Semi-Consistent Mass Matrix Techniques for Solving the Incompressible Navier-Stokes Equations.

摘要

We have described, partially analyzed, and demonstrated three schemes that retain the consistent mass matrix in the Navier-Stokes equations in a cost-effective way relative to either CM everywhere or LM everywhere; viz., by lumping the mass only in the pressure gradient term and solving sequential and uncoupled systems in the context of semi-implicit time integration (implicit for diffusion, explicit for advection) and a pressure Poisson equation. All three deliver nearly the same results as each other and for nearly the same cost as a semi-implicit lumped mass scheme. All three have been shown to be superior to lumped mass methods for flows in which advection accuracy is important. While a relative ranking is difficult, the first of these, PPE, seems to be the worst in that it shows more disadvantages than advantages: (1) it displays an inconsistency at t = O sup +, (2) it delivers non-divergence-free solutions that are both larger than those from the equivalent LM scheme and non-vanishing at steady state, (3) it is slightly less stable, and (4) it is slightly more expensive and slightly more dissipative. Its single advantage seems to be in the application of boundary conditions: they are clearly proper and unambiguous. Projection 1 has the disadvantage that even SS results depend on delta t. While Projection 2 still seems to have somewhat of a ''theoretical'' (or perhaps intuitive) edge over Projection 1, in practice the two schemes have been so close that the simplicity and (very slightly) reduced cost of Projection 1 would seem to favor it. Perhaps more effort on Projection 2 would be fruitful, but we are not sure at this time. 30 refs., 19 figs. (ERA citation 13:057802)