The Minimum Drag Profile in Laminar Flow: A Numerical Way

Abstract

It would be of interest to engineers and scientists to know the shape of the body of a given volume that will have minimum drag when moving through a viscous fluid at constant speed. It would be extremely useful if one could devise an evolution procedure that can evolve the minimum drag body in a logical and an orderly manner. Such a procedure was suggested by Pironneau for laminar flow wherein optimality conditions derived using optimal control theory were used in a non-linear gradient algorithm. The literature cites an attempt of the procedure at high Reynolds number where for each iteration in the evolution process, the flow field required an outer and an inner solution and the calculation of the gradient optimality condition required the solution of the co-state equation, a type of boundary layer equation. This paper addresses the direct simulation of the governing elliptic partial differential equations, viz., the Navier-Stokes and the co-state equations. Even though the latter has no simple mechanical interpretation, capitalizing on its resemblance to the former, this paper shows how the solution to the co-state equation could be obtained by simply adapting an existing Navier-Stokes code. Solution of the flow field and the calculation of the necessary criteria required in the evolution process are also discussed. The novelty of this direct approach is to make the evolution process more general, arbitrary and less complex. The profile evolution is demonstrated for flows at different Reynolds numbers.