Initial Free Surface Research Articles

A new nanometre resolution method for probing densification ratio at nanoindentation sites in glass: Unravelling discrepancies in the literature

The region of permanent densification beneath a Berkovich indentation imprint in silica glass is investigated using a novel chemical dissolution technique. The use of the similitude regime in sharp indentation testing allows one to record reliable data with a good spatial resolution that makes it possible to deal with low loads (typically below 10 mN) and, more importantly, crack-free imprints. The densified zone dissolves more quickly than the non densified regions. The analysis of the results, along the vertical axis, indicates that the densification zone is rather homogeneous with a steep transition to the non densified zone. The size of the densification zone, with respect to the initial free surface, is estimated to be around two times the maximum penetration depth of the instrumented indentation test. These findings are compared with those obtained by numerical simulations using different constitutive equations from the literature. A very good concordance between Raman spectroscopy and chemical probe results is found for imprints made with no or few cracking events during indentation testing.

Numerical investigation on the effects of wave suppression baffles in vehicle-integrated water tanks

The danger of liquid sloshing in pump-driven water tanks has significantly increased due to the growing need to integrate a water tank into new energy vehicles. The best approach to reduce the negative effects of liquid sloshing is to add baffles to the integrated water tank. The addition of baffles to integrated water tanks to reduce liquid sloshing is investigated in this study using the discontinuous mesh approach and the coupled level set and volume of fluid (CLSVOF) method. Comparative experiments and simulations are used to confirm the reliability of the method. In this study, the length ratio, distance to initial free surface, and angle of the added baffle are taken into consideration. The free surface height, magnitude velocity, turbulence energy, and flow field are used to assess the impact of the baffle on the suppression of liquid sloshing. The addition of baffles can improve the liquid sloshing phenomenon caused by the pump excitation conditions in the integrated water tank, increase the volume suction of liquid, inhibit the height of the wave and reduce the liquid flow rate in the tank. Also, at the depth where the baffle has the most obvious suppression effect, the deeper the baffle, the lower the height of the free surface in the tank and the lower the average magnitude velocity in the tank cross-section. The baffle reduces the height of the free surface in the tank to 6.39 mm, and the magnitude of maximum average velocity in the tank cross-section is reduced to 0.12 m/s.

Numerical study on the water-entry characteristics of asynchronous parallel projectiles at an oblique impact angle

This study investigates the water-entry problems of asynchronous parallel projectiles with oblique entry speeds of 200 m/s for different water-entry intervals and two different water-entry sequences numerically. The experiment of the water-entry of the projectile with an oblique impact angle is conducted to verify the numerical method. The influences of the oblique asynchronous parallel water-entry on the evolutions of the cavitating flow and the movement characteristics under different working conditions are studied in detail. It can be found that for the minimum interval, the first launch projectile is not able to generate a complete cavity to encapsulate its body due to the intense compression of the cavity of the following launch projectile, and its movement is seriously affected and ultimately destabilized. Simultaneously, compared with the first launch projectile entering the water from the side close to the initial free surface, the cavity of the first launch projectile entering the water from the side away from the initial free surface is squeezed more, wetted earlier and destabilized faster. With the increase of the interval, the influence of the following launch projectile on the cavity evolution and movement stability of the first launch projectile weakens. Besides, for the following launch projectile, due to the continuous impact of the cavitating flow of the first launch projectile, the profile of the cavity of the following launch projectile presents asymmetry. For the minimum interval, the projectile deflects faster and towards the inner side of the projectile, while for the larger intervals, the projectile deflects to the outer side of the projectile and becomes unstable.

The application of the KdV type equation in engineering simulation

Bores propagating in shallow water transform into undular bores and, finally, into trains of solitons. The observed number and height of these undulations and later discrete solitons are strongly dependent on the propagation length of the bore. Empirical results show that the final height of the leading soliton in the far-field is twice the initial mean bore height. The complete disintegration of the initial bore into a train of solitons requires very long propagation, but unfortunately, these required distances are usually not available in experimental tests of nature. Therefore, the analysis of the bore decomposition for experimental data into solitons is complicated and requires different approaches. Previous studies have shown that by applying the nonlinear Fourier transform based on the Ko- rteweg–de Vries equation (KdV-NFT) to bores and long-period waves propagating in constant depth, the number and height of all solitons can be reliably predicted already based on the initial bore-shaped free surface. Against this background, this study presents the systematic analysis of the leading-soliton amplitudes for non-breaking and breaking bores with different strengths in different water depths to validate the KdV-NFT results for non-breaking bores to show the limitations of wave breaking on the spectral results. The analytical results are compared with data from experimental tests, numerical simulations and other approaches from the literature.

