Research Papers

Development and Use of Machine-Learnt Algebraic Reynolds Stress Models for Enhanced Prediction of Wake Mixing in Low-Pressure Turbines

[+] Author and Article Information
H. D. Akolekar

Department of Mechanical Engineering,
University of Melbourne,
Parkville 3010, VIC, Australia
e-mail: hakolekar@student.unimelb.edu.au

J. Weatheritt, N. Hutchins, R. D. Sandberg

Department of Mechanical Engineering,
University of Melbourne,
Parkville 3010, VIC, Australia

G. Laskowski

General Electric Aviation,
Lynn, MA 01905
e-mail: laskowsk@ge.com

V. Michelassi

Baker Hughes, a GE Company,
Florence 50127, Italy
e-mail: vittorio.michelassi@bhge.com

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received October 3, 2018; final manuscript received October 14, 2018; published online March 7, 2019. Editor: Kenneth Hall.

J. Turbomach 141(4), 041010 (Mar 07, 2019) (11 pages) Paper No: TURBO-18-1280; doi: 10.1115/1.4041753 History: Received October 03, 2018; Revised October 14, 2018

Nonlinear turbulence closures were developed that improve the prediction accuracy of wake mixing in low-pressure turbine (LPT) flows. First, Reynolds-averaged Navier–Stokes (RANS) calculations using five linear turbulence closures were performed for the T106A LPT profile at isentropic exit Reynolds numbers 60,000 and 100,000. None of these RANS models were able to accurately reproduce wake loss profiles, a crucial parameter in LPT design, from direct numerical simulation (DNS) reference data. However, the recently proposed kv2¯ω transition model was found to produce the best agreement with DNS data in terms of blade loading and boundary layer behavior and thus was selected as baseline model for turbulence closure development. Analysis of the DNS data revealed that the linear stress–strain coupling constitutes one of the main model form errors. Hence, a gene-expression programming (GEP) based machine-learning technique was applied to the high-fidelity DNS data to train nonlinear explicit algebraic Reynolds stress models (EARSM), using different training regions. The trained models were first assessed in an a priori sense (without running any RANS calculations) and showed much improved alignment of the trained models in the region of training. Additional RANS calculations were then performed using the trained models. Importantly, to assess their robustness, the trained models were tested both on the cases they were trained for and on testing, i.e., previously not seen, cases with different flow features. The developed models improved prediction of the Reynolds stress, turbulent kinetic energy (TKE) production, wake-loss profiles, and wake maturity, across all cases.

