On the determination of the grad-div criterion

Grad-div stabilization, adding a term −γgraddivu to penalize violation of incompressibility, has proven to be a useful tool in the simulation of incompressible flows. Such a term requires a choice of the coefficient γ and studies have begun appearing with various suggestions for its value. We give an analysis herein that provides a restricted range of possible values for the coefficient in 3d turbulent flows away from walls. If U,L denote the large scale velocity and length respectively and κ is the signal to noise ratio of the body force, estimates suggest that γ should be restricted to the rangeκ224LU≤γ≤κ24ReLU, mesh independent case,κ224LU≤γ≤κ24(hL)−43LU, mesh dependent case.