Research Papers

Simulating Periodic Unsteady Flows Using a Cubic-Spline-Based Time Collocation Method

[+] Author and Article Information
Pengcheng Du

e-mail: dupengcheng22@163.com

Fangfei Ning

e-mail: fangfei.ning@buaa.edu.cn
School of Energy and Power Engineering,
Beihang University,
National Key Laboratory of
Science and Technology on Aero-Engines,
37 Xueyuan Road,
Beijing 100191, China

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received June 30, 2013; final manuscript received July 4, 2013; published online October 23, 2013. Editor: Ronald Bunker.

J. Turbomach 136(4), 041014 (Oct 23, 2013) (12 pages) Paper No: TURBO-13-1120; doi: 10.1115/1.4025203 History: Received June 30, 2013; Revised July 04, 2013

Time-periodical unsteady flows are typical in turbomachinery. Simulating such flows using a conventional time marching approach is the most accurate but is extremely time consuming. In order to achieve a better balance between accuracy and computational expenses, a cubic-spline-based time collocation method is proposed. In this method, the time derivatives in the Navier–Stokes equations are obtained by using the differential quadrature method, in which the periodical flow variables are approximated by cubic splines. Thus, the computation of a time-periodical flow is substituted by several coupled quasi-steady flow computations at sampled instants. The proposed method is then validated against several typical turbomachinery periodical unsteady flows, i.e., transonic compressor rotor flows under circumferential inlet distortions, single stage rotor–stator interactions, and IGV–rotor interactions. The results show that the proposed cubic-spline-based time collocation method with appropriate time sampling can well resolve the dominant unsteady effects, while the computational expenses are kept much less than the traditional time-marching simulation. More importantly, this paper provides a framework on the basis of a time collocation method in which one may choose more compatible test functions for the concerned specific unsteady flows so that better modeling of the flows can be expected.

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Denton, J. D., and Singh, U. K., 1979, “Time Marching Methods for Turbomachinery Flow Calculation,” Application of Numerical Methods to Flow Calculations in Turbomachines (VKI Lecture Series), E.Schmidt, ed., von Kármán Institute for Fluid Dynamics, Rhode-St-Genèse, Belgium.
Adamczyk, J. J., 1985, “Model Equations for Simulating Flows in Multistage Turbomachinery,” ASME Paper No. 85-GT-226.
Erdos, J. I., Alzner, E., and McNally, W., 1977, “Numerical Solution of Periodic Transonic Flow Through a Fan Stage,” AIAA J., 15(11), pp. 1559–1568. [CrossRef]
Giles, M. B., 1990, “Stator/Rotor Interaction in a Transonic Turbine,” J. Propul. Power, 6(5), pp. 621–627. [CrossRef]
He, L., 1990, “An Euler Solution for Unsteady Flows Around Oscillating Blades,” ASME J. Turbomach., 112(4), pp. 714–722. [CrossRef]
Verdon, J. M., and Caspar, J. R., 1984, “A Linearized Unsteady Aerodynamic Analysis for Transonic Cascades,” J. Fluid Mech., 149, pp. 403–429. [CrossRef]
Dufour, G., Sicot, F., Puigt, G., Liauzun, C., and Dugeai, A., 2010, “Contrasting the Harmonic Balance and Linearized Methods for Oscillating-Flap Simulations,” AIAA J., 48(4), pp. 788–797. [CrossRef]
He, L., and Ning, W., 1998, “Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines,” AIAA J., 36(11), pp. 2005–2012. [CrossRef]
Hall, K. C., Thomas, J. P., and Clark, W. S., 2002, “Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique,” AIAA J., 40(5), pp. 879–886. [CrossRef]
Gopinath, A., and Jameson, A., 2005, “Time Spectral Method for Periodic Unsteady Computations Over Two- and Three-Dimensional Bodies,” 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 10–13, AIAA Paper No. 2005-1220. [CrossRef]
Ekici, K., Hall, K. C., and Kielb, R. E., 2010, “Harmonic Balance Analysis of Blade Row Interactions in a Transonic Compressor,” J. Propul. Power, 26(2), pp. 335–343. [CrossRef]
Sicot, F., Dufour, G., and Gourdain, N., 2012, “A Time-Domain Harmonic Balance Method for Rotor/Stator Interactions,” ASME J. Turbomach., 134(1), p. 011001. [CrossRef]
Ekici, K., Hall, K. C., and Dowell, E. H., 2008, “Computationally Fast Harmonic Balance Methods for Unsteady Aerodynamic Predictions of Helicopter Rotors,” J. Comput. Phys., 227(12), pp. 6206–6225. [CrossRef]
Shu, C., 2000, Differential Quadrature and Its Application in Engineering, Springer, London.
Shu, C., and Richards, B. E., 1990, “High Resolution of Natural Convection in Square Cavity by Generalized Differential Quadrature,” Proceedings of the 3rd International Conference on Advances in Numeric Methods in Engineering: Theory and Application, Swansea, UK, pp. 978–985.
Shu, C., and Chew, Y. T., 1997, “Fourier Expansion-Based Differential Quadrature and Its Application to Helmholtz Eigenvalue Problems,” Commun. Numer. Methods Eng., 13(8), pp. 643–653. [CrossRef]
Shu, C., Khoo, B. C., and Yeo, K. S., 1994, “Numerical Solutions of Incompressible Navier-Stokes Equations by Differential Quadrature,” Finite Elem. Anal. Design, 18(1–3), pp. 83–97. [CrossRef]
Shu, C., Chew, Y. T., Khoo, B. C., and Yeo, K. S., 1996, “Solutions of Three-Dimensional Boundary Layer Equations by Global Methods of Generalized Differential-Integral Quadrature,” Int. J. Numer. Methods Heat Fluid Flow, 6(2), pp. 61–75. [CrossRef]
Shu, C., 1996, “An Efficient Approach for Free Vibration Analysis of Conical Shells,” Int. J. Mech. Sci., 38(8–9), pp. 935–949. [CrossRef]
Kouatchou, J., 2003, “Comparison of Time and Spatial Collocation Methods for Heat Equation,” J. Comput. Appl. Math., 150, pp. 129–141. [CrossRef]
Blazek, J., 2001, Computational Fluid Dynamics: Principles and Applications, Elsevier, Oxford, UK, Chap. 2.
Jameson, A., 1991, “Time Dependent Calculations Using Multigrid, With Applications to Unsteady Flows Past Airfoils and Wings,” Tenth Computational Fluid Dynamics Conference, Honolulu, HI, June 24–26, AIAA Paper No. 91-1596. [CrossRef]
Sicot, F., Puigt, G., and Montagnac, M., 2008, “Block-Jacobi Implicit Algorithms for the Time Spectral Method,” AIAA J., 46(12), pp. 3080–3089. [CrossRef]
GopinathA., 2007, “Efficient Fourier-Based Algorithms for Time-Periodic Unsteady Problems,” Ph.D. thesis, Stanford University, Stanford, CA.
Ning, F., and Xu, L., 2001, “Numerical Investigation of Transonic Compressor Rotor Flow Using an Implicit 3D Flow Solver With One-Equation Spalart-Allmaras Turbulence Model,” ASME Paper No. 2001-GT-0359.
Fan, L., Ning, F., and Liu, H., 2008, “Aerodynamics of Compressor Casing Treatment Part I: Experiment and Time-Accurate Numerical Simulation,” ASME Paper No. GT2008-51541. [CrossRef]
Edwards, J. R., 1997, “A Low-Diffusion Flux-Splitting Scheme for Navier-Stokes Calculations,” Comput. Fluids, 26(6), pp. 635–659. [CrossRef]
Spalart, P. R., and Allmaras, S. R., 1992, “A One-Equation Turbulence Transport Model for Aerodynamic Flows,” 30th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 6–9, AIAA Paper No. 92-0439. [CrossRef]
Gorrell, S. E., Okiishi, T. H., and Copenhaver, W. W., 2003, “Stator-Rotor Interactions in a Transonic Compressor—Part 2: Description of a Loss-Producing Mechanism,” ASME J. Turbomach., 125(2), pp. 336–345. [CrossRef]
Snider, A. D., 1972, “An Improved Estimate of the Accuracy of Trigonometric Interpolation,” SIAM J. Numer. Anal., 9(3), pp. 505–508. [CrossRef]
Sharma, A., and Meir, A., 1966, “Degree of Approximation of Spline Interpolation,” J. Math. Mech., 15, pp. 759–767. [CrossRef]


