Time-dependent nonlinear gravity–capillary surface waves with viscous dissipation and wind forcing

In this laboratory investigation, a range of wind and wave conditions was sampled to visualize the airflow and to quantify the mechanisms through which the momentum is transferred towards a wavy surface. The experiments were conducted in the SUrge STructural AIr-sea INteraction facility located at the University of Miami. Three distinct background wave conditions were subjected to wind forcing U 10  of 7 and 14 m/s, to investigate the effects of wave amplitude, frequency, as well as wind forcing on the airflow regime and momentum transfer. We report that under the milder wind forcing (U 10 = 7 m/s ) and the least steep wave, no sign of sheltering was observed. The airflow streamlines follow the shape of surface waves, resulting in a wide area of lower-than-average pressure above the wave crest. In this regime, more than 90% of the air-sea momentum transfer comes from the viscous drag at the surface due to the smooth airflow tightly following a smooth wave surface. Meanwhile, such pressure distribution above the waves is mostly the result of the Bernoulli Effect due to the wave shape, with almost-symmetrical low pressure above the wave crest and high pressure above the wave trough. However, a sole increase in wave frequency, while maintaining the amplitude and the wind forcing, is enough to induce airflow sheltering on the leeward side of the wave due to the steepened wave crest. Further, an increase of wind forcing over the same steepened crest did not alter the airflow regime or the pattern of the airflow pressure distribution, while doubling the momentum transfer magnitude. Here, the sheltered airflow regime is further evidenced by the enhanced turbulent kinetic energy observed on the leeward side of the wave. In these conditions, the viscous drag weakens, and the form drag rises to become the dominant mechanism for the air-sea momentum flux, accounting for over 50% of the total stress. In this regime, the pressure distribution is the result of aerodynamic sheltering, with high pressure on the windward side and low pressure on the leeward side. The results of this work will serve as the first step within a larger effort to develop a new formulation for the wave model wind input function, which will account for the airflow separation physics.

Steep wave, turbulence, and sediment concentration statistics beneath a breaking wave field and their implications for sediment transport

The Rayleigh–Taylor instability of two superposed incompressible fluids of different densities in the presence of small rotation, surface tension and suspended dust particles is investigated. The linearized equations of the problem are constructed and the general dispersion relation is obtained using normal mode analysis by applying the appropriate boundary conditions. The effects of surface tension, the Atwood number, small rotation and suspended dust particles are studied on both conditions of Rayleigh–Taylor instability and growth rate of the unstable Rayleigh–Taylor mode. The numerical calculations have been performed to see the effect of rotation, the Atwood number, relaxation frequency and mass concentration of suspended dust particles. It is found that the growth rate of Rayleigh–Taylor instability depends upon the mass concentration and relaxation frequency of suspended dust particles. The uniform small rotation, relaxation frequency and mass concentration of suspended dust particles all have stabilizing influence on the growth rate of Rayleigh–Taylor instability. It is also found that the Atwood number has destabilizing influence on the growth rate of the considered Rayleigh–Taylor configuration.

