Optimum plane diffusers in laminar flow

The problem of determining the profile of a plane diffuser (of given upstream width and length) that provides the maximum static pressure rise is solved. Two-dimensional, incompressible, laminar flow governed by the steady-state Navier-Stokes equations is assumed through the diffuser. Recent advances in computational resources and algorithms have made it possible to solve the ‘direct’ problem of determining such a flow through a body of known geometry. 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 diffuser profile that leads to a higher pressure rise. The direct as well as the adjoint set of partial differential equations are obtained for Dirichlet-type inflow and outflow conditions. Repeatedly modifying the diffuser geometry with each solution to these two sets of equations with the above boundary conditions would in principle lead to a diffuser with the maximum static pressure rise, also called the optimum diffuser. The optimality condition, that the shear stress all along the wall must vanish for the optimum diffuser, is also recovered from the analysis. It is postulated that the adjoint set of equations continues to hold even if the computationally inconvenient Dirichlet-type outflow boundary condition is replaced by Neumann-type conditions. It is shown that numerical solutions obtained in this fashion do satisfy the optimality condition.