Uncertainty quantification methodologies for cooling systems in gas turbine applications

Historically, the design of turbomachinery components was mainly done through experimental tests; over the years, with the increase of computing resources, there has been increasing use of computational analysis. Numerical simulations are important tools for designers because they allow having a complete understanding of the problem, in relatively short times and with low general costs. Although these analyses have a good predictive level, they are often used when input quantities that characterize the problem are roughly known. These gaps lead to the inclusion of uncertainties within the code, which propagate and eventually influence the solution. In the last fifteen years, statistical aspects have been combined into numerical simulations in order to assess the influence of the unknown parameters in the initial stages of the project. The final common objective is to optimize the various components in order to find out the configuration in which the machine is independent of the uncertainties that may afflict it, thus arriving at a robust design. The aim of this thesis was to explore and apply several methodologies of "uncertainty quantification" (UQ) to numerical codes used in turbomachinery applications, which allow estimating the uncertainties that affect the results of numerical simulations. Both sampling-based methods and stochastic expansion methods were investigated. After an initial benchmarking phase, the software DAKOTA was selected to carry out the UQ analyses. The first part of the work involved a 1-D thermal analysis on a full annular lean-burn aeronautical combustor tested at CIAM during the LEMCOTEC (Low Emissions COre-engine TEChnologies) European project. The analysis was carried out using the one-dimensional code "Therm-1D", developed by DIEF of the University of Florence. Three main uncertainty analyses were investigated depending on the input parameters considered: geometrical, heat transfer coefficient tuning factors, and thermal loads. In particular, the classical Monte Carlo analysis is compared with four stochastic expansion processes: Gauss quadrature, total order with LHS sampling, stochastic collocation, and Smolyak. The analyses proved how these methods give optimum results with a sensible lower amount of simulations. Lastly, an analysis including all the input variables considered was performed and results were compared with experimental data. Working on a 1-D solver has allowed obtaining a large amount of data with modest computational costs: this part was crucial in order to better understand the different methodologies and to have a clearer picture of the potentialities of the software. The second part of the work focused on applying the acquired concepts to a high-fidelity code. Based on an experimental study, a full 3-D computational fluid dynamic (CFD) study using the software ANSYS was carried out in order to assess the film cooling performance of a prismatic gas turbine vane made by additive manufacturing. Both steady and unsteady simulations were performed: the first ones using a RANS approach and the latter using a hybrid LES-RANS approach. For the UQ analysis, only RANS simulations in conjunction with a specific stochastic expansion method were adopted to save computational resources. The influences of the geometric uncertainties of the holes were evaluated: the hole dimension, the streamwise inclination angle and the inlet fillet radius of the hole. Output parameters considered were the film cooling effectiveness, the blowing ratio and the discharge coefficients of the holes. Results will show how a polynomial chaos approach that required 8 evaluations is able to reproduce what the standard Monte Carlo analysis does (with more than 1000 evaluations) with an optimum grade of accuracy. Moreover, results prove how the position tolerance of the holes on the blade, as well as the hole dimension, is extremely important for the film cooling effectiveness, in particular when dealing with additive manufacturing processes.