Capillary Waves Research Articles

On the structure of parasitic gravity-capillary standing waves in the small surface tension limit

We present new numerical solutions for nonlinear standing water waves when the effects of both gravity and surface tension are considered. For small values of the surface tension parameter, solutions are shown to exhibit highly oscillatory capillary waves (parasitic ripples), which are both time- and space-periodic, and which lie on the surface of an underlying gravity-driven standing wave. Our numerical scheme combines a time-dependent conformal mapping together with a shooting method, for which the residual is minimised by Newton iteration. Previous numerical investigations typically clustered gridpoints near the wave crest, and thus lacked the fine detail across the domain required to capture this phenomenon of small-scale parasitic ripples. The amplitude of these ripples is shown to be exponentially small in the zero surface tension limit, and their behaviour is linked to (or explains) the generation of an elaborate bifurcation structure.

СОВЕРШЕНСТВОВАНИЕ МЕТОДИКИ ПРОЕКТИРОВАНИЯ ПОЧВОЗАЩИТНЫХ ТЕХНОЛОГИЙ НА СКЛОНОВЫХ АГРОЛАНДШАФТАХ

The main purpose of the work is to clarify the values of soil erosion resistance. In the existing methods for assessing the erosion resistance of soils, the kinetic and potential energy of the water flow is studied. It is believed that the decrease in total energy in the process of flowing along the channel is associated with the work of the water flow in destroying and washing away the soil. However, this does not take into account the energy of capillary waves existing on the water surface during laminar flow. The frequency of the capillary waves is usually such that the waves are not visible to the naked eye and are therefore not taken into account, although their energy is, in order of magnitude, comparable to the energy of the flow. Refinement of the value of erosion resistance is associated with the ability to evaluate and take into account the energy of capillary waves. The work is related to the development of a methodology for accounting and evaluating the contribution of the energy of capillary waves to the total energy of a water microflow, which has a destructive and eroding effect on the soil of a sloping agricultural landscape. The paper developed a mathematical model for estimating the energy of capillary waves, on the basis of which an expression was obtained to determine the ratio of the energy of capillary waves formed on the surface of a water flow to the kinetic energy of the water flow. To experimentally study the process of development of capillary waves on smooth and rough surfaces and determine the operating parameters, an installation was created in the form of a rectangular-section tray with an adjustable inclination angle, adjustable intensity of water supply and replaceable working surfaces. The geometry of the microflow and the shape of its surface were determined from the results of video filming and the readings of the laser rangefinder. In the range of regime parameters studied in this work, the numerical estimation of the dependence of the energy of capillary waves on the kinetic energy of the water flow for the first time revealed a significant (up to 40–60%) contribution of capillary waves formed on the surface of microflows to the total energy of the water flow. The results obtained in this work indicate the need to take into account the contribution of capillary waves in the energy analysis of the initial stage of water erosion on slopes.

Experimental investigation of three-dimensional free-surface and interfacial sloshing in a vertical cylindrical tank

In the present study, hundreds of experiments have been conducted on the three-dimensional free-surface and interfacial sloshing in a vertical cylindrical tank containing two immiscible liquids. The bounds of different free-surface and interfacial wave regimes are determined by maintaining fixed excitation amplitude and slowly increasing excitation frequency until another type of wave regime began to appear. In general, three types of the free-surface wave regimes are observed when the excitation frequency is in the neighborhood of the lowest natural frequency of the free surface, i.e., planar gravity wave, chaotic gravity wave, and swirling gravity wave. Similarly, when the excitation frequency is near the lowest natural frequency of the internal interface, three types of interfacial wave regimes, i.e., planar gravity wave, chaotic gravity-capillary wave, and swirling gravity-capillary wave, are generated. Besides, it is worth pointing out that when the excitation frequency is near the lowest natural frequency of the internal interface as well as very close to a third of the lowest natural frequency of the free surface, large-amplitude rotating wave motion occurs at both the free surface and the internal interface. This is due to even though the excitation frequency is far away from the natural frequency of the free surface, the secondary resonance can still become dominant and lead to large-amplitude motion of the free-surface rotating wave and subsequently influences the internal interface. This paper reveals that the sloshing behaviors of two-layer liquid in the vertical cylindrical tank are much more complicated than those of single-layer liquid.

