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Interface instabilities in Faraday waves of two-layer liquids with free surface

Published online by Cambridge University Press:  03 May 2022

Dongming Liu Open the ORCID record for Dongming Liu [Opens in a new window]  and
Pengzhi Lin Open the ORCID record for Pengzhi Lin [Opens in a new window]
Dongming Liu
Affiliation:
State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, PR China
Pengzhi Lin*
Affiliation:
State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, PR China
*
†Email address for correspondence: [email protected]
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Abstract

For a fluid system with an interface between two-layer liquids and a free surface, there exist two series of natural frequencies. Under vertical excitations, the interface and the free surface can be excited separately or simultaneously. Laboratory experiments were conducted to excite the first- and second-mode Faraday waves on the free surface, which forced the in-phase motion of the interface. The experimental data were used to validate the extended numerical model NEWTANK. The secondary surface instability in the form of period tripling breaking occurred on the free surface when second-mode surface resonant response was large. Further numerical investigations were conducted to study the resonance of interface and in addition to the primary Faraday instability, the secondary interface instability occurred due to Rayleigh–Taylor (RT) instability. The analysis revealed that this was due to the contribution from the combined effects of vertical fluid acceleration and external excitation, which made the upper layer liquid essentially ‘heavier’ than the lower layer at certain instances, even when the amplitude of acceleration of the external excitation was lower than that of gravity at all times. A dimensionless parameter ‘normalised effective gravitational acceleration’ (NEGA) was proposed to quantify the occurrence criterion of RT instability in a dynamic fluid system. Furthermore, simultaneous resonances at both the interface and the free surface were studied and reverse flows could be formed across the interface as the upper Faraday waves and lower Faraday waves moved in opposite directions, which generated the secondary interface instability of Kelvin–Helmholtz (KH) type. When KH instability took place, more energy was transferred to higher frequency modes as time evolved.

