Nonlinear evolution of two vortex sheets moving separately in uniform shear flows with opposite direction

Chihiro Matsuoka; Chihiro Matsuoka

doi:10.3934/era.2022093

Electronic Research Archive

2022, Volume 30, Issue 5: 1836-1863. doi: 10.3934/era.2022093

Previous Article Next Article

Research article Special Issues

Nonlinear evolution of two vortex sheets moving separately in uniform shear flows with opposite direction

Chihiro Matsuoka ^{1,2,3
,
,}

1.
Laboratory of Applied Mathemetics, Graduate School of Engineering, Osaka City University, Sugimoto, Sumiyoshi, Osaka 5588585, Japan
2.
Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka City University, Sugimoto, Sumiyoshi, Osaka 5580022, Japan
3.
Osaka City University Advanced Mathematical Institute (OCAMI), Sugimoto, Sumiyoshi, Osaka 5580022, Japan

Academic Editor: Lei Wang

Received: 10 December 2021 Revised: 08 March 2022 Accepted: 22 March 2022 Published: 31 March 2022

It has been considered that two close vortex sheets become unstable and evolve simultaneously when sufficiently strong uniform shears exist. However, Moore (Mathematika, 1976) suggested in his linear analysis that a vortex sheet evolves just as if the other vortex sheet were absent under certain conditions. In the current study, we investigate how the two vortex sheets evolve in the nonlinear region when they satisfy Moore's condition. We also consider density stratification, which is not included in Moore's analysis. Moore's estimate is only valid within linear theory; however, a motion suggested by Moore appears even in the nonlinear regime when Moore's condition is satisfied. We found that there is a case that a vortex sheet hardly deforms, even though the other sheet becomes unstable and largely deforms. We also show that there is a case that Moore's analysis is not effective even the condition is satisfied when a density instability exists in the system.
- two vortex sheets,
- multi-layer flow,
- Kelvin-Helmholtz instability,
- density stratification,
- vortex method
Citation: Chihiro Matsuoka. Nonlinear evolution of two vortex sheets moving separately in uniform shear flows with opposite direction[J]. Electronic Research Archive, 2022, 30(5): 1836-1863. doi: 10.3934/era.2022093

Related Papers:

Abstract

It has been considered that two close vortex sheets become unstable and evolve simultaneously when sufficiently strong uniform shears exist. However, Moore (Mathematika, 1976) suggested in his linear analysis that a vortex sheet evolves just as if the other vortex sheet were absent under certain conditions. In the current study, we investigate how the two vortex sheets evolve in the nonlinear region when they satisfy Moore's condition. We also consider density stratification, which is not included in Moore's analysis. Moore's estimate is only valid within linear theory; however, a motion suggested by Moore appears even in the nonlinear regime when Moore's condition is satisfied. We found that there is a case that a vortex sheet hardly deforms, even though the other sheet becomes unstable and largely deforms. We also show that there is a case that Moore's analysis is not effective even the condition is satisfied when a density instability exists in the system.

