The hydraulically smooth limit of flow over surfaces with spanwise heterogeneity

Abstract

Wall slip is conventionally obtained through riblets or grooves. As riblets are macroscopic objects, it is usually not practical to discuss slip patterns at the hydraulically smooth limit. Per Nikuradse, the smooth limit is when the characteristic length of the surface pattern is comparable to the viscous length scale. Recent studies show the possibility of slip at the microscopic scale, making slip patterns at the hydraulically smooth limit relevant. In this study, we leverage a high-fidelity pseudo-spectral code and study flow over surfaces featuring alternating strips of slip and no-slip wall conditions. The wavelength of the surface pattern, denoted as l, varies from 1.5 times the half channel height to 2 viscous units (plus units), eventually approaching the anticipated hydraulically smooth limit. The presence of surface slip gives rise to a slip velocity at the wall, denoted as Us, which contributes to drag reduction. The surface spanwise heterogeneity leads to secondary flows and intensifies turbulent mixing, consequently leading to drag increase. This drag increase effect can be parameterized using the “roughness function” ΔU+. The sum of Us+ and −ΔU+ determines whether the surface increases or reduces drag. Here, the superscript ＋ denotes normalization by the wall units. In most cases, the slip velocity at the wall Us+ predominates over −ΔU+, resulting in drag reduction. However, when l is a few viscous units, the roughness function ΔU+ does not vanish and overwhelms the slip velocity, giving rise to a net drag increase. Considering that the wall is a mixture of slip and no-slip conditions, this drag increase at the anticipated hydraulically smooth limit is unexpected. To gain an insight into the mechanism responsible for this drag increase, we derive a Navier–Stokes-based decomposition of the roughness function. Here, we generalize the definition of the roughness function such that it is a function of the wall-normal coordinate, thereby overcoming the difficulty of measuring the roughness function when the log region is narrow and hard to define. Analysis shows that when l is a few plus units, secondary flows contribute to a slightly positive ΔU+, while turbulent and viscous contributions, by and large, cancel out, ultimately leading to an overall drag increase at the anticipated hydraulically smooth limit. The evidence in the paper suggests that the hydraulically smooth limit does not exist for certain surfaces.