JFM classification

Waves/Free-surface Flows: Faraday waves Instability: Instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 941 , 25 June 2022 , A33
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Abramowitz, M. & Stegun, I.A. 1965 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Bechhoefer, J., Ego, V., Manneville, S. & Johnson, B. 1995 An experimental study of the onset of parametrically pumped surface waves in viscous fluids. J. Fluid Mech. 288, 325–350.CrossRefGoogle Scholar
Benjamin, T.B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. A 225, 505–515.Google Scholar
Cerda, E. & Tirapegui, E. 1997 Faraday's instability for viscous fluids. Phys. Rev. Fluids 78, 859–862.Google Scholar
Cerda, E. & Tirapegui, E. 1998 Faraday's instability in viscous fluids. J. Fluid Mech. 386, 195–228.CrossRefGoogle Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299–340.Google Scholar
Frandsen, J.B. 2004 Sloshing motions in excited tanks. J. Comput. Phys. 196, 53–87.CrossRefGoogle Scholar
Gueyffier, D., Li, J., Nadim, A., Scardovelli, R. & Zaleski, S. 1999 Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows. J. Comput. Phys. 152, 423–456.CrossRefGoogle Scholar
Ibrahim, R.A. 2015 Recent advances in physics of fluid parametric sloshing and related problems. Trans. ASME J. Fluids Engng 137, 090801.CrossRefGoogle Scholar
Jiang, L., Perlin, M. & Schultz, W.W. 1998 Period tripling and energy dissipation of breaking standing waves. J. Fluid Mech. 369, 273–299.CrossRefGoogle Scholar
Jiang, L., Ting, C., Perlin, M. & Schultz, W.W. 1996 Moderate and steep Faraday waves: instabilities, modulation and temporal asymmetries. J. Fluid Mech. 329, 275–307.CrossRefGoogle Scholar
Jin, X., Xue, M. & Lin, P. 2021 Experimental and numerical study of nonlinear modal characteristics of Faraday waves. Ocean Engng 369, 108554.CrossRefGoogle Scholar
Kahouadji, L., Périnet, N., Tuckerman, L.S., Shin, S., Chergui, J. & Juric, D. 2015 Numerical simulation of supersquare patterns in Faraday waves. J. Fluid Mech. 772, R2.CrossRefGoogle Scholar
Kumar, K. & Tuckerman, L.S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.CrossRefGoogle Scholar
Li, X., Li, X. & Liao, S. 2018 a Observation of two coupled Faraday waves in a vertically vibrating Hele-Shaw cell with one of them oscillating horizontally. Phys. Fluids 30, 012108.CrossRefGoogle Scholar
Li, X., Li, J., Liao, S. & Chen, C. 2018 b Effect of depth on the properties of two coupled Faraday waves in a Hele-Shaw cell. Phys. Fluids 30, 102103.CrossRefGoogle Scholar
Lin, P. 2008 Numerical Modeling of Water Waves. Taylor & Francis Co.CrossRefGoogle Scholar
Lin, P. & Li, C.W. 2003 Wave-current interaction with a vertical square cylinder. Ocean Engng 30, 855–876.CrossRefGoogle Scholar
Lioubashevski, O., Fineberg, J. & Tuckerman, L.S. 1997 Scaling of the transition to parametrically driven surface waves in highly dissipative systems. Phys. Rev. E 55, R3832–R3835.CrossRefGoogle Scholar
Liu, D. 2007 Numerical modeling of three-dimensional water waves and their interaction with structures. Ph.D. thesis, National University of Singapore.Google Scholar
Liu, D. & Lin, P. 2008 A numerical study of three-dimensional liquid sloshing in tanks. J. Comput. Phys. 227, 3921–3939.CrossRefGoogle Scholar
Liu, D., Lin, P., Xue, M., Cheng, L. & Lian, J. 2021 Numerical simulation of two-layered liquid sloshing in tanks under horizontal excitations. Ocean Engng 224, 108768.CrossRefGoogle Scholar
McLachlan, N.W. 1947 Theory and application of Mathieu functions. Oxford University.Google Scholar
Miles, J.W. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143–165.CrossRefGoogle Scholar
Murakami, Y. & Chikano, M. 2001 Two-dimensional direct numerical simulation of parametrically excited surface waves in viscous fluid. Phys. Fluids 13, 65–74.CrossRefGoogle Scholar
Périnet, N., Juric, D. & Tuckerman, L.S. 2009 Numerical simulation of Faraday waves. J. Fluid Mech. 635, 1–26.CrossRefGoogle Scholar
Pototsky, A. & Bestehorn, M. 2016 Faraday instability of a two-layer liquid film with a free upper surface. Phys. Rev. Fluids 1, 023901.CrossRefGoogle Scholar
Pucci, G., Ben Amar, M. & Couder, Y. 2013 Faraday instability in floating liquid lenses: the spontaneous mutual adaptation due to radiation pressure. J. Fluid Mech. 725, 402–427.CrossRefGoogle Scholar
Pucci, G., Fort, E., Ben Amar, M. & Couder, Y. 2011 Mutual adaptation of a Faraday instability pattern with its flexible boundaries in floating fluid drops. Phys. Rev. Lett. 106, 024503.CrossRefGoogle ScholarPubMed
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. Mon. Weath. Rev. 91, 99–164.2.3.CO;2>CrossRefGoogle Scholar
Tipton, C.R. & Mullin, T. 2004 An experimental study of Faraday waves formed on the interface between two immiscible liquids. Phys. Fluids 16, 2336–2341.CrossRefGoogle Scholar
Veletsos, A.S. & Shivakumar, P. 1993 Sloshing response of layered liquids in rigid tanks. Earthq. Engng Struct. Dyn. 22, 801–821.CrossRefGoogle Scholar
Waddell, J.T., Niederhaus, C.E. & Jacobs, J.W. 2001 Experimental study of Rayleigh–Taylor instability: low Atwood number liquid systems with single-mode initial perturbations. Phys. Fluids 13, 1263–1273.CrossRefGoogle Scholar
Wright, J., Yon, S. & Pozrikidis, C. 2000 Numerical studies of two-dimensional Faraday oscillations of inviscid fluids. J. Fluid Mech. 402, 1–32.CrossRefGoogle Scholar
Zhao, S., Dietzel, M. & Hardt, S. 2019 Faraday instability of a liquid layer on a lubrication film. J. Fluid Mech. 879, 422–447.CrossRefGoogle Scholar

Liu et al. Supplementary Movie 1

Resonant response of lower layer with Rayleigh-Taylor instability on the interface

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