References

[1]	P. G. Baines, Topographic effects in stratified flows, Cambridge University Press, Cambridge, 1995.
[2]	B. Cushman-Roisin, J. M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, Elsevier, Amsterdam, 2011.
[3]	W. D. Smyth, J. R. Carpenter, Instability in Geophysical flows, Cambridge University Press, Cambridge, 2019.
[4]	B. R. Sutherland, Internal gravity waves, Cambridge University Press, Cambridge, 2010.
[5]	S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, New York, 1981.
[6]	H. Aref, E. D. Siggia, Vortex dynamics of the two-dimensional turbulent shear layer, J. Fluid Mech., 100 (1980), 705–737. https://doi.org/10.1017/S0022112080001371 doi: 10.1017/S0022112080001371
[7]	K. Kamemoto, Formation and interaction of two parallel vortex streets, Bullet. JSME, 19 (1976), 283–290. https://doi 10.1299/jsme1958.19.283 doi: 10.1299/jsme1958.19.283
[8]	H. G. Lee, J. Kim, Two-dimensional Kelvin-Helmholtz instabilities of multi-component fluids, Euro. J. Mech. B/Fluids, 49 (2015), 77–88. https://doi.org/10.1016/j.euromechflu.2014.08.001 doi: 10.1016/j.euromechflu.2014.08.001
[9]	P. Tsoutsanis, I. W. Kokkinakis, L. Könözsy, D. Drikakis, Comparison of structured- and unstructured-grid, compressible and incompressible methods using the vortex pairing problem, Comput. Methods Appl. Mech. Eng., 293 (2015), 207–231. https://doi.org/10.1016/j.cma.2015.04.010 doi: 10.1016/j.cma.2015.04.010
[10]	C. Matsuoka, Motion of unstable two interfaces in a three-layer fluid with a non-zero uniform current, Fluid Dyn. Res., 53 (2021), 055502. https://iopscience.iop.org/article/10.1088/1873-7005/ac2620
[11]	D. W. Moore, The stability of an evolving two-dimensional vortex sheet, Mathematika, 23 (1976), 35–44. https://doi.org/10.1112/S0025579300006124 doi: 10.1112/S0025579300006124
[12]	M. H. Aliabadi, P. Wen, Boundary Element Methods in Engineering And Sciences, World Scientific, 2011.
[13]	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. https://doi.org/10.1017/S0022112092003161 doi: 10.1017/S0022112092003161
[14]	A. J. Chorin, P. S. Bernard, Discretization of a vortex sheet with an example of roll-up, J. Comput. Phys., 13 (1973), 423–429. https://doi.org/10.1016/0021-9991(73)90045-4 doi: 10.1016/0021-9991(73)90045-4
[15]	C. Börgers, On the numerical solution of the regularized Birkhoff equations, Math. Comput., 187 (1989), 141–156. https://www.jstor.org/stable/2008353
[16]	R. Krasny, A study of singularity formation in a vortex sheet by the point vortex approximation, J. Fluid Mech., 167 (1986), 65–93. https://doi.org/10.1017/S0022112086002732 doi: 10.1017/S0022112086002732
[17]	R. Krasny, Computation of vortex sheet roll-up in the Trefftz plane, J. Fluid Mech., 184 (1987), 123–155. https://doi.org/10.1017/S0022112087002830 doi: 10.1017/S0022112087002830
[18]	G. R. Baker, J. Beale, Vortex blob methods applied to interfacial motion, J. Compt. Phys., 196 (2003), 233–258. https://doi.org/10.1016/j.jcp.2003.10.023 doi: 10.1016/j.jcp.2003.10.023
[19]	G. R. Baker, L. Pham, A comparison of blob methods for vortex sheet roll-up, J. Fluid Mech., 547 (2006), 297–316. https://doi.org/10.1017/S0022112005007305 doi: 10.1017/S0022112005007305
[20]	G-H. Cottet, P. D. Koumoutsakos, Vortex methods: theory and practice, Cambridge University Press, Cambridge, 2000.
