A cubic-spline-based time collocation method applied to simulate periodic transonic flows

Abstract

A cubic-spline-based time collocation method is used to solve periodic transonic flows for aeroelastic analysis. The time periodical flow variables are first approximated by cubic splines, and then the time-derivative terms are represented by a source term, which couples the flow solutions at all the sampled time instants. Simulation of two transonic external unsteady flows, including the oscillating NACA 0012 aerofoil and pitching rectangular supercritical wing, were carried out. The cubic-spline-based time collocation method with relatively small number of time instants sampled demonstrate high accuracy, whilst it is several times faster than the conventional time-marching method. The effect of different basis function used in the time collocation methods are also studied by comparing the results to that of the state-of-the-art time spectral method. This method was also validated against aeroelastic experimental data of the vibrating STCF 11 turbine blades. Only single passage computational mesh is involved by adopting the phase-lag periodic conditions. Good accuracy of the proposed time collocation method is also achieved and about one order of speed-up is obtained as compared with time-marching simulation.

Keywords

Time collocation method periodic transonic flow cubic-spline oscillating aerofoil aeroelastic

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References

Verdon

Caspar

. A linearized unsteady aerodynamic analysis for transonic cascades. J Fluid Mech 1984; 149: 403–429.

Dufour

Sicot

Puigt

. Contrasting the harmonic balance and linearized methods for oscillating-flap simulations. AIAA J 2010; 48: 788–797.

Ning

. Efficient approach for analysis of unsteady viscous flows in turbomachines. AIAA J 1998; 36: 2005–2012.

Hall

Thomas

Clark

. Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. AIAA J 2002; 40: 879–886.

Gopinath A and Jameson A. Time spectral method for periodic unsteady computations over two- and three-dimensional bodies. In: 43rd AIAA aerospace sciences meeting and exhibit, Reno, NV, USA, 10–13 January 2005, paper no. AIAA 2005-1220. New York: AIAA.

Ekici

Hall

Kielb

. Harmonic balance analysis of blade row interactions in a transonic compressor. J Propul Power 2010; 26: 335–343.

Sicot

Dufour

Gourdain

. A time-domain harmonic balance method for rotor/stator interactions. ASME J Turbomach 2012; 134: 011001–011001-13.

Ekici

Hall

Dowell

. Computationally fast harmonic balance methods for unsteady aerodynamic predictions of helicopter rotors. J Comput Phys 2008; 227: 6206–6225.

Woodgate

Badcock

. Implicit harmonic balance solver for forced motion transonic flow. AIAA J 2008; 47: 893–901.

10.

Sicot

Puigt

Montagnac

. Block-Jacobi implicit algorithms for the time spectral method. AIAA J 2008; 46: 3080–3089.

11.

Da Ronach

McCracken

Badcock

. Linear frequency domain and harmonic balance predictions of dynamic derivatives. J Aircraft 2013; 50: 694–707.

12.

Sicot

Gomar

Dufour

. Time-domain harmonic balance method for turbomachinery aeroelasticity. AIAA J 2014; 52: 62–71.

13.

Ning

. Simulating periodic unsteady flows using a cubic-spline based time collocation method. ASME J Turbomach 2014; 136: 041014–041014-12.

14.

Shu

. Differential quadrature and its application in engineering, London: Springer, 2000.

15.

Shu C and Richards BE. High resolution of natural convection in square cavity by generalized differential quadrature. In: Proceedings of the 3rd international conference on advances in numeric methods in engineering: Theory and application, Swansea, UK, 1990, pp.978–985.

16.

Shu

Chew

. Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problem. Commun Numer Meth Eng 1997; 13: 643–653.

17.

Niu

Zheng

Yang

. Chebyshev spectral method for unsteady axisymmetric mixed convection heat transfer of power law fluid over a cylinder with variable transport properties. Int J Numer Anal Model 2014; 11: 525–540.

18.

Choi

McClure

. Mapped Chebyshev pseudospectral method for unsteady flow analysis. AIAA J 2015; 53: 3805–3820.

19.

