Curvature Of Free Surface Research Articles

Mathematical modeling and numerical simulation of wave-front flow on a vertical wall with surfactant effects

The modeling of thin-layer flow in the presence of surfactant is considered. The classical problem of wave-front flow down a vertical wall, as discussed by Tuck and Schwartz (SIAM Rev 32:453, 1990), is extended to demonstrate the effects of an assumed insoluble surfactant. Results using the thin-layer approximation (“lubrication theory”) are compared with numerical solutions of the full Navier–Stokes problem using a volume-of-fluid method. The surfactant takes the form of either (i) a concentrated clump, or ‘bolus,’ or (ii) an initially uniform distribution. The basic problem is two-dimensional downhill flow from thick to thin wetting layers. If no surfactant is present, or if the surfactant is entirely passive, and if the layers are long, a steadily propagating solution exists. A discrete quantity of surfactant, when deposited on the liquid surface will be transported to the neighbourhood of the wave front. An apparently stable steady-state will ultimately arise. Similar surfactant transport will occur for uniform surfactant, although no steady-state solution is possible then because the quantity of surfactant becomes unbounded. The volume-of-fluid calculations confirm the validity of all the qualitative features revealed by the lubrication analysis. Quantitative discrepancies appear when the evolving free-surface slope or curvature is no longer small, even in the absence of surfactant effect.

쇄파의 초기단계 생성조건과 수치시뮬레이션

Abstract ： The steady breakers at an infant stage are investigated through the numerical simulation. The appearing condition and characteristics of the sub-breaking waves are reviewed by analysing bow waves. The instability analysis is po ssibly done through the relationship between the free-surface curvature and circumferen tial force, which is obtained from the momentum equations. Navier-Stokes equations a re solved by a finite difference method where the body-fitted coordinate system, the wall function and the advanced mesh system are invoked. The numerical result shows th at the gradient of M/U s is greatly influenced by the Froude number and the decrease of M/U s indicates that the flows are unstable. Additionally flows with plunging o r spilling are simulated successfully, but the application of breakers to the severely b roken wave still remains to be settled in the future. Key words ：Steady breaker(정상쇄파), Sub-breaking(쇄파), Finite difference method( 유한차분법), Free surface flow(자유표면유동), Navier-Stokes(나비에스톡스)

The flip-through of a plane inviscid jet with a free surface

The growth of a two-dimensional liquid jet is modelled, in which an inviscid incompressible fluid is in irrotational flow. On the moving free surface, the pressure is constant. The flow is symmetric with respect to the y-axis. The free-surface fluid particle that lies on the y-axis is Q and it has position y = Y(t). A velocity potential is presented that describes the local features of the flow near the centreline, and which contains essentially two unknowns: the velocity V(t) = d Y/d t of Q, and a length L(t). Near Q the free surface is a time-dependent parabola, whose curvature is directly proportional to a third unknown, F(t). The kinematic and dynamic free-surface boundary conditions constrain V, L and F to satisfy three ordinary differential equations. The solutions are a one-parameter family that separate into five types. For one type the free-surface curvature changes sign during the motion, so that the free surface changes from concave to convex—it executes flip-through. Soon after flip-through, there is a local maximum in the pressure, with respect to time and space. A short time later there is a maximum in the acceleration of Q. As t tends to infinity, V tends to a constant, V ∞. An estimate is made of the time scale for flip-through, based on pressure changes, and it is found to decrease as V ∞ increases. When V ∞ is chosen very large, the values of the large maxima in pressure and acceleration become sensitive to small changes in the initial conditions. The results focus on the highly transient pressure field. The findings may help to describe the flip-through of a steep fronted water wave meeting a plane impermeable wall.

