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.
The ultimate state of turbulent permeable-channel flow
