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Computing Optimal Forcing Using Laplace Preconditioning

Part of: Numerical linear algebra Hydrodynamic stability Basic methods in fluid mechanics

Published online by Cambridge University Press:  31 October 2017

M. Brynjell-Rahkola ,
L. S. Tuckerman ,
P. Schlatter  and
D. S. Henningson
M. Brynjell-Rahkola*
Affiliation:
Linné FLOW Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
L. S. Tuckerman*
Affiliation:
PMMH (UMR 7636 CNRS – ESPCI – UPMC Paris 6 – UPD Paris 7), 10 rue Vauquelin, 75005 Paris, France
P. Schlatter*
Affiliation:
Linné FLOW Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
D. S. Henningson*
Affiliation:
Linné FLOW Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
*
*Corresponding author. Email addresses:[email protected](M. Brynjell-Rahkola), [email protected](L. S. Tuckerman), [email protected](P. Schlatter), [email protected](D. S. Henningson)
*Corresponding author. Email addresses:[email protected](M. Brynjell-Rahkola), [email protected](L. S. Tuckerman), [email protected](P. Schlatter), [email protected](D. S. Henningson)
*Corresponding author. Email addresses:[email protected](M. Brynjell-Rahkola), [email protected](L. S. Tuckerman), [email protected](P. Schlatter), [email protected](D. S. Henningson)
*Corresponding author. Email addresses:[email protected](M. Brynjell-Rahkola), [email protected](L. S. Tuckerman), [email protected](P. Schlatter), [email protected](D. S. Henningson)
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Abstract

For problems governed by a non-normal operator, the leading eigenvalue of the operator is of limited interest and a more relevant measure of the stability is obtained by considering the harmonic forcing causing the largest system response. Various methods for determining this so-called optimal forcing exist, but they all suffer from great computational expense and are hence not practical for large-scale problems. In the present paper a new method is presented, which is applicable to problems of arbitrary size. The method does not rely on timestepping, but on the solution of linear systems, in which the inverse Laplacian acts as a preconditioner. By formulating the search for the optimal forcing as an eigenvalue problem based on the resolvent operator, repeated system solves amount to power iterations, in which the dominant eigenvalue is seen to correspond to the energy amplification in a system for a given frequency, and the eigenfunction to the corresponding forcing function. Implementation of the method requires only minor modifications of an existing timestepping code, and is applicable to any partial differential equation containing the Laplacian, such as the Navier-Stokes equations. We discuss the method, first, in the context of the linear Ginzburg-Landau equation and then, the two-dimensional lid-driven cavity flow governed by the Navier-Stokes equations. Most importantly, we demonstrate that for the lid-driven cavity, the optimal forcing can be computed using a factor of up to 500 times fewer operator evaluations than the standard method based on exponential timestepping.

Keywords

Hydrodynamic stabilityoptimal forcingLaplace preconditioneriterative methodseigenvalue problemGinzburg-Landau equationlid-driven cavity flow

