4][5][6][7]). While finite difference, finite element and finite volume methodologies have been extensively developed, the majority of the early fundamental research work in this field was based on finite difference formulations, whereas the finite volume method dominates more recent activity and the commercial codes used in industry.…”

“…Generally these authors, and other papers using vorticity-streamfunction formulations, do not discuss how to calculate the pressure field. Zhu [7] presented a new method based on a velocity-vorticity formulation for unsteady incompressible viscous flows. In this method, both the velocity and pressure are solved in integral formulations and the general numerical method is based on the standard finite volume scheme.…”