Effects of a subsonic gas flow on the wave blocking and energy of interfacial waves with a constant vorticity

Negative energy waves play a significant role in hydrodynamic instabilities of flow and are involved in all kinds of instability in flows because they can withdraw energy from the flow. Effects of subsonic gas flow and vorticity on the existence of negative energy for water waves as well as the occurrence of capillary-gravity wave blocking have been studied. The existence of blocking points and variation in group velocity for different flow parameters such as the vorticity and subsonic gas factor is studied. The results are in agreement with a previous experiment [Suastika et al., “Experimental study of wave blocking,” Coastal Eng. 1, 227 (2000)]. The present study aims to deal with the combined effect of a subsonic gas flow parallel to solid boundaries and a moving liquid layer with linear velocity profile to establish the existing condition for negative energy waves and the blocking waves, especially for waves propagating in the opposite direction of the current. As a limiting case of the present model, in the absence of aerodynamic effect, it is found that the incident and reflected interfacial waves propagate inside the stability domain; this means that the transformed wave is generated at the expense of the energy of incident wave.

Dynamics of general higher-order rogue waves in the two-component long wave–short wave model of Newell type

An integrable two component long-wave–short-wave (2-LS) model of Newell type governing the resonant interaction between two capillary waves with a common gravity wave is considered. The general higher order rogue wave (RW) solutions of this system are obtained by employing the bilinear KP hierarchy reduction method. Our study explores three different families of RWs depending upon the root structure of an algebraic equation characterizing them. Those are the Nth-order bounded rogue waves, mixed bounded (N1,N2)th-order rogue waves and degradable bounded (N¯1,N¯2)th-order rogue waves, respectively, where N,N1,N2 are arbitrary positive integer and N¯1,N¯2 are arbitrary non-negative integers. It is interesting to note that the fundamental rogue wave in one short wave (SW) component can attain a peak amplitude as high as more than six times the background amplitude while it is three times the background amplitude in the other SW component. We identify that the Nth-order bounded rogue waves are composed of N(N+1)2 fundamental rogue waves, the mixed bounded (N1,N2)th-order rogue waves comprise N1th-order bounded rogue waves mixing with another type N2th-order bounded rogue waves existing in different state, and the degradable bounded (N¯1,N¯2)th-order rogue waves consist of N¯12+N¯22−N¯1(N¯2−1) fundamental rogue waves. Non-trivial super rouge waves are reported.

Agglomeration Drives the Reversed Fractionation of Aqueous Carbonate and Bicarbonate at the Air-Water Interface.

In the course of our investigations of the adsorption of ions to the air-water interface, we previously reported the surprising result that doubly charged carbonate anions exhibit a stronger surface affinity than singly charged bicarbonate anions. In contrast to monovalent, weakly hydrated anions, which generally show enhanced concentrations in the interfacial region, multivalent (and strongly hydrated) anions are expected to show a much weaker surface propensity. In the present work, we use resonantly enhanced deep-UV second-harmonic generation spectroscopy to measure the Gibbs free energy of adsorption of both carbonate (CO32-) and bicarbonate (HCO3-) anions to the air-water interface. Contrasting the predictions of classical electrostatic theory and in support of our previous findings from X-ray photoelectron spectroscopy, we find that carbonate anions do indeed exhibit much stronger surface affinity than do the bicarbonate anions. Extensive computer simulations reveal that strong ion pairing of CO32- with the Na+ countercation in the interfacial region results in the formation of near-neutral agglomerate clusters, consistent with a theory of interfacial ion adsorption based on hydration free energy and capillary waves. Simulated X-ray photoelectron spectra predict a 1 eV shift in the carbonate spectra compared to that of bicarbonate, further confirming our experiments. These findings not only advance our fundamental understanding of ion adsorption chemistry but also impact important practical processes such as ocean acidification, sea-spray aerosol chemistry, and mammalian respiration physiology.

Geochemical analysis of SAR backscattering (Sentinel-1) on global ocean oil spill cases

ABSTRACT The oil spill is one of the most impactful sources of marine pollution on the ocean surface, detected by the SAR sensors as dark areas, regions with low backscatter values. Due to the complex mixture of hydrophobic hydrocarbons, mineral oil spills change the water surface tension dampening the capillary gravity waves and provoking a specular reflection. In this work, we associated the geochemical oil characteristics, such as density, viscosity, API, and molecular composition with the backscatter values for each oil spill case. We identified the relationship between the oil weathering processes, with the changes in the backscattering values of ocean oil spills. The method designed zonal sections over the oil spills detected in the SAR images, to extract the backscatter values for each pixel along the section. The lowest backscatter average was observed by the heavy oil spill in the Corsica Island study (−29,99 dB). The highest level of weathering had the highest backscatter averages. Damping rates ranged between 4,12 and 7,07 dB and the backscatter values may be related to low oil layer thickness. Furthermore, low wind speeds may have reduced the contrast between water and oil spills, resulting in low damping ratios in all events.

