A Fixed Point Algorithm for the Inverse Solution of Fluid Flow Equations

The development of several fluid mechanics applications lead to inverse problems. Given a required distribution of flow variables one has to find the corresponding geometry which provides such distribution. Since the flow governing equations do not allow explicit inversion, iterative methods are used. A linear auxiliary equation, which is a simplified model of the flow governing equations, can be used to develop a fixed point iterative method. Such equation is used to compute the geometrical correction required to minimize the difference between the required and actual flow variable distributions. The auxiliary inverse problem is coupled to a flow solver (from potential flow to Navier-Stokes, the method is solver independent) to iterate the correction until convergence. An auxiliary equation method, the Modified Garabedian McFadden, is analyzed. It involves a certain number of arbitrary parameters whose choice affects the rate of convergence. The present work describes how to find adequate parameters for different families of airfoils in transonic flow. A 2D Euler/Navier-Stokes flow solver will be used as an analysis tool and a series of studies, which demonstrate the accuracy and robustness of the technique, are presented. NOMENCLATURE A, B, C, D coefficients of the auxiliary equation αy changes in vertical coordinate Q magnitude of velocity q state vector of conserved properties E, F inviscid fluxes R, S viscous fluxes Re Reynolds Number ± fluid density u , v cartesian components of velocity e total energy T temperature p pressure Cv specific heat at constant volume • x , • y, ♣x ,♣y metrics terms J Jacobian of the transformation U, V contravarient components of velocity ≥ viscous stress χR , χS viscous dissipation terms αt time step A , B Jacobian matrices of the fluxes ≤ difference operator DI , DE implicit and explicit dissipation ⁄I , ⁄E dissipation coefficients M freestream Mach number C , R residual INTRODUCTION Computational Fluid Dynamics (CFD) has been proven an important analysis tool in a vast amount of applications: from aircraft design to weather prediction. Its ability to handle realistic flow conditions is an enormous advantage when compared to other methods. On the other hand the computational cost associated to the solution of the NavierStokes equations still severely limits its application to design. This issue has received the a lot of attention from researchers over the years, especially in aerospace engineering. Historically, a successful technique in airfoil design is the inverse approach. Based on designer's experience to judge