The Saffman--Taylor finger problem is to predict the shape and, in particular, width of a finger of fluid travelling in a Hele--Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem sufficiently difficult that asymptotic or numerical methods must be used. We adapt numerical methods used to solve the Saffman--Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele--Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as for the surface tension problem. References P. G. Saffman and G. Taylor. The penetration of a fluid into a porous medium or Hele--Shaw cell containing a more viscous liquid. Proc. R. Soc. London, Ser. A , 245:312--329, 1958. doi:10.1098/rspa.1958.0085 J. W. McLean and P. G. Saffman. The effect of surface tension on the shape of fingers in a Hele--Shaw cell. J. Fluid. Mech. , 102:455--469, 1981. doi:10.1017/S0022112081002735 J. M. Vanden-Broeck. Fingers in a Hele--Shaw cell with surface tension. Phys. Fluids , 26(8):2033--2034, 1983. doi:10.1063/1.864406 S. J. Chapman. On the role of Stokes lines in the selection of Saffman--Taylor fingers with small surface tension. Eur. J. Appl. Math. , 10(06):513--534, 1999. doi:10.1017/S0956792599003848 R. Combescot, T. Dombre, V. Hakim, Y. Pomeau, and A. Pumir. Shape selection of Saffman--Taylor fingers. Phys. Rev. Lett. , 56(19):2036--2039, May 1986. doi:10.1103/PhysRevLett.56.2036 R. Combescot, V. Hakim, T. Dombre, Y. Pomeau, and A. Pumir. Analytic theory of the Saffman--Taylor fingers. Phys. Rev. A , 37(4):1270--1283, February 1988. doi:10.1103/PhysRevA.37.1270 M. Reissig, S. V. Rogosin, and F. Hubner. Analytical and numerical treatment of a complex model for Hele--Shaw moving boundary value problems with kinetic undercooling regularization. Eur. J. Appl. Math. , 10(6):561--579, 1999. doi:10.1017/S0956792599003939 S. J. Chapman and J. R. King. The selection of Saffman--Taylor fingers by kinetic undercooling. J. Eng. Math. , 46(1):1--32, 2003. doi:10.1023/A:1022860705459 D. Bensimon. Stability of viscous fingering. Phys. Rev. A , 33(2):1302--1309, 1986. doi:10.1103/PhysRevA.33.1302 D. A. Kessler and H. Levine. Theory of the Saffman--Taylor ``finger'' pattern. {I}. Phys. Rev. A , 33(4):2621--2633, 1986. doi:10.1103/PhysRevA.33.2621 D. A. Kessler and H. Levine. Theory of the Saffman--Taylor ``finger'' pattern. {II}. Phys. Rev. A , 33(4):2634--2639, 1986. doi:10.1103/PhysRevA.33.2634 A. J. Degregoria and L. W. Schwartz. A boundary integral method for two-phase displacement in Hele--Shaw cells. J. Fluid Mech. , 164:383--400, 1986. doi:10.1017/S0022112086002604 T. Y. Hou, Z. Li, S. Osher, and H. Zhao. A hybrid method for moving interface problems with application to the Hele--Shaw flow. J. Comput. Phys. , 134(2):236--252, 1997. doi:10.1006/jcph.1997.5689 G. Tryggvason and H. Aref. Numerical experiments on Hele--Shaw flow with a sharp interface. J. Fluid Mech. , 136:1--30, 1983. doi:10.1017/S0022112083002037 N. Whitaker. Some numerical methods for the Hele--Shaw equations. J. Comput. Phys. , 111(1):81--88, 1994. doi:10.1006/jcph.1994.1046

Nonlinear interaction of a vortex pair with clean and surfactant-covered free surfaces

The author presents a study on the development of approximate theoretical forms of the distributions of wave steepness and surf parameter. The approach is based on assuming the random wave process to be long crested and narrow banded. The results are compared with data from measurements at sea representing two severe storms. Both parameters are lognormal distributed. The resulting statistics for the surf parameter are applied to breakers at normal incidence on sloping beaches. As stated by the author, the joint statistics of wave steepness with wave heights or crest heights, or the wave steepness with wave heights or crest heights above a specified threshold may be more appropriate in formulating risks of capsizing of vessels, overtopping, and slamming forces on seawalls, etc. However, the wave steepness is of interest by itself, particularly in relation with many of the surf-zone processes. This discussion points out that other data sets may result in other distributions. This is exemplified by using the data referred to by the author—the data used by Myrhaug and Kjeldsen 1984, 1987 , Myrhaug and Rue 1993 , and Myrhaug and Kvalsvold 1995 . These papers discuss various aspects of wave steepness statistics using data from a large population of waves obtained by waverider buoys at three different deep water locations on the Norwegian continental shelf. Fig. 1 here shows the cdf of the normalized wave steepness s=S /Srms in Weibull scale, where Srms is as defined in Eq. 20 of the original paper. The pluses represent the Myrhaug and Kjeldsen 1984 data. The full line represents the combined lognormal and Weibull distribution obtained as a best fit by eye to the data; the pdf of the lognormal distribution is given by Eq. 17 of the original paper with , = −0.257,0.523 for s 1.2; the pdf of the Weibull distribution is given by Eq. 14 with , = 1.40,0.84 for s 1.2. It appears that this model gives a good representation of the data. Myrhaug and Rue 1993 used the Weibull model to study the statistics of two successive wave steepness parameters with the focus on steep waves S 0.10 . Myrhaug and Kjeldsen 1984 , Myrhaug and Kvalsvold 1995 , and Myrhaug 1994 discussed the joint distribution of wave height and wave steepness by taking f h ,s = f s h f h where f s h is given by the lognormal distribution in Myrhaug and Kjeldsen 1984, Eq. 4 , with x=s, Eq. 18 , and Eq. 19 , and f h is given by the Weibull distribution in Myrhaug and Kjeldsen

