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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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Figures

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Fig. 1

Illustration of the time collocation method

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Fig. 2

Spectral radius of the weighting coefficient matrix

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Fig. 3

Computational mesh

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Fig. 4

Convergence histories of the TCM computations

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Fig. 5

Unsteady rotor efficiency

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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. 10

Close-spacing computational mesh

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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. 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. 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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Fig. 24

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

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