Modelling spanwise heterogeneous roughness through a parametric forcing approach

Abstract

Inhomogeneous rough surfaces in which strips of roughness alternate with smooth-wall strips are known to generate large-scale secondary motions. Those secondary motions are strongest if the strip width is of the order of the half-channel height and they generate a spatial wall shear stress distribution whose mean value can significantly exceed the area-averaged mean value of a homogeneously smooth and rough surface. In the present paper it is shown that a parametric forcing approach (Busse & Sandham, J. Fluid Mech., vol. 712, 2012, pp. 169–202; Forooghi et al., Intl J. Heat Fluid Flow, vol. 71, 2018, pp. 200–209), calibrated with data from turbulent channel flows over homogeneous roughness, can capture the topological features of the secondary motion over protruding and recessed roughness strips (Stroh et al., J. Fluid Mech., vol. 885, 2020, R5). However, the results suggest that the parametric forcing approach roughness model induces a slightly larger wall offset when applied to the present heterogeneous rough-wall conditions. Contrary to roughness-resolving simulations, where a significantly higher resolution is required to capture roughness geometry, the parametric forcing approach can be applied with usual smooth-wall direct numerical simulation resolution resulting in less computationally expensive simulations for the study of localized roughness effects. Such roughness model simulations are employed to systematically investigate the effect of the relative roughness protrusion on the physical mechanism of secondary flow formation and the related drag increase. It is found that strong secondary motions present over spanwise heterogeneous roughness with geometrical height difference generally lead to a drag increase. However, the physical mechanism guiding the secondary flow formation, and the resulting secondary flow topology, is different for protruding roughness strips and recessed roughness strips separated by protruding smooth surface strips.

Simulation of turbulent flow over roughness strips

“…Irrespective of the reference definition, i.e. as arithmetic mean or according to (5.1), the data of Schäfer et al (2022) indicate a much higher contribution of the secondary flow to drag increase than the present data. At the same time the isovel curvature due to the secondary flow is similar for the present data and non-protruding roughness strips of Schäfer et al (2022) (see their figure 5(c,j,k) or the Appendix).…”

“…as arithmetic mean or according to (5.1), the data of Schäfer et al (2022) indicate a much higher contribution of the secondary flow to drag increase than the present data. At the same time the isovel curvature due to the secondary flow is similar for the present data and non-protruding roughness strips of Schäfer et al (2022) (see their figure 5(c,j,k) or the Appendix). We suspect that this difference is related to drag effects on the protruding lateral sides of the smooth surface patches that were introduced in the IBM resolved and PFA modelled roughness DNS (Stroh et al 2020;Schäfer et al 2022).…”

“…In particular, the qualitative agreement is achieved for the case of non-protruding roughness strips (cf. figure 8c,k in Schäfer et al (2022) vs figure 11b). Note that the spanwise wavelengths are different: Schäfer et al (2022) present results for s/δ = 0.5.…”

“…However, if this scale separation does not hold, the influence of an effective channel height has to be taken into account as discussed in e.g. Schäfer et al (2022).…”

Linearised Reynolds-averaged predictions of secondary currents in turbulent channels with topographic heterogeneity

“…(2020 b ) and Schäfer et al. (2022). However, modelling the physics of the flow–surface interaction within a linear framework is less straightforward than modelling a modulation of the surface height, as in the present case.…”