0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Montomoli, F. , Massini, M. , Salvadori, S. , and Martelli, F. , 2012, “ Geometrical Uncertainty and Film Cooling: Fillet Radii,” ASME J. Turbomach., 134(1), p. 011019. [CrossRef]
Bunker, R. S. , 2009, “ The Effects of Manufacturing Tolerances on Gas Turbine Cooling,” ASME J. Turbomach., 131(4), p. 041018. [CrossRef]
Montomoli, F. , Ammaro, A. , and Uchida, S. , 2013, “ Uncertainty Quantification and Conjugate Heat Transfer: A Stochastic Analysis,” ASME J. Turbomach., 135(3), p. 031014. [CrossRef]
Oladyshkin, S. , and Nowak, W. , 2012, “ Data-Driven Uncertainty Quantification Using the Arbitrary Polynomial Chaos Expansion,” Reliab. Eng. Syst. Saf., 106, pp. 179–190. [CrossRef]
Jaynes, E. T. , 1982, “ On the Rationale of Maximum-Entropy Methods,” Proc. IEEE, 70(9), pp. 939–952. [CrossRef]
Witteveen, J. A. S. , Sarkar, S. , and Bijl, H. , 2007, “ Modeling Physical Uncertainties in Dynamic Stall Induced Fluid-Structure Interaction of Turbine Blades Using Arbitrary Polynomial Chaos,” Comput. Struct., 85(11–14), pp. 866–878. [CrossRef]
Oladyshkin, S. , Class, H. , Helmig, R. , and Nowak, W. , 2011, “ A Concept for Data-Driven Uncertainty Quantification and Its Application to Carbon Dioxide Storage in Geological Formations,” Adv. Water Resour., 34(11), pp. 1508–1518. [CrossRef]
Oladyshkin, S. , Schröder, P. , Class, H. , and Nowak, W. , 2013, “ Chaos Expansion Based Bootstrap Filter to Calibrate CO2 Injection Models,” Energy Proc., 40, pp. 398–407. [CrossRef]
Oladyshkin, S. , Class, H. , Helmig, R. , and Nowak, W. , 2011, “ An Integrative Approach to Robust Design and Probabilistic Risk Assessment for CO2 Storage in Geological Formations,” Comput. Geosci., 15(3), pp. 565–577. [CrossRef]
Ernst, O. G. , Mugler, A. , Starkloff, H.-J. , and Ullmann, E. , 2012, “ On the Convergence of Generalized Polynomial Chaos Expansions,” Math. Modell. Numer. Anal., 46(2), pp. 317–339. [CrossRef]
Soize, C. , and Ghanem, R. , 2004, “ Physical Systems With Random Uncertainties: Chaos Representations With Arbitrary Probability Measure,” SIAM J. Sci. Comput., 26(2), pp. 395–410. [CrossRef]
Carnevale, M. , Montomoli, F. , Ammaro, A. , Salvadori, S. , and Martelli, F. , 2013, “ Uncertainty Quantification: A Stochastic Method for Heat Transfer Prediction Using LES,” ASME J. Turbomach., 135(5), p. 051021. [CrossRef]
Loeven, G. J. A. , 2010, “ Efficient Uncertainty Quantification in Computational Fluid Dynamics,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands. https://repository.tudelft.nl/islandora/object/uuid%3A8313a3bd-701c-458a-bf99-e819e1276084
Montomoli, F. , Amirante, D. , Hills, N. , Shahpar, S. , and Massini, M. , 2014, “ Uncertainty Quantification, Rare Events, and Mission Optimization: Stochastic Variations of Metal Temperature During a Transient,” ASME J. Eng. Gas Turbines Power, 137(4), p. 042101. [CrossRef]
Mysovskih, I. P. , 1968, “ On the Construction of Cubature Formulas With Fewest Nodes,” Dokl. Akad. Nauk SSSR, 178(6), pp. 1252–1254.
Golub, G. H. , and Welsch, J. H. , 1969, “ Calculation of Gauss Quadrature Rules,” Math. Comput., 23(106), pp. 221–230. [CrossRef]
Brack, S. , and Muller, Y. , 2015, “ Probabilistic Analysis of the Secondary Air System of a Low-Pressure Turbine,” ASME J. Eng. Gas Turbines Power, 137(2), p. 022602. [CrossRef]
Schnell, R. , Lengyel-Kampmann, T. , and Nicke, E. , 2014, “ On the Impact of Geometric Variability on Fan Aerodynamic Performance, Unsteady Blade Row Interaction, and Its Mechanical Characteristics,” ASME J. Turbomach., 136(9), p. 091005. [CrossRef]
Garzon, V. E. , 2003, “ Probabilistic Aerothermal Design of Compressor Airfoils,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA. https://dspace.mit.edu/handle/1721.1/16995
Duffner, J. D. , 2008, “ The Effects of Manufacturing Variability on Turbine Vane Performance,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA. https://dspace.mit.edu/handle/1721.1/44933
