Application of Sweep to Transonic Compressor Rotor Blades for Low-order Statistical Moment Averaging in Robust Design

Robust design optimization involves finding the loworder statistical moments, i.e. the maximization of some mean quantity of interest and minimization of its variance. The question arises as to when a mean or variance value can be considered to be converged to an acceptable level of certainty. A designer naturally seeks to keep the number of function evaluations as low as possible when converging statistics. There is no general answer to the question of how many CFD simulations need to be carried out in order to obtain reliable estimators and which sampling methods perform better. Furthermore, multi-fidelity optimization techniques such as Co-Kriging can be used to combine different convergence levels and the question remains as to how many functions evaluations should be carried out. Practical guidelines applicable for the robust design optimization of turbomachinery blades are provided here. The applied methodology involves the freely available NASA Rotor 37 geometry and 3D steady-state RANS-based CFD with the Spalart-Allmaras turbulence model. The numerical CFD results are validated against actual experimental results. A uniformly distributed sweep uncertainty applied at the tip of the blade is propagated using Monte Carlo and Quasi-Monte Carlo-based sampling (low-discrepancy Halton and randomized Sobol sequence) for comparisons. Statistical postprocessing of the results is based on 500 CFD runs for each sampling strategy. As an indicator of the error bounds, standard deviation and confidence intervals for the converging sample means of all quantities of interest are calculated. The required number of iterations is estimated. INTRODUCTION Modern aircraft engines and their individual components such as turbine and compressor blades are highly-optimized designs and subject to many uncertainties. Blade geometries in service now are the result of multi-objective optimization processes, involving a plethora of design variables, structural and performance constraints and many conflicting objectives. Such designs can be very sensitive to inevitable uncertainties, for example due to manufacturing tolerances, in-service deterioration or varying operating conditions. It has been shown in literature (e.g. [4, 14]) that such uncertainties, even very small perturbations on the order of tenths of millimetres, can lead to significant engine performance degradation. For compressor blades, aerodynamic performance considerations are of prime importance whereas for turbine blades the metal temperature is more crucial. Next generation computational modelling and scientific research therefore requires engineers and researchers to incorporate variations into the design process. This approach is termed robust design and typically involves the low-order statistical moments, that is maximization of some mean performance measure, such as the adiabatic efficiency, and simultaneous minimization of its variance. However, it is unclear how many simulations need to be carried out in order to obtain reliable estimators (i.e. converged to an acceptable margin of error), what errors can be considered acceptable and what sampling methods perform better than others by requiring fewer function evaluations for the same level of accuracy. Furthermore, different levels can be combined with multi-fidelity robust design optimization techniques such as Co-Kriging. How many simulations should be carried out for each level of fidelity? In the literature, CoKriging was for example used to optimize a 2D section of a Rolls-Royce compressor blade [6]. 100 LPτ samples were used to calculate more accurate, but computationally expensive converged sample means and 5 LPτ points were used to calculate less accurate but cheap estimates of the mean. Was this a good combination or is there a computationally cheaper, more efficient combination to get results with the same accuracy? Answers to these questions are sought in this paper. METHODOLOGY A commonly used 3D engineering parameter for turbomachinery blades, sweep is introduced as an uncertainty at the blade tip and 3D Reynolds-averaged Navier-Stokes