0
Research Papers

Employing the Time-Domain Unsteady Discrete Adjoint Method for Shape Optimization of Three-Dimensional Multirow Turbomachinery Configurations

[+] Author and Article Information
Georgios Ntanakas

Rolls-Royce Deutschland,
Blankenfelde-Mahlow 15827, Germany;
Parallel CFD & Optimization Unit,
Lab of Thermal Turbomachines,
School of Mechanical Engineering,
National Technical University of Athens,
Athens 15780, Greece
e-mail: gntanak@central.ntua.gr

Marcus Meyer

Rolls-Royce Deutschland,
Blankenfelde-Mahlow 15827, Germany
e-mail: marcus.meyer@rolls-royce.com

Kyriakos C. Giannakoglou

Parallel CFD & Optimization Unit,
Lab of Thermal Turbomachines,
School of Mechanical Engineering,
National Technical University of Athens,
Athens 15780, Greece
e-mail: kgianna@central.ntua.gr

Manuscript received December 3, 2017; final manuscript received June 12, 2018; published online July 26, 2018. Assoc. Editor: Li He.

J. Turbomach 140(8), 081006 (Jul 26, 2018) (11 pages) Paper No: TURBO-17-1229; doi: 10.1115/1.4040564 History: Received December 03, 2017; Revised June 12, 2018