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Mayle, R. E. , 1991, “ The 1991 IGTI Scholar Lecture: The Role of Laminar-Turbulent Transition in Gas Turbine Engines,” ASME J. Turbomach., 113(4), pp. 509–536.
Hodson, H. P. , and Howell, R. J. , 2005, “ Bladerow Interactions, Transition, and High-Lift Aerofoils in Low-Pressure Turbines,” Annu. Rev. Fluid Mech., 37(1), pp. 71–98.
Stieger, R. D. , and Hodson, H. P. , 2003, “ The Transition Mechanism of Highly-Loaded LP Turbine Blades,” ASME Paper No. GT2003-38304.
Pacciani, R. , Marconcini, M. , Arnone, A. , and Bertini, F. , 2013, “ Predicting High-Lift Low-Pressure Turbine Cascades Flow Using Transition-Sensitive Turbulence Closures,” ASME J. Turbomach., 136(5), p. 051007.
Keadle, K. , and Mcquilling, M. , 2013, “ Evaluation of RANS Transition Modeling for High Lift LPT Flows at Low Reynolds Number,” ASME Paper No. GT2013–95069.
Praisner, T. J. , Clark, J. P. , Nash, T. C. , Rice, M. J. , and Grover, E. A. , 2006, “ Performance Impacts Due to Wake Mixing in Axial Flow Turbomachinery,” ASME Paper No. GT2006-90666.
Schmitt, G. , 2007, “ About Boussinesq's Turbulent Viscosity Hypothesis: Historical Remarks and a Direct Evaluation of Its Validity,” C. R. Mec. Elsevier Masson, 335(9–10), pp. 617–627.
Leschziner, M. , 2015, Statistical Turbulence Modelling for Fluid Dynamics - Demystified: An Introductory Text for Graduate Engineering Students, World Scientific, London.
Rodi, W. , 1976, “ A New Algebraic Relation for Calculating the Reynolds Stresses,” Z. Angew. Math. Mech., 56, pp. T219–T221.
Gatski, T. , and Speziale, C. , 1993, “ On Explicit Algebraic Stress Models for Complex Turbulent Flows,” J. Fluid Mech., 254(1), pp. 59–78.
Wallin, S. , and Johansson, A. V. , 2000, “ An Explicit Algebraic Reynolds Stress Model for Incompressible and Compressible Turbulent Flows,” J. Fluid Mech., 403, pp. 89–132.
Wang, J.-X. , Wu, J.-L. , and Xiao, H. , 2017, “ A Physics Informed Machine Learning Approach for Reconstructing Reynolds Stress Modeling Discrepancies Based on DNS Data,” Phys. Rev. Fluids, 2(3), p. 034603.
Duraisamy, K. , Zhang, Z. J. , and Singh, A. P. , 2015, “ New Approaches in Turbulence and Transition Modeling Using Data-Driven Techniques,” AIAA Paper No. AIAA 2015-1284.
Tracey, B. D. , Duraisamy, K. , and Alonso, J. J. , 2015, “ A Machine Learning Strategy to Assist Turbulence Model Development,” AIAA Paper No. AIAA 2015-1287.
Ferreira, C. , 2001, “ Gene Expression Programming: A New Adaptive Algorithm for Solving Problems,” Complex Syst., 13(2), pp. 87–129.
Weatheritt, J. , and Sandberg, R. D. , 2016, “ A Novel Evolutionary Algorithm Applied to Algebraic Modifications of the RANS Stress-Strain Relationship,” J. Comput. Phys., 325, pp. 22–37.
Weatheritt, J. , and Sandberg, R. D. , 2017, “ The Development of Algebraic Stress Models Using a Novel Evolutionary Algorithm,” Int. J. Heat Fluid Flow, 68, pp. 298–318.
Weatheritt, J. , Pichler, R. , Sandberg, R. D. , Laskowski, G. , and Michelassi, V. , 2017, “ Machine Learning for Turbulence Model Development Using a High Fidelity HPT Cascade Simulation,” ASME Paper No. GT2017–63497.
Lopez, M. , and Walters, D. K. , 2016, “ Prediction of Transitional and Fully Turbulent Flow Using an Alternative to the Laminar Kinetic Energy Approach,” J. Turbul., 17(3), pp. 253–273.
Michelassi, V. , Chen, L. W. , Pichler, R. , and Sandberg, R. D. , 2015, “ Compressible Direct Numerical Simulation of Low Pressure Turbine—Part II: Effect of Inflow Disturbances,” ASME J. Turbomach., 137(7), p. 071005.
Pope, S. B. , 1975, “ A More General Effective-Viscosity Hypothesis,” J. Fluid Mech., 72(2), pp. 331–340.
Sandberg, R. D. , Michelassi, V. , Pichler, R. , Chen, L. , and Johnstone, R. , 2015, “ Compressible Direct Numerical Simulation of Low-Pressure Turbines—Part I: Methodology,” ASME J. Turbomach., 137(5), p. 51011.
Stadtmüller, P. , and Fottner, L. , 2001, “ A Test Case for the Numerical Investigation of Wake Passing Effects on a Highly Loaded LP Turbine Cascade Blade,” ASME Paper No. 2001-GT-0311.
Menter, F. R. , 1994, “ Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications,” AIAA J., 32(8), pp. 1598–1605.
Spalart, P. R. , and Allmaras, S. R. , 1992, “ A One-Equation Turbulence Model for Aerodynamic Flows,” 30th Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan. 6–9, p. 439.
Langtry, R. B. , and Menter, F. R. , 2009, “ Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes,” AIAA J., 47(12), pp. 2894–2906.
Walters, D. K. , and Cokljat, D. , 2008, “ A Three-Equation Eddy-Viscosity Model for Reynolds-Averaged Navier-Stokes Simulations of Transitional Flow,” ASME J. Fluids Eng., 130(12), p. 121401.
Mayle, R. , and Schulz, A. , 1997, “ The Path to Predicting Bypass Transition,” ASME J. Turbomach., 119(3), pp. 405–411.
Dick, E. , and Kubacki, S. , 2017, “ Transition Models for Turbomachinery Boundary Layer Flows: A Review,” Int. J. Turbomach., Propuls. Power, 2(2), p. 4.
Lopez, M. , and Walters, D. K. , 2017, “ A Recommended Correction to the kT-kL-Omega Transition-Sensitive Eddy-Viscosity Model,” ASME J. Fluids Eng., 139(2), p. 024501.
Parneix, S. , Laurence, D. , and Durbin, P. A. , 1998, “ A Procedure for Using DNS Databases,” ASME J. Fluids Eng., 120(1), pp. 40–46.
Bode, C. , Aufderheide, T. , Friedrichs, J. , and Kozulovic, D. , 2014, “ Improved Turbulence and Transition Prediction for Turbomachinery Flows,” ASME Paper No. IMECE2014–36866.
Akolekar, H. D. , Sandberg, R. D. , Hutchins, N. , Michelassi, V. , and Laskowski, G. , 2018, “ Machine-Learnt Turbulence Closures for LPTs With Unsteady Inflow Conditions,” 15th ISUAAAT, Oxford, UK, Sept. 24–29, Paper No. ISUAAAT-019.


