  Prandtl's secondary flows of the second kind generated by laterally-varying
roughness are studied in this work using the linearised Reynolds-Averaged
Navier-Stokes approach recently proposed. The momentum equations are coupled to
the SA transport model for the turbulent viscosity and the surface roughness is
captured by adapting established strategies for homogeneous rough walls to
spanwise inhomogeneous surfaces. Linearisation of the governing equations and
of the roughness model yields a linear framework that elucidates the linear
mechanisms that play a role in the generation of secondary flows. In addition,
the framework allows exploring the parameter space associated with
heterogeneous rough surfaces. Channel flow is considered, with high and low
roughness strips arranged symmetrically on the two channel walls. The strip
width is varied systematically to ascertain the role played by the length scale
characterising the heterogeneity on the size and intensity of secondary flows.
It is found that linear mechanisms, i.e. whereby secondary flows are
interpreted as the output response of the turbulent mean flow subjected to a
forcing localised at the wall, may be sufficient to explain such a role. In
fact the model predicts that secondary flows are most intense when the strip
width is about 0.7 times the half-channel height, in excellent agreement with
available data. Further, a unified framework to analyse combinations of
heterogeneous roughness properties and laterally-varying topographies, common
in engineering applications, is discussed. This framework yields two separate
secondary-flow inducing source mechanisms, i.e. the lateral variation of the
virtual origin from which the turbulent structure is allowed to develop and the
lateral variation of the streamwise velocity slip, capturing the
acceleration/deceleration perceived by the bulk flow over troughs and crests of
the topography.
