On the accuracy of the rotation form in simulations of the Navier–Stokes equations

The rotation form of the Navier–Stokes equations nonlinearity is commonly used in high Reynolds number flow simulations. It was pointed out by a few authors (and not widely known apparently) that it can also lead to a less accurate approximate solution than the usual u · ∇ u form. We give a different explanation of this effect related to (i) resolution of the Bernoulli pressure, and (ii) the scaling of the coupling between velocity and pressure error with respect to the Reynolds number. We show analytically that (i) the difference between the two nonlinearities is governed by the difference in the resolution of the Bernoulli and kinematic pressures, and (ii) a simple, linear grad–div stabilization ameliorates much of the bad scaling of the velocity error with respect to Re. The rotation form does have superior conservation properties to the alternatives and it appears to be amenable to more efficient preconditioners. Thus, the rotational form with grad–div stabilization is a promising method. We also give experiments that show bad velocity approximation is tied to poor pressure resolution in either form.