Evaluation of Nearby Flows by a Shape Sensitivity Equation Method

Abstract

This paper studies the use of the Continuous Sensitivity Equation Method (CSEM) for the Navier-Stokes equations to compute nearby solutions, in the particular case of shape parameters. The o w and sensitivity elds are solved using an adaptive nite-elemen t method. A new approach is presented to extract accurate o w derivatives at the boundary, as they appear in the sensitivity boundary conditions for shape parameters. High order Taylor series expansions are used on layered patches in conjunction with a constrained least- squares procedure to evaluate accurate rst and second derivatives of the o w variables at the boundary. The proposed methodology is rst veried on a problem with a closed form solution obtained by the Method of Manufactured Solutions. The methodology is then applied to airfoil o ws, the CSEM yielding fast o w evaluation for such shape parameters as airfoil thickness, angle of attack and camber. tiate approach (often called Discrete Sensitivity Equation Method), the discrete form of the o w equations are dieren tiated and the total derivative of the o w discretization with respect to the design parameters is calculated. In the dieren tiate-then-approximate approach (known as the Continuous Sensitivity Equation Method CSEM), partial dieren tial equations for the o w sensitivities are obtained by implicit dieren tiation of the equations governing the o w. They are then approximated numerically. The CSEM is preferred for the present study, because it oers several advantages over the discrete sensitivity approach. In particular, since dieren tiation occurs before any discretization, the delicate com- putation of mesh sensitivities and all of the overhead associated with them are avoided. Consequently, the CSEM requires less memory and is computationaly less expensive than automatic dieren tiation, as shown by Borggaard and Verma. 3 Moreover, the CSEM is a natural approach when using adaptive methods: since the topology of the mesh changes with adaptation, mesh derivatives do not exist, making the discrete sen- sitivity method ill-suited. Another advantage is that there is no requirement to use the same algorithm to approximate the CSE and the original PDE model. Thus, special algorithms can be constructed to take advantage of the linear structure of the CSE. However, the main dicult y with the CSEM arises when one deals with shape parameters. In this particular case, o w gradients of the PDE solution are required as source terms in the CSE and they also appear as coecien ts in the boundary conditions for the CSE. Flow gradients in the interior of the computational domain can be computed relatively easily and accurately by a local projection technique (which is already used for error estimation). However, the accuracy of such reconstructed derivatives degrades near the boundary. 4 This induces errors in the boundary conditions that results in poor solutions for the sensitivity elds. The current study presents a new approach to obtain accurate boundary conditions for the CSE. It uses high order Taylor series expansions in conjunction with a constrained least-squares procedure. The