Are there higher-order Betchov homogeneity constraints for incompressible isotropic turbulence?

TL;DR

This paper proves that the Betchov homogeneity constraints are the only such relations for incompressible, isotropic turbulence, and extends these results to include other quantities like the pressure Hessian.

Contribution

It establishes the uniqueness of Betchov homogeneity constraints for isotropic turbulence and derives new relations involving the velocity gradient and pressure Hessian.

Findings

Betchov constraints are the only homogeneity relations for isotropic turbulence

Derived new homogeneity relations involving pressure Hessian

Extended understanding of turbulence kinematic relations

Abstract

Incompressible and 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 the velocity gradient are among the most well known and extensively employed homogeneity relations. These homogeneity relations have far-reaching implications for the coupled dynamics of strain and vorticity, as well as for the turbulent energy cascade. Whether the Betchov homogeneity constraints are the only possible ones or whether additional homogeneity relations exist has not been proven yet. Here we show that the Betchov homogeneity constraints are the only homogeneity constraints for incompressible and statistically isotropic velocity gradient fields. We also extend our results to derive homogeneity relations involving the velocity gradient and other dynamically relevant quantities, such as the pressure Hessian.