Please cite this article in press as: R. Ahlfeld et al., SAMBA: Sparse approximation of moment-based arbitrary polynomial chaos, J. Comput. Phys. (2016), http://dx.doi.org/10.1016/j. jcp.2016.05.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. SAMBA AbstractA new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability.The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of
This paper presents an investigation of the aerothermal performance of a modern unshrouded high-pressure (HP) aero-engine turbine subject to nonuniform inlet temperature profile. The turbine used for this study was the MT1 turbine installed in the QinetiQ turbine test facility based in Farnborough (UK). The MT1 turbine is a full scale transonic HP turbine, and is operated in the test facility at the correct nondimensional conditions for aerodynamics and heat transfer. Datum experiments of aerothermal performance were conducted with uniform inlet conditions. Experiments with nonuniform inlet temperature were conducted with a temperature profile that had a nonuniformity in the radial direction defined by (Tmax−Tmin)/T¯=0.355, and a nonuniformity in the circumferential direction defined by (Tmax−Tmin)/T¯=0.14. This corresponds to an extreme point in the engine cycle, in an engine where the nonuniformity is dominated by the radial distribution. Accurate experimental area surveys of the turbine inlet and exit flows were conducted, and detailed heat transfer measurements were obtained on the blade surfaces and end-walls. These results are analyzed with the unsteady numerical data obtained using the in-house HybFlow code developed at the University of Firenze. Two particular aspects are highlighted in the discussion: prediction confidence for state of the art computational fluid dynamics (CFD) and impact of real conditions on stator-rotor thermal loading. The efficiency value obtained with the numerical analysis is compared with the experimental data and a 0.8% difference is found and discussed. A study of the flow field influence on the blade thermal load has also been detailed. It is shown that the hot streak migration mainly affects the rotor pressure side from 20% to 70% of the span, where the Nusselt number increases by a factor of 60% with respect to the uniform case. Furthermore, in this work, it has been found that a nonuniform temperature distribution is beneficial for the rotor tip, contrary to the results found in open literature. Although the hot streak is affected by the pressure gradient across the tip gap, the radial profile (which dominates the temperature profile being considered) is not fully mixed out in passing through the HP stage, and contributes significantly to cooling the turbine casing. A design approach not taking into account these effects will underestimate the rotor life near the tip and the thermal load at midspan. The temperature profile that has been used in both experiments and CFD is the first simulation of an extreme cycle point (more than twice the magnitude of distortion of all previous experimental studies): It represents an engine-take-off condition combined with the full combustor cooling. This research was part of the EU funded Turbine AeroThermal External Flows 2 program.
PrefaceIt is no wonder that Uncertainty Quantification has become more and more of an actuality in the last decade as the modelling capability jointly with computational power has increased a lot. In the past, the capability to predict flow field and performance in aero engines as well as in turbomachinery was of great support to the design. However, the range of errors in such results was so large as to suggest the use of CFD, mainly to understand the direction of trends and improvements more than the exact evaluation of thermo-fluid-dynamic parameters, which could affect performance, reliability and life of the engine components.Recently, we have seen two different but relevant matters:
Computational fluid dynamics (CFD) prediction of the unsteady aerothermal interaction in the HP turbine stage, with inlet temperature nonuniformity, requires appropriate unsteady modeling and a low diffusive numerical scheme coupled with suitable turbulence models. This maybe referred to as high fidelity CFD. A numerical study has been conducted by the University of Florence in collaboration with ONERA to compare capabilities and limitations of their CFD codes for such flows. The test vehicle used for the investigation is a turbine stage of three-dimensional design from the QinetiQ turbine facility known as MT1. This stage is a high pressure transonic stage that has an unshrouded rotor, configured, and uncooled with 32 stators and 60 rotor blades. Two different CFD solvers are compared that use different unsteady treatments of the interaction. A reduced count ratio technique has been used by the University of Florence with its code HYBFLOW, while a phase lag model has been used by ONERA in their code, ELSA. Four different inlet conditions have been simulated and compared with focus on the experimental values provided by QinetiQ in the frame of TATEF and TATEF2 EU Sixth Framework Projects. The differences in terms of performance parameters and hot fluid redistribution, as well as the time- and pitch-averaged radial distributions on a plane downstream of the rotor blade, have been underlined. Special attention was given to the predictions of rotor blade unsteady pressure and heat transfer rates.
In computational fluid dynamics (CFD), it is possible to identify namely two uncertainties: epistemic, related to the turbulence model, and aleatoric, representing the random-unknown conditions such as the boundary values and or geometrical variations. In the field of epistemic uncertainty, large eddy simulation (LES and DES) is the state of the art in terms of turbulence closures to predict the heat transfer in internal channels. The problem is still unresolved for the stochastic variations and how to include these effects in the LES studies. In this paper, for the first time in literature, a stochastic approach is proposed to include these variations in LES. By using a classical uncertainty quantification approach, the probabilistic collocation method is coupled to numerical large eddy simulation (NLES) in a duct with pin fins. The Reynolds number has been chosen as a stochastic variable with a normal distribution. The Reynolds number is representative of the uncertainties associated with the operating conditions, i.e., velocity and density, and geometrical variations such as the pin fin diameter. This work shows that assuming a Gaussian distribution for the Reynolds number of ±25%, it is possible to define the probability to achieve a specified heat loading under stochastic conditions, which can affect the component life by more than 100%. The same method, applied to a steady RANS, generates a different level of uncertainty. New methods have been proposed based on the different level of aleatoric uncertainties which provides information on the epistemic uncertainty. This proves, for the first time, that the uncertainties related to the unknown conditions, aleatoric, and those related to the physical model, epistemic, are strongly interconnected. This result, which is idealized for this specific issue, can be extrapolated, and has direct consequences in uncertainty quantification science and not only in the gas turbine world.
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