Computing time-periodic solutions of a model for the vortex sheet with surface tension
Authors:
David M. Ambrose, Mark Kondrla, Jr. and Michael Valle
Journal:
Quart. Appl. Math. 73 (2015), 317-329
MSC (2010):
Primary 35B10; Secondary 37M20, 76B45
DOI:
https://doi.org/10.1090/S0033-569X-2015-01364-8
Published electronically:
March 20, 2015
MathSciNet review:
3357496
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Abstract: We compute time-periodic solutions of a simple model for the vortex sheet with surface tension. The model has the same dispersion relation as the full system of evolution equations, and it also has the same destabilizing nonlinearity (if the surface tension parameter were to be set to zero, then this nonlinearity would cause an analogue of the Kelvin-Helmholtz instability). The numerical method uses a gradient descent algorithm to minimize a functional which measures whether a solution of the system is time periodic. We find continua of genuinely time-periodic solutions bifurcating from equilibrium.
References
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- Jorge Nocedal and Stephen J. Wright, Numerical optimization, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. MR 2244940
- P. I. Plotnikov and J. F. Toland, Nash-Moser theory for standing water waves, Arch. Ration. Mech. Anal. 159 (2001), no. 1, 1–83. MR 1854060, DOI https://doi.org/10.1007/PL00004246
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- Walter A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 671–694. MR 2721042, DOI https://doi.org/10.1090/S0273-0979-2010-01302-1
- D. Viswanath, Recurrent motions within plane Couette turbulence, J. Fluid Mech. 580 (2007), 339–358. MR 2326295, DOI https://doi.org/10.1017/S0022112007005459
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- V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics 9 (1968), 190–194.
References
- David M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal. 35 (2003), no. 1, 211–244 (electronic). MR 2001473 (2005g:76006), DOI https://doi.org/10.1137/S0036141002403869
- David M. Ambrose, Singularity formation in a model for the vortex sheet with surface tension, Math. Comput. Simulation 80 (2009), no. 1, 102–111. MR 2573270 (2010k:76053), DOI https://doi.org/10.1016/j.matcom.2009.06.014
- David M. Ambrose and Jon Wilkening, Global paths of time-periodic solutions of the Benjamin-Ono equation connecting pairs of traveling waves, Commun. Appl. Math. Comput. Sci. 4 (2009), 177–215. MR 2565874 (2010j:35437), DOI https://doi.org/10.2140/camcos.2009.4.177
- David M. Ambrose and Jon Wilkening, Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension, Proceedings of the National Academy of Sciences 107 (2010), no. 8, 3361–3366.
- David M. Ambrose and Jon Wilkening, Computation of time-periodic solutions of the Benjamin-Ono equation, J. Nonlinear Sci. 20 (2010), no. 3, 277–308. MR 2639896 (2011d:65146), DOI https://doi.org/10.1007/s00332-009-9058-x
- Uri M. Ascher, Steven J. Ruuth, and Brian T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal. 32 (1995), no. 3, 797–823. MR 1335656 (96j:65076), DOI https://doi.org/10.1137/0732037
- T. Brooke Benjamin and Thomas J. Bridges, Reappraisal of the Kelvin-Helmholtz problem. I. Hamiltonian structure, J. Fluid Mech. 333 (1997), 301–325. MR 1437021 (98g:76037a), DOI https://doi.org/10.1017/S0022112096004272
- J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5 (1995), no. 4, 629–639. MR 1345016 (96h:35011), DOI https://doi.org/10.1007/BF01902055
- Jean Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices 11 (1994), 475ff., approx. 21 pp. (electronic). MR 1316975 (96f:58170), DOI https://doi.org/10.1155/S1073792894000516
- K. M. Case, Benjamin-Ono-related equations and their solutions, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 1, 1–3. MR 516140 (82c:76106)
- Walter Craig and C. Eugene Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math. 46 (1993), no. 11, 1409–1498. MR 1239318 (94m:35023), DOI https://doi.org/10.1002/cpa.3160461102
- F. de la Hoz, M. A. Fontelos, and L. Vega, The effect of surface tension on the Moore singularity of vortex sheet dynamics, J. Nonlinear Sci. 18 (2008), no. 4, 463–484. MR 2429683 (2009g:76022), DOI https://doi.org/10.1007/s00332-008-9020-3
- Frédéric Dias and Thomas J. Bridges, The numerical computation of freely propagating time-dependent irrotational water waves, Fluid Dynam. Res. 38 (2006), no. 12, 803–830. MR 2270649 (2007h:76017), DOI https://doi.org/10.1016/j.fluiddyn.2005.08.007
- James M. Hamilton, John Kim, and Fabian Waleffe, Regeneration mechanisms of near-wall turbulence structures, Journal of Fluid Mechanics 287 (1995), 317–348.
