Linear models of strip-type roughness

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.