Transport equations for the normalized nth-order moments of velocity derivatives in grid turbulence

Abstract

Transport equations for the normalized moments of the longitudinal velocity derivative ${F_{n + 1}}$ (here, $n$ is $1, 2, 3\ldots$ ) are derived from the Navier–Stokes (N–S) equations for shearless grid turbulence. The effect of the (large-scale) streamwise advection of ${F_{n + 1}}$ by the mean velocity on the normalized moments of the velocity derivatives can be expressed as $C_1 {F_{n + 1}}/Re_\lambda$ , where $C_1$ is a constant and $Re_\lambda$ is the Taylor microscale Reynolds number. Transport equations for the normalized odd moments of the transverse velocity derivatives ${F_{y,n + 1}}$ (here, $n$ is 2, 4, 6), which should be zero if local isotropy is satisfied, are also derived and discussed in sheared and shearless grid turbulence. The effect of the (large-scale) streamwise advection term on the normalized moments of the velocity derivatives can also be expressed in the form $C_2 {F_{y,n + 1}}/Re_\lambda$ , where $C_2$ is a constant. Finally, the contribution of the mean shear in the transport equation for ${F_{n + 1}}$ can be modelled as $15 B/Re_\lambda$ , where $B$ ( $=S^*{S_{s,n + 1}}$ ) is the product of the non-dimensional shear parameter $S^*$ and the normalized mixed longitudinal-transverse velocity derivatives ${{S_{s,n + 1}}}$ ; if local isotropy is satisfied, $S_{s,n + 1}$ should be zero. These results indicate that if ${F_{n + 1}}$ , ${F_{y,n + 1}}$ and $B$ do not increase as rapidly as $Re_\lambda$ , then the effect of the large-scale structures on small-scale turbulence will disappear when $Re_\lambda$ becomes sufficiently large.