Optimum design of minimum drag bodies in incompressible laminar flow using a control theory approach

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Abstract
The problem of modifying the shape of a two-dimensional body to reduce its drag while maintaining its section area (volume per unit span) constant is addressed. Two-dimensional, incompressible, laminar flow governed by the steady-state Navier-Stokes equations is assumed about the body. In this paper, a set of “adjoint” equations is obtained, the solution to which permits the calculation of the direction and relative magnitude of change in the body profile that leads to a lower viscous drag. The direct as well as the adjoint set of partial differential equations are obtained for Dirichlet-type far-field conditions. Repeatedly modifying the body shape with each solution to these two sets of equations with the above boundary conditions, would lead to a body with minimum drag, for a specified section area. For such a body it is shown that the product of shear and the “adjoint” shear is constant everywhere along the body. Even though the viscous terms are retained in the direct and the adjoint equations, in order to determine farfield boundary conditions, the characteristics of the adjoint set of equations are examined in the absence of the viscous terms. This is done in order to obtain computationally tractable boundary conditions at far-field instead of the Dirichlet type conditions required by analysis. Numerical solutions to the direct and the adjoint equations are obtained using these modified boundary conditions. Preliminary results obtained using an ellipse as the initial body shape demonstrate that the approach does provide meaningful shape modification information.