Riblets in the viscous sublayer : Optimal Shape Design of Microstructures

Abstract

Previous research has established that a smooth surface has not necessarily minimal drag: Many experiments by different laboratories, e.g. NASA and DLR Berlin, indicate that an extra surface layer with tiny grooves aligned in the stream-wise direction can be used to reduce the drag. The aim of this project is to find the optimal shape of such microstructures on surfaces of submerged bodies. We assume that these microstructures remain in the viscous sublayer where the flow equations are the 3D incompressible, steady state Navier-Stokes equations with a Couette in- and outflow determinated through two boundary conditions, the no-slip condition on the lower boundary and the friction condition on the upper one. The objective function of our optimization problem is the tangential drag force, which we want to minimize. Solving this problem is difficult because of the rough boundary, which causes a big amount of data. We apply homogenization theory and replace the rough boundary by a smooth one, where the right boundary conditions have been determined. Furthermore, our optimization problem can be simplified using this approximation and we end up minimizing a scalar size, the Navier constant, which is calculated using the velocity of an auxiliary boundary layer equation. To solve the optimization problem we use sensitivity-based optimization methods. The sensitivity is calculated analytically and we use it to determine the gradient of the cost function with respect to the design variable. A minimum is sought by using the steepest descent algorithm with step size according to Armijo rule. The necessary optimality conditions are derived and a sequence of admissible domains is built which tends to the optimal solution. The state equations are solved numerically using finite elements on unstructured grids and multigrid algorithms. The results obtained with this approach give us a drag reduction of approximately 2-6% relative to the drag of the smooth configuration.