Volume integrals of the QA-RA invariants of the velocity gradient tensor in incompressible flows

Abstract

Volume integrals of the second and third invariants, i.e. QA and RA, respectively, of the velocity gradient tensor Aij over an incompressible flow domain are shown to vanish for certain combinations of boundary conditions used in a large variety of direct numerical simulations of turbulent flows. For these turbulent flows, the dissipation of total kinetic energy is directly proportional to the total enstrophy. Betchov [J. Fluid Mech. 1 (1956) 497] showed that for homogeneous flows an increase in mean enstrophy implies that it is more likely that the intermediate rate-of-strain is positive rather than negative. This paper shows that using the framework of the invariants of the velocity gradie..