7 results
Jet from a very large, axisymmetric, surface-gravity wave
- Lohit Kayal, Ratul Dasgupta
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- Journal:
- Journal of Fluid Mechanics / Volume 975 / 25 November 2023
- Published online by Cambridge University Press:
- 16 November 2023, A22
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We demonstrate that gravity acting alone at large length scales (compared to the capillary length) can produce a jet from a sufficiently steep, axisymmetric surface deformation imposed on a quiescent, deep pool of liquid. Mechanistically, the jet owes it origin to the focusing of a concentric, surface wave towards the axis of symmetry, quite analogous to such focusing of capillary waves and resultant jet formation observed during bubble collapse at small scales. A weakly nonlinear theory based on the method of multiple scales in the potential flow limit is presented for a modal (single-mode) initial condition representing the solution to the primary Cauchy–Poisson problem. A pair of novel, coupled, amplitude equations are derived governing the modulation of the primary mode. For moderate values of the perturbation parameter $\epsilon$ (a measure of the initial perturbation steepness), our second-order theory captures the overshoot (incipient jet) at the axis of symmetry quite well, demonstrating good agreement with numerical simulation of the incompressible, Euler equation with gravity (Popinet 2014, Basilisk. http://basilisk.fr) and no surface tension. We demonstrate that the underlying wave focusing mechanism may be understood in terms of radially inward motion of nodal points of a linearised, axisymmetric, standing wave. This explanation rationalises the ubiquitous observation of such jets accompanying cavity collapse phenomena, spanning length scales from microns to several metres. Expectedly, our theory becomes inaccurate as $\epsilon$ approaches unity. In this strongly nonlinear regime, slender jets form with surface accelerations exceeding gravity by more than an order of magnitude. In this inertial regime, we compare the jets in our simulations with the inertial, self-similar, analytical solution by Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121) and find qualitative agreement with the same. This analysis demonstrates, from first principles, an example of a jet created purely under gravity from a smooth initial perturbation and provides support to the analytical model of Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121).
Dimples, jets and self-similarity in nonlinear capillary waves
- Lohit Kayal, Saswata Basak, Ratul Dasgupta
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- Journal:
- Journal of Fluid Mechanics / Volume 951 / 25 November 2022
- Published online by Cambridge University Press:
- 08 November 2022, A26
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Numerical studies of dimple and jet formation from a collapsing cavity often model the initial cavity shape as a truncated sphere, mimicking a bursting bubble. In this study, we present a minimal model containing only nonlinear inertial and capillary forces, which produces dimples and jets from a collapsing capillary wave trough. The trough in our simulation develops from a smooth initial perturbation, chosen to be an eigenmode to the linearised ${O}(\epsilon )$ problem ($\epsilon$ is the non-dimensional amplitude). We explain the physical mechanism of dimple formation and demonstrate that, for moderate $\epsilon$, the sharp dimple seen in simulations is well captured by the weakly nonlinear ${O}(\epsilon ^3)$ theory developed here. For $\epsilon \gg 1$ the regime is strongly nonlinear, spreading surface energy into many modes, and the precursor dimple now develops into a sharply rising jet. Here, simulations reveal a novel localised window (in space and time) where the jet evolves self-similarly following inviscid (Keller & Miksis, SIAM J. Appl. Maths, vol. 43, issue 2, 1983, pp. 268–277) scales. We develop an analogy of this regime to a self-similar solution of the first kind, for linearised capillary waves. Our first-principles study demonstrates that, at sufficiently small scales, dimples and jets can form from radial inward focusing of capillary waves, and the formation of this may be described by a relatively simple model employing (nonlinear) inertial and capillary effects. Viscosity and gravity can, however, significantly influence the focusing process, either intensifying the singularity or weakening it (Walls et al., Phys. Rev. E, vol. 92, issue 2, 2015, 021002; Gordillo & Rodríguez-Rodríguez, J. Fluid Mech., vol. 867, 2019, pp. 556–571). This leads, in particular, to critical values of Ohnesorge and Bond numbers, which cannot be obtained from our minimal model.
