A compact monotonic discretization scheme for solving second-order vorticity–velocity equations

• Published in: Computer methods in applied mechanics and engineering Vol. 190; no. 11-12; pp. 1407 - 1424
• Main Authors: Sheu, Tony W.H., Chiang, T.P., Liou, S.M.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier 08.12.2000
• Subjects: Boundary-integral methods; Computational methods in fluid dynamics; Computational techniques; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Physics; Second order equation; Transport equation; Peclet number; Stokes equation; Digital simulation; Boundary conditions; Boundary integral method; Advection diffusion equation; Convection diffusion equation; Numerical method; Steady state; Convection; Incompressible flow; Equations of state; Discretization; Differential equations; Vorticity; Flow velocity; Incompressible fluid; Diffusion; Transport theory; Navier-Stokes equations
• ISSN: 0045-7825
• DOI: 10.1016/S0045-7825(00)00169-9

This paper presents a numerical method for solving the steady-state Navier-Stokes equations for incompressible fluid flows using velocities and vorticity as working variables. The method involves solving a second-order differential equation for the velocity and a convection-diffusion equation for the vorticity in Cartesian grids. The key to the success of the numerical simulation of this class of flow equations depends largely on proper simulation of vorticity transport equation subject to proper boundary vorticity. In this paper, we present a monotonic advection-diffusion multi-dimensional scheme and a theoretically rigorous implementation of vorticity boundary conditions. While the derivation of the proposed integral vorticity boundary condition is more elaborate and is more difficult to solve than conventional local approaches, the present approach offers significant advantages. In this study, both lid-driven and backward-facing step problems have been selected for comparison and validation purposes.