0
Research Papers

Unsteady Adjoint of Pressure Loss for a Fundamental Transonic Turbine Vane

[+] Author and Article Information
Chaitanya Talnikar

Aerospace Computational Design Laboratory,
Department of Aerospace and Astronautics,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: talnikar@mit.edu

Qiqi Wang

Aerospace Computational Design Laboratory,
Department of Aerospace and Astronautics,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: qiqi@mit.edu

Gregory M. Laskowski

Engineering Technologies,
GE Aviation,
Lynn, MA 01910
e-mail: laskowski@ge.com

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received August 29, 2016; final manuscript received September 9, 2016; published online November 8, 2016. Editor: Kenneth Hall.

J. Turbomach 139(3), 031001 (Nov 08, 2016) (10 pages) Paper No: TURBO-16-1217; doi: 10.1115/1.4034800 History: Received August 29, 2016; Revised September 09, 2016

High-fidelity simulations, e.g., large eddy simulation (LES), are often needed for accurately predicting pressure losses due to wake mixing and boundary layer development in turbomachinery applications. An unsteady adjoint of high-fidelity simulations is useful for design optimization in such aerodynamic applications. In this paper, we present unsteady adjoint solutions using a large eddy simulation model for an inlet guide vane from von Karman Institute (VKI) using aerothermal objectives. The unsteady adjoint method is effective in capturing the gradient for a short time interval aerothermal objective, whereas the method provides diverging gradients for long time-averaged thermal objectives. As the boundary layer on the suction side near the trailing edge of the vane is turbulent, it poses a challenge for the adjoint solver. The chaotic dynamics cause the adjoint solution to diverge exponentially from the trailing edge region when solved backward in time. This results in the corruption of the sensitivities obtained from the adjoint solutions. An energy analysis of the unsteady compressible Navier–Stokes adjoint equations indicates that adding artificial viscosity to the adjoint equations can dissipate the adjoint energy while potentially maintaining the accuracy of the adjoint sensitivities. Analyzing the growth term of the adjoint energy provides a metric for identifying the regions in the flow where the adjoint term is diverging. Results for the vane obtained from simulations performed on the Titan supercomputer are demonstrated.