The mean and turbulent flow structure of a weak hydraulic jump

The turbulent air–water interface and flow structure of a weak, turbulent hydraulic jump are analyzed in detail using particle image velocimetry measurements. The study is motivated by the need to understand the detailed dynamics of turbulence generated in steady spilling breakers and the relative importance of the reverse-flow and breaker shear layer regions with attention to their topology, mean flow, and turbulence structure. The intermittency factor derived from turbulent fluctuations of the air–water interface in the breaker region is found to fit theoretical distributions of turbulent interfaces well. A conditional averaging technique is used to calculate ensemble-averaged properties of the flow. The computed mean velocity field accurately satisfies mass conservation. A thin, curved shear layer oriented parallel to the surface is responsible for most of the turbulence production with the turbulence intensity decaying rapidly away from the toe of the breaker (location of largest surface curvature) with both increasing depth and downstream distance. The reverse-flow region, localized about the ensemble-averaged free surface, is characterized by a weak downslope mean flow and entrainment of water from below. The Reynolds shear stress is negative in the breaker shear layer, which shows that momentum diffuses upward into the shear layer from the flow underneath, and it is positive just below the mean surface indicating a downward flux of momentum from the reverse-flow region into the shear layer. The turbulence structure of the breaker shear layer resembles that of a mixing layer originating from the toe of the breaker, and the streamwise variations of the length scale and growth rate are found to be in good agreement with observed values in typical mixing layers. All evidence suggests that breaking is driven by a surface-parallel adverse pressure gradient and a streamwise flow deceleration at the toe of the breaker. Both effects force the shear layer to thicken rapidly, thereby inducing a sharp free surface curvature change at the toe.

Computational predictions for viscoelastic filament stretching flows: ALE methods and free-surface techniques (CM and VOF)

This article considers the dynamics of filament stretching for viscoelastic liquids. We investigate the consequences of employing both fe and fv spatial discretisations within an incremental pressure–correction scheme, considering various mesh movement and free-surface tracking techniques. Finite element discretisation is employed for the momentum and continuity equation, whilst a pure-upwinding cell-vertex finite volume representation is utilised for the hyperbolic stress equation. Compressive mesh procedures outperform volume-of-fluid counterparts, and when coupled to an ALE-formulation governing mesh movement, provide a powerful technique to access impressively large Hencky-strains. If precision in determination of free-surface curvature is demanded, a particle-tracking approach is shown to be preferable to a kinematic condition for surface-level. Results of the study lie in close agreement with the literature in trends and measures of Trouton ratio, minimum radius and extensional viscosity predictions.

Singularity of free-surface curvature in convergent flow: Cusp or corner?

The nature of singularities that arise in the mathematical modeling of free-surface flows and the ways of their analysis and regularization aimed at removing the physically unacceptable features is one of fundamental issues in theoretical fluid dynamics. The present work considers the type of the free-surface curvature singularity emerging in the steady two-dimensional convergent flow of a Newtonian fluid near a free boundary. The unphysical singularities in the flow field, unavoidable in the conventional model, are removed by describing this flow as a particular case of the interface formation/disappearance process in the framework of an earlier developed macroscopic theory of such processes which is applied without any ad hoc alterations. The near-field asymptotic analysis of the problem shows that at finite capillary numbers the singularity of the free-surface curvature is always a sharp corner, not a cusp.

Oscillatory thermocapillary convection in liquid bridges with highly deformed free surfaces: Experiments and energy-stability analysis

Laboratory experimentation, numerical simulation, and energy-stability theory are used to examine the effect of interface deformation on the onset of oscillatory thermocapillary convection in half zones. Experiments are performed to map the stability boundaries marking the onset of oscillatory flow, modifying the free-surface deformation by adjusting the volume of liquid in the bridge. The stability results presented here along with those of other researchers [Monti et al., Proceedings of the 43rd Cong. Int. Artro. Fed. (1992); Hu et al., J. Cryst. Growth 142, 379 (1994)] show that free-surface curvature can have a pronounced influence on flow stability. Steady, axisymmetric flow simulations are computed using the commercial code FIDAP to model the conditions of the experiments, and reveal that flow structure near the stability boundary is sensitive to several parameters. Energy theory is applied to these simulations to determine sufficient conditions for stability. Comparisons between the theoretical and experimental results show nonconservative energy limits falling above the experimentally determined stability boundaries for bridges of various liquid volumes. While the trend of the experimental data is predicted for zones of large volume ratio (bulging zones), the same cannot be said for those with small volume ratio (necked-down zones). In addition, energy-stability limits for some undeformed-free-surface cases were determined which are above the linear-stability limits determined by other researchers, in clear contradiction of the roles of the respective theories.

