Research Papers

Comparison of Two Methods for Sensitivity Analysis of Compressor Blades

[+] Author and Article Information
Robin Schmidt

Institute of Fluid Mechanics,
Technische Universität Dresden,
Dresden D-01062, Germany
e-mail: robin.schmidt@tu-dresden.de

Matthias Voigt, Konrad Vogeler

Institute of Fluid Mechanics,
Technische Universität Dresden,
Dresden D-01062, Germany

Marcus Meyer

Rolls-Royce Deutschland Ltd & Co KG,
Blankenfelde-Mahlow D-15827, Germany
e-mail: marcus.meyer@rolls-royce.com

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received May 14, 2017; final manuscript received June 5, 2017; published online August 16, 2017. Editor: Kenneth Hall.

J. Turbomach 139(11), 111006 (Aug 16, 2017) (8 pages) Paper No: TURBO-17-1065; doi: 10.1115/1.4037127 History: Received May 14, 2017; Revised June 05, 2017

This paper will compare two approaches of sensitivity analysis, namely (i) the adjoint method which is used to obtain an initial estimate of the geometric sensitivity of the gas-washed surfaces to aerodynamic quantities of interest and (ii) a Monte Carlo type simulation with an efficient sampling strategy. For both approaches, the geometry is parameterized using a modified NACA parameterization. First, the sensitivity of those parameters is calculated using the linear (first-order) adjoint model. Since the effort of the adjoint computational fluid dynamics (CFD) solution is comparable to that of the initial flow CFD solution and the sensitivity calculation is simply a postprocessing step, this approach yields fast results. However, it relies on a linear model which may not be adequate to describe the relationship between relevant aerodynamic quantities and actual geometric shape variations for the derived amplitudes of shape variations. Second, in order to better capture nonlinear and interaction effects, a Monte Carlo type simulation with an efficient sampling strategy is used to carry out the sensitivity analysis. The sensitivities are expressed by means of the coefficient of importance (CoI), which is calculated based on modified polynomial regression and therefore able to describe relationships of higher order. The methods are applied to a typical high-pressure compressor (HPC) stage. The impact of a variable rotor geometry is calculated by three-dimensional (3D) CFD simulations using a steady Reynolds-averaged Navier–Stokes model. The geometric variability of the rotor is based on the analysis of a set of 400 blades which have been measured using high-precision 3D optical measurement techniques.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Jameson, A. , 2003, Aerodynamic Shape Optimization Using the Adjoint Method (Lecture Series), Von Karman Institute for Fluid Dynamics, Brussels, Belgium.
Giebmanns, A. , Backhaus, J. , Frey, C. , and Schnell, R. , 2013, “ Compressor Leading Edge Sensitivities and Analysis With an Adjoint Flow Solver,” ASME Paper No. GT2013-94427.
Pini, M. , Persico, G. , Pasquale, D. , and Rebay, S. , 2014, “ Adjoint Method for Shape Optimization in Real-Gas Flow Applications,” ASME J. Eng. Gas Turbines Power, 137(3), p. 032604. [CrossRef]
Walther, B. , and Nadarajah, S. , 2012, “ Constrained Adjoint-Based Aerodynamic Shape Optimization of a Single-Stage Transonic Compressor,” ASME J. Turbomach., 135(2), p. 021017. [CrossRef]
Walther, B. , and Nadarajah, S. , 2015, “ Optimum Shape Design for Multirow Turbomachinery Configurations Using a Discrete Adjoint Approach and an Efficient Radial Basis Function Deformation Scheme for Complex Multiblock Grids,” ASME J. Turbomach., 137(8), p. 081006. [CrossRef]
Garzon, V . E. , and Darmofal, D. L. , 2003, “ Impact of Geometric Variability on Axial Compressor Performance,” ASME J. Turbomach., 125(4), pp. 692–703. [CrossRef]
Lange, A. , Voigt, M. , Vogeler, K. , Schrapp, H. , Johann, E. , and Gümmer, V. , 2010, “ Probabilistic CFD Simulation of a High-Pressure Compressor Stage Taking Manufacturing Variability Into Account,” ASME Paper No. GT2010-22484.
Lange, A. , Voigt, M. , Vogeler, K. , Schrapp, H. , Johann, E. , and Gümmer, V. , 2012, “ Impact of Manufacturing Variability and Nonaxisymmetry on High-Pressure Compressor Stage Performance,” ASME J. Eng. Gas Turbines Power, 134(3), p. 032504.
Lange, A. , Voigt, M. , Schrapp, H. , Johann, E. , Vogeler, K. , and Gümmer, V. , 2012, “ Impact of Manufacturing Variability on Multistage High-Pressure Compressor Performance,” ASME J. Eng. Gas Turbines Power, 134(11), p. 112601. [CrossRef]
