Abstract
This study uses the diffusion analogy (Miyake, Sci. Rep., 5R-6, 1965, Univ. of Washington, Seattle, USA) to predict the full growth behaviour of internal boundary layers (IBLs) induced by a roughness change for neutrally – and especially stably – stratified boundary layers with finite thickness. The physics of the diffusion analogy shows that the streamwise variation of the IBL thickness is dictated by
$\sigma _w/U$
at the interface, where
$\sigma _w$
and
$U$
represent wall-normal Reynolds stress and mean streamwise velocity, respectively. The existing variants of the model, summarised by Savelyev & Taylor (2005, Boundary-Layer Meteorol., vol. 115, pp. 1–25), are tailored to IBLs confined within the constant shear stress layer. To extend the applicability of the model to the outer region, we investigate the relation between
$\sigma _w/U$
and
$U/U_\infty$
in the outer region across varying stratification, where
$U_\infty$
is the free-stream velocity. Our analysis reveals that wind tunnel data from a number of facilities collapse onto a master curve when
$\sigma _w/U$
is premultiplied by a height-independent parameter, which is a function of the ratio of Monin–Obukhov length to the boundary layer thickness. The scaled
$\sigma _w/U$
decreases inversely with
$U/U_\infty$
in the surface layer, transitioning to a linear decrease as
$U/U_\infty$
increases. The new model, which integrates these findings, along with the effects of streamline displacement and acceleration, captures the complete characteristics of IBLs as they develop within turbulent boundary layers of finite thickness.