[21]	N. J. Zabusky, M. H. Hughrs, K. V. Roberts, Contour dynamics for the Euler equations in two dimensions, J. Comput. Phys., 293 (1979), 96–106. https://doi.org/10.1016/0021-9991(79)90089-5 doi: 10.1016/0021-9991(79)90089-5
[22]	C. Pozrikidis, J. J. L. Higdon, Nonlinear Kelvin-Helmholtz instability of a finite vortex layer, J. Fluid Mech., 157 (1985), 225–263. https://doi.org/10.1017/S0022112085002361 doi: 10.1017/S0022112085002361
[23]	C. Pozrikidis, J. J. L. Higdon, Instability of compound vortex layers and wakes, Phys. Fluids, 30 (1987), 2965–2975. https://doi.org/10.1063/1.866074 doi: 10.1063/1.866074
[24]	G. Birkhoff, Helmholtz and Taylor instability, Proc. Symp. Appl. Maths. Soc., 13 (1962), 55–76.
[25]	N. Rott, Diffraction of a weak shock with vortex generation, J. Fluid Mech., 1 (1956), 111–128. https://doi.org/10.1017/S0022112056000081 doi: 10.1017/S0022112056000081
[26]	P. G. Saffman, Vortex Dynamics, Cambridge University Press, Cambridge, 1992.
[27]	R. E. Caflisch, J. S. Lowengrub, Convergence of the vortex method for vortex sheets, SIAM J. Numer. Anal., 26 (1989), 1060–1080. https://doi.org/10.1137/0726059 doi: 10.1137/0726059
[28]	T. Y. Hou, J. S. Lowengrub, R. Krasny, Convergence of a point vortex method for vortex sheets, SIAM J. Numer. Anal., 28 (1979), 308–320. https://www.jstor.org/stable/2157815
[29]	A. Sidi, M. Israeli, Quadrature methods for periodic singular and weakly singular Fredholm integral equations, J. Sci. Compt., 3 (1988), 201–231. https://link.springer.com/article/10.1007/BF01061258
[30]	D. W. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet, Proc. Roy. Soc. A, 365 (1979), 105–119. https://doi.org/10.1098/rspa.1979.0009 doi: 10.1098/rspa.1979.0009
[31]	C. Matsuoka, K. Nishihara, Vortex core dynamics and singularity formations in incompressible Richtmyer-Meshkov instability, Phys. Rev. E., 73 (2006), 026304. https://doi.org/10.1103/PhysRevE.73.026304 doi: 10.1103/PhysRevE.73.026304
[32]	C. Matsuoka, K. Nishihara, Fully nonlinear evolution of a cylindrical vortex sheet in incompressible Richtmyer-Meshkov instability, Phys. Rev. E., 73 (2006), 055304(R). https://doi.org/10.1103/PhysRevE.73.055304 doi: 10.1103/PhysRevE.73.055304
[33]	C. Matsuoka, K. Nishihara, Analytical and numerical study on a vortex sheet in incompressible Richtmyer-Meshkov instability in cylindrical geometry, Phys. Rev. E., 74 (2006), 066303. https://doi.org/10.1103/PhysRevE.74.066303 doi: 10.1103/PhysRevE.74.066303
[34]	C. Matsuoka, Vortex sheet motion in incompressible Richtmyer-Meshkov and Rayleigh-Taylor instabilities with surface tension, Phys. Fluids, 21 (2009), 092107. https://doi.org/10.1063/1.3231837 doi: 10.1063/1.3231837
[35]	C. Matsuoka, K. Nishihara, T. Sano, Nonlinear Dynamics of Non-uniform Current-Vortex Sheets in Magnetohydrodynamic Flows, J. Nonlinear Sci., 27 (2017), 531–572. https://link.springer.com/article/10.1007/s00332-016-9343-4
[36]	C. Matsuoka, K. Nishihara, Nonlinear interaction between bulk point vortices and an unstable interface with nonuniform velocity shear such as Richtmyer-Meshkov instability, Phys. Plasmas, 27 (2020), 052305. https://doi.org/10.1063/1.5131701 doi: 10.1063/1.5131701
[37]	C. Matsuoka, K. Nishihara, F. Cobos-Campos, Linear and nonlinear interactions between an interface and bulk vortices in Richtmyer-Meshkov instability, Phys. Plasmas, 27 (2020), 112301. https://doi.org/10.1063/5.0016553 doi: 10.1063/5.0016553
