The turbulent eddy viscosity is obtained from the computed mean flowfield by integration of the turbulent field equations. An implicit, factorized, upwind-biased numerical scheme is used for the integration of the compressible, Reynolds-averaged Navier-Stokes equations. The ability of one- and two- equation turbulence models to predict unsteady separated flows over airfoils is evaluated. Mean flow field and turbulence field results are presented for an NACA 0012 airfoil at zero and nonzero incidence angles of Reynolds number up to one million and low subsonic Mach numbers.Ĭomputation of oscillating airfoil flows with one- and two- equation turbulence models The transition turbulence model is based upon the turbulence kinetic energy equation and predicts regions of laminar, transitional, and turbulent flow. A nonorthogonal body-fitted coordinate system is used which has maximum resolution near the airfoil surface and in the region of the airfoil leading edge. The equations are solved by a consistently split linearized block implicit scheme. Extensive comparisons are made with available data, particularly for a Mach number of 1, and with existing solutions.Ī compressible solution of the Navier-Stokes equations for turbulent flow about an airfoilĪ compressible time dependent solution of the Navier-Stokes equations including a transition turbulence model is obtained for the isolated airfoil flow field problem. At a Mach number of 1 general expressions are given for the pressure distribution on an airfoil of specified geometry and for the shape of an airfoil having a prescribed pressure distribution. Results are obtained in closed analytic form for a large and significant class of nonlifting airfoils. Solutions are found for two-dimensional flows at a Mach number of 1 and for purely subsonic and purely supersonic flows. Thin airfoil theory based on approximate solution of the transonic flow equationĪ method is presented for the approximate solution of the nonlinear equations transonic flow theory. Results are presented for both the inverse problem and drag minimization problem. Here the control law serves to provide computationally inexpensive gradient information to a standard numerical optimization method. Therefore, we have developed methods which can address airfoil design through either an analytic mapping or an arbitrary grid perturbation method applied to a finite volume discretization of the Euler equations. The goal of our present work is to extend the development to treat the Euler equations in two-dimensions by procedures that can readily be generalized to treat complex shapes in three-dimensions. In our previous work it was shown that control theory could be employed to devise effective optimization procedures for two-dimensional profiles by using the potential flow equation with either a conformal mapping or a general coordinate system. This paper describes the implementation of optimization techniques based on control theory for airfoil design. Control theory based airfoil design using the Euler equations
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