The ultimate state of turbulent permeable-channel flow

TL;DR

This study uses direct numerical simulations to explore heat and momentum transfer in turbulent permeable-channel flow, revealing a transition from classical to ultimate scaling regimes at high Reynolds numbers due to large-scale spanwise rolls.

Contribution

It identifies the emergence of large-scale spanwise rolls at high Reynolds numbers and their role in achieving the ultimate heat transfer state in permeable-channel flow.

Findings

Classical Blasius law observed at Re_b < 10^4.

Ultimate scaling with Re_b^0 occurs at Re_b > 10^4.

Large-scale spanwise rolls induce ultimate heat transfer.

Abstract

Direct numerical simulations have been performed for heat and momentum transfer in internally heated turbulent shear flow with 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). The temperature is supposed to be a passive scalar, and the Prandtl number is set to unity. Turbulent heat and momentum transfer in permeable-channel flow for $\beta u_{b}=0.5$ has been found to exhibit distinct states depending on the Reynolds number $Re_b=2h u_b/\nu$. At $Re_{b}\lesssim 10^4$, the classical Blasius law of the friction coefficient and its similarity to the Stanton number, $St\approx c_{f}\sim Re_{b}^{-1/4}$, are observed, whereas at $Re_{b}\gtrsim 10^4$, the so-called ultimate scaling, $St\sim Re_b^0$ and $c_{f}\sim Re_b^0$, is found. The ultimate state is attributed to the appearance of large-scale intense spanwise rolls with the length scale of $O(h)$ arising from the Kelvin-Helmholtz type of shear-layer instability over the permeable walls. The large-scale rolls can induce large-amplitude velocity fluctuations of $O(u_b)$ as in free shear layers, so that the Taylor dissipation law $\epsilon\sim u_{b}^{3}/h$ (or equivalently $c_{f}\sim Re_b^0$) holds. In spite of strong turbulence promotion there is no flow separation, and thus large-amplitude temperature fluctuations of $O(\theta_b)$ can also be induced similarly. 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.