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.
SUMMARYThe total drag force on the surface of a body, which is the sum of the form drag and the skin friction drag in a 2D domain, is numerically evaluated by integrating the energy dissipation rate in the whole domain for an incompressible Stokes fluid. The finite element method is used to calculate both the energy dissipation rate in the whole domain as well as the drag on the boundary of the body. The evaluation of the drag and the energy dissipation rate are post-processing operations which are carried out after the velocity field and the pressure field for the flow over a particular profile have been obtained. The results obtained for the flow over three different but constant area profiles -a circle, an ellipse and a cross-section of a prolate spheroid-with uniform inlet velocity are presented and it is shown that the total drag force times the velocity is equal to the total energy dissipation rate in the entire finite flow domain. Hence, by calculating the energy dissipation rate in the domain with unit velocity specified at the far-field boundary enclosing the domain, the drag force on the boundary of the body can be obtained.
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