Blazek

. Computational fluid dynamics: Principles and applications, Oxford: Elsevier, 2001.

20.

Jameson A. Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. In: AIAA 10th computational fluid dynamics conference, Honolulu, HI, USA, 24–26 June 1991, paper no. AIAA 91-1596. New York: AIAA.

21.

De Boer

Van der Schoot

Bijl

. Mesh deformation based on radial basis function interpolation. Comput Struct 2007; 85: 784–795.

22.

Gopinath AK. Efficient Fourier-based algorithms for time-periodic unsteady problems. PhD Thesis, Stanford University, USA, 2007.

23.

Edwards

. A low-diffusion flux-splitting scheme for Navier-Stokes calculations. Comput Fluids 1997; 26: 635–659.

24.

Spalart

Allmaras

. A one-equation turbulence transport model for aerodynamic flows. Recher Aerosp 1994; 1: 5–21.

25.

Ning F and Xu L. Numerical investigation of transonic compressor rotor flow using an implicit 3D flow solver with one-equation Spalart-Allmaras turbulence model. In: ASME turbo expo 2001: Power for land, sea, and air, New Orleans, LA, 4–7 June 2001, paper no. 2001-GT-0359, pp.V001T03A054. New York: ASME.

26.

Ning F. MAP: A CFD package for turbomachinery flow simulation and aerodynamic design optimization. In: ASME turbo expo 2014: Turbine technical conference and exposition, Düsseldorf, Germany, 16–20 June 2014, paper no. GT2014-26515, pp.V02BT39A030.

27.

Landon RH. NACA 0012 oscillating and transient pitching, data Set 3 in AGARD-R-702, compendium of unsteady aerodynamic measurements. Report, August 1982.

28.

Green LL and Newman PA. Wall-interference assessment and corrections for transonic NACA 0012 airfoil data from various wind tunnels. NASA TP-3070, April 1991.

29.

Heeg J, Chwalowski P, Florance JP, et al. Overview of the aeroelastic prediction workshop. In: 51st AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, Grapevine, TX, USA, 7–10 January 2013, paper no. AIAA 2013-0783. New York: AIAA.

30.

Bennett RM and Walker CE. Computational test cases for a rectangular supercritical wing undergoing pitching oscillations. NASA/TM-1999-209130, 1999.

31.

Structure Dynamics Technical Committee. Rectangular supercritical wing, https://c3.nasa.gov/dashlink/static/media/other/RSW_base_legacy.htm (2011, accessed 10 September 2015).

32.

NASA DASH link. Aeroelastic prediction workshop, https://c3.nasa.gov/dashlink/resources/468/ (2011, accessed 10 September 2015).

33.

Heeg J, Chawlowski P and Wieseman CD, et al. Lessons learned in the selection and development of test cases for the aeroelastic prediction workshop: rectangular supercritical wing. In: 51st AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, Grapevine, TX, USA, 7–10 January 2013, paper no. AIAA 2013-0784. New York: AIAA.

34.

Schuster DM, Wieseman C and Chwalowski P. Analysis of test case computations and experiments for the aeroelastic prediction workshop. In: 51st AIAA aerospace sciences meeting including the new horizons forum and aerospace exposition, Grapevine, TX, USA, 7–10 January 2013, paper no. AIAA 2013-0788. New York: AIAA.

35.

Fransson

Jöcker

Bölcs

. Viscous and inviscid linear/nonlinear calculations versus quasi 3D experimental cascade data for a new aeroelastic turbine standard configuration. ASME J Turbomach 1999; 121: 717–725.

36.

Fransson TH. Standard configurations for unsteady flow through vibrating axial-flow turbomachine-cascades (STCF), http://www.energy.kth.se/proj/projects/Markus%20Joecker/STCF/STCF11/stcf11.htm (1998, accessed 20 September 2015).

37.

. An Euler solution for unsteady flows around oscillating blades. ASME J Turbomach 1990; 112: 714–722.

38.

Cinnella

De Palma

Pascazio

. A numerical method for turbomachinery aeroelasticity. ASME J Turbomach 2004; 126: 310–316.