Energy amplification in channel flow over riblets

A number of methods of reducing the drag over airfoils have been investigated. This paper considers the use of riblets, which are structures that run parallel to one another that are positioned longitudinally to the flow. It has been shown experimentally that when the shape, spacing and height of the riblets are optimized, the drag coefficient can be reduced by up to 10%. These results have been confirmed by direct numerical simulation studies. However, the mechanism of drag reduction is unclear. 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. Spectral methods are used to discretize these equations, leading to a large-scale state space model. The energy amplification is calculated for the streamwise constant component of the flow from the controllability grammian and the associated Lyapunov equation is solved efficiently by exploiting the structure of the problem. Results of preliminary investigations show that introducing riblets into the flow reduces the amplification associated with the largest mode.