In turbomachinery, the steady adjoint method has been successfully used for the computation of derivatives of various objective functions with respect to design variables in gradient-based optimization. However, the continuous advances in computing power and the accuracy limitations of the steady-state assumption lead toward the transition to unsteady computational fluid dynamics (CFD) computations in the industrial design process. Previous work on unsteady adjoint for turbomachinery applications almost exclusively rely upon frequency-domain methods, for both the flow and adjoint equations. In contrast, in this paper, the development the discrete adjoint to the unsteady Reynolds-averaged Navier–Stokes (URANS) solver for three-dimensional (3D) multirow applications, in the time-domain, is presented. The adjoint equations are derived along with the adjoint to the five-stage Runge–Kutta scheme. Communication between adjacent rows is achieved by the adjoint sliding interface method. An optimization workflow that uses unsteady flow and adjoint solvers is presented and tested in two cases, with objective functions accounting for the transient flow in a turbine vane and the periodic flow in a compressor three-row setup.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Giles, M. B. , and Pierce, N. A. , 2000, “ An Introduction to the Adjoint Approach to Design,” Flow, Turbul. Combust., 65(3–4), pp. 393–415. [CrossRef]
Jameson, A. , and Pierce, N. , and L. M. , 1988, “ Aerodynamic Design Via Control Theory,” Theor. Comput. Fluid Dyn., 3(3), pp. 233–260.
Vanderplaats, G. N. , Hicks, R. M. , and Murman, E. M. , 1975, “ Application of Numerical Optimization Techniques to Airfoil Design,” NASA Conference on Aerodynamic Analyses Requiring Advanced Computers, Paper No. NASA SP-347.
Baysal, O. , and Eleshaky, M. E. , 1991, “ Aerodynamic Sensitivity Analysis Methods for the Compressible Euler Equations,” ASME J. Fluids Eng., 113(4), pp. 681–688. [CrossRef]
Jameson, A. , Shankaran, S. , and Martinelli, L. , 2008, “ Continuous Adjoint Method for Unstructured Grids,” AIAA J., 46(5), pp. 1226–1239. [CrossRef]
Papoutsis-Kiachagias, E. M. , and Giannakoglou, K. C. , 2016, “ Continuous Adjoint Methods for Turbulent Flows, Applied to Shape and Topology Optimization: Industrial Applications,” Arch. Comput. Methods Eng., 23(2), pp. 255–299. [CrossRef]
Giannakoglou, K. C. , Papadimitriou, D. I. , Papoutsis-Kiachagias, E. M. , and Kavvadias, I. S. , 2015, “ Aerodynamic Shape Optimization Using “Turbulent” Adjoint and Robust Design in Fluid Mechanics,” Engineering and Applied Sciences Optimization, Springer International Publishing, Cham, Switzerland, pp. 289–309. [CrossRef]
Giles, M. B. , Duta, M. C. , Müller, J.-D. , and Pierce, N. A. , 2003, “ Algorithm Developments for Discrete Adjoint Methods,” AIAA J., 41(2), pp. 198–205. [CrossRef]
Mavriplis, D. J. , 2007, “ Discrete Adjoint-Based Approach for Optimization Problems on Three-Dimensional Unstructured Meshes,” AIAA J., 45(4), pp. 741–750. [CrossRef]
Peter, J. E. V. , and Dwight, R. P. , 2010, “ Numerical Sensitivity Analysis for Aerodynamic Optimization: A Survey of Approaches,” Comput. Fluids, 39(3), pp. 373 –391. [CrossRef]
Denton, J. D. , 1992, “ The Calculation of Three-Dimensional Viscous Flow Through Multistage Turbomachines,” ASME J. Turbomach., 114(1), pp. 18–26. [CrossRef]
He, L. , and Wang, D. X. , 2010, “ Concurrent Blade Aerodynamic-Aero-Elastic Design Optimization Using Adjoint Method,” ASME J. Turbomach., 133(1), p. 011021. [CrossRef]
Ma, C. , Su, X. , and Yuan, X. , 2016, “ An Efficient Unsteady Adjoint Optimization System for Multistage Turbomachinery,” ASME J. Turbomach., 139(1), p. 011003. [CrossRef]
Duta, M. C. , Giles, M. B. , and Campobasso, M. S. , 2002, “ The Harmonic Adjoint Approach to Unsteady Turbomachinery Design,” Int. J. Numer. Methods Fluids, 40(3–4), pp. 323–332. [CrossRef]
Yi, J. , and Capone, L. , “ Adjoint-Based Sensitivity Analysis for Unsteady Bladerow Interaction Using Space-Time Gradient Method,” ASME J. Turbomach., 139(11), p. 111008. [CrossRef]
Nemili, A. , Özkaya, E. , Gauger, N. R. , Kramer, F. , Hoell, T. , and Thiele, F. , 2014, “ Optimal Design of Active Flow Control for a Complex High-Lift Configuration,” AIAA Paper No. 2014-2515.
Thomas, J. P. , Hall, K. C. , and Dowell, E. H. , 2005, “ Discrete Adjoint Approach for Modeling Unsteady Aerodynamic Design Sensitivities,” AIAA J., 43(9), pp. 1931–1936. [CrossRef]
Mani, K. , and Mavriplis, D. J. , 2009, “ Adjoint-Based Sensitivity Formulation for Fully Coupled Unsteady Aeroelasticity Problems,” AIAA J., 47(8), pp. 1902–1915. [CrossRef]
Mishra, A. , Mavriplis, D. , and Sitaraman, J. , 2016, “ Time-Dependent Aeroelastic Adjoint-Based Aerodynamic Shape Optimization of Helicopter Rotors in Forward Flight,” AIAA J., 54(12), pp. 3813–3827. [CrossRef]
Nielsen, E. J. , Diskin, B. , and Yamaleev, N. K. , 2010, “ Discrete Adjoint-Based Design Optimization of Unsteady Turbulent Flows on Dynamic Unstructured Grids,” AIAA J., 48(6), pp. 1195–1206. [CrossRef]
Nielsen, E. J. , and Diskin, B. , 2013, “ Discrete Adjoint-Based Design for Unsteady Turbulent Flows on Dynamic Overset Unstructured Grids,” AIAA J., 51(6), pp. 1355–1373. [CrossRef]
Ntanakas, G. , and Meyer, M. , 2014, “ Towards Unsteady Adjoint Analysis for Turbomachinery Applications,” Sixth European Conference on Computational Fluid Dynamics (ECFD), Barcelona, Spain, July 20–25, pp. 5071–5081. http://congress.cimne.com/iacm-eccomas2014/admin/files/fileabstract/a1557.pdf
Ntanakas, G. , and Meyer, M. , 2015, “ The Unsteady Discrete Adjoint Method for Turbomachinery Applications,” 14th International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (SUAAAT'14), Stockholm, Sweden, Sept. 8–11, pp. 1–8. https://zenodo.org/record/823136#.WzC2l1UzbZ4
Lapworth, L. , 2004, “ Hydra-CFD: A Framework for Collaborative CFD Development,” International Conference on Scientific and Engineering Computation, Singapore.
Spalart, P. R. , and Allmaras, S. , 1992, “ A One-Equation Turbulence Model for Aerodynamic Flows,” AIAA Paper No. 1992-0439.
Wesseling, P. , and Oosterlee, C. W. , 2001, “ Geometric Multigrid With Applications to Computational Fluid Dynamics,” J. Comput. Appl. Math., 128(1–2), pp. 311–334. [CrossRef]
Martinelli, L. , 1987, “ Calculations of Viscous Flows With a Multigrid Method,” Ph.D. thesis, Princeton University, Princeton, NJ. http://adsabs.harvard.edu/abs/1987PhDT........45M
Jameson, A. , Schmidt, W. , and Turkel, E. , 1981, “ Numerical Solution of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping Schemes,” AIAA Paper No. 1981-1259.
Moinier, P. , 1999, “ Algorithm Developments for an Unstructured Viscous Flow Solver,” Ph.D. thesis, University of Oxford, Oxford, UK. http://people.maths.ox.ac.uk/gilesm/files/pierre_thesis.pdf
Giles, M. B. , 2000, “ On the Use of Runge-Kutta Time-Marching and Multigrid for the Solution of Steady Adjoint Equations,” Oxford University Computing Laboratory, Oxford, UK, Technical Report No. NA00/10. https://ora.ox.ac.uk/objects/uuid:8866ebe2-c76f-41d4-817e-04aa078d1858
Griewank, A. , and Walther, A. , 2008, Evaluating Derivatives, Society for Industrial and Applied Mathematics, Philadelphia, PA. [CrossRef]
Hascoet, L. , and Pascual, V. , 2013, “ The Tapenade Automatic Differentiation Tool: Principles, Model, and Specification,” ACM Trans. Math. Software, 39(3), pp. 20:1–20:43. [CrossRef]
Milli, A. , and Shahpar, S. , 2012, “ PADRAM: Parametric Design and Rapid Meshing System for Complex Turbomachinery Configurations,” ASME Paper No. GT2012-69030.
Hills, N. , 2007, “ Achieving High Parallel Performance for an Unstructured Unsteady Turbomachinery CFD Code,” Aeronaut. J., 111(1117), pp. 185–193. [CrossRef]
Rogers, S. E. , Suhs, N. E. , and Dietz, W. E. , 2003, “ Pegasus 5: An Automated Preprocessor for Overset-Grid Computational Fluid Dynamics,” AIAA J., 41(6), pp. 1037–1045. [CrossRef]
Crumpton, P. I. , and Giles, M. B. , 1996, “ Multigrid Aircraft Computations Using the OPlus Parallel Library,” Parallel Comput. Fluid Dyn., pp. 339–346.
Nimmagadda, S. , Economon, T. D. , Alonso, J. J. , Silva, C. , Zhou, B. Y. , and Albring, T. , 2018, “ Low-Cost Unsteady Discrete Adjoints for Aeroacoustic Optimization Using Temporal and Spatial Coarsening Techniques,” AIAA Paper No. 2018-1911.
Meyer, C. D. , 2000, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, pp. 429–430. [CrossRef]
Yamaleev, N. , Diskin, B. , and Nielsen, E. , 2010, “ Local-in-Time Adjoint-Based Method for Design Optimization of Unsteady Flows,” J. Comput. Phys., 229(14), pp. 5394–5407. [CrossRef]
Schönweitz, D. , Voges, M. , Goinis, G. , Enders, G. , and Johann, E. , 2013, “ Experimental and Numerical Examinations of a Transonic Compressor-Stage With Casing Treatment,” ASME Paper No. GT2013-95550.
Schmitz, A. , Aulich, M. , Schönweitz, D. , and Nicke, E. , 2012, “ Novel Performance Prediction of a Transonic 4.5 Stage Compressor,” ASME Paper No. GT2012-69003.