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Fig. 1

Sketch of the T106A blade geometry and domain boundaries

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Fig. 2

Pressure coefficient distribution across the blade at Re2is=60,000. Inset showed zoomed view of suction side (0.7 ≤x/Cax≤1).

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Fig. 3

Wall shear stress on the suction side at Re2is=60,000. Inset shows zoomed view of (0.79 ≤x/Cax≤1).

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Fig. 4

Displacement thickness, momentum thickness, and shape factor on the suction side at Re2is=60,000

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Fig. 5

Magnitude of velocity contours depicting open and closed separation bubbles at (a) Re2is=60,000 and (b) Re2is=100,000, generated using the kv2¯ω model

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Fig. 6

Wake loss profiles for (a) Re2is = 60,000 and (b) Re2is = 100,000, at 40% chord downstream of the trailing edge

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Fig. 7

GEP algorithm flow depicting inputs and output

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Fig. 8

Training regions for model development

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Fig. 9

Histogram of the GEP cost function (J) at (a) Re2is = 60,000 and (b) Re2is = 100,000

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Fig. 10

(a) Alignment of anisotropy and negative strain for the linear Boussinesq approximation and (b) alignment contour with the linear Boussinesq approximation and 60k-Wake model applied in the regions x/C < 0.95 and x/C ≥ 0.95 respectively, at Re2is = 60,000. Additional contour at γ = 0.75.

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Fig. 11

Shear component of Reynolds stress at 40% chord downstream for (a) Re2is=60,000 and (b) Re2is=100,000

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Fig. 12

Production of TKE at 40% chord downstream for (a) Re2is=60,000 and (b) Re2is=100,000

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Fig. 13

Wake loss profiles at 40% chord downstream for (a) Re2is = 60,000 and (b) Re2is = 100,000

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Fig. 14

Nondimensionalized wake maturity for (a) Re2is = 60,000 and (b) Re2is = 100,000



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