The long-time viscous decay of large-amplitude bubble oscillations is considered in an incompressible Newtonian fluid, based on the Rayleigh–Plesset equation. At large Reynolds numbers, this is a multi-scaled problem with a short time scale associated with inertial oscillation and a long time scale associated with viscous damping. A multi-scaled perturbation method is thus employed to solve the problem. The leading-order analytical solution of the bubble radius history is obtained to the Rayleigh–Plesset equation in a closed form including both viscous and surface tension effects. Some important formulae are derived including the following: the average energy loss rate of the bubble system during each cycle of oscillation, an explicit formula for the dependence of the oscillation frequency on the energy, and an implicit formula for the amplitude envelope of the bubble radius as a function of the energy. Our theory shows that the energy of the bubble system and the frequency of oscillation do not change on the inertial time scale at leading order, the energy loss rate on the long viscous time scale being inversely proportional to the Reynolds number. These asymptotic predictions remain valid during each cycle of oscillation whether or not compressibility effects are significant. A systematic parametric analysis is carried out using the above formula for the energy of the bubble system, frequency of oscillation, and minimum/maximum bubble radii in terms of the Reynolds number, the dimensionless initial pressure of the bubble gases, and the Weber number. Our results show that the frequency and the decay rate have substantial variations over the lifetime of a decaying oscillation. The results also reveal that large-amplitude bubble oscillations are very sensitive to small changes in the initial conditions through large changes in the phase shift.

In a recent paper the effect of variations in calibrated wave parameters on wave crest and height distributions was analyzed (OMAE2010-20304, [1]). Theoretical distribution functions were compared to wave measurements with a variation in water depth, wave seed (group spectrum) and location of measurement for the same initial power spectrum. The wave crest distribution of the shallow water waves exceeded both second-order and Rayleigh distribution. Whereas, in intermediate water depth the measured crests followed the second order distribution. The distributions of the measured waves showed that different wave seeds result in the same wave height and crest distributions. Measured wave heights were lower closer to the wave maker. In this paper the results of the continued statistical analysis of basin waves are presented with focus on wave steepness and their influence on wave height and wave crest distributions. Furthermore, the sampling variability of the presented cases is assessed.

Capillary effect on vertically excited surface wave in circular cylindrical vessel

The phase split behavior of air–water slug flow at branching micro‐T‐junctions was studied by the volume of fluid model. The effects of velocity, surface tension, and viscosity on phase split results have been tested by slug units of fixed length. The split of a gas bubble at the T‐junction shows obstructing and nonobstructing modes. Through comparison, we found that the appearance of tunnels changes the driving force of bubble breakup from upstream pressure to viscous shearing force. Split features are influenced by the length of slug unit and the flow ratio in the obstructing mode. Nevertheless, the viscous force and surface tension should not be slighted for nonobstructing mode. Moreover, tendency of the flow maldistribution of nonobstructing mode is more pronounced with higher velocity and smaller surface tension. The liquid slug length has a negligible effect on the split results of gas phase. Thus, the results of two‐phase split could be obtained from the gas split ratio solely. Pattern diagram for the transition between the 2 modes was proposed. Eventually, the importance of the length information of slug unit was also verified by the predictive model of phase split that formulated with the regression analysis of numerical data.

