Design of minimum drag bodies in incompressible laminar flow

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 study, a set of “adjoint” equations are solved which permits the calculation of the direction and relative magnitude of change in the body profile that leads to a lower 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 the product of “direct” shear and the “adjoint” shear is constant everywhere along the body. Even though viscous terms are retained in the direct and the adjoint equations, far field boundary conditions are obtained ...