Evaluation of the onset of oscillatory thermocapillary convection in liquid bridge with empirical correlations in terms of a modified Marangoni number

For the onset of oscillatory Marangoni convection in liquid bridge, the present paper proposes an empirical correlation in terms of the modified Marangoni number that was proposed in order to take into account the effect of the free-surface axial curvature by the present authors [Yamamoto, Torii, J. Crystal Growth 182 (1997) 485]. The proposed equation can correlate the available experimental data within 22% error for Ma R<3×10 4 and within 24% error for Ma R>3×10 4. It is found that the critical Marangoni number for the onset of oscillatory convection in liquid bridge can be well correlated in terms of three parameters of the aspect ratio, the Prandtl number and the curvature parameter ( C-parameter).

The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures

AbstractWe present the spectral element method to simulate elastic-wave propagation in realistic geological structures involving complieated free-surface topography and material interfaces for two- and three-dimensional geometries. The spectral element method introduced here is a high-order variational method for the spatial approximation of elastic-wave equations. The mass matrix is diagonal by construction in this method, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multi-corrector format. Long-term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The associated Courant condition behaves as ΔtC &amp;lt; O (nel−1/ndN−2), with nel the number of elements, nd the spatial dimension, and N the polynomial order. In practice, a spatial sampling of approximately 5 points per wavelength is found to be very accurate when working with a polynomial degree of N = 8. The accuracy of the method is shown by comparing the spectral element solution to analytical solutions of the classical two-dimensional (2D) problems of Lamb and Garvin. The flexibility of the method is then illustrated by studying more realistic 2D models involving realistic geometries and complex free-boundary conditions. Very accurate modeling of Rayleigh-wave propagation, surface diffraction, and Rayleigh-to-body-wave mode conversion associated with the free-surface curvature are obtained at low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves by three-dimensional (3D) surface topographies and the associated local effects on strong ground motion. Complex amplification patterns, both in space and time, are shown to occur even for a gentle hill topography. Extension to a heterogeneous hill structure is considered. The efficient implementation on parallel distributed memory architectures will allow to perform real-time visualization and interactive physical investigations of 3D amplification phenomena for seismic risk assessment.

Nonlinear shallow-water oscillations in a parabolic channel: exact solutions and trajectory analyses

A new exact solution of the nonlinear shallow-water equations is presented. The solution corresponds to divergent and non-divergent free oscillations in an infinite straight channel of parabolic cross-section on the rotating Earth. It provides a description of the one-dimensional subclass of shallow-water flows in paraboloidal basins considered by Ball (1964), Thacker (1981), Cushman-Roisin (1987) and others in which the velocity field varies linearly and the free-surface displacement varies quadratically with the spatial coordinates. In contrast to the previous exact solutions describing divergent oscillations in circular and elliptic paraboloidal basins, the oscillation frequency of the divergent oscillation in the parabolic channel is found to depend, in part, on the amplitudes of the relative vorticity and free-surface curvature. This result is consistent with Thacker's (1981) numerical finding that when the free surface in parabolic channel flow is curved, the oscillation frequency depends on the amplitude of the motion. Solutions for parcel trajectories are also presented. The exact solution provides a rare description of a class of nonlinear flows and is potentially valuable as a validation test for numerical shallow-water models in Eulerian and Lagrangian frameworks.

Numerical Study of Vortical Flows beneath the Free Surface around Struts

Characteristics of vortical flows around free surface piercing struts are studied by a numerical simulation method solving 3-D laminar incompressible Navier-Stokes and continuity equations. The simulation shows that vorticity is generated on the curved free surface to satisfy a no-shearing stress condition. The vorticity, whose strength is proportional to the free surface curvature, leads the generation of the vortical motions beneath the free surface in front of a bow. To investigate curvature effect of the body, three different struts having NACA 0005, NACA 0008 and NACA 0012 sections are used. The effect of free surface boundary conditions and grid density around the free surface are also discussed.

