A semi-empirical model is presented that describes the development of a fully developed turbulent boundary layer in the presence of surface roughness with length scale
$k_{s}$
that varies with streamwise distance
$x$
. Interest is centred on flows for which all terms of the von Kármán integral relation, including the ratio of outer velocity to friction velocity
$U_{\infty }^{+}\equiv U_{\infty }/u_{\unicode[STIX]{x1D70F}}$
, are streamwise constant. For
$Re_{x}$
assumed large, use is made of a simple log-wake model of the local turbulent mean-velocity profile that contains a standard mean-velocity correction for the asymptotic fully rough regime and with assumed constant parameter values. It is then shown that, for a general power-law external velocity variation
$U_{\infty }\sim x^{m}$
, all measures of the boundary-layer thickness must be proportional to
$x$
and that the surface sand-grain roughness scale variation must be the linear form
$k_{s}(x)=\unicode[STIX]{x1D6FC}x$
, where
$x$
is the distance from the boundary layer of zero thickness and
$\unicode[STIX]{x1D6FC}$
is a dimensionless constant. This is shown to give a two-parameter
$(m,\unicode[STIX]{x1D6FC})$
family of solutions, for which
$U_{\infty }^{+}$
(or equivalently
$C_{f}$
) and boundary-layer thicknesses can be simply calculated. These correspond to perfectly self-similar boundary-layer growth in the streamwise direction with similarity variable
$z/(\unicode[STIX]{x1D6FC}x)$
, where
$z$
is the wall-normal coordinate. Results from this model over a range of
$\unicode[STIX]{x1D6FC}$
are discussed for several cases, including the zero-pressure-gradient (
$m=0$
) and sink-flow (
$m=-1$
) boundary layers. Trends observed in the model are supported by wall-modelled large-eddy simulation of the zero-pressure-gradient case for
$Re_{x}$
in the range
$10^{8}{-}10^{10}$
and for four values of
$\unicode[STIX]{x1D6FC}$
. Linear streamwise growth of the displacement, momentum and nominal boundary-layer thicknesses is confirmed, while, for each
$\unicode[STIX]{x1D6FC}$
, the mean-velocity profiles and streamwise turbulent variances are found to collapse reasonably well onto
$z/(\unicode[STIX]{x1D6FC}x)$
. For given
$\unicode[STIX]{x1D6FC}$
, calculations of
$U_{\infty }^{+}$
obtained from large-eddy simulations are streamwise constant and independent of
$Re_{x}$
when this is large. The present results suggest that, in the sense that
$U_{\infty }^{+}(\unicode[STIX]{x1D6FC},m)$
is constant, these flows can be interpreted as the fully rough limit for boundary layers in the presence of small-scale linear roughness.