Low-order moments of the velocity gradient in homogeneous compressible turbulence

We derive from first principles analytic relations for the second- and third-order moments of Formula Presented, the spatial gradient of fluid velocity Formula Presented, Formula Presented, in compressible turbulence, which generalize known relations in incompressible flows. These relations, although derived for homogeneous flows, hold approximately for a mixing layer. We also discuss how to apply these relations to determine all the second- and third-order moments of the velocity gradient experimentally for isotropic compressible turbulence.