Journal of Mathematical Fluid Mechanics

, Volume 12, Issue 2, pp 243–265 | Cite as

The Optimal Shape of Riblets in the Viscous Sublayer



Our aim is to find the optimal shape of periodically distributed microstructures on surfaces of swimming bodies in order to reduce their drag. The model describes the flow in the viscous sublayer of the boundary layer of a turbulent flow. The microscopic optimization problem is reduced applying homogenization. In the reduced so-called macroscopic optimization problem we minimize the Navier constant subject to the boundary layer equations which are solved in a very small part of the original domain. Under the assumptions that the microstructures can be represented as smooth functions the sensitivity can be determined analytically. The optimization problem is then solved by a sensitivity based method (steepest descent with optimal step size) and the state equations are solved in each iteration with an external software. Our reduced model is validated by comparing the results from the homogenized model with those obtained by simulating the whole rough channel. An improved shape is found and a drag reduction up to 10% can be shown.

Mathematics Subject Classification (2000).

Primary 35B27 secondary 76D10, 76D55, 49Q12 


Drag minimization homogenization shape optimization Navier–Stokes equations Couette flow boundary layer sensitivity analysis 


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Copyright information

© Birkhäuser Verlag, Basel 2008

Authors and Affiliations

  1. 1.Interdisciplinary Center for Scientific Computing (IWR)University HeidelbergHeidelbergGermany

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