On the determination of the grad-div criterion

Abstract Grad-div stabilization, adding a term − γ 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 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 3 d 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 range κ 2 24 L U ≤ γ ≤ κ 2 4 R e L U , mesh independent case , κ 2 24 L U ≤ γ ≤ κ 2 4 ( h L ) − 4 3 L U , mesh dependent case.