Design of minimum drag bodies in incompressible laminar flow

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 from examination of the characteristics of the adjoint set of equations in the absence of the viscous terms. These modified boundary conditions at far-field are used in actual numerical solutions to the direct and the adjoint equations instead of the Dirichlet type conditions required by analysis. A series of two-dimensional minimum drag bodies are obtained for a range of Reynolds numbers varying from 20 to 105. A geometrical constraint of fixed sectional area was imposed in these calculations. The optimization process monotonically reduces total (skin friction plus pressure) drag along with an associated reduction in flow separation. It is shown that a minimum drag body obtained for near transition albeit laminar flow Reynolds number demonstrates low drag characteristics even under turbulent flow conditions.