Jet formation and large amplitude oscillations from a severely deformed air-water interface

In this two-part talk, I will present theoretical and computational work on surface waves from an initially severely deformed, air-water interface in a cylindrical container. For modal shaped (Bessel function J0) delocalised interfacial deformations dominated by gravity (negligible capillary effects), it will be shown that a thin jet arises at the axis of symmetry when the modal amplitude is sufficiently large. This can rise to a height significantly exceeding the maximum amplitude of the initial deformation. A second-order, weakly-nonlinear, potential flow theory will be presented, which can describe the inception of this jet. In the strongly nonlinear regime where strong jets are seen in simulations, perturbative theory is inadequate. In this regime, we will compare the jets in simulations with an exact analytical solution of a hyperbolic inertial jet by Longuet-Higgins, 1984. In the second part, I will present a study of converging waves in the same geometry as above (cylindrical container). Here, we restrict the length scale of initial deformation to be such that gravity and capillarity are nearly equally strong. The initial deformation is chosen to be shaped as a localised cavity, at the air-water interface. The relaxation of this produces a train of waves subsequently. The reflection of these from the cylinder wall generates a radially inward propagating wave-train focussing towards the symmetry axis of the container. For a sufficiently shallow-cavity, the linearised potential flow equations are adequate for describing the wave-focusing. These, however, become qualitatively inaccurate when the initial cavity is sufficiently deep. It will be shown that nonlinear effects become prominent around the symmetry axis of the container during the focusing. I will describe a second-order potential-flow theory, which can describe this focussing quite well.

References:
1. Bubbles, breaking waves and hyperbolic jets at a free-surface, M. S. Longuet-Higgins, J. Fluid Mech. 1983.
2. Axisymmetric, viscous interfacial oscillations – theory and simulations, Farsoiya, Mayya & Dasgupta, J. Fluid Mech. 2017
3. Jetting in finite-amplitude, free, capillary-gravity waves, Basak, Farsoiya & Dasgupta, J. Fluid Mech. 2020
4. Jet from a very large, axisymmetric, surface-gravity wave, Kayal & Dasgupta, J. Fluid Mech. 2023
5. Focussing of concentric, free-surface waves, Kayal, Sanjay, Yewale, Kumar & Dasgupta (under review), 2024