Only two Betchov homogeneity constraints exist for isotropic turbulence. (English) Zbl 1526.76026
Summary: Statistically homogeneous flows obey exact kinematic relations. The R. Betchov homogeneity constraints [J. Fluid Mech. 1, 497–504 (1956; Zbl 0071.40603)] 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. Our analysis also applies to compressible/perceived velocity gradients, and it allows the derivation of homogeneity relations involving the velocity gradient and other dynamically relevant quantities, such as the pressure Hessian and viscous stresses.
MSC:
| 76F05 | Isotropic turbulence; homogeneous turbulence |
Keywords:
incompressible statistically isotropic velocity gradient field; pressure Hessian; viscous stressCitations:
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