Third‐order sensitivity analysis for robust aerodynamic design using continuous adjoint

Robust design problems in aerodynamics are associated with the design variables, which control the shape of an aerodynamic body, and also with the so‐called environmental variables, which account for uncertainties. In this kind of problems, the set of design variables, which leads to optimal performance, taking into account possible variations in the environmental variables, is sought. One of the possible ways to solve this problem is by means of the second‐order second‐moment approach, which requires first‐order and second‐order derivatives of the objective function with respect to the environmental variables. Should the minimization problem be solved using a gradient‐based method, algorithms for the computation of up to third‐order sensitivity derivatives (twice with respect to the environmental variables and once with respect to the shape controlling design variables) must be devised. In this paper, a combination of the continuous adjoint variable method and direct differentiation to compute the third‐order sensitivities is proposed. This is shown to be the most efficient among all alternative methods provided that the environmental variables are much less than the design ones. Apart from presenting the method formulation, this paper focuses on the assessment of the so‐computed up‐to third‐order mixed derivatives through comparison with costly finite‐difference schemes. To this end, the robust design of a two‐dimensional duct is performed. Then, using the validated method, the robust design of a two‐dimensional cascade airfoil is demonstrated. Although both cases are handled as inverse design problems, the method can be extended to other objective functions or three‐dimensional problems in a straightforward manner. Copyright © 2012 John Wiley & Sons, Ltd.     

Evaluation of RBFs for volume data interpolation in CFD

A greedy method for choosing an optimum reduced set of control points is integrated with RBF interpolation and evaluated for the purpose of interpolating large‐volume data sets in CFD. Given a function defined at a set of points, the greedy method selects a small subset of these points that is sufficient to keep the interpolation error at all the remaining points below a chosen bound. This is equivalent to a type of data compression and would have useful storage, post‐processing, and computational applications in CFD. To test the method in terms of both the point selection scheme and the suitability of reduced control point volume interpolation, a trial application of the interpolation to velocity fields in CFD volume meshes is considered. To optimise the point selection process, and attempt to be able to capture multiple length scales, a variable support radius formulation has also been included. Structured and unstructured mesh cases are considered for aerofoils, a wing case and a wing‐body case. For smooth volume functions, the method is shown to work well, producing accurate velocity interpolations using a very small number of the cells in the mesh. For general complex fields including large gradients, the method is still shown to be effective, although large gradients require more interpolation points to be used.Copyright © 2013 John Wiley & Sons, Ltd.     

On the uncertainty of CFD in sail aerodynamics

A verification and validation procedure for yacht sail aerodynamics is presented. Guidelines and an example of application are provided. The grid uncertainty for the aerodynamic lift, drag and pressure distributions for the sails is computed. The pressures are validated against experimental measurements, showing that the validation procedure may allow the identification of modelling errors. Lift, drag and L₂ norm of the pressures were computed with uncertainties of the order of 1%. Convergence uncertainty and round‐off uncertainty are several orders of magnitude smaller than the grid uncertainty. The uncertainty due to the dimension of the computational domain is computed for a flat plate at incidence and is found to be significant compared with the other uncertainties. Finally, it is shown how the probability that the ranking between different geometries is correct can be estimated knowing the uncertainty in the computation of the value used to rank. Copyright © 2013 John Wiley & Sons, Ltd.     

A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind‐biased discretization of the Euler equations, written in arbitrary Lagrangian–Eulerian form and integrated in time by means of a dual‐time steeping technique. In order to exploit the meshless potential of the method, a domain deformation approach based on the spring network analogy is implemented, and h‐adaptivity is also employed in the computations. Typical movable boundary problems in transonic flow regime are solved to assess the performance of the proposed technique. In addition, an application to a fluid–structure interaction problem involving static aeroelasticity illustrates the capability of the method to deal with practical engineering analyses. The computational cost and multi‐core performance of the proposed technique is also discussed through the examples provided. Copyright © 2013 John Wiley & Sons, Ltd.     

PIV measurement of the flow past a generic car body with wheels at LES applicable Reynolds number

Experiments by using 2D–2C Particle Image Velocimetry (PIV) were carried out and reported concerning the flow field past a generic car body (modified Ahmed body) which is equipped with wheels and wheel-arches. The Reynolds number was chosen to not exceed 2E+5 based on the height of the Ahmed body which makes it possible to investigate the same configuration by means of Large Eddy Simulation (LES). The wheels were rotating but the ground was stationary. The wheel-ground contact was realized by means of small rectangular openings below the wheels in the ground plane in which the wheels were immersed. The transition contour of the immersed wheels and the ground, as well as the rectangular openings below the wheels were properly sealed to prevent parasite flow and to provide well defined boundary conditions for an upcoming LES investigation.The flow field was measured in several planes with normal vectors pointing towards the directions normal to the free stream. Statistical characteristics of the flow are provided and discussed.     