- T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, The long-time motion of vortex sheets with surface tension, Phys. Fluids 9 (1997), no. 7, 1933–1954. MR 1455083 (98d:76033), DOI https://doi.org/10.1063/1.869313
- Thomas Y. Hou, John S. Lowengrub, and Michael J. Shelley, Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys. 114 (1994), no. 2, 312–338. MR 1294935 (95e:76069), DOI https://doi.org/10.1006/jcph.1994.1170
- G. Iooss, P. I. Plotnikov, and J. F. Toland, Standing waves on an infinitely deep perfect fluid under gravity, Arch. Ration. Mech. Anal. 177 (2005), no. 3, 367–478. MR 2187619 (2007a:76017), DOI https://doi.org/10.1007/s00205-005-0381-6
- Genta Kawahara and Shigeo Kida, Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst, J. Fluid Mech. 449 (2001), 291–300. MR 1871646 (2002j:76063), DOI https://doi.org/10.1017/S0022112001006243
- Robert Krasny, A study of singularity formation in a vortex sheet by the point-vortex approximation, J. Fluid Mech. 167 (1986), 65–93. MR 851670 (87g:76028), DOI https://doi.org/10.1017/S0022112086002732
- D. W. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet, Proc. Roy. Soc. London Ser. A 365 (1979), no. 1720, 105–119. MR 527594 (80b:76006), DOI https://doi.org/10.1098/rspa.1979.0009
- Jorge Nocedal and Stephen J. Wright, Numerical optimization, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. MR 2244940 (2007a:90001)
- P. I. Plotnikov and J. F. Toland, Nash-Moser theory for standing water waves, Arch. Ration. Mech. Anal. 159 (2001), no. 1, 1–83. MR 1854060 (2002k:76019), DOI https://doi.org/10.1007/PL00004246
- M. J. Shelley, A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method, J. Fluid Mech. 244 (1992), 493–526. MR 1192255 (93g:76035), DOI https://doi.org/10.1017/S0022112092003161
- Michael Siegel, A study of singularity formation in the Kelvin-Helmholtz instability with surface tension, SIAM J. Appl. Math. 55 (1995), no. 4, 865–891. MR 1341530 (96d:76015), DOI https://doi.org/10.1137/S0036139994262659
- Walter A. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 671–694. MR 2721042 (2012b:76022), DOI https://doi.org/10.1090/S0273-0979-2010-01302-1
- D. Viswanath, Recurrent motions within plane Couette turbulence, J. Fluid Mech. 580 (2007), 339–358. MR 2326295 (2008f:76100), DOI https://doi.org/10.1017/S0022112007005459
- Jon Wilkening, Breakdown of self-similarity at the crests of large-amplitude standing water waves, Phys. Rev. Lett. 107 (2011), 184501.
- V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics 9 (1968), 190–194.
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Additional Information
David M. Ambrose
Affiliation:
Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
MR Author ID:
720777
Email:
ambrose@math.drexel.edu
Mark Kondrla, Jr.
Affiliation:
Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
Email:
mKondrla415@gmail.com
Michael Valle
Affiliation:
Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, Pennsylvania 19104
Email:
mikau16@hotmail.com
Keywords:
vortex sheet,
surface tension,
time-periodic,
gradient descent
Received by editor(s):
March 30, 2013
Published electronically:
March 20, 2015
Additional Notes:
The authors gratefully acknowledge support from the National Science Foundation through NSF grants DMS-1008387 and DMS-1016267. We also gratefully acknowledge support in the form of Research Co-Op Funding from the Drexel University Office of the Provost and the Steinbright Career Development Office.
Article copyright:
© Copyright 2015
Brown University