Interaction of a turbulent round jet with the free surface

The interaction of a turbulent round jet with the free surface was investigated experimentally. Flow visualization, free-surface curvature measurements and hot-film velocity measurements were used to study this flow. It is shown that surface waves are generated by the large-scale vortical structures in the jet flow as they interact with the free surface. These waves propagate at an angle with respect to the flow direction which increases as the Froude number is increased. Propagation of the waves in the flow direction is suppressed by the surface current produced by the jet. Farther downstream the surface motions are caused by the large-scale vortical structures. Characteristic dark circular features are observed in shadowgraph images associated with concentrated vorticity normal to the free surface. The normal vorticity is believed to be the result of vortex line reconnection processes in the turbulent flow. Measurements of the mean velocity and turbulence intensity are reported. Owing to the confinement by the free surface, the decay rate of the maximum mean velocity is reduced by a factor of √2 compared to an unconfined jet.

Experimental characterization of viscous film flows over complex surfaces

Ordered packings are made of a variety of materials such as ceramics, metal and plastics, and shaped to a wide spectrum of macro- and micro-structures. Macro-structures, usually in the form of corrugations, sustain good liquid-vapor contact with low overall pressure drop. Micro-structures in the way of surface treatment, metal gauze and surface indentations, help to increase the stability of the liquid film by preventing breakups and dry patches. An experimental research program was carried out in order to characterize the mechanics of viscous flows over model complex surfaces. Film thickness profiles, streamline patterns and free-surface velocities were measured for a variety of surface shapes and fluids. These results are important in the understanding of the interaction of capillary, viscous and gravity forces on the shaping of the film free surface. The position of the liquid film interface was found, in general, to have the same period as the wavy solid but amplitude and phase-shift vary with flow parameters. Free-surface profiles are patterned around the shape of the solid surfaces but their amplitude decreased with increasing Nusselt film thickness. Free-surface velocity measurements show peaks of velocity larger than the maximum predicted for vertical films. These peaks are associated to a minimum in film thickness and are detected in regions where there is an inflexion point in free surface curvature. Three parameters, Nusselt film thickness, Reynolds number and Capillary number, are necessary to characterize accurately these liquid film flows.

THERMOCAPILLARY CAVITY CONVECTION IN WETTING AND NONWETTING LIQUIDS

A numerical study has been conducted to examine the influence of free surface curvature, due to the presence of various contact angles, on thermocapillary convection within a cavity. The results indicate that the system hydrodynamics are changed by a combination of surface temperature gradient modification due to the presence of the adiabatic meniscus and variation in the length of the free surface, as well as viscous force dependence on the wetting angle. Local and overall heat transfer rates are, in turn, altered. Wetting liquids exhibit decreased surface fluid velocities and convective heat transfer rates, while non-wetting liquids are characterized by more complicated behavior with local increases and decreases in surface fluid velocities and an increase in local heat transfer near the free surface.

Thermal cavity nucleation at intergranular inclusions in high temperature creep

The role of intergranular inclusions at the thermal cavity nucleation in high temperature creep is analysed in detail. The theoretical treatment is generalized for cavities the free surface curvature radius of which is comparable with the inclusion size and for cavities forming with inclusions the equilibrium systems. Special attention is devoted to the influence of the interface energies, the inclusions volume and the internal stress concentrations to the cavity nucleation kinetics. Each kinetics of cavity nucleation observed is qualitatively explained on the basis of the thermally activated heterogeneous cavity nucleation at intergranular inclusions.