The effect of surface tension on the onset of convection in horizontal double-diffusive layer was studied both experimentally and by linear stability analysis. The experiments were conducted in a rectangular tank with base dimension of 25×13 cm and 5 cm in height. A stable solute (NaCl) stratification was first established in the tank, and then a vertical temperature gradient was imposed. Vertical temperature and concentration profiles were measured using a thermocouple and a conductivity probe and the flow patterns were visualized by a schlieren system. Two types of experiments were carried out which illustrate the effect of surface tension on the onset of convection. In the rigid–rigid experiments, when the critical thermal Rayleigh number, RT, is reached, large double-diffusive plumes were seen simultaneously to rise from the heated bottom and descend from the cooled top. In the rigid–free experiments, owing to surface tension effects, the first instability onset was of the Marangoni type. Well-organized small plumes were seen to emerge and persist close to the top free surface at a relatively small RTM (where subscript M denotes ‘Marangoni’). At larger RTt > RTM (where subscript t denotes ‘top’) these plumes evolved into larger double-diffusive plumes. The onset of double-diffusive instability at the bottom region occurred at a still higher RTb > RTt (where subscript b denotes ‘bottom‘). A series of stability experiments was conducted for a layer with an initial top concentration of 2 wt% and different concentration gradients. The stability map shows that in the rigid–free case the early Marangoni instability in the top region reduces significantly the critical RT for the onset of double-diffusive convection. Compared with the rigid–rigid case, the critical RT in the top region is reduced by about 60% and in the bottom region by about 30%. The results of the linear stability analysis, which takes into account both surface tension and double-diffusive effects, are in general agreement with the experiments. The analysis is then applied to study the stability characteristics of such a layer as gravity is reduced to microgravity level. Results show that even at 10 −4g0, where g0 is the gravity at sea level, the double-diffusive effect is of equal importance to the Marangoni effect.

Bubbles directly link sea surface structure to the dissipation rate of turbulence in the ocean surface layer through wave breaking, and they are an important vehicle for air–sea transfer of heat and gases and important for understanding both hurricanes and global climate. Adequate parameterization of bubble dynamics, especially in high winds, requires simultaneous measurements of surface waves and breaking-induced turbulence; collection of such data would be hazardous in the field, and they are largely absent from laboratory studies to date. We therefore present data from a series of laboratory wind-wave tank experiments designed to observe bubble size distributions in natural seawater beneath hurricane conditions and connect them to surface wave statistics and subsurface turbulence. A shadowgraph imager was used to observe bubbles in three different water temperature conditions. We used these controlled conditions to examine the role of stability, surface tension, and water temperature on bubble distributions. Turbulent kinetic energy dissipation rates were determined from subsurface ADCP data using a robust inertial-subrange identification algorithm and related to wind input via wave-dependent scaling. Bubble distributions shift from narrow to broadbanded and toward smaller radius with increased wind input and wave steepness. TKE dissipation rate and shear were shown to increase with wave steepness; this behavior is associated with a larger number of small bubbles in the distributions, suggesting shear is dominant in forcing bubbles in hurricane wind-wave conditions. These results have important implications for bubble-facilitated air–sea exchanges, near-surface ocean mixing, and the distribution of turbulence beneath the air–sea interface in hurricanes. Significance Statement Bubbles are a vehicle for the flux of heat, momentum, and gases between the atmosphere and ocean. These fluxes contribute to the energy budgets of hurricanes, climate, and upper-ocean biology. Few to no simultaneous measurements of surface waves, bubbles, and turbulence have been made in hurricane conditions. To improve numerical model representation of bubbles, we performed laboratory experiments to parameterize bubble size distributions using physical variables including wind and waves. Bubble distributions were found to become broadbanded and shift toward smaller radius with increased wind stress and wave steepness. Turbulence dissipation rate and shear were shown to increase with wave steepness. Our results give the first physically based bubble distribution parameterization from naturally breaking waves in hurricane-force conditions.

