The ultimate state of turbulent permeable-channel flow

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 = +/- h is assumed to be proportional to the local pressure fluctuations, i.e. v = +/-beta p/rho (Jimenez 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 = 2hu(b)/nu. At Re-b less than or similar to 10(4), the classical Blasius law of the friction coefficient and its similarity to the Stanton number, St approximate to c(f) similar to Re-b(-1/4), are observed, whereas at Re-b greater than or similar to 10(4), the so-called ultimate scaling, St similar to Re-b(0) and c(f) similar to 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 similar to u(b)(3)/h (or equivalently c(f) similar to 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 similar to Re-b(0)) independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction.