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Research Papers

A Single Formulation for Uncertainty Propagation in Turbomachinery: SAMBA PC

[+] Author and Article Information
Richard Ahlfeld

Uncertainty Quantification Lab,
Department of Aeronautics,
Imperial College of London,
London SW7 2AZ, UK

Francesco Montomoli

Uncertainty Quantification Lab,
Department of Aeronautics,
Imperial College of London,
London SW7 2AZ, UK
e-mail: f.montomoli@imperial.ac.uk

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received August 3, 2016; final manuscript received July 25, 2017; published online August 23, 2017. Editor: Kenneth Hall.

J. Turbomach 139(11), 111007 (Aug 23, 2017) (10 pages) Paper No: TURBO-16-1184; doi: 10.1115/1.4037362 History: Received August 03, 2016; Revised July 25, 2017

This work newly proposes an uncertainty quantification (UQ) method named sparse approximation of moment-based arbitrary polynomial chaos (SAMBA PC) that offers a single solution to many current problems in turbomachinery applications. At the moment, every specific case is characterized by a variety of different input types such as histograms (from experimental data), normal probability density functions (PDFs) (design rules) or fat tailed PDFs (for rare events). Thus, the application of UQ requires the adaptation of ad hoc methods for each individual case. A second problem is that parametric PDFs have to be determined for all inputs. This is difficult if only few samples are available. In gas turbines, however, the collection of statistical information is difficult, expensive, and having scarce information is the norm. A third critical limitation is that if using nonintrusive polynomial chaos (NIPC) methods, the number of required simulations grows exponentially with increasing numbers of input uncertainties: the so-called “curse of dimensionality.” It is shown that the fitting of parametric PDFs to small data sets can lead to large bias and the direct use of the available data is more accurate. This is done by propagating uncertainty through several test functions and the computational fluid dynamics (CFD) simulation of a diffuser, highlighting the impact of different PDF fittings on the output. From the results, it is concluded that the direct propagation of the experimental data set is preferable to the fit of parametric distributions if data is scarce. Thus, the suggested method offers an alternative to the maximum entropy theorem to handle scarce data. SAMBA simplifies the mathematical procedure for many different input types by basing the polynomial expansion on moments. Its moment-based expansion automatically takes care of arbitrary combinations of different input data. It is also numerically efficient compared to other UQ implementations. The relationship between the number of random variables and number of simulation is linear (only 21 simulations for ten input random variables are required). It is shown in this paper that SAMBA's algorithm can propagate a high number of input distributions through a set of nonlinear analytic test functions. Doing this, the code needs a very small number of simulations and preserve a 5% error margin. SAMBA's flexibility to handle different forms of input distributions and a high number of input variables is shown on a low-pressure turbine (LPT) blade-based on H2 profile. The relative importance of manufacturing errors in different location of the blade is analyzed.

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References

Figures

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Fig. 1

Schematic of SAMBA algorithm

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Fig. 2

Optimal Gaussian collocation points for uniform, fat-tailed student-t, and Weibull (last two both not Askey scheme) and for various mixed and multimodal histograms

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Fig. 3

Number of collocation points for level 2 Smolyak and third order full tensor

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Fig. 4

Histogram based Gaussian Smolyak grids at level 3

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Fig. 5

SAMBA convergence for increasing sample size

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Fig. 6

Comparison of SAMBA and Gaussian PDF fitting for growing histogram sample size

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Fig. 7

Mean velocity of CFD diffuser flow with coarse mesh for better visualization

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Fig. 8

Random data describing the stochastic variation of the taper angle α

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Fig. 9

Various parametric PDFs fitted to the random samples of the diffuser taper angle

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Fig. 10

Normalized measurements of geometric manufacturing variations

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Fig. 11

Schematic of H2 airfoil, not in scale, with six sections and two points varied according to six different measurement sets and two PDFs given in Fig. 10 and isentropic Mach number

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Fig. 12

Histogram of pressure losses ω of H2 profile for the six manufacturing uncertainty histograms and two assumed PDFs

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