Geodesic
Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes
D. J. Mavriplis1 ; Z. Yang1
1 Department of Mechanical Engineering, University of Wyoming, Laramie WY 82072, USA
Mathematical modelling of natural phenomena, Tome 6 (2011) no. 3, pp. 213-236.

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For flows with strong periodic content, time-spectral methods can be used to obtain time-accurate solutions at substantially reduced cost compared to traditional time-implicit methods which operate directly in the time domain. However, these methods are only applicable in the presence of fully periodic flows, which represents a severe restriction for many aerospace engineering problems. This paper presents an extension of the time-spectral approach for problems that include a slow transient in addition to strong periodic behavior, suitable for applications such as transient turbofan simulation or maneuvering rotorcraft calculations. The formulation is based on a collocation method which makes use of a combination of spectral and polynomial basis functions and results in the requirement of solving coupled time instances within a period, similar to the time spectral approach, although multiple successive periods must be solved to capture the transient 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 which minimizes the modifications required to an existing steady-state solver. For dynamically deforming mesh cases, a formulation which preserves discrete conservation as determined by the Geometric Conservation Law is derived and implemented. A fully implicit approach which takes into account the coupling between the various time instances is implemented and shown to preserve the baseline steady-state multigrid convergence rate as the number of time instances is increased. Accuracy and efficiency are demonstrated for periodic and non-periodic problems by comparing the performance of the method with a traditional time-stepping approach using a simple two-dimensional pitching airfoil problem, a three-dimensional pitching wing problem, and a more realistic transitioning rotor problem.
DOI : 10.1051/mmnp/20116309

D. J. Mavriplis 1 ; Z. Yang 1

1 Department of Mechanical Engineering, University of Wyoming, Laramie WY 82072, USA 
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D. J. Mavriplis; Z. Yang. Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes. Mathematical modelling of natural phenomena, Tome 6 (2011) no. 3, pp. 213-236. doi : 10.1051/mmnp/20116309. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20116309/

[1] C. Canuto, M. .Y. Hussaini, A. Quarteroni, T. A. Zang. Spectral methods in fluid dynamics. Springer, 1987.

[2] P. Geuzaine, C. Grandmont, C. Farhat J. Comput. Phys. 2003 206 227

[3] A.K. Gopinath, A. Jameson. Time spectral method for periodic unsteady computations over two- and three- dimensional bodies. AIAA Paper 2005-1220, Jan. 2005.

[4] 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.

[5] K. C. Hall, E. F. Crawley AIAA Journal 1989 777 787

[6] K. C. Hall, J. P. Thomas, W. S. Clark AIAA Journal 2002 879 886

[7] J. Hesthaven, S. Gottlieb, D. Gottlieb. Spectral methods for time-dependent problems. Cambridge Monographs on Applied and Computational Mathematics, 2007.

[8] C. Lanczos. Discourse on Fourier series. Hafner, New York, 1966.

[9] D. J. Mavriplis J. Comput. Phys. 1998 141 165

[10] D. J. Mavriplis, S. Pirzadeh AIAA Journal of Aircraft 1999 987 998

[11] D. J. Mavriplis, V. Venkatakrishnan International Journal of Computational Fluid Dynamics 1997 247 263

[12] D. J. Mavriplis, Z. Yang J. Comput. Phys. 2006 557 573

[13] M. McMullen, A. Jameson, J. J. Alonso. Acceleration of convergence to a periodic steady state in turbomachineary flows. AIAA Paper 2001-0152, 2001.

[14] 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.

[15] E. J. Nielsen, B. Diskin, N. K. Yamaleev AIAA Journal 2010 1195 1206

[16] F. Sicot, G. Puigt, M. Montagnac AIAA Journal 2008 3080 3089

[17] P. R. Spalart, S. R. Allmaras La Recherche Aérospatiale 1994 5 21

[18] E. van der Weide, A. K. Gopinath, A. Jameson. Turbomachineary applications with the time spectral method. AIAA Paper 2005-4905, 2005.

[19] A. H. Van Zuijlen, A. De Boer, H. Bijl J. Comput. Phys. 224 414 430

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