Jetting Dynamics of Viscous Droplets on Superhydrophobic Surfaces.

We investigated the dynamics of liquid jets engendered by the impact of droplets on a fractal superhydrophobic surface. Depending on the impact conditions, jets emanate from the free liquid surface with several different shapes and velocities, sometimes accompanied by droplet ejection. Experimental outcomes exhibit two different regimes: the singular jet and columnar jet. We found that droplet impacts at a lower impact velocity and low viscosity result in singular jets, attaining a maximum velocity nearly 20-fold higher than the impact velocity. The high-speed video frames reveal that the formation and subsequent collapse of the cylindrical air cavities within the droplet favor the formation of these high-speed singular jets. In contrast, the capillary wave focusing engenders columnar jets at a moderate to high impact velocity. With an increase in viscosity, singular jets are suppressed at lower impact velocities, whereas columnar jets are seen regularly. The columnar jets ascend and grow over time, feeding a bulbous mass, and subsequently the bulb separates itself from the parent jet due to capillary pinch-off phenomena. The quantitative analysis shows that columnar jets' top jet drop size varies nonmonotonically and is influenced by preceding jetting dynamics. At moderate viscosity, the drop size varies with jet velocity, following a power-law scaling. At very high viscosities, both singular and columnar jetting events are inhibited. The results are relevant to several recent technologies, including microdispensing, thermal management, and disease transmission.

Exact Solution for Capillary Waves on the Surface of a Liquid of Finite Depth

Exact Solution for Capillary Waves on the Surface of a Liquid of Finite Depth

Quantized vortex nucleation in collisions of superfluid nanoscopic helium droplets at zero temperature.

We address the collision of two superfluid 4He droplets at non-zero initial relative velocities and impact parameters within the framework of liquid 4He time-dependent density functional theory at zero temperature. Despite the small size of these droplets (1000 He atoms in the merged droplet) imposed by computational limitations, we have found that quantized vortices may be readily nucleated for reasonable collision parameters. At variance with head-on collisions, where only vortex rings are produced, collisions with a non-zero impact parameter produce linear vortices that are nucleated at indentations appearing on the surface of the deformed merged droplet. Whereas for equal-size droplets, vortices are produced in pairs, an odd number of vortices can appear when the colliding droplet sizes are different. In all cases, vortices coexist with surface capillary waves. The possibility for collisions to be at the origin of vortex nucleation in experiments involving very large droplets is discussed. An additional surprising result is the observation of the drops coalescence even for grazing and distal collisions at relative velocities as high as 80 and 40m/s, respectively, induced by the long-range van der Waals attraction between the droplets.

Surface tension of steel at high temperatures

Surface tension is a material property that is needed to describe fluid behaviour, which impacts industrial processes, in which molten material is created, such as thermal cutting, welding and Additive Manufacturing. In particular when using metals, the material properties at high temperatures are often not known. This is partly because of limited possibilities to measure those properties, limitations of temperature measurement methods and a lack of theoretical models that describe the circumstances at such high temperatures sufficiently. When using beam heat sources, such as a laser beam, temperatures far above the melting temperature are reached. Therefore, it is mandatory to know the material properties at such high temperatures in order to describe the material behaviour in models and gain understanding of the occurring effects. Therefore, in this work, an experimental surface wave evaluation method is suggested in combination with thermal measurements in order to derive surface tension values of steel at higher temperatures than reported in literature. The evaluation of gravity-capillary waves in high-speed video recordings shows a steeper decrease of surface tension values than the extrapolation of literature values would predict, while the surface tension values seem not to decrease further above boiling temperature. Using a simplified molecular dynamic model based on pair correlation, a similar tendency of surface values was observed, which indicates that the surface tension is an effect requiring at least two atomic layers. The observed and calculated decreasing trend of the surface tension indicates an exponential relation between surface tension and temperature.

