This work presents a framework for predicting unknown input distributions for turbomachinery applications starting from scarce experimental measurements. The problem is relevant to turbomachinery where important parameters are obtained using indirect measurements. In this paper a scroll compressor is used as example but the suggested framework is completely general and can be used to infer missing data on material composition (carbon fiber properties, laser melted specimens for additive manufacturing etc) or input data (such as the turbine inlet temperature).

Scroll compressors are small devices with a very complex geometry that is difficult to measure. Moreover these compressors are highly sensitive to manufacturing errors and clearances. For these reasons we have chosen this example as an ideal candidate to prove the effectiveness of the framework. An input probability distribution for the scroll height is recovered based on a scarce, synthetic data set. The scroll height is used as an example of a missing distribution for a geometric parameter as it has the highest variance and is challenging to measure experimentally.

The framework consists of two main building blocks: an equivalence in a probabilistic sense and a Non-Intrusive Polynomial Chaos formulation able to deal with scarce data. The probabilistic equivalence is defined by a Probability Density Function (PDF) matching approach in which the statistical distance between probability distributions is quantified by either the Kolmogorov-Smirnov (KS) distance or the Kullback-Leibler (KL) divergence. By representing the missing inputs with a generalised Polynomial Chaos Expansion (gPCE) the back-calculation problem can be recast as an optimisation problem in which an arbitrary Polynomial Chaos (aPC) formulation was used to propagate the uncertain input distributions through a computational model of the system and generate a probability distribution for the Quantity of Interest (QoI).

The framework has been tested with multiple non-Askey scheme distributions to prove the generality of the proposed approach.

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