A note on kinetic energy, dissipation and enstrophy

The dissipation rate of a Newtonian fluid with constant shear viscosity can be shown to include three constituents: dilatation, vorticity, and surface strain. The last one is found to make no contributions to the change of kinetic energy. These dissipation constituents are used to identify typical compact turbulent flow structures at high Reynolds numbers. The incompressible version of the simplified kinetic-energy equation is then cast to a novel form, which is free from the work rate done by surface stresses but in which the full dissipation reenters. 1. Introduction. We make a general theoretical examination on the relation between kinetic energy transport, dissipation, and enstrophy. First, we derive an exact but simplified transport equation for the kinetic energy of viscous compressible flow, in which the full dissipation function is replaced by the squares of vorticity and dilatation. This forms a local counterpart of the classic Bobyleff-Forsyth formula (1) on the integral equivalence of incompressible dissipation and enstrophy under special boundary condition, and provides a physical clarification on dynamic processes really involved in the evolution of kinetic energy. The dissipation constituents are used to identify typical compact flow structures. Second, we cast the incom- pressible version of the simplified kinetic-energy equation to a novel form, of which the local spatical average does not explicitly depend on boundary conditions, but the full dissipation reappears along with enstrophy. This result is of relevance to some current studies of intermittency and scaling laws in inhomogeneous and anisotropic turbulence.