Linearized Reynolds-averaged Navier-Stokes model for the prediction of secondary flows on heterogeneous surfaces

A rapid predictive tool based on the linearised Reynolds-averaged Navier-Stokes equations is proposed in this thesis to investigate secondary currents generated by streamwise-independent surface roughness heterogeneity 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 roughness heterogeneity, are derived by assuming that surface modulations are shallow. Since the equations are linear, the superposition principle holds and the flow response is obtained by combining appropriately the elementary responses over a sinusoidal modulation of the surface properties, such as the modulation amplitude or the sand-grain roughness height, at multiple spanwise length scales. The tool permits a rapid exploration of large parameter spaces characterising the surface topographies and topologies examined in the literature. Channels with both ridge-type and strip-type roughness heterogeneity are considered. Regarding the topographical modulation and for {undulated} 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. The generation of the tertiary structures strongly depends on the shape of the ridges. This aspect is described for trapezoidal ridges that combine the main characteristics of triangular and rectangular ridges. The position of the secondary vortices and how these latter structures influence the strength of the tertiary flows are finally outlined. Similar behaviour has been observed for the strip-type roughness where the strength of the secondary flows is a function of the width of the high- and low-roughness regions. Finally, the combination of both types of roughness heterogeneity has been discussed in order to better understand which roughness type is dominant.