On the determination of the grad-div criterion

Grad-div stabilization, adding a term -gamma grad div u 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 gamma 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 kappa is the signal to noise ratio of the body force, estimates suggest that gamma should be restricted to the range k(2)/24 LU <= gamma <= kappa(2)/4 ReLU, mesh independent case, k(2)/24 LU <= gamma <= kappa(2)/4(h/L)(-4/3) LU, mesh independent case. (C) 2018 Elsevier Inc. All rights reserved.