Adaptive mesh refinement and coarsening for aerodynamic flow simulations

A new mesh refinement technique for unstructured grids is discussed. The new technique presents the important advantage of maintaining the original grid skewness, thanks to the capability of handling hanging nodes. The paper also presents an interpretation of MacCormack's method in an unstructured grid context. Results for a transonic convergent–divergent nozzle, for a convergent nozzle with a supersonic entrance and for transonic flow over a NACA 0012 airfoil are presented and discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

The lift computation for an oscillating flat plate in incompressible potential flow

The initial aim of this work was the estimation of the lift acting on a flat plate performing small oscillations in a plane uniform stream by means of a simplified model based on one or at the most two lumped vortices, and the assessment of its results by comparison to those that were exact. The model was found to work well up to a reduced frequency of about 1 or 2, above which the results diverged from those that were correct. In order to improve the model, its behaviour at very high frequencies was then investigated, discovering: (i) that if the number of lumped vortices is greater than one the possibility to impose all boundary conditions is subject to certain geometrical constraints; (ii) that the asymptotical behaviour is not the right one. A straightforward extension of this conclusion to the exact case of a continuous, vorticity distribution simulating the motion of the plate and to the classical equation describing it leads apparently to an incorrect result. The reason for the discrepancy is found in the singularity displayed by the integral equation which cannot be reproduced by the discrete model. It this therefore concluded that the latter can be trusted at low and middle frequencies but its extension to higher ones is fundamentally uncorrect.

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Sommario Lo scopo iniziale di questo lavoro era il calcolo approssimato della portanza agente su una lamina piana soggetta a piccole oscillazioni in una corrente bidimensionale uniforme per mezzo di un modello semplificato basato su uno o al più due vortici concentrati, e il confronto dei risultati con quelli esatti. Il modello risulta funzionare bene per frequenze ridotte inferiori ad 1 o 2, sopra le quali, tuttavia, i risultati si allontanano da quelli corretti. Per migliorarlo si è allora studiato il suo comportamento alle frequenze molto alte, scoprendo che: (i) la possibilità di imporre tutte le condizioni al contorno quando il numero dei vortici concentrati è superiore a uno è soggetta a certe limitazioni sulla configurazione geometrica; (ii) che il comportamento asindotico non è quello corretto. Un'estensione automatica di questa conclusione al caso esatto in cui il moto della lamina è simulato da una distribuzione continua di vorticità e alla classica equazione che lo descrive sembra condurre ad un risultato errato. La ragione di questa discrepanza viene individuata nella singolarità contenuta nell'equazione integrale, che non può essere riprodotta dal modello discreto. Se ne conclude perciò che esso è utilizzabile alle basse e medie frequenze ma che una sua estensione alle alte è fondamentalmente errata.

     

Remarks on the numerical solution of the adjoint quasi‐one‐dimensional Euler equations

We examine the numerical solution of the adjoint quasi‐one‐dimensional Euler equations with a central‐difference finite volume scheme with Jameson‐Schmidt‐Turkel (JST) dissipation, for both the continuous and discrete approaches. First, the complete formulations and discretization of the quasi‐one‐dimensional Euler equations and the continuous adjoint equation and its counterpart, the discrete adjoint equation, are reviewed. The differences between the continuous and discrete boundary conditions are also explored. Second, numerical testing is carried out on a symmetric converging–diverging duct under subsonic flow conditions. This analysis reveals that the discrete adjoint scheme, while being manifestly less accurate than the continuous approach, gives nevertheless more accurate flow sensitivities. Copyright © 2011 John Wiley & Sons, Ltd.     

A rod-airfoil experiment as a benchmark for broadband noise modeling

A low Mach number rod-airfoil experiment is shown to be a good benchmark for numerical and theoretical broadband noise modeling. The benchmarking approach is applied to a sound computation from a 2D unsteady-Reynolds-averaged Navier–Stokes (U-RANS) flow field, where 3D effects are partially compensated for by a spanwise statistical model and by a 3D large eddy simulation. The experiment was conducted in the large anechoic wind tunnel of the Ecole Centrale de Lyon. Measurements taken included particle image velocity (PIV) around the airfoil, single hot wire, wall pressure coherence, and far field pressure. These measurements highlight the strong 3D effects responsible for spectral broadening around the rod vortex shedding frequency in the subcritical regime, and the dominance of the noise generated around the airfoil leading edge. The benchmarking approach is illustrated by two examples: