# Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes

Mathematical Modelling of Natural Phenomena (2011)

- Volume: 6, Issue: 3, page 213-236
- ISSN: 0973-5348

## Access Full Article

top## Abstract

top## How to cite

topMavriplis, D. J., and Yang, Z.. "Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes." Mathematical Modelling of Natural Phenomena 6.3 (2011): 213-236. <http://eudml.org/doc/222305>.

@article{Mavriplis2011,

abstract = {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.
},

author = {Mavriplis, D. J., Yang, Z.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {time spectral; periodic; unstructured mesh},

language = {eng},

month = {5},

number = {3},

pages = {213-236},

publisher = {EDP Sciences},

title = {Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes},

url = {http://eudml.org/doc/222305},

volume = {6},

year = {2011},

}

TY - JOUR

AU - Mavriplis, D. J.

AU - Yang, Z.

TI - Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes

JO - Mathematical Modelling of Natural Phenomena

DA - 2011/5//

PB - EDP Sciences

VL - 6

IS - 3

SP - 213

EP - 236

AB - 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.

LA - eng

KW - time spectral; periodic; unstructured mesh

UR - http://eudml.org/doc/222305

ER -

## References

top- C. Canuto, M. .Y. Hussaini, A. Quarteroni, T. A. Zang. Spectral methods in fluid dynamics. Springer, 1987.
- P. Geuzaine, C. Grandmont, and C. Farhat. 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.
- 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.
- K. C. Hall, E. F. Crawley. Calculation of unsteady flows in turbomachinery using the linearized Euler equations. AIAA Journal, 27 (1989), No. 6, 777–787.
- K. C. Hall, J. P. Thomas, W. S. Clark. Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. AIAA Journal, 40 (2002), No. 5, 879–886.
- 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.
- D. J. Mavriplis. Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes. J. Comput. Phys., 145 (1998), No. 1, 141–165.
- D. J. Mavriplis, S. Pirzadeh. Large-scale parallel unstructured mesh computations for 3D high-lift analysis. AIAA Journal of Aircraft, 36 (1999), No. 6, 987–998.
- D. J. Mavriplis, V. Venkatakrishnan. A unified multigrid solver for the Navier-Stokes equations on mixed element meshes. International Journal of Computational Fluid Dynamics, 8 (1997), 247–263.
- D. J. Mavriplis, Z. Yang. 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.
- 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.
- E. J. Nielsen, B. Diskin, N. K. Yamaleev. Discrete adjoint-based design optimization of unsteady turbulent flows on dynamic unstructured grids. AIAA Journal, 48 (2010), No. 6, 1195–1206.
- F. Sicot, G. Puigt, M. Montagnac. Block-Jacobi implicit algorithm for the time spectral method. AIAA Journal, 46 (2008), No. 12, 3080–3089.
- P. R. Spalart, S. R. Allmaras. A one-equation turbulence model for aerodynamic flows. La Recherche Aérospatiale, 1 (1994), 5–21.
- E. van der Weide, A. K. Gopinath, A. Jameson. Turbomachineary applications with the time spectral method. AIAA Paper 2005-4905, 2005.
- A. H. van Zuijlen, A. de Boer, H. Bijl. High order time integration through smooth mesh deformation for 3D fluid–structure interaction simulations. J. Comput. Phys., No. 2007 (224), No. 1, 414–430.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.