Scalings of turbulence dissipation in space and time for turbulent channel flow

Abstract

We investigate scalings of turbulence dissipation and turbulence length/time scales in the fully developed turbulent channel flow region of wall distances $y$ where the ratio of turbulence production to turbulence dissipation oscillates close to 1. First, we study averages over both time and wall-parallel streamwise ( $x$ ) and spanwise ( $z$ ) planes at $y$ . Turbulent channel flow data with friction velocity $u_{\tau }$ , and global Reynolds number $Re_{\tau }$ ranging from $550$ to $5200$ , suggest that the integral length scales of streamwise fluctuating velocities along the streamwise direction, and of wall-normal fluctuating velocities along the transverse direction, tend towards scaling with $y$ , and that the respective turbulence dissipation coefficients tend towards being constant with increasing $Re_{\tau }$ . However, the data for integral lengths of transverse fluctuating velocities in the transverse direction suggest that these lengths obey an asymptotic scaling $\sqrt {\delta y}$ (where $\delta$ is the channel half-width) with increasing $Re_{\tau }$ . The corresponding turbulence dissipation's scaling seems to tend towards $\sqrt {Re_{\tau }}/Re_{\lambda }$ , which is reminiscent of the non-equilibrium turbulence dissipation scaling found in boundary-free turbulent flows, $Re_{\lambda }$ being a $y$ -local Taylor-length-based Reynolds number. The data do not exclude minor corrections from these asymptotic scalings, and in fact, suggest finite Reynolds number deviations to them. Second, we remove time averaging and study time-fluctuating averages over wall-parallel planes at $y$ . We find that the time fluctuations of the turbulence dissipation coefficients and the Taylor-length-based Reynolds number are very strongly anti-correlated at all wall distances $y$ considered, reflecting a dominance of turbulent kinetic energy fluctuations at the lower frequencies, but a dominance of both turbulent kinetic energy and turbulence dissipation at the higher frequencies. In the case of the turbulence dissipation coefficient corresponding to the integral length of the wall-normal velocity along the transverse direction, it is possible to determine the cross-over frequency $f_c^*$ between these two behaviours, and we find $f_c^{*}\sim u_{\tau }/y$ for $Re_{\tau } = 950$ , but $f_c^* \sim u_{\tau }/\delta$ for $Re_{\tau }=2000$ , where there is evidence of very-large-scale motions.