The ultimate state of turbulent permeable-channel flow

Shingo Motoki, Kentaro Tsugawa, Masaki Shimizu, Genta Kawahara

Published 2021-06-15Version 1

Direct numerical simulations have been performed for heat and momentum transfer in internally heated turbulent shear flow of constant bulk mean velocity and temperature, $u_{b}$ and $\theta_{b}$, between parallel, isothermal, no-slip and permeable walls. The wall-normal transpiration} velocity on the walls $y=\pm h$ is assumed to be proportional to the local pressure fluctuations, i.e. $v=\pm \beta p/\rho$ (Jim\'enez et al., J. Fluid Mech., vol. 442, 2001, pp.89-117). Here $\rho$ is the mass density of the fluid, and the property of the permeable wall is given by the dimensionless parameter $\beta u_{b}$. The temperature is supposed to be a passive scalar, and the Prandtl number is set to unity. In the permeable channel for $\beta u_{b}=0.5$, we have found the critical transition of the scaling of the Stanton number St and the friction coefficient $c_{f}$ from the Blasius empirical law $St\approx c_{f}\sim Re_{b}^{-1/4}$ to the so-called ultimate state represented by $St\sim Re_{b}^{0}$ and $c_{f}\sim Re_{b}^{0}$ at the bulk Reynolds number $Re_{b}\sim10^{4}$. The ultimate state is attributed to significant heat and momentum transfer enhancement without flow separation by intense large-scale spanwise rolls with the length scale of $O(h)$ arising from the Kelvin-Helmholtz instability over the permeable walls. The large-scale rolls can induce large-amplitude velocity fluctuations of $O(u_{b})$, so that the Taylor dissipation law $\epsilon\sim u_{b}^{3}/h$ (or equivalently $c_{f}\sim Re_{b}^{0}$) holds, and similarly the temperature fluctuations of $O(\theta_{b})$. As a consequence, the ultimate heat transfer is achieved, i.e., a wall heat flux scales with $u_{b}\theta_{b}$ (or equivalently $St\sim Re_{b}^{0}$) independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction.