The authors investigate two-dimensional airfoil optimization problems under a range of operating conditions. The geometry of the airfoil is parameterized through B-splines. The B-spline control points and the angle of attack are the design variables. The compressible Navier- Stokes equations are solved with a Newton-Krylov method in which the linear system arising at each Newton iteration is solved using the generalized minimal residual method (GMRES) preconditioned with an incomplete lower-upper factorization with limited fill. The Spalart-Allmaras turbulence model is used to compute the eddy viscosity. The gradient is calculated using the discrete-adjoint method; solution of the adjoint equation is accomplished through the same preconditioned Krylov method. Geometric constraints are added to the objective function as penalty terms. The design variables are updated using a quasi-Newton optimizer in which an estimate of the inverse Hessian based on the BFGS (Broyden-Fanno-Goldfarb-Shannon) rank-two update formula is used to compute a search direction. Each time a new shape is calculated, the initial grid is perturbed using a simple algebraic technique. An automated procedure for choosing sampling points within the operating range and their weights to obtain the desired performance over the range Mach numbers. A constant drag coefficient is assumed to be desired over the Mach number range. A number of sampling points and weights is chosen within the Mach interval. The objective function is given by the weighted sum of the objective functions at each sampling mach number. The weights are updated automatically using a formular which depends on the number of samples and the drag coefficient at each sample Mach number. When the evaluation of the drag coefficient over the complete range of Mach numbers reveals a significant local maximum an additional sampling point is added with an initial weight of zero. Tradeoffs and their implications for the formulation of multipoint problems are also presented and discussed, for example, the optimization of performance at off-design conditions. Whenever an airfoil is designed with a region of natural laminar flow, one must consider the possibility that transition will occur earlier than expected for some reason, such as frost, roughness, damage, etc. On the other hand, when an analysis is performed with the assumption of fully turbulent flow of the airfoil optimized under the assumption of free transition, the endurance factor (which is chosen as target function here) drops considerably. Optimizing the design under fully turbulent conditions leads to a higher endurance factor. The tradeoffs between good performance under fully turbulent conditions and free transition can be assessed using a Pareto front, calculated from the weighted sum of the endurance factors for fully turbulent and free-transition condition.
Aerodynamic optimization under a range of operating conditions
AIAA Journal (online) ; 44 , 11 ; 2787-2792
2006
6 Seiten, 6 Bilder, 3 Tabellen, 12 Quellen
Aufsatz (Zeitschrift)
Englisch
FLUID DYNAMICS - Aerodynamic Optimization Under a Range of Operating Conditions
Online Contents | 2006
|AIAA 2006-1053 On Aerodynamic Optimization Under a Range of Operating Conditions
British Library Conference Proceedings | 2006
|Engineering Index Backfile | 1964
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