[38]	R. D. Richtmyer, Taylor instability in shock acceleration of compressible fluids, Commun. Pure Appl. Math., 13 (1960), 297–319. https://www.osti.gov/biblio/4272289
[39]	E. E. Meshkov, Instability of the interface of two gases accelerated by a shock wave, Sov. Fluid Dynamics, 4 (1969), 101–108. https://link.springer.com/article/10.1007/BF01015969
[40]	C. Matsuoka, Nonlinear dynamics of double-layer unstable interfaces with non-uniform velocity shear, Phys. Fluids, 32 (2020), 102109. https://doi.org/10.1063/5.0023558 doi: 10.1063/5.0023558
[41]	K. O. Mikaelian, Normal modes and symmetries of the Rayleigh-Taylor instability in stratified fluids, Phys. Rev. Lett., 19 (1982), 1365–1368. https://doi.org/10.1103/PhysRevLett.48.1365 doi: 10.1103/PhysRevLett.48.1365
[42]	K. O. Mikaelian, Rayleigh-Taylor instabilities in stratified fluids, Phys. Rev. A, 26 (1982), 2140–2158. https://doi.org/10.1103/PhysRevA.26.2140 doi: 10.1103/PhysRevA.26.2140
[43]	K. O. Mikaelian, Time evolution of density perturbation in accelerating stratified fluids, Phys. Rev. A, 28 (1983), 1637–1646. https://doi.org/10.1103/PhysRevA.28.1637 doi: 10.1103/PhysRevA.28.1637
[44]	K. O. Mikaelian, Richtmyer-Meshkov instabilities in stratified fluids, Phys. Rev. A, 31 (1985), 410–419. https://doi.org/10.1103/PhysRevA.31.410 doi: 10.1103/PhysRevA.31.410
[45]	L. J. Liu, L. J. Yang, H. Y. Ye, Weakly nonlinear varicose-mode instability of planar liquid sheets, Phys. fluids, 28 (2016), 034105. https://doi.org/10.1063/1.4942994 doi: 10.1063/1.4942994
[46]	W. Liu, X. Li, C. Yu, Y. Fu, P. Wang, L. Wang, et al., Theoretical study on finite-thickness effect on harmonics in Richtmyer-Meshkov instability for arbitrary Atwood numbers, Phys. Plasmas, 25 (2018), 122103. https://doi.org/10.1063/1.5053766 doi: 10.1063/1.5053766
[47]	A. D. D. Craik, J. A. Adam, Explosive' resonant wave interactions in a three-layer fluid flow, J. Fluid Mech., 92 (1979), 15–33. https://doi.org/10.1017/S0022112079000501 doi: 10.1017/S0022112079000501
[48]	G. R. Baker, D. I. Meiron, S. A. Orszag, Generalized vortex methods for free surface flow problems, J. Fluid Mech., 123 (1982), 477–501. https://doi.org/10.1017/S0022112082003164 doi: 10.1017/S0022112082003164
[49]	R. M. Kerr, Simulation of Rayleigh-Taylor flows using vortex blobs, J. Comput. Phys., 76 (1988), 48–84. https://doi.org/10.1016/0021-9991(88)90131-3 doi: 10.1016/0021-9991(88)90131-3
[50]	T. Y. Hou, J. S. Lowengrub, M. J. Shelley, Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys., 114 (1994), 312–338. https://doi.org/10.1006/jcph.1994.1170 doi: 10.1006/jcph.1994.1170
[51]	J. W. Jacobs, J. M. Sheeley, Experimental study of incompressible Richtmyer-Meshkov instability, Phys. Fluids, 8 (1996), 405–415. https://doi.org/10.1063/1.868794 doi: 10.1063/1.868794
[52]	M. Nitsche, R. Krasny, A numerical study of vortex ring formation at the edge of a circular tube, J. Fluid Mech., 276 (1994), 139–161. https://doi.org/10.1017/S0022112094002508 doi: 10.1017/S0022112094002508
[53]	N. J. Zabusky, Vortex paradigm for accelerated inhomogeneous flows: visiometrics for the Rayleigh-Taylor and Richtmyer-Meshkov environments, Annu. Rev. Fluid Mech., 31 (1999), 495–536. https://doi.org/10.1146/annurev.fluid.31.1.495 doi: 10.1146/annurev.fluid.31.1.495

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)