The ultimate state of turbulent permeable-channel flow

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. $v=\pm \beta p/\rho$ (Jiménez 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 the dimensionless permeability parameter $\beta u_b=0.5$ 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.

Reynolds number dependence of turbulent flows over a highly permeable wall

“…where C 1 and C 2 are the proportionality constants for d p and h r with √ K, respectively. The Reynolds number independence of C p f is consistent with observations in experimental (Suga et al 2010;Esteban et al 2022) and DNS (Motoki et al 2022) studies.…”

“…Given that Re K in table 2 is significantly higher than the threshold value 3.0 (Suga et al 2010), an increase in C p f is attributed to the combined effects of wall roughness and permeability. Another observation is that C p f for thick-wall cases tends to converge to a certain value at a higher Re τ , which is similar to the DNS results for turbulent channel flows with modelled permeable walls (Motoki et al 2022) as well as experiments for ceramic foams (Suga et al 2010) and metal foams (Esteban et al 2022).…”

“…A similar observation was reported in a recent DNS study using modified wall boundary conditions (Motoki et al. 2022), where the wall-normal transpiration velocity is assumed to be proportional to the local pressure fluctuation, as reported by Jiménez et al. (2001).…”

“…They found that C p f depends on the characteristics of the porous substrates and reaches a plateau beyond Re τ 5000. A similar observation was reported in a recent DNS study using modified wall boundary conditions (Motoki et al 2022), where the wall-normal transpiration velocity is assumed to be proportional to the local pressure fluctuation, as reported by Jiménez et al (2001). They reported that C p f takes a constant value in the sufficiently high Reynolds number regime, where spanwise rollers with large-scale wall blowing and suction dominate the boundary layer.…”

“…Notably, for cases 626-thick and 955-thick, the energetic regions λ x /δ p 4 and λ z /δ p 3 remain unchanged by the Reynolds number. The corresponding large-scale motions for case 955-thick are visualized in figure 6, where the spanwise average over 0 < z < 3δ p is applied to the velocity fluctuations to facilitate clear inspection of large-scale motions (Suga et al 2018;Motoki et al 2022). We observe large-scale spanwise roller motion with a streamwise extent of approximately λ x /δ p 3-4.…”