An investigation on the impact of two vertically aligned drops on a liquid surface

The dynamics of two vertically coalescing drops and a pool of the same liquid have been investigated using a Coupled Level Set and Volume of Fluid (CLSVOF) method. Such a configuration enables us to study the dynamic interaction of an arbitrary-shaped liquid conglomerate, formed owing to drop–drop coalescence, with a pool. Similar to drop–pool and drop–drop interactions, partial coalescence is observed when a conglomerate interacts with a pool. The presence of the pool below the father drop is found to influence the coalescence characteristic of the two drops. At the same time, the movement of the capillary waves resulting from the interaction of two drops governs the coalescence dynamics of the conglomerate with the pool. As liquid interfaces interact and generate capillary waves at multiple locations, complex trajectories of capillary waves are observed, which play a crucial role in determining the pinch-off characteristics of the satellite during conglomerate–pool interaction. We examine the effect of the ratio of the diameters of the lower/father drop to the upper/mother drop (Dr) on the coalescence dynamics while maintaining the size of the mother drop constant. The variation in the coalescence dynamics due to change in Dr is quantified in terms of the residence time (τr), pinch-off time (τp) and the satellite diameter to conglomerate diameter ratio (Ds/Dc). The coalescence dynamics of the conglomerate is then compared with that of an equivalent spherical drop of the same volume and also with that of a drop initialized with the same shape as that of the conglomerate. Finally, the regions of complete and partial coalescence for the conglomerate–pool interactions are demarcated on the Weber number–diameter ratio (We−Dr) space.

Singular jets during droplet impact on superhydrophobic surfaces

Hypothesis: The impact of droplets is prevalent in numerous applications, and jetting during droplet impact is a critical process controlling the dispersal and transport of liquid. New jetting dynamics are expected in different conditions of droplet impact on super-hydrophobic surfaces, such as new jetting phenomena, mechanisms, and regimes.Experiments: In this experimental study of droplet impact on super-hydrophobic surfaces, the Weber number and the Ohnesorge number are varied in a wide range, and the impact process is analyzed theoretically.Findings: We identify a new type of singular jets, i.e., singular jets induced by horizontal inertia (HI singular jets), besides the previously studied singular jets induced by capillary deformation (CD singular jets). For CD singular jets, the formation of the cavity is due to the propagation of capillary waves on the droplet surface; while for HI singular jets, the cavity formation is due to the large horizontal inertia of the toroidal edge during the retraction of the droplet after the maximum spreading. Key steps of the impact process are analyzed quantitatively, including the spreading of the droplet, the formation and the collapse of the spire, the formation and retraction of the cavity, and finally the formation of singular jets. A regime map for the formation of singular jets is obtained, and scaling relationships for the transition conditions between different regimes are analyzed.

Numerical Study of Free-Surface Electro-Hydrodynamic Wave Turbulence

Direct numerical simulation of threedimensional chaotic motion of a dielectric liquid with a free surface under the action of external horizontal electric field is carried out. The numerical model takes into account the effects of surface tension, viscosity, and external isotropic random forcing acting at large scales. A transition from dispersive capillary wave turbulence to quasi-isotropic nondispersive EHD surface turbulence with increase of the external electric field strength is numerically observed for the first time. At the regime of developed EHD wave turbulence, the total electrical energy is found to be much greater than the energy of capillary waves, i.e., electrohydrodynamic effects play a dominant role. At the same time, anisotropic effects are detected that lead to the generation of capillary wave packets traveling perpendicular to the external field direction. Despite the revealed anisotropy, the calculated spectrum of EHD wave turbulence is in very good agreement with the analytical spectrum obtained on the basis of dimensional analysis of weak turbulence spectra.

Nonlinear evolution of viscoplastic film flows down an inclined plane.

In this article, we experimentally investigate the nonlinear behaviour of a viscoplastic film flow down an inclined plane. We focus on the nonlinear instabilities that appear as roll waves. Roll waves are generated by perturbing a permanent flow of Herschel-Bulkley fluid (Carbopol 980) at low frequencies. To determine the local thickness of the film, we used a laser sensor and a camera to globally capture the transverse shape of the waves. For a regular forcing, the results show the existence of different regimes. First, we observe primary instabilities below the cut-off frequency at the entrance of the channel. After the exponential growth of the wave in the linear regime, we recognise the nonlinear dynamics with the existence of finite amplitude waves. This finite amplitude depends on the frequency, the Reynolds number and the inclination angle. The results show that this instability is supercritical. At moderate Reynolds numbers, the finite 2-D waves become sensitive to transverse perturbations, due to a secondary instability, and become 3-D waves. The experimental results illustrate a phenomenology of viscoplastic film flows similar to Newtonian fluids, except for the capillary waves.

