Vorticity Transport on a Lagrangian Tetrahedral Mesh

An integral vorticity method for computation of incompressible, three-dimensional, viscous fluid flows is introduced which is based on a tetrahedral mesh that is fit to Lagrangian computational points. A fast method for approximation of Biot–Savart type integrals over the tetrahedral elements is introduced, which uses an analytical expression for the nearest few elements, Gaussian quadratures for moderately distant elements, and a multipole expansion acceleration procedure for distant elements. Differentiation is performed using a moving least-squares procedure, which maintains between first- and second-order accuracy for irregularly spaced points. The moving least-squares method is used to approximate the stretching and diffusion terms in the vorticity transport equation at each Lagrangian computational point. A new algorithm for the vorticity boundary condition on the surface of an immersed rigid body is developed that accounts for the effect of boundary vorticity values both on the total vorticity contained within tetrahedra attached to boundary points and on vorticity diffusion from the surface during the time step. Sample computations are presented for uniform flow past a sphere at Reynolds number 100, as well as computations for validation of specific algorithms.