Optimum plane diffusers in laminar flow

Abstract

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.

Effect of shroud on the performance of horizontal axis hydrokinetic turbines

“…The maximum half angle for a conical diffuser in order to be able to maintain a steady attached flow throughout the inner surface is 7° reported in several studies (Azad, 1996;Ghose and Kline, 1978;Savkar, 1980;Strawn and Kline, 1983). Flow in such a diffuser has enough energy to overcome the viscous dissipation and adverse pressure gradient and hence stay attached to the diffuser wall (Cabuk and Modi, 1992;Chang, 1976;Gad-el-Hak and Bushnell, 1991;Lai et al, 1989;Madsen et al, 2000). For a diffuser augmented wind turbine on the other hand, a compact diffuser with a large included angle is preferred.…”

Turbine Airfoil Optimization Using Quasi‐3D Analysis Codes

“…This is a direct design approach in which the sensitivities of the objective function to the airfoil shape are computed by solving an adjoint formulation of the flow equations where the sensitivities can be obtained in a single solution of the flow equations. It has been used for applications such as wing design [14] shape optimization of a planar diffuser [15] and aerodynamic optimization [16]. This approach requires changes to the Navier-Stokes solver, and each time a change is made to the optimization, problem the formulation needs to be modified.…”

Optimal Shape Design of Compact Heat Exchangers Based on Adjoint Analysis of Momentum and Heat Transfer

“…Over the decades, remarkable advances in computational power have made it more practical to employ such variational methods, which need vast amount of computational resources. Various applications of the optimal shape design method to fluidic devices have been reported [8][9][10][11]. Although the application of such methods to heat-transfer related problems have not been reported so far, these are expected to provide useful information also in designing high-performance heat exchangers.…”