# Uncertainty Quantification Approach on Numerical Simulation for Supersonic Jets Performance
## 1 Introduction  
In recent years, an important part of the research in numerical simulations, in many engineering sectors, has been dedicated to the performance prediction of systems and components in off-design conditions. The behavior of a system or a component in conditions different from those for which it was designed is not only related to the deterministic variation of the input parameters, but also to the aleatory uncertainty which can characterize the input data and the geometrical tolerances. Hence, in order to improve the accuracy and the reliability of the numerical predictions, it is necessary to understand how the uncertainties can affect the results of the problem under investigation. This is one of the main targets of Uncertainty Quantification (UQ) analysis with direct positive fall-out on engineering problems. Computational Fluid Dynamics (CFD) is one of the disciplines where UQ is increasingly applied within a simulation environment. The UQ analysis is becoming an effective approach for industrial use thanks to the concurrent development of both soft-computing methods and computer performance. The use of CFD coupled to optimization algorithms for the automatic design optimization of industrial components is nowadays a mature technology [1; 2]. More recently,the development of Response Surface Methodology (RSM) has significantly enhanced the effectiveness of the design optimization based on simulation [3; 4], as an industrial standard or for data mining and diagnostic applications [5; 6]. The same RSM approach can be used to develop very efficient frameworks for UQ analysis [7]. A contribution to this field is given in the paper with reference to one of the most interesting examples of compressible flow thanks to its high non-linearity: the adiabatic flow in a variable section duct. It is a relevant case for its physical and mathematical background, and it has also a wide range of engineering applications. The best known application is for spacecraft propulsion in aircraft engines or in supersonic wind tunnel [8]. Convergent-divergent nozzles are also applied in gas burners [9; 10] and in supersonic water separators for natural gas purification [11; 12]. The preliminary design of a supersonic nozzle (De Laval nozzle) is based on the basic gas-dynamic equations in order to have a reference, ideal, operative condition of a fully expanded jet [8]. However, the theory is based on some assumptions, such as isentropic flow conditions, that can affect the actual nozzle performance. Moreover, the real operating conditions (pressure, temperature, gas mixture composition, etc.) can cause a significant change in the flow structure and in the shock waves generation with direct outcomes on nozzle performance. The main goal of this paper is to answer to the following questions: a small variation of some input parameters can remarkably affect the performance of a designed convergent-divergent nozzle? Can UQ methods help to quantify the above effect and give more insight into the nozzle behavior? The uncertainties propagation through the CFD model of the nozzle was performed using an automated procedure developed by the authors within the Dakota open source platform. The UQ approach is a'surrogate-based' approach: a response surface was generated from a DoE, and then, the Latin Hypercube Sampling (LHS) was applied to the meta-model to perform the UQ analysis. The use of a surrogate model is highly effective in reducing calculation times when a sampling-based UQ method is used: in this case, in fact, a high number of response function evaluations is required to generate converging statistics. The LHS was performed both for the DoE generation and for the UQ analysis. In LHS, the design space (with dimensions equal to the problem variables) is subdivided into an orthogonal grid with N elements per parameter. Within the grid, N sub-volumes are located so that along each row and column of the grid, only one sub-volume is chosen. Inside each sub-volume, a sample is chosen randomly [13].