The phase-averaged characteristics of the turbulent velocity fields beneath steep short wind waves are investigated. A scheme was developed to compute the phase of individual wind waves using spatial surface displacement data. This information was used to analyze the two-dimensional velocity data acquired using particle image velocimetry (PIV) in a wind-wave tank. The experiments were conducted at a fetch of 5.5m and at wind speeds that ranged from 4 to 10ms−1. Under these conditions previous studies have shown that a significant percentage of the waves are microscale breaking waves. An analysis of the phase-averaged results suggests under these conditions (short fetches and moderate wind speeds) a wind-driven water surface can be divided into three regions based on the intensity of the turbulence. These are the crests of microscale breaking waves, the crests of non-breaking waves and the troughs of all waves. The turbulence is most intense beneath the crests of microscale breaking waves. In the crest region of microscale breaking waves coherent structures were observed that were stronger and occurred more frequently than beneath the crests of non-breaking waves. Beneath the crests of non-breaking waves the turbulence is a factor of two to three times weaker and beneath the wave troughs it is a factor of six weaker. These findings provide additional support for the hypothesis that approximately two-thirds of the gas and heat fluxes occur across the turbulent wakes produced by microscale breaking waves.

Laboratory measurements of the fine space-time structure of short gravity-capillary waves, as well as Ku-band scattering at grazing and moderate incidence from wind waves in the large Delft Hydraulics Laboratory wind-wave channel are presented. This study was stimulated by the need to verify the processes that significantly contribute to scattering at grazing and moderate incidence. A scanning laser slope gauge was used for measuring capillary waves from 2-mm to 2-cm wavelengths and frequencies ranging up to 100 Hz. A dual-polarized (vertical, VV, and horizontal, HH), coherent, pulsed Ku-band scatterometer with good temporal resolution (3 ns) was used to obtain simultaneous Doppler spectra and the absolute cross section of scattered signals for grazing angles 6 and 25/spl deg/ and for winds in the range 2.5-12.5 m/s. Two-dimensional (2D) filtering and bispectral analyses were used to separate and study the influence of free and bound surface waves. The results of this study demonstrate that the frequency-wavenumber spectra of capillary waves consist of two parts. The first corresponds to free capillary waves, which satisfy the dispersion relationship. The second corresponds to bound parasitic capillary waves, which are located near the crests of steep wind waves. The phase velocity of these capillary waves is approximately equal to the phase velocity of the steep waves. Measurements of the Doppler frequency of the scattered signals show that the Doppler spectra also have a bimodal structure. While the first low-frequency part of the spectrum corresponds to the Bragg scattering from the free capillary waves, the highfrequency part is associated with Bragg scattering from the bound capillary waves on the crests of the steep waves. This type of scattering is predominant for the upwind direction of illumination (especially for HH-polarization).

Sea spray droplets play an important role in the momentum, heat and mass transfer in the marine atmospheric boundary layer. We have developed a new direct numerical simulation method to study the generation and transport mechanisms of spume droplets by wind blowing over breaking waves, with the wave breaking process taken into account explicitly. In this new computational framework, the air and water are simulated as a coherent system on fixed Eulerian grid with the density and viscosity varying with the fluid phase. The air-water interface is captured accurately using a coupled level-set and volume-of-fluid method. The trajectories of sea spray droplets are tracked using a Lagrangian particle-tracking method. The generation of droplets is captured by comparing the fluid particle velocity of water and the phase speed of the wave surface. From the simulation data, we obtain for the first time a detailed description of the instantaneous distribution of droplets at different stages of wave breaking. Furthermore, the time histories of the droplet number and its generation and disappearance rates are analyzed. Simulation cases with different parameters are performed to study the effects of wave age and wave steepness. The flow and droplet fields obtained from simulation provided a detailed physical picture of the problem of interest. It is found that plunging breakers generate more droplets than spilling breakers. Droplets are generated near the wave crest at young and intermediate wave ages, but at old wave ages, droplets are generated both near and behind the wave crest. It is also elucidated that the large-scale spanwise vortex induced by the wave plunging event plays an important role in suspending droplets. Our simulation result of the vertical profile of sea spray concentration is consistent with laboratory measurement reported in the literature.