Kinematics of the First Wave Faraday Mode on the Side Wall of a Rectangular Vessel

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Abstract

New results of experiments on studying the first Faraday wave mode on the surface of a shallow liquid in a rectangular vessel are given. For regular waves, resonance dependences were measured and the wave profiles were analyzed. It is shown that the presence of a moving local surface elevation in the form of a hump is associated with the nonlinearity of wave oscillations of liquid. A comparison is made with a theoretical model of nonlinear gravity waves. The mechanism of breaking the first Faraday wave mode consisting in the formation of a plane jet ejection on the side wall of the vessel as a result of focusing fluid flows in the growing crest and surface hump has been studied.

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About the authors

V. A. Kalinichenko

Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences

Author for correspondence.
Email: kalin@ipmnet.ru
Russian Federation, Moscow

References

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2. Fig. 1. Profiles of regular waves with a time step of a quarter period: a-c - n = 1, (h = 5, 7.5, 10 cm, T = 1.356, 1.224, 1.110 s, H = 4.0, 5.8, 7.9 cm); d - n = 2, T = 0.814 s, H = 4.4 cm. The time interval for the three profiles on a-d corresponds to a quarter of the wave period.

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3. Fig. 2. a) Resonance dependences of H (Ω) for the first 1-4 and second 5 modes of regular waves at the water surface of different depth h: 1-4 - n = 1, h = 5, 7.5, 10, 20 cm; 5 - n = 2, h = 5 cm; solid curves - calculated dependence of H (Ω) for Faraday gravity waves [15, 17]; b) dependence of the wave steepness of the limiting height on the dimensionless depth of the liquid (n = 1).

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4. Fig. 3. Regular wave on the free surface of water (h = 10 cm; T = 1.058 s; Ω = 11.87 s-1; H = 10.8 cm; ω = 5.938 s-1; s = 0.7 cm): a) sequence of snapshots of the free surface during the half-period of the wave (video recording at a frame rate of 1000 fps); b) profiles and velocities of the free surface particles calculated by (2.1).

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5. Fig. 4. Regular wave profiles and trajectories of tracer particles (h = 10 cm; T = 1.058 s; H = 10.8 cm; ω = 5.938 s-1; s = 0.7 cm): 1-6 - t = 0, 40, 80, 120, 160, 240 ms; right half of the vessel. Frames 1 and 4 show the velocity field of tracer particles.

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6. Fig. 5. Sequence of frames (from left to right) showing the process of the second wave mode collapse: a) - collapse of a cavern in the centre of the vessel and formation of a jet burst at the stage of crest formation; b) - two caverns and formation of two flat jets (arrows) on the side walls of the vessel at the stage of wave trough formation; n = 2; h = 5 cm; s = 1. 9 cm; Ω = 15.70 s-1; time step 0.04 s; video recording, 120 k/s; H1 - height of the surge.

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7. Fig. 6. Decaying first wave mode at the free water surface (h = 10 cm; T = 1.026 s; Ω = 12.24 s-1; ω = 6.12 s-1; s = 0.7 cm).

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8. Fig. 7. Detail of the process of flat jet formation on the side wall: 1-6 - t = 0, 40, 80, 120, 160, 200 ms; overlay of 20 video frames (20 ms). The experimental parameters are the same as in the caption to Fig. 6.

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9. Fig. 8. Breakdown of the first mode on the sidewall during video angles: 1-6 - t = 0, 40, 80, 120, 160, 200 ms. The experimental parameters are the same as in the caption of Fig. 6.

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10. Fig. 9. a) Time dependence of the jet burst height on the side wall for the first wave mode: 1, 2 - T = 1.058, 1.026 s (wave period); 3 - calculation according to (2.1); h = 10 cm; s = 0.7 cm; ω = 5.938 s-1 (based on the results of video recording at a speed of 1000 k/s); b) Time dependence of the jet burst height in the dimensionless form H1* (t*): 1-4 - (h, s, Ω / 2, Hlim) = (5, 1.9, 4.88, 4), (7.5, 1.9, 5.51, 5.5), (10, 1.9, 5.72, 7), (10, 0.7, 6.12, 7).

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