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Time Spectral Method for Periodic and Quasi-Periodic UnsteadyComputations on Unstructured Meshes

Published online by Cambridge University Press:  16 May 2011

D. J. Mavriplis  and
Z. Yang
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Abstract

For flows with strong periodic content, time-spectral methods can be used to obtaintime-accurate solutions at substantially reduced cost compared to traditionaltime-implicit methods which operate directly in the time domain. However, these methodsare only applicable in the presence of fully periodic flows, which represents a severerestriction for many aerospace engineering problems. This paper presents an extension ofthe time-spectral approach for problems that include a slow transient in addition tostrong periodic behavior, suitable for applications such as transient turbofan simulationor maneuvering rotorcraft calculations. The formulation is based on a collocation methodwhich makes use of a combination of spectral and polynomial basis functions and results inthe requirement of solving coupled time instances within a period, similar to the timespectral approach, although multiple successive periods must be solved to capture thetransient behavior.

The implementation allows for two levels of parallelism, one in the spatial dimension,and another in the time-spectral dimension, and is implemented in a modular fashion whichminimizes the modifications required to an existing steady-state solver. For dynamicallydeforming mesh cases, a formulation which preserves discrete conservation as determined bythe Geometric Conservation Law is derived and implemented. A fully implicit approach whichtakes into account the coupling between the various time instances is implemented andshown to preserve the baseline steady-state multigrid convergence rate as the number oftime instances is increased. Accuracy and efficiency are demonstrated for periodic andnon-periodic problems by comparing the performance of the method with a traditionaltime-stepping approach using a simple two-dimensional pitching airfoil problem, athree-dimensional pitching wing problem, and a more realistic transitioning rotor problem.

Keywords

time spectralperiodicunstructured mesh
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 3: Computational aerodynamics , 2011 , pp. 213 - 236
Copyright
© EDP Sciences, 2011

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References

Références

C. Canuto, M. .Y. Hussaini, A. Quarteroni, T. A. Zang. Spectral methods in fluid dynamics. Springer, 1987.
Geuzaine, P., Grandmont, C., and Farhat, C.. Design and analysis of ALE schemes with provable second-order time-accuracy for inviscid and viscous flow simulations. J. Comput. Phys., 191 (2003), No. 1, 206–227. CrossRefGoogle Scholar
A.K. Gopinath, A. Jameson. Time spectral method for periodic unsteady computations over two- and three- dimensional bodies. AIAA Paper 2005-1220, Jan. 2005.
D. Gottlieb, S. A. Orszag. Numerical analysis of spectral methods: theory and applications. CBMS-26, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1977.
Hall, K. C., Crawley, E. F.. Calculation of unsteady flows in turbomachinery using the linearized Euler equations. AIAA Journal, 27 (1989), No. 6, 777–787. CrossRefGoogle Scholar
Hall, K. C., Thomas, J. P., Clark, W. S.. Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. AIAA Journal, 40 (2002), No. 5, 879–886. CrossRefGoogle Scholar
J. Hesthaven, S. Gottlieb, D. Gottlieb. Spectral methods for time-dependent problems. Cambridge Monographs on Applied and Computational Mathematics, 2007.
C. Lanczos. Discourse on Fourier series. Hafner, New York, 1966.
Mavriplis, D. J.. Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes. J. Comput. Phys., 145 (1998), No. 1, 141–165. CrossRefGoogle Scholar
Mavriplis, D. J., Pirzadeh, S.. Large-scale parallel unstructured mesh computations for 3D high-lift analysis. AIAA Journal of Aircraft, 36 (1999), No. 6, 987–998. CrossRefGoogle Scholar
Mavriplis, D. J., Venkatakrishnan, V.. A unified multigrid solver for the Navier-Stokes equations on mixed element meshes. International Journal of Computational Fluid Dynamics, 8 (1997), 247–263. CrossRefGoogle Scholar
Mavriplis, D. J., Yang, Z.. Construction of the discrete geometric conservation law for high-order time accurate simulations on dynamic meshes. J. Comput. Phys., 213 (2006), No. 2, 557–573. CrossRefGoogle Scholar
M. McMullen, A. Jameson, J. J. Alonso. Acceleration of convergence to a periodic steady state in turbomachineary flows. AIAA Paper 2001-0152, 2001.
M. McMullen, A. Jameson, J. J. Alonso. Application of a non-linear frequency domain solver to the Euler and Navier-Stokes equations. AIAA Paper 2002-0120, 2002.
Nielsen, E. J., Diskin, B., Yamaleev, N. K.. Discrete adjoint-based design optimization of unsteady turbulent flows on dynamic unstructured grids. AIAA Journal, 48 (2010), No. 6, 1195–1206. CrossRefGoogle Scholar
Sicot, F., Puigt, G., Montagnac, M.. Block-Jacobi implicit algorithm for the time spectral method. AIAA Journal, 46 (2008), No. 12, 3080–3089. CrossRefGoogle Scholar
Spalart, P. R., Allmaras, S. R.. A one-equation turbulence model for aerodynamic flows. La Recherche Aérospatiale, 1 (1994), 5–21. Google Scholar
E. van der Weide, A. K. Gopinath, A. Jameson. Turbomachineary applications with the time spectral method. AIAA Paper 2005-4905, 2005.
van Zuijlen, A. H., de Boer, A., Bijl, H.. High order time integration through smooth mesh deformation for 3D fluid–structure interaction simulations. J. Comput. Phys., No. 2007 (224), No. 1, 414–430. Google Scholar