Dispersion Of Capillary Waves Research Articles

A coupled immersed interface and level set method for simulation of interfacial flows steered by surface tension

This work presents a methodology to address the problems of bubbles and drops evolving in incompressible, viscous flows due to the effect of surface tension. It is based on the combination of a recently developed immersed interface method to resolve discontinuities, with the level set (LS) method to reproduce the evolving interfaces. The paramount feature of this immersed interface method is the use of Lagrange interpolation enclosing grid points positioned in the vicinity of the interface and few exceptional grid points positioned on the interface. Different problems are considered to assert the accurateness of the proposed methodology, involving both simple and complex interface geometries. Precisely, the following problems are addressed: circular flow with a fixed interface, the dispersion of capillary waves, initially circular, ellipse, star shaped bubbles oscillating to an equilibrium state, and circular drops deforming in shear flows. The transient evolution of bubbles/drops in terms of their shapes, pressure profiles, velocity vectors, deformation ratios of major and minor axis is analyzed to observe the effect of surface tension. The proposed methodology is seen to recover the exact numerical equilibrium between the surface tension and pressure gradient in the vicinity of complex interface geometries as well, while recreating the flow physics with an adequate level of accuracy with well representation of overall trends. Moreover, the numerical results yield a good level of agreement with the reference data.

A hybrid [formula omitted]-[formula omitted] HOC approach for surface tension driven flows in level set framework

In this study, a higher order compact (HOC) finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of Navier–Stokes (N–S) equations with the level set method. Unlike most of the existing methods for multiphase flows that solve the incompressible Navier–Stokes equations in the velocity–pressure form or the stream-function vorticity form, we recast the governing equations into the biharmonic form of the N–S equations which carries stream-function ψ as the only variable to be computed. The coupling of this formulation with level set (LS) enables the simulation of two phase flows by the core computation of only the single variable ψ free from discontinuities in the whole physical domain. In the process, we also propose a novel hybrid HOC algorithm for post-processing the pressure on either side of the interface including the jump across it. The robustness of this method is established through a series of benchmark problems such as dispersion of capillary waves, static bubble and oscillating bubble at equilibrium in a fluid at rest, and deforming circular drops in shear flows. We compare our numerical results with available benchmark results and excellent match is obtained in all the cases along with accurate representation of overall trends. Our approach is proved to be very efficient in terms of its easy implementation, relatively low computational time and high order of accuracy.

Effect of viscosity on dispersion of capillary–gravity waves

The effect of viscosity on dispersion of capillary–gravity waves becomes significant when the attenuation coefficient is greater than about 2.5% of the wave number. For low viscosity fluids such as water this condition is met at frequencies greater than about 5 kHz in which case direct measurement of wavelength is difficult. For higher viscosity fluids the effect appears at much lower frequencies but direct measurement of wavelength becomes difficult since viscosity causes severe attenuation of surface waves. We have overcome the measurement difficulties by using a new miniature laser interferometer, which directly measures the wavelength of standing capillary waves with the requisite precision to yield reliable dispersion data for viscous fluids. Here we review the effect of viscosity on the dispersion relation and present new experimental data on dispersion of capillary waves in several water-glycerol mixtures. Our data provides direct experimental verification of the theoretical analysis.

Direct measurement of the dispersion relation of capillary waves by laser interferometry

Surface waves on fluids with wavelengths in the millimeter range are known as capillary waves. Surface tension determines the propagation and dispersion of capillary waves while gravity plays a minor role. We describe a simple method for generating standing capillary waves of known frequency on water and introduce a novel noncontact technique based on laser interferometry to measure the wavelength of capillary waves with great precision. The data gives the dispersion relation of capillary waves and provides an accurate method for determining the surface tension of fluids.

Properties of Superfluid Fog Produced by an Ultrasonic Transducer

A dense fog consisting of superfluid 4 He droplets can be generated in helium vapor up -to a height of a few cm from the helium surface by driving a piezo transducer plate immersed under the liquid. The average droplet size is measured with a long-distance microscope and CCD camera, and ranges from about 100 μm at a drive frequency of 1 kHz to 10 μm at 100 kHz. The dependence of the droplet size on frequency is quite consistent with the capillary-wave dispersion relation of the helium surface, with the droplets ejected from the surface being about one wavelength in diameter. The initial vertical velocity of the droplets emerging from the surface is found to increase linearly with the driving voltage being applied to the piezo.

Capillary waves in−43He solutions and the properties of localized electronson the helium surface

We derive the dispersion of capillary waves (ripplons) in a microstratified $^{3}\mathrm{He}$ phase above a $^{3}\mathrm{\ensuremath{-}}^{4}$He solution substrate. Both the real and imaginary parts of the ripplon frequencies are evaluated by taking into account the essential process of viscous damping. The influence of these surface modes on the well-known system formed by electrons trapped at the surface of liquid helium is investigated. We show that the low-damped capillarylike surface mode, found in the microstratified solution, interacting with the surface electrons determines their dynamical properties at low temperatures.

Renormalization-group relaxational dynamics of interfaces in4−εdimensions

We present a dynamic renormalization-group calculation in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions of the capillary wave dispersion relation ${\ensuremath{\omega}}_{q}$ for the interface of an Ising-like system driven by relaxational dynamics (model A). The dispersion relation is of the form ${\ensuremath{\omega}}_{q}=\ensuremath{-}i\ensuremath{\Gamma}{q}^{z}\ensuremath{\Omega}(q\ensuremath{\xi})$, with $\ensuremath{\Omega}(x)$ universal and $z=2+O({\ensuremath{\epsilon}}^{2})$, and satisfies the Goldstone theorem for the spontaneously broken Euclidean symmetry.

Fluctuating hydrodynamics and capillary waves

We derive an equation of motion for the instantaneous position of a planar liquid surface using fluctuating hydrodynamics. For the fluid constrained to be in a nonequilibrium steady state by a small temperature gradient, we then obtain the dynamic structure factor ${S}_{\mathrm{ss}}(q,\ensuremath{\omega})$, which represents the spectrum of thermal ripplons (capillary waves) on the liquid surface. Our analysis is done for both classical and quantum fluids. We find that there is an asymmetry in the heights of the two ripplon peaks in ${S}_{\mathrm{ss}}(q,\ensuremath{\omega})$ due to broken time-reversal symmetry. Our numerical estimates of the effect for liquid helium II, suggest that it could be measurable. We have undertaken a moment analysis of ${S}_{\mathrm{ss}}(q,\ensuremath{\omega})$; the first frequency moment is long ranged ($\ensuremath{\sim}\frac{1}{{q}^{2}}$) in the steady state, though it vanishes in equilibrium. From the zeroth frequency moment we recover the familiar result for the infrared logarithmic divergence of the mean-square displacement of the interface; the analysis explicitly involves the coupling of the surface to the bulk fluid. We modify the so-called capillary-wave model by including the surface-bulk coupling, and thereby make it consistent with the capillary-wave dispersion.