On vorticity-based numerical methods for incompressible viscous flows

Vorticity-based numerical methods for incompressible viscous flows are studied at a fundamental level. Important issues concerning efficient yet accurate implementation of vorticity-based methods for unsteady flows are examined from both theoretical and numerical points of view. Among them, local boundary con- ditions for vorticity at a solid wall and the methods of ensuring the divergence-free condition of both vorticity and velocity fields are the key subjects. The main theme is to develop the building blocks for future development of vorticity-based methods instead of a specific algorithm.

The boundary conditions of vorticity are studied based on the physical mechanism of vorticity generation from a solid wall. We observe that the dynamic condition, represented by the boundary vorticity flux, leads to a clear physical picture and robust numerical schemes.

The coupling between the boundary vorticity flux and tangential pressure gradient has been a troublesome problem in using the flux as boundary condition, since the pressure is unknown and in general has to be solved in a global way. However, it is proved that the coupling is weak for high Reynolds number flows; local decoupled approximations are therefore developed. If necessary, the coupling can be resumed by simple iteration. Numerical experiments on one-dimensional unidirectional flows and two-dimensional flow over a circular cylinder confirm our theoretical predictions, and compare favorably with exact solutions or benchmark numerical solutions, as well as fine experiments. The dynamic condition of vorticity is also extended to three-dimensional problems. Other local boundary conditions of vorticity are tested for comparison.

In vorticity-velocity formulation, the divergence-free conditions of vorticity and velocity are not automatically satisfied by numerical schemes, especially in three dimensions. In order to impose these conditions, differential and integral methods of solving the kinematic equations of velocity are studied. An efficient method of obtaining divergence-free velocity from arbitrary vorticity distribution is established. The method also leads to an efficient scheme to project vorticity onto the divergence-free space at a minimum cost. Meanwhile, we discover the general principles of constructing true divergence-free (finite difference) schemes for solving the vorticity transport equation, i.e., a divergence-free vorticity solu- tion is obtained without projection. The construction is thus no longer limited to central differences as in the literature; a variety of more advanced techniques are now applicable.

Three-dimensional numerical tests using the new divergence-free and projection methods are conducted for the lid-driven cavity flow at several Reynolds numbers. For high Reynolds-number unsteady flows, ensuring divergence-free con- dition of vorticity is crucial for obtaining physically correct results; whereas for steady-state flows it gives better accuracy. The projection method is very flexible, and is found to be valuable even for divergence-free schemes if the computation is done with relatively large roundoff error. The results are compared with other computations and experiments with good agreement.