Dynamic stabilisation of Rayleigh–Plateau modes on a liquid cylinder
- Sagar Patankar, Saswata Basak, Ratul Dasgupta
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- Journal:
- Journal of Fluid Mechanics / Volume 946 / 10 September 2022
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- 02 August 2022, A2
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We demonstrate dynamic stabilisation of axisymmetric Fourier modes susceptible to the classical Rayleigh–Plateau (RP) instability on a liquid cylinder by subjecting it to a radial oscillatory body force. Viscosity is found to play a crucial role in this stabilisation. Linear stability predictions are obtained via Floquet analysis demonstrating that RP unstable modes can be stabilised using radial forcing. We also solve the linearised, viscous initial-value problem for free-surface deformation obtaining an equation governing the amplitude of a three-dimensional Fourier mode. This equation generalizes the Mathieu equation governing Faraday waves on a cylinder derived earlier in Patankar et al. (J. Fluid Mech., vol. 857, 2018, pp. 80–110), is non-local in time and represents the cylindrical analogue of its Cartesian counterpart (Beyer & Friedrich, Phys. Rev. E, vol. 51, issue 2, 1995, p. 1162). The memory term in this equation is physically interpreted and it is shown that, for highly viscous fluids, its contribution can be sizeable. Predictions from the numerical solution to this equation demonstrate the predicted RP mode stabilisation and are in excellent agreement with simulations of the incompressible Navier–Stokes equations (up to the simulation time of several hundred forcing cycles).
Jetting in finite-amplitude, free, capillary-gravity waves
- Saswata Basak, Palas Kumar Farsoiya, Ratul Dasgupta
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- Journal:
- Journal of Fluid Mechanics / Volume 909 / 25 February 2021
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- 17 December 2020, A3
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We present a theoretical and computational study of the mechanics of a jet formed from a large amplitude, axisymmetric, capillary-gravity wave on the surface of a liquid pool in a cylindrical container. Jetting can cause a pronounced overshoot of the interface at the axis of symmetry. A linear theory presented earlier in Farsoiya et al. (J. Fluid Mech., vol. 826, 2017, pp. 797–818) was shown to be incapable of describing this jet. To understand its mechanics, we present here the inviscid, weakly nonlinear solution to the initial value problem where the initial surface perturbation is a single Fourier–Bessel mode on quiescent fluid. The theory predicts that energy injected into a primary (Bessel) mode initially is transferred nonlinearly to a spectrum of modes. The extent of the theoretically predicted energy transfer is found to be very accurate for modes up to the second harmonic. We show using numerical simulations that the jet originates as a small dimple formed at the trough of the wave, analogous to similar observations in bubble cavity collapse (Duchemin et al., Phys. Fluids, vol. 14, 2002, pp. 3000–3008; Lai et al., Phys. Rev. Lett. vol. 121, 2018, 144501). The theory is able to describe the jet overshoot and the velocity and pressure fields in the liquid qualitatively, but does not capture the temporal evolution of the dimple or the thinning of the jet neck leading to pinchoff. Modal analysis shows that the latter phenomenon requires higher-order approximations, beyond the second order presented here. The nonlinear theory yields explicit analytical expressions without any fitting parameters which are systematically tested against numerical simulations of the incompressible Euler equation. The theory contains cylindrical analogues of the singularities corresponding to second harmonic resonance (Wilton, Lond. Edinb. Dubl. Phil. Mag. J. Sci., vol. 29, 1915, pp. 688–700). The connection of these to triadic resonant interactions among capillary-gravity waves in a cylindrical confined geometry is discussed.
Azimuthal capillary waves on a hollow filament – the discrete and the continuous spectrum
- Palas Kumar Farsoiya, Anubhab Roy, Ratul Dasgupta
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- Journal:
- Journal of Fluid Mechanics / Volume 883 / 25 January 2020
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- 25 November 2019, A21
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We study the temporal spectrum of linearised, azimuthal, interfacial perturbations imposed on a cylindrical gaseous filament surrounded by immiscible, viscous, quiescent fluid in radially unbounded geometry. Linear stability analysis shows that the base state is stable to azimuthal perturbations of standing wave form. Normal mode analysis leads to a viscous dispersion relation and shows that in addition to the discrete spectrum, the problem also admits a continuous spectrum. For a given azimuthal Fourier mode and Laplace number, the discrete spectrum yields two eigenfunctions which decay exponentially to zero at large radii and thus cannot represent far field perturbations. In addition to these discrete modes, we find an uncountably infinite set of eigenmodes which decay algebraically to zero. The completeness theorem for perturbation vorticity may be expressed as a sum over the discrete modes and an integral over the continuous ones. We validate our normal mode results by solving the linearised, initial value problem (IVP). The initial perturbation is taken to be an interfacial, azimuthal Fourier mode with zero perturbation vorticity. It is shown that the expression for the time dependent amplitude of a capillary standing wave (in the Laplace domain, $s$) has poles and branch points on the complex $s$ plane. We show that the residue at the poles yields the discrete spectrum, while the contribution from either side of the branch cut provides the continuous spectrum contribution. The particular initial condition treated here in the IVP, has projections on the discrete as well as the continuous spectrum eigenmodes and thus both sets are excited initially. Consequently the time evolution of the standing wave amplitude and the perturbation vorticity field have the form of a sum over discrete exponential contributions and an integral over a continuous range of exponential terms. The solution to the IVP leads to explicit analytical expressions for the standing wave amplitude and the vorticity field in the fluid outside the filament. Linearised analytical results are validated using direct numerical simulations (DNS) conducted using a code developed in-house for solving the incompressible, Navier–Stokes equations with an interface. For small perturbation amplitude, analytical predictions show excellent agreement with DNS. Our analysis complements and extends earlier results on the discrete and the continuous spectrum for interfacial viscous, capillary waves on unbounded domain.