A new criterion of coalescence-induced microbubble detachment in three-dimensional microfluidic channel

This work is motivated by an experiment of microbubble transport in a polymer microfluidic gas generation device where coalescence-induced detachment exhibits. We numerically study three-dimensional microbubble coalescence using the graphics processing unit accelerating free energy lattice Boltzmann method with cubic polynomial boundary conditions. The focus is on the coalescence-induced microbubble detachment (CIMD) in microfluidics. From the experimental observation, we identified that size inequality between two-parent bubbles and the size of the father (large) bubble are key factors to determine if a CIMD will occur. First, the analytical relationship between equilibrium contact angle and dimensionless wetting potential and experimental results of coalescence with and without CIMD are employed for the verification and validation, respectively. From eighteen experimental and computational cases, we derive a new criterion for CIMD: CIMD occurs when the two-parent bubbles are (nearly) equal with a relatively large radius. The underlying mechanism behind this criterion is explored by the time evolution of the velocity vector field, vorticity field, and kinetic energy in the entire coalescence. It is found that the symmetric capillary force drives the formation of vertical flow stream to the horizontal alignment of parent bubbles and the blockage of the downward stream due to the solid interface promotes the intensity of the upward stream. Meanwhile, large-sized parent bubbles transfer a large amount of kinetic energy from the initial free surface energy, which is essential to lead a CIMD in the post-coalescence stage. Such a new criterion is expected to impact the design and optimization of microfluidics in various applications.

Stagnant peaked free surface released at a sloping beach

A stagnant free-surface flow is an instantaneous flow field of pure acceleration with zero velocity and a deformed surface. There exists a potential-flow acceleration field. With zero velocity and the acceleration field given, there is a limiting free-surface position which possesses one peak at its point of highest elevation. By complex analysis, it can be shown that the surface peak has a right angle. We elaborate on an elementary model of two-dimensional stagnant free-surface flow with a peak. Our model may serve to describe a situation of maximal single-wave run-up with a given energy at a uniformly sloping beach. The highest possible run-up of an incoming solitary wave corresponds to zero kinetic energy. It encompasses an idealized situation where the kinetic wave energy is converted into potential energy in a water mass piling up along the slope to become stagnant at one single moment. Multipoles with singularities outside the fluid domain may give rise to a smooth and gradual deceleration needed for a non-breaking run-up process. A pair of dipoles with an orientation perpendicular to a given slope represents the stagnant acceleration fields with the highest surface peak spatially concentrated along the slope. Thereby, a one-parameter family of surface shapes is constituted, only dependent on the slope angle. The initial flow field, the initial free surface, the initial isobars and the geometric parameters are all calculated for different slope angles.

A Numerical Investigation on the Flooding Process of Multiple Compartments Based on the Volume of Fluid Method

A detailed description of the flooding process is crucial to analyze the complex hydrodynamic behaviors and enhance the survivability of the damaged ship. In this paper, through establishing three typical damage scenarios with various locations, the commercial software CD Adapco STAR-CCM+ based on the Reynolds-Averaged Navier-Stokes (RANS) solver is applied to simulate the flooding process involving multiple compartments. The basic computational fluid dynamics (CFD) models and specific simulation settings are elaborated. The volume of fluid (VOF) method combined with the user defined field function is developed to distribute the initial free surface. The captured flooding process indicates that the air compression due to the restricted ventilation decreases the flooding amount. The obtained flooding time can provide necessary data to support for appropriate rescue management and evacuation options.

Temporal and Topographic Source Effects on Tsunami Generation

AbstractWe present a systematic study of the influence on tsunami waves generated by the uplift of a rectangular plug, of the rise time of the deformation, and of its topographic details (e.g., the presence of a sill). We are motivated by the fact that most simulation codes use an instantaneous deformation of a flat ocean floor as an initial condition of the problem, although Hammack (1973, http://resolver.caltech.edu/CaltechAUTHORS:HAMjfm73) performed pioneering laboratory studies as well as analytical computations featuring variable rise times. Here, we consider three 2‐D source shapes, including a flat seafloor, a simple elevated piston, and additional trapezoidal sill on top of it, all with variable rise times, and simulate the resulting waves using the fully nonlinear smooth particle hydrodynamic model graphics processing unit smooth particle hydrodynamic. We validate our results against Hammack's (1973) laboratory measurements and analytical results. We find that a relatively large sill, with height and width of more than half of the local depth and width of the source, has a profound effect on the spatiotemporal structure of the generated free surface wavefield. Specifically, we show that the maximum water surface elevation over the source region is not always the same as the bottom displacement, as assumed in most tsunami propagation models. Next, we obtain simple scaling relationships to predict the maximum height of the generated tsunami over and outside the source, based on the geometry of the sill and the nondimensional bed rise time. Last, we show that inertial effects may lead to an initial free surface displacement over the generation region greater than the maximum vertical displacement of the displaced seabed.

Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations

This paper is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, we can consider solutions such that the curvature of the initial free surface does not belong to $L^2$. The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and H\older estimates. We first prove tame estimates in Sobolev spaces depending linearly on H\older norms and then we use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these H\older norms.

The experimental investigation on dynamic response of free surface for non-isothermal liquid bridge with the varying shear airflow

Through high-speed camera and the self-developed software package for interface recognition, the space-time evolution of free surface has been investigated for non-isothermal liquid bridge with shear airflow, and the dynamic response of free surface to the shear airflow has also been analyzed. When the shear airflow is induced from the upper disk (u < 0), the convex part of free surface is oppressed downward to the gas side by the shear airflow, which increases the curvature of concave part on the free surface. Under the condition of u > 0, the convex part of free surface is pulled upward to the gas side by the shear airflow, which also exacerbates the curvature of concave part on the free surface. The experimental results show that the shear airflow is introduced from upper disk increasing the probability of dam break for the liquid bridge. The deformation of free surface is intensified with the accelerated shear airflow. Due to the different initial free surface shapes (different volume ratios), the direction and intensity of shear stress vary at the different positions of free surface, and the dynamic response law of the free surface deformation is also different. For a liquid bridge with the volume ratio less than 1 (V = 0.802, V = 0.899), the deformation of free surface presents a certain sinusoidal rule. When the volume ratio of liquid bridge is larger than 1 (V = 1.071), under shear airflow induced from upper disk, the deformation of free surface is still presents sinusoidal rule. When the velocity of shear airflow is u > 0 (u = 1 m/s, u = 1.5 m/s, u = 2.0 m/s), the convex part of free surface moves up by the lifting action of shear airflow, and the shape of free surface presents multi-peak structure. For the liquid bridge with the large aspect ratio (Γ = 1.4), the convex region occupies most part of the interface, and there is no change for the free surface shape under the effect of shear airflow. The free surface shape still presents upper concave and lower convex. With the decreasing aspect ratio (Γ = 1.2), the deformation of free surface is intensified, and the shear airflow can excite obvious fluctuation on the free surface.

Comparative simulations of microjetting using atomistic and continuous approaches in the presence of viscosity and surface tension

We compare, at similar scales, the processes of microjetting and ejecta production from shocked roughened metal surfaces by using atomistic and continuous approaches. The atomistic approach is based on very large scale molecular dynamics (MD) simulations with systems containing up to 700 × 106 atoms. The continuous approach is based on Eulerian hydrodynamics simulations with adaptive mesh refinement; the simulations take into account the effects of viscosity and surface tension, and the equation of state is calculated from the MD simulations. The microjetting is generated by shock-loading above its fusion point a three-dimensional tin crystal with an initial sinusoidal free surface perturbation, the crystal being set in contact with a vacuum. Several samples with homothetic wavelengths and amplitudes of defect are simulated in order to investigate the influence of viscosity and surface tension of the metal. The simulations show that the hydrodynamic code reproduces with very good agreement the profiles, calculated from the MD simulations, of the ejected mass and velocity along the jet. Both codes also exhibit a similar fragmentation phenomenology of the metallic liquid sheets ejected, although the fragmentation seed is different. We show in particular, that it depends on the mesh size in the continuous approach.

On the impact of free surfaces on the measurement of diffusion coefficients in metallic binary alloys using shear cells

A new methodology, based on the numerical resolution of the nondimensional momentum and mass transport equations, is applied here in order to analyze the impact of free surfaces on the measurement of the diffusion coefficients using shear cells at high temperatures. If solutal Marangoni convection acts during all the process, calculations suggest that the percentage deviation of the measured coefficient quadratically scales with the solutal Marangoni number. Also, the orientation between the gradient of concentration and the direction of the residual gravity appreciably influences the accuracy of the values, independently on the initial free surface extension. In the cases in which, by oxidation or reduction, the extension of the free surfaces decreases or increases as a function of time, the percentage deviation seems to increase also in a quadratic way as a function of the characteristic time, an indicator of the period of activity of the time dependent Marangoni convection. The orientation between the gravity vector and the gradient of concentration maintains the same tendency in both cases.

