Semiconsistent mass matrix techniques for solving the incompressible NavierStokes equations
Abstract
We have described, partially analyzed, and demonstrated three schemes that retain the consistent mass matrix in the NavierStokes equations in a costeffective 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 semiimplicit 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 semiimplicit 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(+), (2) it delivers nondivergencefree solutions that are both larger than those from the equivalent LM scheme and nonvanishing 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 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.
 Publication:

Presented at the 1st International Conference on Computational Methods in Flow Analysis
 Pub Date:
 September 1988
 Bibcode:
 1988cmfa.confR....G
 Keywords:

 Incompressible Flow;
 Mass Flow;
 Matrices (Mathematics);
 NavierStokes Equation;
 Continuity Equation;
 Cost Effectiveness;
 Poisson Equation;
 Pressure Gradients;
 Fluid Mechanics and Heat Transfer