Faraday waves on a cylindrical fluid filament – generalised equation and simulations
- Sagar Patankar, Palas Kumar Farsoiya, Ratul Dasgupta
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- Journal:
- Journal of Fluid Mechanics / Volume 857 / 25 December 2018
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- 19 October 2018, pp. 80-110
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We perform linear stability analysis of an interface separating two immiscible, inviscid, quiescent fluids subject to a time-periodic body force. In a generalised, orthogonal coordinate system, the time-dependent amplitude of interfacial perturbations, in the form of standing waves, is shown to be governed by a generalised Mathieu equation. For zero forcing, the Mathieu equation reduces to a (generalised) simple harmonic oscillator equation. The generalised Mathieu equation is shown to govern Faraday waves on four time-periodic base states. We use this equation to demonstrate that Faraday waves and instabilities can arise on an axially unbounded, cylindrical capillary fluid filament subject to radial, time-periodic body force. The stability chart for solutions to the Mathieu equation is obtained through numerical Floquet analysis. For small values of perturbation and forcing amplitude, results obtained from direct numerical simulations (DNS) of the incompressible Euler equation (with surface tension) show very good agreement with theoretical predictions. Linear theory predicts that unstable Rayleigh–Plateau modes can be stabilised through forcing. This prediction is borne out by DNS results at early times. Nonlinearity produces higher wavenumbers, some of which can be linearly unstable due to forcing and thus eventually destabilise the filament. We study axisymmetric as well as three-dimensional perturbations through DNS. For large forcing amplitude, localised sheet-like structures emanate from the filament, suffering subsequent fragmentation and breakup. Systematic parametric studies are conducted in a non-dimensional space of five parameters and comparison with linear theory is provided in each case. Our generalised analysis provides a framework for understanding free and (parametrically) forced capillary oscillations on quiescent base states of varying geometrical configurations.
Axisymmetric viscous interfacial oscillations – theory and simulations
- Palas Kumar Farsoiya, Y. S. Mayya, Ratul Dasgupta
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- Journal:
- Journal of Fluid Mechanics / Volume 826 / 10 September 2017
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- 15 August 2017, pp. 797-818
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We study axisymmetric, free oscillations driven by gravity and surface tension at the interface of two viscous, immiscible, radially unbounded fluids, analytically and numerically. The interface is perturbed as a zeroth-order Bessel function (in space) and its evolution is obtained as a function of time. In the linearised approximation, we solve the initial value problem (IVP) to obtain an analytic expression for the time evolution of wave amplitude. It is shown that a linearised Bessel mode temporally evolves in exactly the same manner as a Fourier mode in planar geometry. We obtain novel analytical expressions for the time varying vorticity and pressure fields in both fluids. For small initial amplitudes, our analytical results show excellent agreement with those obtained from solving the axisymmetric Navier–Stokes equations numerically. We also compare our results with the normal mode approximation and find the latter to be an accurate representation at very early and late times. The deviation between the normal mode approximation and the IVP solution is found to increase as a function of viscosity ratio. The vorticity field has a jump discontinuity at the interface and we find that this jump depends on the viscosity and the density ratio of the two fluids. Upon increasing the initial perturbation amplitude in the simulations, nonlinearity produces qualitatively new features not present in the analytical IVP solution. Notably, a jet is found to emerge at the axis of symmetry rising to a height greater than the initial perturbation amplitude. Increasing the perturbation amplitude further causes the jet to undergo end pinch off, giving birth to a daughter droplet. This can happen either for an advancing or a receding jet, depending on the viscosity ratio. A relation is found between the maximum jet height and the perturbation amplitude. Hankel transform of the interface demonstrates that at large perturbation amplitudes higher wavenumbers emerge, sharing some of the energy of the lowest mode. When these additional higher modes are present, the interface has pointed crests and rounded troughs.