 Near-wall radiative heating layer

For a wall shear $u\simeq sz$, heating over streamwise length $L$ gives an outer temperature rise proportional to $L/(sz)$. Its transverse thermal conduction grows as $z^{-3}$ relative to the local source. If $\chi$ is thermal diffusivity, comparing conduction with heating gives $\delta \sim (\chi L/s{)}^{1/3}$, up to a numerical constant depending on the precise comparison. The same scale follows by comparing transverse diffusion time $z^2/\chi$ with residence time $L/(sz)$.