A note on Kolmogorov's third-order structure-function law, the local isotropy hypothesis and the pressure-velocity correlation

We show that Kolmogorov's (1941b) inertial-range law for the third-order structure function can be derived from a dynamical equation including pressure terms and mean flow gradient terms. A new inertial-range law, relating the two-point pressure-velocity correlation to the single-point pressure-strain tensor, is also derived. This law shows that the two-point pressure-velocity correlation, just like the third-order structure function, grows linearly with the separation distance in the inertial range. The physical meaning of both this law and Kolmogorov's law is illustrated by a Fourier analysis. An inertial-range law is also derived for the third-order velocity-enstrophy structure function of two-dimensional turbulence. It is suggested that the second-order vorticity structure function of two-dimensional turbulence is constant and scales with epsilon(omega)(2/3) in the enstrophy inertial range, epsilon(omega) being the enstrophy dissipation. Owing to the constancy of this law, it does not imply a Fourier-space inertial-range law, and therefore it is not equivalent to the k(-1) law for the enstrophy spectrum, suggested by Kraichnan (1967) and Batchelor (1969).