Figures

Grahic Jump Location
Fig. 1

Simplified two-dimensional sliding interface at a radial (a) and an axial (b) cut (thick dashed line: interior sliding plane (donors), thick line: exterior sliding plane (receivers), arrows: interpolation direction)

Grahic Jump Location
Fig. 2

Schematic of the temporal coarsening technique used for the unsteady adjoint solver

Grahic Jump Location
Fig. 3

Turbine vane: geometry and blade-to-blade grid at midspan

Grahic Jump Location
Fig. 4

Turbine vane: inlet and outlet flow condition profiles

Grahic Jump Location
Fig. 5

Turbine vane: (a) instantaneous pressure ratio values and the L2 norm of the adjoint field. The two vertical dotted lines define the time interval over which the unsteady objective function is defined (from the 750th to the 800th time-step). (b) Typical convergence plot of the flow and adjoint equations in pseudo-time at an arbitrarily selected time-step.

Grahic Jump Location
Fig. 6

Turbine vane: (a) comparison of gradients computed using the unsteady adjoint method and finite differences and (b) optimization of the total pressure ratio

Grahic Jump Location
Fig. 7

Turbine vane: baseline (light) versus improved (dark) vane geometries: (a) pressure side and (b) suction side

Grahic Jump Location
Fig. 8

Rig250 compressor: cross section of the compressor and the selected configuration

Grahic Jump Location
Fig. 9

Rig250 compressor: geometry and blade-to-blade grids at midspan

Grahic Jump Location
Fig. 10

Rig250 compressor: inlet and outlet flow condition profiles

Grahic Jump Location
Fig. 11

Rig250 compressor: Steady (mixing interface; (a)) and unsteady (sliding interface; (b)) Spalart–Allmaras variable fields at midspan

Grahic Jump Location
Fig. 12

Rig250 compressor: steady (mixing interface; (a)) and unsteady (sliding interface; (b)) adjoint Spalart–Allmaras variable fields at midspan. Objective function: axial force on the R3 blade.

Grahic Jump Location
Fig. 13

Rig250 compressor: steady (a), mean (b), and instantaneous unsteady (c) sensitivity maps plotted over the pressure side of an S2 blade. Objective function: axial force on the R3 blade.

Grahic Jump Location
Fig. 14

Rig250 compressor: value change (%) of objective and constraint functions during the first three optimization steps. Objective function: axial force on rotor blade. Constraints: exit capacity and total pressure coefficient.

Grahic Jump Location
Fig. 15

Rig250 compressor: baseline (gray) versus improved (green) stator blade and rotor blade geometries: (a) stator blade (S2; view from the leading edge) and (b) rotor blade (R3; view from the trailing edge)

Grahic Jump Location
Fig. 16

Rig250 compressor: Baseline versus improved blade's profiles near hub (5% blade's span) and tip (95% blade's span): (a) S2 near tip, (b) S2 near hub, (c) R3 near tip, and (d) R3 near hub

Grahic Jump Location
Fig. 17

Rig250 compressor: gradients computed with the temporal coarsening method compared with the reference gradients. Objective function: axial force on the R3 blade.

Tables

Errata

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