Approach towards local isotropy in statistically stationary turbulent shear flows

Abstract

We analyse the approach towards local isotropy in statistically stationary turbulent shear flows using the transport equations for the fourth-order moments of the velocity derivative. It is found that terms of these equations representing the large-scale contribution associated with the uniform mean velocity gradient gradually decrease as the Taylor microscale Reynolds number $Re_\lambda$ increases, and finally disappear when $Re_\lambda$ is sufficiently large. This gradual weakening of the large-scale effect is accompanied by a gradual approach towards local isotropy of the small-scale motion. The rate at which local isotropy is approached depends on the weakening of the large-scale forcing, which is controlled by the magnitude of the non-dimensional velocity shear parameter $S^*$ ( $\equiv \overline {u_1^2}({{\partial {{\bar U}_1}}}/{{\partial {x_2}}})/{\bar {\varepsilon }_{iso}}$ , where $\bar {\varepsilon }_{iso}$ is the isotropic mean turbulent energy dissipation rate, $\overline {u_1^2}$ is the streamwise velocity variance, and ${\partial {{\bar U}_1}/\partial {x_2}}$ is the uniform mean velocity gradient in the transverse direction). In particular, we show that the approach towards local isotropy can be recast in the form $C\, Re_\lambda ^{-1}$ , where $C$ is the product of $S^*$ and a ratio of transverse-to-streamwise velocity derivative variances. This is consistent with the behaviour of the normalized third-order moments of transverse velocity derivatives. With the further use of the transport equations for the eighth- and twelfth-order velocity derivative moments, it is found that the even moments of transverse velocity derivatives can significantly affect the rate at which local isotropy is approached, especially for higher orders.