Modelling channel flow over riblets: Calculating the energy amplification

Riblets have been considered as a passive method for drag reduction. Riblets are structures on a surface that run parallel to one another, which are aligned longitudinally to the flow. It has been shown experimentally that when the shape, spacing and height of the riblets are optimized, the drag coefficient over the surface can be reduced by up to 10%. These results have also been confirmed by direct numerical simulation studies. Although the benefits of riblets have been known since the early 1980's, the mechanism of drag reduction is still not fully understood. This paper examines the effect of riblet structures on the amplification of background noise within channel flow between two parallel plates (Poiseuille flow), where riblets are present on the surface of one of the plates. A linearized version of the Navier-Stokes equation about the steady flow is developed and through a coordinate transformation, the boundary conditions associated with the riblets are transferred into the partial differential equations. Previous work has used spectral methods to discretize these equations, leading to a large-scale state space model, and the energy amplification was calculated for the streamwise constant component of the flow from the controllability gramian. However, solving the associated Lyapunov equation can be computationally prohibitive, which limits the density of the discretized grid. This paper shows how the problem can be transformed to decouple the system, so that the gramian can be obtained by solving a set of smaller Lyapunov equations, which has the potential to allow the energy amplification to be calculated for systems with a dense discretization grid.