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Fig. 2

Spectral radius of the weighting coefficient matrix

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Fig. 1

Illustration of the time collocation method

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Fig. 5

Unsteady rotor efficiency

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Fig. 4

Convergence histories of the TCM computations

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Fig. 3

Computational mesh

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Fig. 6

Snapshots of normalized total pressure: (a) DTS, (b) TCM-NT7, and (c) TSM-NT7

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Fig. 7

Square-wave form inlet distortion

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Fig. 8

Unsteady blade forces variation

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Fig. 9

Comparison of instantaneous total pressure: (a) 50% axial chord upstream the rotor and (b) 50% axial chord downstream the rotor

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Fig. 11

Blade row interface interpolation process

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Fig. 12

Time-averaged efficiency with respect to different time sampling at close spacing

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Fig. 13

Unsteady efficiency at close spacing

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Fig. 14

Time-averaged performances: (a) efficiency and (b) total pressure ratio

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Fig. 10

Close-spacing computational mesh

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Fig. 15

Snapshots of entropy (J/K) at close spacing: (a) DTS, (b) TCM-NT15, and (c) TSM-NT11

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Fig. 16

Amplitude of the unsteady static pressure on blade surface: (a) rotor and (b) stator

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Fig. 17

Residual convergence histories

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Fig. 18

Computation speed-up factor

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Fig. 19

Whole annulus computational mesh

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Fig. 20

Residual convergence histories

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Fig. 24

Normalized unsteady pressure on IGV blade: (a) real part, (b) imaginary part, and (c) amplitude

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Fig. 21

Time-averaged efficiency with respect to different time sampling at near stall

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Fig. 22

Time-averaged performances: (a) efficiency and (b) total pressure ratio

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Fig. 23

Snapshots of normalized static pressure (normalized by inlet total pressure): (a) DTS, (b) TCM-NT13, and (c) TSM-NT11



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