On Measuring the Rotationality of Turbulence

Motivated and inspired by Truesdell's seminal article [``Two measures of vorticity," Journal of Rational Mechanics and Analysis {\bf 2}, 173--217 (1953)], recently the present author has introduced the turbulence kinematical vorticity number $\widetilde{\cal V}_{K}$ to measure the mean rotationality of turbulence [``On the classical Bradshaw--Richardson number: Its generalized form, properties, and application in turbulence," Physics of Fluids {\bf 30}, 125110 (2018)]. In this work, first, within the general framework of the Cauchy equation of motion, we derive the general equation of motion for the turbulence kinematical vorticity number $\widetilde{\cal V}_{K}$ in turbulent flows of incompressible non-Newtonian fluids, which depicts the underlying dynamical character of $\widetilde{\cal V}_{K}$ and in laminar flows reduces to the general equation of motion for the kinematical vorticity number---the Truesdell number ${\cal V}_{K}$. Second, we obtain an inequality which places the relevant dynamical restriction upon the mean Cauchy stress tensor, the Reynolds stress tensor, and the mean body force density vector in the ensemble-averaged Cauchy equation of motion for turbulence modelling. Moreover, we derive the general Reynolds stress transport equation for turbulence modelling of incompressible non-Newtonian fluids based on Cauchy's laws of motion, which includes as a special case the classical Reynolds stress transport equation for an incompressible Newtonian fluid derived from the Navier--Stokes equation.