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

References

Gourdain, N. , Gicquel, L. Y. , and Collado, E. , 2012, “ Comparison of RANS and LES for Prediction of Wall Heat Transfer in a Highly Loaded Turbine Guide Vane,” J. Propul. Power, 28(2), pp. 423–433. [CrossRef]
Jameson, A. , 1995, “ Optimum Aerodynamic Design Using CFD and Control Theory,” AIAA Paper No. 95-1729.
Lyu, Z. , and Martins, J. R. , 2014, “ Aerodynamic Design Optimization Studies of a Blended-Wing-Body Aircraft,” J. Aircr., 51(5), pp. 1604–1617. [CrossRef]
Economon, T. D. , Palacios, F. , and Alonso, J. J. , 2013, “ A Viscous Continuous Adjoint Approach for the Design of Rotating Engineering Applications,” AIAA Paper No. 2013-2580.
Wang, Q. , and Gao, J.-H. , 2013, “ The Drag-Adjoint Field of a Circular Cylinder Wake at Reynolds Numbers 20, 100 and 500,” J. Fluid Mech., 730, pp. 145–161. [CrossRef]
Blonigan, P. , Chen, R. , Wang, Q. , and Larsson, J. , 2012, “ Towards Adjoint Sensitivity Analysis of Statistics in Turbulent Flow Simulation,” Stanford Center of Turbulence Research Summer Program, p. 229.
Aceves, A. , Adachihara, H. , Jones, C. , Lerman, J. C. , McLaughlin, D. W. , Moloney, J. V. , and Newell, A. C. , 1986, “ Chaos and Coherent Structures in Partial Differential Equations,” Phys. D: Nonlinear Phenom., 18(1), pp. 85–112. [CrossRef]
Wilcox, D. C. , 1998, Turbulence Modeling for CFD, Vol. 2, DCW Industries, La Canada, CA.
Arts, T. , and de Rouvroit, M. L. , 1992, “ Aero-Thermal Performance of a Two-Dimensional Highly Loaded Transonic Turbine Nozzle Guide Vane: A Test Case for Inviscid and Viscous Flow Computations,” ASME J. Turbomach., 114(1), pp. 147–154. [CrossRef]
Lea, D. J. , Allen, M. R. , and Haine, T. W. , 2000, “ Sensitivity Analysis of the Climate of a Chaotic System,” Tellus A, 52(5), pp. 523–532. [CrossRef]
Thuburn, J. , 2005, “ Climate Sensitivities Via a Fokker–Planck Adjoint Approach,” Q. J. R. Meteorol. Soc., 131(605), pp. 73–92. [CrossRef]
Ruelle, D. , 2008, “ Differentiation of SRB States for Hyperbolic Flows,” Ergodic Theory Dyn. Syst., 28(02), pp. 613–631. [CrossRef]
Ruelle, D. , 2009, “ A Review of Linear Response Theory for General Differentiable Dynamical Systems,” Nonlinearity, 22(4), p. 855. [CrossRef]
Blonigan, P. , Gomez, S. , and Wang, Q. , 2014, “ Least Squares Shadowing for Sensitivity Analysis of Turbulent Fluid Flows,” preprint arXiv:1401.4163.
Garnier, E. , Adams, N. , and Sagaut, P. , 2009, Large Eddy Simulation for Compressible Flows, Springer Science & Business Media, Medford, MA.
Moeng, C.-H. , and Wyngaard, J. C. , 1989, “ Evaluation of Turbulent Transport and Dissipation Closures in Second-Order Modeling,” J. Atmos. Sci., 46(14), pp. 2311–2330. [CrossRef]
Macdonald, C. B. , 2003, “ Constructing High-Order Runge-Kutta Methods With Embedded Strong-Stability-Preserving Pairs,” Ph.D. thesis, Simon Fraser University, Burnaby, BC.
Roe, P. L. , 1981, “ Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes,” J. Comput. Phys., 43(2), pp. 357–372. [CrossRef]
Choi, H. , and Moin, P. , 2012, “ Grid-Point Requirements for Large Eddy Simulation: Chapman’s Estimates Revisited,” Phys. Fluids (1994-Present), 24(1), p. 011702. [CrossRef]
Bastien, F. , Lamblin, P. , Pascanu, R. , Bergstra, J. , Goodfellow, I . J. , Bergeron, A. , Bouchard, N. , and Bengio, Y. , 2012, “ Theano: New Features and Speed Improvements,” Deep Learning and Unsupervised Feature Learning NIPS 2012 Workshop, p. 5590.
Bergstra, J. , Breuleux, O. , Bastien, F. , Lamblin, P. , Pascanu, R. , Desjardins, G. , Turian, J. , Warde-Farley, D. , and Bengio, Y. , 2010, “ Theano: A CPU and GPU Math Expression Compiler,” Python for Scientific Computing Conference (SciPy).
Abarbanel, S. , and Gottlieb, D. , 1981, “ Optimal Time Splitting for Two-and Three-Dimensional Navier-Stokes Equations With Mixed Derivatives,” J. Comput. Phys., 41(1), pp. 1–33. [CrossRef]
Ford, W. F. , and Sidi, A. , 1987, “ An Algorithm for a Generalization of the Richardson Extrapolation Process,” SIAM J. Numer. Anal., 24(5), pp. 1212–1232. [CrossRef]
Celik, I. , and Zhang, W.-M. , 1995, “ Calculation of Numerical Uncertainty Using Richardson Extrapolation: Application to Some Simple Turbulent Flow Calculations,” ASME J. Fluids Eng., 117(3), pp. 439–445. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Turbine vane geometry

Grahic Jump Location
Fig. 2

Contour plot of shear stress on the surface of the vane. Nondimensionalized with respect to flow velocity Mach 0.9. Reynolds number 1 × 106, turbulent intensity 1%.

Grahic Jump Location
Fig. 3

Stagnation pressure on a vertical spanwise cross section 10 mm downstream of the trailing edge of the vane. Nondimensionalized with respect to inlet stagnation pressure.

Grahic Jump Location
Fig. 4

Instantaneous and time-averaged stagnation pressure loss coefficient for the vane. X-axis denotes the time normalized by the time it takes for the flow to pass from the inlet to outlet.

Grahic Jump Location
Fig. 5

Turbine vane computational domain

Grahic Jump Location
Fig. 6

A visualization of the density adjoint field from halfway through a short time 3D unsteady adjoint simulation

Grahic Jump Location
Fig. 7

Growth of energy norm of adjoint fields for different simulations. Y-axis shows energy norm of a dimensional conservative adjoint field.

Grahic Jump Location
Fig. 8

Contour plot of divergence indicator σ1 for the turbine vane, normalized by inverse of a single flow through time

Grahic Jump Location
Fig. 9

Growth of energy norm of adjoint for various scaling factors. X-axis is time normalized by a single flow through time.

Grahic Jump Location
Fig. 10

Error in adjoint sensitivity for various scaling factors

Grahic Jump Location
Fig. 11

Density adjoint solution at t = 0

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