  A rapid predictive tool based on the linearised Reynolds-averaged
Navier-Stokes equations is proposed in this work to investigate secondary
currents generated by streamwise-independent surface topography modulations in
turbulent channel flow. The tool is derived by coupling the Reynolds-averaged
momentum equation to the Spalart-Allmaras transport equation for the turbulent
eddy viscosity, using a nonlinear constitutive relation for the Reynolds
stresses to capture correctly secondary motions. Linearised equations,
describing the steady flow response to arbitrary surface modulations, are
derived by assuming that surface modulations are shallow. Since the equations
are linear, the superposition principle holds and the flow response induced by
an arbitrary modulation can be obtained by combining appropriately the
elementary responses obtained over sinusoidal modulations at multiple spanwise
length scales. The tool permits a rapid exploration of large parameter spaces
characterising structured surface topographies previously examined in the
literature. Here, channels with sinusoidal walls and with longitudinal
rectangular ridges are considered. For sinusoidal walls, a large response is
observed at two spanwise wavelengths scaling in inner and outer units
respectively, mirroring the amplification mechanisms in turbulent shear flows
observed from transient growth analysis. For longitudinal rectangular ridges,
the model suggests that the analysis of the response and the interpretation of
the topology of secondary structures is facilitated when the ridge width and
the gap between ridges are used instead of other combinations proposed in the
literature.