Panizza, A. , Bonini, A. , and Innocenti, L. , 2015, “ Uncertainty Quantification of Hot Gas Ingestion for a Gas Turbine,” ASME Paper No. GT2015-42679.
Pečnik, R. , Witteveen, J. A. S. , and Iaccarino, G. , 2011, “ Uncertainty Quantification for Laminar-Turbulent Transition Prediction in RANS Turbomachinery Applications,” AIAA Paper No. 2011-660.
Gopinathrao, N. P. , Mabilat, C. , and Alizadeh, S. , 2009, “ Non-Deterministic Thermo-Fluid Analysis of a Compressor Rotor-Stator Cavity,” AIAA Paper No. 2009-2278.
Loeven, G. J. A. , and Bijl, H. , 2010, “ The Application of the Probabilistic Collocation Method to a Transonic Axial Flow Compressor,” AIAA Paper No. 2010-2923.
Weitzman, M. , 2011, “ Fat-Tailed Uncertainty in the Economics of Catastrophic Climate Change,” Rev. Environ. Econ. Policy, 5(2), pp. 275–292. [CrossRef]
Suárez-Lledó, J. , 2011, The Black Swan: The Impact of the Highly Improbable, Vol. 25, Random House, New York.
Ahlfeld, R. , Belkouchi, B. , and Montomoli, F. , 2016, “ SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos,” J. Comput. Phys., 320, pp. 1–16. [CrossRef]
Rutishauser, H. , 1963, “ On a Modification of the QD-Algorithm With Graeffe-Type Convergence,” Z. Angew. Math. Phys., 13(5), pp. 493–496. https://doi.org/10.1007/BF01601077
Eldred, M. , and Burkardt, J. , 2009, “ Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification,” AIAA Paper No. 2009-976.
Smolyak, S. A. , 1963, “ Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,” Dokl. Akad. Nauk SSSR, 4, pp. 240–243.
Judd, K. L. , Maliar, L. , Maliar, S. , and Valero, R. , 2014, “ Smolyak Method for Solving Dynamic Economic Models: Lagrange Interpolation, Anisotropic Grid and Adaptive Domain,” J. Econ. Dyn. Control, 44, pp. 92–123. [CrossRef]
Chandavari, V. , and Palekar, S. , 2014, “ Diffuser Angle Control to Avoid Flow Separation,” Tech. Res. Appl., 2(5), pp. 16–21. http://www.ijtra.com/view/diffuser-angle-control-to-avoid-flow-separation.pdf
DalBello, T. , Dippold, V. , and Georgiadis, N. J. , 2005, “ Computational Study of Separating Flow in a Planar Subsonic Diffuser,” National Aeronautics and Space Administration, Glenn Research Center, Cleveland, OH, Report No. 2005-213894. https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20050237896.pdf
Thakur, N. , Keane, A. , and Nair, P. B. , 2008, “ Capture of Manufacturing Uncertainty in Turbine Blades Through Probabilistic Techniques,” Association for Structural and Multidisciplinary Optimization (ASMO), Bath, UK, July 7–8, pp. 1–10. https://eprints.soton.ac.uk/64276/
Montomoli, F. , Hodson, H. , and Haselbach, F. , 2010, “ Effect of Roughness and Unsteadiness on the Performance of a New Low Pressure Turbine Blade at Low Reynolds Numbers,” ASME J. Turbomach., 132(3), p. 431018.
Ahlfeld, R. , Montomoli, F. , Scalas, E. , and Shahpar, S. , 2017, “ Uncertainty Quantification for Fat-Tailed Probability Distributions in Aircraft Engine Simulations,” J. Propul. Power, 33(4), pp. 881–890. https://doi.org/10.2514/1.B36278

Figures

Grahic Jump Location
Fig. 1

Schematic of SAMBA algorithm

Grahic Jump Location
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

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

Histogram based Gaussian Smolyak grids at level 3

Grahic Jump Location
Fig. 5

SAMBA convergence for increasing sample size

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
Fig. 8

Random data describing the stochastic variation of the taper angle α

Grahic Jump Location
Fig. 9

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

Grahic Jump Location
Fig. 10

Normalized measurements of geometric manufacturing variations

Grahic Jump Location
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

Grahic Jump Location
Fig. 12

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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In