Kumar, A. , 2006, “ Robust Design Methodologies: Application to Compressor Blades,” Ph.D. thesis, University of Southampton, Southampton, UK. https://eprints.soton.ac.uk/72037/
Dow, E. A. , and Wang, Q. , 2015, “ The Implications of Tolerance Optimization on Compressor Blade Design,” ASME J. Turbomach., 137(10), p. 101008. [CrossRef]
Iooss, B. , and Lematre, P. , 2014, “ A Review on Global Sensitivity Analysis Methods,” preprint arXiv:1404.2405. https://arxiv.org/abs/1404.2405
Lange, A. , Vogeler, K. , Gmmer, V. , Schrapp, H. , and Clemen, C. , 2009, “ Introduction of a Parameter Based Compressor Blade Model for Considering Measured Geometry Uncertainties in Numerical Simulation,” ASME Paper No. GT2009-59937.
Heinze, K. , 2015, Probabilistische Untersuchung zum Einfluss von Fertigungsstreuungen auf die hochzyklische Ermdung von Verdichterschaufeln, TUDpress, Dresden, Germany.
Moinier, P. , Mller, J. D. , and Giles, M. B. , 2002, “ Edge-Based Multigrid and Preconditioning for Hybrid Grids,” AIAA J., 40(10), pp. 1954–1960. [CrossRef]
Moinier, P. , 1999, “ Algorithm Developments for an Unstructured Viscous Flow Solver,” Ph.D. thesis, Oxford University, Oxford, UK. http://people.maths.ox.ac.uk/gilesm/files/pierre_thesis.pdf
Moinier, P. , and Giles, M. B. , 1998, “ Preconditioned Euler and Navier–Stokes Calculations on Unstructured Meshes,” Numerical Methods for Fluid Dynamics VI, Oxford University, Oxford, UK.
Martinelli, L. , 1987, “ Calculations of Viscous Flows With a Multigrid Method,” Ph.D. thesis, Princeton University, Princeton, NJ. https://www.osti.gov/scitech/biblio/7029684
Giles, M. B. , Duta, M. C. , Mller, J.-D. , and Pierce, N. A. , 2003, “ Algorithm Developments for Discrete Adjoint Methods,” AIAA J., 41(2), pp. 198–205. [CrossRef]
Ntanakas, G. , and Meyer, M. , 2015, “ The Unsteady Discrete Adjoint Method for Turbomachinery Applications,” International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Stockholm, Sweden, Sept. 8–11, pp. 1–8.
Nielsen, E. J. , and Park, M. A. , 2006, “ Using an Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design,” AIAA J., 44(5), pp. 948–953. [CrossRef]
Campobasso, M. S. , Duta, M. C. , and Giles, M. B. , 2003, “ Adjoint Calculation of Sensitivities of Turbomachinery Objective Functions,” J. Propul. Power, 19(4), pp. 693–703. [CrossRef]
Schmidt, R. , Voigt, M. , and Vogeler, K. , 2014, “ Extension of Latin Hypercube Samples While Maintaining the Correlation Structure,” 12th International Probabilistic Workshop, Weimar, Germany, Nov. 5–8, pp. 297–314. http://docplayer.net/45534900-12-th-international-probabilistic-workshop.html
McKay, M. D. , Beckman, R. J. , and Conover, W. J. , 2000, “ A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code,” Technometrics, 42(1), pp. 55–61. [CrossRef]
Stein, M. , 1987, “ Large Sample Properties of Simulations Using Latin Hypercube Sampling,” Technometrics, 29(2), pp. 143–151. [CrossRef]
Bucher, C. , 2009, Computational Analysis of Randomness in Structural Mechanics (Structures and Infrastructures Series), Vol. 3, CRC Press/Balkema, Leiden, The Netherlands. [CrossRef]
Most, T. , and Will, J. , 2011, “ Sensitivity Analysis Using the Metamodel of Optimal Prognosis,” Weimar Optimization and Stochastic Days, Weimar, Germany, Nov. 24–25, pp. 1–16. https://www.researchgate.net/publication/228532931_Sensitivity_analysis_using_the_Metamodel_of_Optimal_Prognosis
Beschorner, A. , Voigt, M. , and Vogeler, K. , 2014, “ Monte Carlo Cross-Validation for Response Surface Benchmark,” 12th International Probabilistic Workshop, Weimar, Germany, Nov. 5–8, pp. 43–54. https://tu-dresden.de/ing/maschinenwesen/ism/tfa/forschung/publikationen/2014/monte_carlo_cross_validation_for_response_surface_benchmark/document_view?set_language=en
Papadimitriou, D. , and Giannakoglou, K. , 2008, “ Computation of the Hessian Matrix in Aerodynamic Inverse Design Using Continuous Adjoint Formulations,” Comput. Fluids, 37(8), pp. 1029–1039. [CrossRef]


Grahic Jump Location
Fig. 1

Model of two-stage compressor

Grahic Jump Location
Fig. 2

Pressure coefficient over axial chord

Grahic Jump Location
Fig. 3

Sensitivity ranking of the adjoint method

Grahic Jump Location
Fig. 4

Anthill plot of total pressure loss of stator 3 over rotor 3

Grahic Jump Location
Fig. 5

Pitchwise-averaged axial whirl angle over blade height: (a) and (b) rotor 3

Grahic Jump Location
Fig. 6

Sensitivity ranking of the CoI

Grahic Jump Location
Fig. 7

Comparison of the sensitivity ranking for adjoint method and MCS + PA



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