Introduction A very efficient formulation for the inverse design of two-dimensional turbomachinery blades in turbulent transonic flow conditions is presented. The proposed optimization technique employs a highly accurate space discretization scheme for the accurate computation of the flow solution and of the objective function. The sensitivity derivatives are evaluated very efficiently by using a discrete adjoint formulation. The use of an auxiliary dissipative flow solver in the adjoint equations, rather than the accurate basic flow solver, allows a robust evaluation of the sensitivity derivatives, even in presence of shocks and of numerical noise. Moreover, if the variable prescribed on the blade surface is the pressure, as usual, a simplified inviscid formulation of the adjoint problem can be used, even in the real case of turbulent flows. The entire procedure is termed progressive optimization, because both the flow computation and the iterative solution of the adjoint problem, based on the addition of an artificial time-dependent term in the adjoint equations, are brought to convergence while approaching the optimum. The idea of the simultaneous convergence of all iterative solutions suggests also the use of progressively finer grids for the progressive computation of the flow field; the cost of the adjoint problem is strongly reduced by solving the adjoint equations always at the coarsest level. The proposed approach has been tested on the inverse design of a two-dimensional turbine blade in turbulent transonic flow conditions, both with fixed stagger angle and with fixed outlet angle of the camberline. The numerical results demonstrate that the methodology is robust and highly efficient, with a converged design optimization produced in no more than the amount of computational work needed to perform from 0.7 to 1.3 converged flow analyses on the finest grid. 'Associate Professor. ^Professor and Department Head, AIAA senior member. Copyright © 2000 by L. A. Catalano and A. Dadone. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Until few years ago, the optimization of the aerodynamic efficiency of turbomachinery bladings has been mostly approached by subsequent improvements based on the experience of the designers, each one of them followed by expensive and timeconsuming wind tunnel tests. For such a reason, the recent development of automatic tools based on the numerical solution of the flow equations and on an optimization process for the shape changes has greatly attracted the interest of turbomachinery designers. Due to the complex flow patterns about turbomachinery bladings, these numerical methods have been first developed and applied to the inverse and the direct design of airfoils; some of them have been later extended to the optimization of turbine and compressor blades. Nevertheless, many of the inverse design methods for turbomachinery bladings currently used are based on the use of a transpiration model', which cannot be extended to the direct design of the blade. For such a reason, the authors find very attractive to extend more general metodologies, already developed and tested for the optimization of airfoils or wings, to the design of turbomachinery bladings. In the application of these methods to transonic flows, typical of high-load bladings, a major difficulty commonly arises: the objective function is generally noisy or non-smooth, so that the convergence of standard optimization techniques may be compromised or even prevented, as shown in Ref. 3 for the case of a transonic nozzle. In such situations a standard finite-difference approach for design sensitivities or an alternative route based on the solution of a set of sensitivity equations' may result in an optimization process hung up at one of the many local minima artificially created by the shock shifting. The adjoint procedure proposed by Jameson et al overcomes this problem by effectively smoothing the objective function, which however may change the result of the optimization process, as outlined in Ref. 7. An alternative smoothing procedure which operates only on the design sensitivities and not on the objective function itself is proposed in Ref. 8, where it is (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. shown to be effective and efficient in designing transonic diffusers and airfoils by using two-dimensional inviscid flow equations. A second, very important challenge is the development of an optimization technique able to converge to the optimum in a computational work corresponding to few flow analyses. Indeed, most existing methodologies are based on a sequence of global iterations, each one of them requiring a computation of the converged flow field, a solution for the sensitivity derivatives and an appropriate update of the design variables. The repeated solution of the flow equations typical of such a serial approach results in a high computational cost of the entire design procedure. The efficiency of the smoothing procedure developed in Ref. 8 has been recently improved by introducing a progressive optimization strategy''', whereby the optimization process is based on partially converged flow solutions with the aim to converge the flow solution while converging the design problems. In particular, in Ref. 11 this efficient approach is applied to the inverse design of turbine blades in inviscid transonic flow conditions. In Refs. 9 — 1 2 , the adjoint equations are solved using a direct solver, appropriate for two-dimensional flow problems, but not effective for three-dimensional flow problems. The extension to three dimensions has been considered in Ref. 13, where the progressive procedure suggested in Ref. 9 is modified by adding a time derivative in the adjoint equations, and by solving them iteratively. in a progressive manner quite similar to the one employed for the flow equations. The smoothing procedure and the use of partially converged flow solutions as well as of partially converged adjoint equation solutions in the progressive optimization implies that the optimization process is based on approximate values of the gradient of the objective function. Another example of approximate design sensitivity has been discussed by Matsuzawa and Hafez, where adjoint equations based on an inviscid flow formulation have been used successfully to determine approximate gradients of the objective function for airfoil inverse design in laminar flow conditions. The aim of the present paper is to apply all of the aforementioned strategies that contribute to improve the robustness and the efficiency of the optimization process to the inverse design of two-dimensional cascades in turbulent flow conditions. The final goal is to validate the robustness and efficiency of the outlined procedure also for this important class of design problems and, more importantly, to perform a first important step towards the development of an automatic tool for the direct optimization of turbomachinery bladings in real flow conditions. The adjoint equations will be derived using an inviscid solver involving a traditional centered scheme with added numerical viscosity. The inviscid formulation will be used in order to preserve the simplicity of the adjoint equations. Moreover, an additional time derivative will be introduced in order to solve the time dependent adjoint equations progressively in time. Finally, the flow equations will be solved on a progressively finer mesh, while the adjoint equations will be always solved on the coarsest mesh in order to reduce the cost of the computation of the gradient of the objective function to a few percents of the total cost of the optimization procedure. The following sections will be devoted to the description of the main ingredients of the proposed approach, that will be tested on the inverse design of a two-dimensional turbine blade in turbulent transonic flow conditions, both with fixed stagger angle and with fixed outlet angle of the camberline.
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