Gas Turbines and Uncertainty Quantification: Impact of PDF Tails on UQ Predictions, the Black Swan

Abstract

In the last five years Uncertainty Quantification (UQ) techniques became popular to predict gas turbine performances. Taking into account the uncertainties in the input parameters it is possible to evaluate the impact of random variations and to overcome the limitations of deterministic studies. These methods, that only recently have been widely used in computational fluid dynamics, have some limitations that must be considered. One of the most important limitations is that these models cannot predict a “Black Swan” (BS) event. In probability a Black Swan is an event rare, possible and with serious consequences. A reliable design requires a correct evaluation of the probabilities of occurrence of the Black Swan that could strongly affect the life of the turbine. Black Swans are generated by the variability of the input parameters in the “tail” of the statistical distributions. Being far from the mean value design geometry/condition, these events have a low probability of occurrence. In this paper is shown that the use of the Gaussian distribution for the input parameters could strongly underestimate the probability of occurrence of a Black Swan event. Despite that most of the models used in UQ for aerodesign are neglecting the problem. As an example of Black Swan, the hot gas ingestion across a stator is analysed. The gaps have been assumed to be affected by uncertainty with a variation of +/-50% of the nominal value. By using a Monte Carlo simulation with 108 realizations and a Gauss distribution as input, the configuration is initially considered reliable. The six sigma criterion is also satisfied and the probability to have a failure is only 2.54 10−4%. However if a “fat tail” for the input distribution is used instead, the probability to have hot gas ingestion becomes 2.33%, 104 times higher. Most of the methods used in literature aim to have an accurate reproduction of the PDF moments such as mean, standard deviation, skew and kurtosis. However the “tail” of the distribution affects the gas turbine life and must be considered. In particular “fat tails”, the mathematical origin of Black Swans events, can have serious consequences, but in modern stochastic models used for computational fluid dynamics they are not accounted for.