Spark-generated bubble collapse near or inside a circular aperture and the ensuing vortex ring and droplet formation

This paper aims to study the oscillation of a sparkgenerated submerged bubble located near or inside a circular aperture made in a flat plate using high-speed visualization technique. In the case of a bubble oscillating near an aperture the initial free surface of the water was set at the bottom surface of the plate. The effects of aperture size and bubble-free surface distance on the bubble behavior as well as on the ensuing droplet dynamics are investigated. It was found that the direction of the bubble reentrant jet was towards the aperture or away from it respectively when the normalized aperture size was smaller or greater than a certain critical value. In addition, a toroidal vortex ring was observed to form, which rotated inwards as it moved away from the aperture. It was also found that if the bubble was incepted at a distance sufficiently away from a supercritical size aperture a single droplet could be produced. In the case of a bubble initiated in the middle of a circular aperture submerged just beneath the water free surface, the bubble was found to take the shape of an ellipsoid during its expansion. Then a reentrant jet was initiated and pierced the bubble from its top side.

Application of NEM in seepage analysis with a free surface

As one kind of meshless methods, the natural element method (NEM) constructs shape functions based on the Voronoi diagrams, and it has advantages of both the conventional meshless method and the finite element method. Since the nodes are independent of the integral mesh, it is more suitable for the analysis of seepage with a free surface than the finite element method. In addition, its shape functions satisfy the Kronecker δ conditions, therefore, its boundary conditions can be dealt with much easily than those of such meshless method as element-free Galerkin method (EFGM). In this paper, the NEM was used in the seepage analysis of dams. The initial free surface was assumed first in the calculations, and the location of the free surface was adjusted according to the calculation results. The examples showed that the natural element method lead to satisfactory results.

Numerical Investigation of Thin Film Spreading Driven by Surfactant Using Upwind Schemes

Numerical solutions of a coupled system of nonlinear partial differential equations modelling the effects of surfactant on the spreading of a thin film on a horizontal substrate are investigated. A CFL condition is obtained from a von Neumann stability analysis of a linearised system of equations. Numerical solutions obtained from a Roe upwind scheme with a third-order TVD Runge-Kutta approximation to the time derivative are compared to solutions obtained with a Roe-Sweby scheme coupled to a minmod limiter and a TVD approximation to the time derivative. Results from both of these schemes are compared to a Roe upwind scheme and a BDF approximation to the time derivative. In all three cases high-order approximations to the spatial derivatives are employed on the interior points of the spatial domain. The Roe-BDF scheme is shown to be an efficient numerical scheme for capturing sharp changes in gradient in the free surface profile and surfactant concentration. Numerical simulations of an initial exponential free surface profile coupled with initial surfactant concentrations for both exogenous and endogenous surfactants are considered.

Numerical analysis of the hydroelastic behavior of a vertical plate due to solitary waves

Interactions of a vertical elastic plate with fully nonlinear water waves were simulated. Utilizing the mixed Eulerian Lagrangian method for the free-surface flow and the finite element method for the deflection of an elastic plate, a fully coupled scheme for accurately determining fluid–plate motions was developed. Using this scheme, some modifications to the solvers for both fluid and plate were made. A hybrid wave-absorbing beach was installed to prevent wave reflection from the end of the wave tank. A fourth-order Runge–Kutta time-marching scheme with a uniform time step was applied to achieve numerical stability. The method was validated by simulating the wave generated by the initial deformation of a vertical plate and comparing the result with the corresponding analytical solution. For further validation, the hydroelastic behavior of a vertical plate induced by a pulse-type wave (where the initial pulse-type elevation of the free surface is specified) was computed, and the result was compared with another numerical result from a mode-expansion method. The interaction of a surface-piercing plate with nonzero initial free surface was then simulated, and the result was compared with the corresponding linear analytical solution. Finally, the hydroelastic response of a surface-piercing vertical plate due to a solitary wave (generated by actuating the vertical plate at the right end of the tank only at the beginning) was computed and investigated systematically.

