A monotonic, non-kernel density variant of the density-matching technique for optimization under uncertainty is developed. The approach is suited for turbomachinery problems which, by and large, tend to exhibit monotonic variations in the circumferentially and radially mass-averaged quantities-such as pressure ratio, efficiency and capacity-with common aleatory turbomachinery uncertainties. The method is successfully applied to de-sensitize the effect of an uncertainty in rear-seal leakage flows on the fan stage of a modern jet engine. IntroductionAero-engine manufacturers are becoming increasingly reliant on computational predictions of compressor and turbine flows to guide their design decisions. As this trend will continue in the years to come, there will be a growing emphasis on leveraging these methods within optimization routines. In tandem, there will also be a demand for the application of uncertainty quantification (UQ) methods, which seek to compute error bars, probability density functions (pdfs) and probabilities of failure for a given compressor or turbine design. Computational simulations of turbomachinery flows can be used to yield these statistics in an efficient manner. Central to this endeavor is the identification of operational uncertainties and their variations. These may be of the form of operational variations in the tip gap, blade and endwall manufacturing deviations and leakage flows between rotating and stationary components-to name but a few.Finally, to close the loop for redesign, this statistical data needs to be fed to an optimizer in a suitable manner. Here one would like to depart from expensive multi-objective methods and opt for computationally tractable alternatives. A framework that can leverage adjoint-based design sensitivities is also extremely favorable. These perspectives give rise to the main focus area of this research. In this paper, a monotonic variant of a probability density-matching technique for optimization under uncertainty (OUU) is presented. A brief background on density-matching and its advantages over existing robust and reliability-based design optimization techniques is given in Section 2. In Section 3, the mathematics behind monotonic density-matching and the motivation for this technique is presented. Finally, in Section 4, the method is successfully applied to a real turbomachinery problem: fan sub-system design under rear-seal leakage uncertainty. Robust design optimization (RDO)Robust design methodologies seek to optimize the mean performance metric while minimizing its variance [1]. Typically RDO lends itself to a multi-objective optimization strategy where there is a trade-off between these two objectives. This approach yields a Pareto set of feasible design solutions from which the designer will select a single design that offers a reasonable trade-off between the two objectives. By and large, multi-objective genetic algorithms are the optimizer of choice for these problems. Aerospace applications range from the design of compressor blades [2] to en...

Our investigation raises an important question that is of relevance to the wider turbomachinery community: how do we estimate the spatial average of a flow quantity given finite (and sparse) measurements? This paper seeks to advance efforts to answer this question rigorously. In this paper, we develop a regularized multivariate linear regression framework for studying engine temperature measurements. As part of this investigation, we study the temperature measurements obtained from the same axial plane across five different engines yielding a total of 82 data-sets. The five different engines have similar architectures and therefore similar temperature spatial harmonics are expected. Our problem is to estimate the spatial field in engine temperature given a few measurements obtained from thermocouples positioned on a set of rakes. Our motivation for doing so is to understand key engine temperature modes that cannot be captured in a rig or in computational simulations, as the cause of these modes may not be replicated in these simpler environments. To this end, we develop a multivariate linear least squares model with Tikhonov regularization to estimate the 2D temperature spatial field. Our model uses a Fourier expansion in the circumferential direction and a quadratic polynomial expansion in the radial direction. One important component of our modeling framework is the selection of model parameters, i.e. the harmonics in the circumferential direction. A training-testing paradigm is proposed and applied to quantify the harmonics.

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