MSC classification

Secondary: 76M25: Other numerical methods 65F08: Preconditioners for iterative methods 76E15: Absolute and convective instability and stability 76E09: Stability and instability of nonparallel flows 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 5 , November 2017 , pp. 1508 - 1532
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Åkervik, E., Ehrenstein, U., Gallaire, F., and Henningson, D. S.. Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. B-Fluids, 27(5):501–513, 2008.CrossRefGoogle Scholar
[2] Alizard, F., Cherubini, S., and Robinet, J.-C.. Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids, 21(6), 2009.Google Scholar
[3] Auteri, F., Parolini, N., and Quartapelle, L.. Numerical investigation on the stability of singular driven cavity flow. J. Comput. Phys., 183(1):1–25, 2002.CrossRefGoogle Scholar
[4] Bagheri, S., Henningson, D. S., Hoepffner, J., and Schmid, P. J.. Input-output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev., 62:020803, 2009.CrossRefGoogle Scholar
[5] Barkley, D., Blackburn, H.M., and Sherwin, S. J.. Direct optimal growth analysis for timesteppers. Int. J. Numer. Meth. Fl., 57(9):1435–1458, 2008.CrossRefGoogle Scholar
[6] Barkley, D. and Tuckerman, L. S.. Stokes preconditioning for the inverse power method. In Kutler, P., Flores, J., and Chattot, J.-J., editors, Lecture Notes in Physics: Proc. of the Fifteenth Int’l. Conf. on Numerical Methods in Fluid Dynamics, pages 75–76. Springer, New York, 1997.Google Scholar
[7] Batiste, O., Knobloch, E., Alonso, A., and Mercader, I.. Spatially localized binary-fluid convection. J. Fluid Mech., 560:149–158, 2006.Google Scholar
[8] Beaume, C.. Adaptive Stokes preconditioning for steady incompressible flows. Commun. Comput. Phys., 22(2):494–516, 2017.Google Scholar
[9] Bergeon, A., Henry, D., BenHadid, H., and Tuckerman, L. S.. Marangoni convection in binary mixtures with Soret effect. J. Fluid Mech., 375:143–177, 1998.Google Scholar
[10] Butler, K. M. and Farrell, B. F.. Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A, 4(8):1637–1650, 1992.Google Scholar
[11] Campbell, S. L., Ipsen, I. C. F., Kelley, C. T., and Meyer, C. D.. GMRES and the minimal polynomial. BIT Numer. Math., 36(4):664–675, 1996.Google Scholar
[12] Canuto, C., Hussaini, Y., Quarteroni, A., and Zang, T. A.. Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer Berlin Heidelberg, 1988.CrossRefGoogle Scholar
[13] Chevalier, M., Schlatter, P., Lundbladh, A., and Henningson, D. S.. SIMSON – A Pseudo-Spectral Solver for Incompressible Boundary Layer Flows. Technical Report TRITA-MEK 2007:07, KTH Mechanics, 2007.Google Scholar
[14] Chomaz, J. M.. Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech., 37:357–392, 2005.CrossRefGoogle Scholar
[15] Chomaz, J. M., Huerre, P., and Redekopp, L. G.. Models of hydrodynamic resonances in separated shear flows. In Proceedings of the 6th Symp. Turb. Shear Flows, September 1987.Google Scholar
[16] Chorin, A. J. and Marsden, J. E.. A Mathematical Introduction to Fluid Mechanics. Texts in Applied Mathematics. Springer New York, 2000.Google Scholar
[17] Cossu, C. and Chomaz, J.M.. Globalmeasures of local convective instabilities. Phys. Rev. Lett., 78:4387–4390, Jun 1997.Google Scholar
[18] Driscoll, T. A., Toh, K. C., and Trefethen, L. N.. From potential theory to matrix iterations in six steps. SIAM Rev., 40(3):547–578, 1998.CrossRefGoogle Scholar
[19] Embree, M.. How descriptive are GMRES convergence bounds? Technical Report 99/08, Oxford University Computing Laboratory, June 1999.Google Scholar
[20] Fischer, P. F., Lottes, J. W., and Kerkemeier, S. G.. nek5000 Web page, 2008. http://nek5000.mcs.anl.gov.Google Scholar
[21] Greenbaum, A., Pták, V, and Strakoš, Z.. Any nonincreasing convergence curve is possible for GMRES. SIAM J. Matrix Anal. Appl., 17(3):465–469, 1996.CrossRefGoogle Scholar
[22] Huepe, C., Métens, S., Borckmans, P., and Brachet, M. E.. Decay rates in attractive Bose-Einstein condensates. Phys. Rev. Lett., 82(8):1616–1619, 1999.Google Scholar
[23] Huepe, C., Tuckerman, L.S., Métens, S., and Brachet, M. E.. Stability and decay rates of nonisotropic attractive Bose-Einstein condensates. Phys. Rev. A, 68:023609, 2003.Google Scholar
[24] Maday, Y., Patera, A. T., and Rønquist, E. M.. The ℙ N ×ℙ N−2 method for the approximation of the Stokes problem. Technical report, Department of Mechanical Engineering, MIT, Cambridge, MA.Google Scholar
[25] Mamun, C. K. and Tuckerman, L. S.. Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids, 7(1):80–91, 1995.Google Scholar
[26] Monokrousos, A., Åkervik, E., Brandt, L., and Henningson, D. S.. Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech., 650:181–214, 5 2010.Google Scholar
[27] Parks, M. L., de Sturler, E., Mackey, G., Johnson, D. D., and Maiti, S.. Recycling Krylov subspaces for sequences of linear systems. SIAM J. Sci. Comput., 28(5):1651–1674, 2006.Google Scholar
[28] Parlett, B.. The Symmetric Eigenvalue Problem. Society for Industrial and Applied Mathematics, 1998.Google Scholar
[29] Patera, A. T.. A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys., 54(3):468–488, 1984.Google Scholar
[30] Reddy, S. C., Schmid, P. J., Baggett, J. S., and Henningson, D. S.. On stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech., 365:269–303, 1998.Google Scholar
[31] Reddy, S. C., Schmid, P. J., and Henningson, D. S.. Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl.Math., 53(1):15–47, 1993.Google Scholar
[32] Saad, Y. and Schultz, M. H.. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3):856–869, 1986.CrossRefGoogle Scholar
[33] Sipp, D.. Open-loop control of cavity oscillations with harmonic forcings. J. Fluid Mech., 708:439–468, 2012.CrossRefGoogle Scholar
[34] Sipp, D. and Marquet, O.. Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comp. Fluid Dyn., 27(5):617–635, 2013.Google Scholar
[35] Sipp, D., Marquet, O., Meliga, P., and Barbagallo, A.. Dynamics and control of global instabilities in open-flows: a linearized approach. Appl.Mech. Rev., 63(3):030801, 2010.Google Scholar
[36] Toh, K. C. and Trefethen, L. N.. Calculation of pseudospectra by the Arnoldi iteration. SIAM J. Sci. Comp., 17(1):1–15, 1996.Google Scholar
[37] Trefethen, L. N. and Embree, M.. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, 2005.Google Scholar
[38] Tuckerman, L. S.. Laplacian preconditioning for the inverse Arnoldi method. Commun. Comput. Phys., 18:1336–1351, 2015.Google Scholar
[39] Tuckerman, L. S. and Barkley, D.. Bifurcation analysis for timesteppers. In Doedel, E. and Tuckerman, L. S., editors, NumericalMethods for Bifurcation Problems and Large-Scale Dynamical Systems. Springer, New York, 2000.Google Scholar
[40] van der Vorst, H.. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 13(2):631–644, 1992.Google Scholar
[41] Weideman, J. A. and Reddy, S. C.. A MATLAB differentiation matrix suite. ACM Trans. Math. Software, 26(4):465–519, 2000.CrossRefGoogle Scholar