Rayleigh-Taylor instability at large times

The potential flow of an inviscid incompressible heavy fluid lying above a light one is investigated. The asymptotic stage is described by an unsteady discontinuity, which approximates the flow in the neighborhood of the tongue, and by a steady flow outside this narrow region. Consequences of the conservation laws which make it possible to check the accuracy of the solution of the steady-state problem are obtained. A steady-state solution is constructed for Froude numbers 0<Fr<0.37. The position of the discontinuity, the corresponding values of the complex potential and the pressure, the average acceleration of the tongue, and the dependence of the parameters of the limiting regime on the values of the conservation integrals (in particular, the Froude numbers) and the level of the initial energy perturbation are determined from the conservation laws. At the apex of the bubble the derivative of the free-surface curvature has a discontinuity when Fr≠0.23, which makes it possible to attribute the choice of solution obtained in a number of studies to the presence of an artificial surface tension.

Torque measurements on a stationary axially positioned sphere partially and fully submerged beneath the free surface of a slowly rotating viscous fluid

Experimental measurements are presented for the hydrodynamic torque exerted on a stationary sphere situated at the axis of a slowly rotating viscous liquid at small rotary sphere Reynolds numbers (Re &lt; 0.1) as a function of depth of submersion of the sphere below the free surface. Effects of free-surface proximity on the torque furnished the impetus for the study. Experiments were performed for different depths of sphere immersion beneath the free surface, varying from full to partial submersion. Rotation rates were maintained sufficiently low to approximate a planar interface. Torque measurements agreed well with existing theoretical predictions for both the interface-straddling and fully submerged sphere cases. In particular, the predicted continuity of the torque and its derivative at the interface-penetration point (where the sphere first starts to protrude through the free surface) was observed. Free-surface curvature as well as meniscus-curvature effects upon the torque were found to be negligible in the experiments, including even the extreme case where the sphere was in the almost fully withdrawn configuration.

Torsional oscillations of a non-Newtonian fluid with a free surface

A study is made of the flow and the shape of the free surface of a non-Newtonian fluid contained in a flat-bottomed cylindrical vessel performing torsional oscillations of small amplitude. For the case where the fluid depth is small compared with the vessel radius, the solution is shown to have a simple radial dependence except in a boundary-layer region near the side wall. It is shown that under certain circumstances the mean steady second-order components of both free surface curvature and radial–axial flow may be in the reverse direction to those for a Newtonian fluid. It is found that flow reversal may occur at any frequency of oscillation, but that surface curvature reversal cannot occur at low frequencies.

The lower symmetric regime and its transition to waves in rotating annulus convection

Measurements of the temperature and zonal velocity fields which develop in a rotating annulus of fluid with an upper surface, differentially heated from the inner to outer cylinder, are described for the lower symmetric regime (small radial temperature differences). The temperature field is essentially conductive for moderate to large rotation rates, Ω (>1.0 sec−1). The zonal velocity field is poorly approximated by the thermal wind equation. Measurements of the transition to waves from the lower symmetric regime at very large rotation rates are presented for positive and negative radial temperature differences. They suggest that the centrifugal buoyancy force and the free surface curvature may be important factors for the lower symmetric-wave transition at large Ω. By varying the stratification of the fluid over a range of 103 independently of the radial temperature difference, ΔrwT, it is conclusively shown that several theories are correct in predicting that the lower symmetric transition is independent of the stratification at small ΔrwT > 0 for large enough Ω.

Wave propagation in a uniformly rotating fluid

An unsteady flow generated by a harmonically oscillating pressure distribution of frequency ω acting on the paraboloidal free surface of an inviscid, incompressible fluid rotating with uniform angular velocity Ω has been investigated. It is shown that case (i), ω>2 Ω, corresponds to the usual surface waves, and case (ii), ω<2 Ω, in contrast to the surface waves, corresponds to the inertial waves which are originated entirely due to rotation and have no counterpart in a non-rotating fluid motion. An explicit solution of the problem related to the above cases are obtained by the joint Laplace and Hankel transforms treatment in conjunction with asymptotic methods. The significant effects of the Coriolis force and the curvature of the free surface on the wave motions have been investigated. A comparison is made between the solutions of the problems with the horizontal and the paraboloidal free surface curvature. The analysis is concluded by exihibiting the characteristic features of the wave motions.