Effects of surface shape on the geometry and surface topography of the melt pool in low‐power density laser melting

AbstractThe quantitative correlations between workpiece volume and melt pool geometry, as well as the flow and thermal features of the melt pool are established. Thermocapillary convections in melt pool with a deformable free surface are investigated with respect to surface shape and laser intensity. When the contact angle between the tangent to the top surface and the vertical wall at the hot center is acute, the free surface flattens, compared with that of the initial free surface. Otherwise, the free surface forms a bowl‐like shape with a deep crater and a low peripheral rim when the contact angle at the hot center is obtuse. Increasing the workpiece volume at a fixed laser intensity and a negative radial height gradient cause linear decreases in the geometric size and magnitude of flow and temperature of the melt pool. Conversely, linear increases are observed with a positive radial height gradient. © 2011 American Institute of Chemical Engineers AIChE J, 2012

Some aspects of the flip-through phenomenon: A numerical study based on the desingularized technique

Flip-through is known as a rapidly focusing phenomenon at a wall leading to high loads without impact of liquid. In order to simulate numerically these highly nonlinear waves, the boundary value problem is formulated in potential theory without surface tension. A desingularized technique is used to compute the velocity potential. Conformal mappings of the fluid domain simplify the formulation of the solution. As shown by many contributors to the method of fundamental equations (another name to denote desingularized methods), the suitable desingularizing distance must be chosen with care. Here the criteria for choosing it follow from energy and mass conservation laws. This study shows what is the influence of an arbitrary additive constant to the velocity potential regarding conservation laws. Validation tests are performed on a focused wave. Recommendations are given regarding the choice of the desingularizing distance and the additive constant as well. In order to better control the initiation of flip-through, the simulations start from an initial free surface deformation in a rectangular tank, with or without varying bathymetry. The subsequent jet running along the wall, is described and the corresponding loads are discussed. In particular in the present configuration, it is shown that, along the wall, the maximum acceleration precedes the maximum of pressure contrary to the findings of previous studies. The sensitivity of the results with regard to the shape of the initial deformation and the local bathymetry is discussed.

Generation and propagation of nonlinear tsunamis in shallow water by a moving topography

The objective of the present work is to simulate the nonlinear effects of a local underwater earthquake on the water wave propagation. We consider an ideal homogeneous fluid layer, with a free surface, occupying an infinite channel of finite depth. The bottom of the channel is horizontal except for an area of a small horizontal extent, where the bottom has the form of an irregular topography set in a bounded motion in the neighborhood of its position at rest. The nonlinear waves, generated by the motion of the topography inside the fluid layer, are studied within the frame of the shallow water theory. Double series representations are proposed for the velocity potential and the free surface elevation in terms of a small parameter characterizing the horizontal extent of the moving part of the bottom. This representation is used to obtain a solution for the problem of the second order in the small parameter. Higher order solutions are shown to comprise a secular term vanishing at the origin of the coordinate system, taken over the topography on the initial free surface. This secular term increases indefinitely with the increase of the distance from the origin. Such a result is not physically acceptable and has been remedied by using a multiple scale transformation of coordinate and time, along with suitable expansions for the velocity potential and the free surface elevation. This procedure reduces the problem of the determination of the nonlinear waves to the solution of the well-known Korteweg and de Vries (KdV) equation. Numerical solution for this nonlinear partial differential equation is carried out for a certain particular case of physical interest and the obtained results are analyzed and discussed. The establishment of the proposed mathematical procedure requires the solution of certain dual trigonometric series equations. A closed form analytical solution, thought to be new, for such dual equations is given in Appendix.

Time-dependent free surface Stokes flow with a moving contact line. I. Flow over plane surfaces

We study a gravity driven two-dimensional viscous flow with a moving contact line. Using a boundary integral method, the Stokes equations are written as a system of algebraic equations for the flow variables on the domain boundary. The equations are solved for an initial free surface profile, providing the local velocity of points on the free surface, which are then kinematically advected to obtain the new profile at the next time step; repeated time-stepping represents the flow profile evolution. We have studied the time evolution and fully developed steady solutions of the viscous flow for relatively high capillary numbers, Ca∼O(0.1), and a wide range of the contact angle, φ. Our results show a profile with a ridge, followed by a capillary wave decaying to the incoming parallel flow, whose size is a function of both Ca and φ. For a range of high contact angles and high Ca we find steady solutions with an overhang or nose and have characterized these in the (Ca,φ) plane. Finally, we find that the profiles exhibit an apparent or dynamic contact angle that varies with Ca and φ, and which agrees with Cox–Voinov-type relationships.