Waveguide Waves on the Water Surface in an Open Channel with an Built-in Source and a Capillary Wave Resonator

Waveguide Waves on the Water Surface in an Open Channel with an Built-in Source and a Capillary Wave Resonator

On the Interaction of Azimuthal Modes of Capillary Waves on the Surface of an Elliptic Jet in a Homogeneous Electrostatic Field

On the Interaction of Azimuthal Modes of Capillary Waves on the Surface of an Elliptic Jet in a Homogeneous Electrostatic Field

Melt pond detection on landfast sea ice using dual co-polarized Ku-band backscatter

The onset of melt ponds on landfast first-year and multiyear sea ice is detected using dual co-polarized Ku-band spaceborne scatterometer data. Twenty-nine locations in the Canadian Arctic Archipelago are used to develop and test the methodology. The daily ScatSat-1 sigma-nought backscatter composite product with a spatial resolution of 2.216 km is used. Vertical (VV) and horizontal (HH) polarizations of these data are used to derive the co-polarized ratio (VV/HH). An increase in VV/HH is associated with an increase in melt pond fraction on the sea ice. This occurs because there is a larger increase in resonant Bragg scattering for VV than for HH when melt pond surfaces exhibit wind-generated capillary waves. VV backscatter magnitude is evaluated in time series using a peak-finding methodology to initiate detection of melt ponds with VV/HH. The peak-finding is based on geophysical controls on backscatter from snow-covered sea ice during the melt season. Daily time-series data from ScatSat-1 are used in conjunction with ERA-5 reanalysis data (2-m air temperature and wind) and a semi-empirical snowmelt model. Air temperature data are used to identify snow melt onset and to force the snowmelt model. Wind data are used to evaluate if the wind speed is sufficient for capillary wave formation. The NSCAT-4DS geophysical model function is used to model Ku-band backscatter from water, which is then used in a sea ice mixture model to simulate the VV/HH magnitude for a range of pond fractions and wind speeds. The potential for detecting pond onset is evaluated in relation to the noise inherent in the ScatSat-1 VV/HH sea ice measurements, which is found to be approximately 0.16 dB. When the melt pond fraction is sufficiently large to cause VV/HH to exceed the noise, pond onset is detected. Cloud-free Sentinel-2 optical satellite imagery is used for validation; specifically, the near-infrared albedo. We find an overall mean bias of −0.5 days, and an absolute deviation of 1.4 days, when comparing our method to a near-infrared method for detecting pond onset. The mean biases (absolute deviations) for first-year and multiyear sea ice are −0.3 (1.5) and −0.8 (1.2) days, respectively. Results show that the temporal resolution, spatial resolution, and efficacy of this method can provide robust sea ice melt pond onset timing for landfast sea ice regions of the Arctic.

Comprehensive Perturbation Approach to Nonlinear Viscous Gravity–Capillary Surface Waves at Arbitrary Wavelengths in Finite Depth

This study presents a comprehensive analysis of the second-order perturbation theory applied to the Navier–Stokes equations governing free surface flows. We focus on gravity–capillary surface waves in incompressible viscous fluids of finite depth over a flat bottom. The amplitude of these waves is regarded as the perturbation parameter. A systematic derivation of a nonlinear-surface-wave equation is presented that fully takes into account dispersion, while nonlinearity is included in the leading order. However, the presence of infinitely many over-damped modes has been neglected and only the two least-damped modes are considered. The new surface-wave equation is formulated in wave-number space rather than real space and nonlinear terms contain convolutions making the equation an integro-differential equation. Some preliminary numerical results are compared with computational-modelling data obtained via open source CFD software OpenFOAM.

NONLINEAR SELF-MODULATION OF GRAVITY-CAPILLARY WAVES ON SHEAR CURRENTS IN FINITE DEPTH

AbstractA nonlinear evolution equation correct to fourth order is developed for gravity-capillary waves on linear shear currents in finite water depth. Therefore, this equation covers both effects of depth uniform currents and uniform vorticity. Starting from this equation, an instability analysis is then made for narrow banded uniform Stokes waves. The notable feature is that our investigation due to fourth order shows a remarkable improvement compared with the third-order one, and produces an excellent result compatible with the exact result of Longuet-Higgins. We observe that linear shear currents considerably change the modulational instability properties of capillary-gravity waves, such as the growth rate and bandwidth of instability.