Modeling turbulent flow over sharp riblets via change of coordinates*

Numerical and experimental studies have revealed the drag reducing trends of small-sized riblets in wall-bounded flows. Recent efforts in reduced-order modeling have demonstrated the efficacy of the eddy-viscosity-enhanced linearized Navier-Stokes equations in capturing such trends. In these works, the immersed boundary method was used to capture the effect of riblets on the flow momentum. Such methods, which are complemented by constructive turbulence modeling procedures exhibit deficiencies in capturing intricate flow mechanisms that gain strength above large and sharp riblet structures. We promote the use of an alternative method in accounting for riblet-shaped boundaries that involves a transformation of the coordinate system to the generalized curvilinear coordinates. While the change of coordinates reflects the spatial periodicity of the boundary conditions onto the differential operators and thereby convolutes the structure of the linearized dynamics, it allows for an accurate representation without the need for a stretched mesh or excessive parametric tuning of the immersed boundary. We further demonstrate how perturbation analysis in the height of riblets yields a sequence of computationally efficient equations that can be solved to capture the flow physics. We use this simulation-free approach to predict drag reduction and energy suppression due to semi-circular riblets, as well as the emergence of flow mechanisms such as Kelvin-Helmholtz instability for larger riblets.