Only two Betchov homogeneity constraints exist for isotropic turbulence

Statistically homogeneous flows obey exact kinematic relations. The Betchov homogeneity constraints (Betchov, J. Fluid Mech. , vol. 1, 1956, pp. 497–504) for the average principal invariants of velocity gradient are among most well-known and extensively employed These relations have far-reaching implications coupled dynamics strain vorticity, as well turbulent energy cascade. Whether only possible ones or whether additional exist has not been proven yet. Here we show that incompressible statistically isotropic fields. Our analysis also applies to compressible/perceived gradients, it allows derivation involving other dynamically relevant quantities, such pressure Hessian viscous stresses.