Plane Rigid Wall Research Articles

А particle model of interaction between weakly non-spherical bubbles

A mathematical model of interaction of weakly non-spherical gas bubbles in liquid is proposed. It is a system оf the second order ODEs in the radii of bubbles, the position vectors of their centers and the amplitudes of their small deformations in the form of spherical harmonics. Compared to the available analogs, the proposed model equations are more accurate in terms of the ratio of the radii of bubbles to the distance between their centers. This allows one to study bubble-bubble interaction at smaller distances between the bubbles. The model equations are quite compact, which simplifies their theoretical analysis and construction of algorithms of their solution. Five problems (dynamics of a single bubble under harmonic excitation, bubble collapse near a plane rigid wall, interaction of three synchronized and nonsynchronized bubbles located in line, and interaction between sixteen bubbles with the centers located on a plane surface) are considered for validation. It is shown that in all cases the proposed model results well agree with the corresponding available experimental data and numerical solutions. For demonstration of the applicability, the proposed model is used to analyze radial oscillations, spatial movements and surface deformations of the bubbles in rectilinear, planar and spatial clusters under harmonic forcing. Initially 125, 121 and 125 bubbles of the clusters are located at the nodes of 1D, 2D and 3D uniform grids, respectively. It is found that the monotonically decaying radial oscillations of the 1D cluster bubbles during one period of forcing are quite similar to those of a single bubble. Unlike this, the attenuation of the radial oscillations of the central bubbles in the 2D and 3D clusters is essentially non-monotonic. The maximum pressures in the central bubbles of the 2D and 3D clusters are significantly higher than in the single and 1D cluster bubbles. In all clusters the maximum deformations of bubbles are in the form of the second spherical harmonics, except for the central bubble of the 3D cluster, the maximum deformations of which are in the form of the fourth harmonics.

Bubble dynamics near a locally curved region of a plane rigid wall

The dynamics of a bubble near a local axisymmetric convexity or concavity, cosine-like in the axial section, on a plane rigid wall is studied under conditions of formation of a cumulative liquid jet directed to the wall. The study is performed by the boundary element method. Our main attention is focused on the bubble collapse characteristics at the moment tc when the jet impacts on the bubble surface part near the wall, namely, on the jet tip velocity and radius, the internal bubble pressure, the distance between the bubble and the uneven region of the wall, and so on. These characteristics are of interest since they, to a large extent, determine the destructive potential of cavitation. The dependences of these characteristics on the radius of the non-plane region of the wall, the depth of the concave region, the height of the convex region, and the local character of the wall unevenness have been investigated. To estimate the effect of the local character of the wall unevenness, the cases of completely concave and convex walls in the form of blunt conical surfaces are used for comparison. It has been found, in particular, that with changing radius of the convex region of the wall, the bubble shape at the moment tc varies significantly: the shape diversity includes the profiles with a gas microcavity in front of the impacting jet tip, an emerging second oppositely directed jet, and a relatively large area of the jet tip impact at its beginning.

Experimental study on local scour and onset of VIV of a pipeline on a silty seabed under steady currents

The experiments described here are composed of two parts: the 2D scour below a fixed pipeline on a silty seabed under steady currents and the vortex-induced vibration (VIV) of an elastic-mounted pipeline over the scour trench. In the former, the equilibrium scour depth on s silty seabed under clear-water conditions is investigated, specifically assessing the effects of flow velocity and pipeline position. The equilibrium scour depth on a silty seabed under currents is approximately 1/3 smaller than that on a sandy seabed under the same conditions, indicating that the existing formula initially proposed for a sandy seabed overestimates when it is applied to a silty seabed. An empirical formula is proposed, based on which the equilibrium scour depth below a pipeline on a silty seabed can be estimated from the existing formulae for a sandy seabed by applying a reduction coefficient. Then, the effects of the pipeline position and depth of the scour trench on the onset reduced velocity of VIV for a pipeline over a scour trench were investigated. The onset reduced velocity for the VIV of a pipeline over a scour trench is larger than that for either a wall-free pipeline or a pipeline near a plane rigid wall with the same gap, due to the sheltering effect of the scour trench on the formation of regular vortex shedding behind the pipeline. The onset reduced velocity decreases with deeper scour trench and lower pipeline position. Then, an empirical formula for the influential factors was proposed based on present and published results.

Numerical Study of the Dynamics of a Gas Bubble Near a Wall Under Ultrasound Excitation

A numerical study of the dynamics of a gas bubble in water near a plane rigid wall under ultrasound excitation has been performed. The bubble dynamics is excited at the natural frequency of the radial oscillations of the bubble in the absence of the wall. The dynamics of the bubble with the formation on its surface of a cumulative liquid jet directed to the wall is considered. A regime of the ultrasound excitation is investigated, at which the cumulative jet impact against the bubble surface part closer to the wall is realized at the moment of the beginning of the second expansion of the bubble. At this moment two known mechanisms of emitting pressure pulses into liquid by a bubble start working simultaneously. The first one is due to the bubble expansion. The second one results from the impact of the cumulative jet. The boundary element method is used. The study has revealed the dependencies of the ultrasound excitation amplitude, at which the indicated regime is realized, on the distance between the bubble and the wall, as well as the influence of the bubble-to-wall distance on the jet impact velocity and the pressure inside the bubble corresponding to that regime.

Influence of the Initial Shape of a Gas Bubble on Its Dynamics near a Wall under Acoustic Excitation

Influence of initial non-sphericity of a gas bubble on its dynamics at various distances from a plane rigid wall under harmonically varying liquid pressure is investigated. The initial shape of the bubble is taken to be a spheroid with the axis of symmetry orthogonal to the wall. The liquid pressure is varied with the own frequency of the radial oscillations of the bubble in the absence of the wall, beginning with the negative acoustic pressure phase. The liquid is ideal and incompressible, its flow is potential. The velocity of the bubble surface is determined by the boundary element method, its displacement is calculated by the Euler method. It is shown that under the adopted conditions, the bubble collapses either with its transformation into a toroidal one, as a result of impact of the cumulative liquid jet on the bubble surface part close to the wall, or with its disintegration into two smaller bubbles, which results from closing of a narrow neck formed by the bubble surface in the vicinity of the axis of symmetry. Some features of the influence of the initial non-sphericity of the bubble on the shape of the jet, the velocity of its tip and the pressure inside the bubble at the moment of the jet impact on the bubble surface part close to the wall are determined.

Stokes velocity generated by a point force in various geometries

.In this short review, we visualize the fluid velocity generated by a point force close to a plane free surface or a plane rigid wall. We present separately contributions from all the multipoles which form the corresponding classical systems of images. Such graphical images might be useful in the theoretical and numerical modeling of the dynamics of micro-objects moving close to an interface.Graphical abstract

Numerical simulaion of dynamics of a gas bubble in liquid near a rigid wall during its growth and collapse

A numerical technique of calculating the dynamics of a cavitation bubble near a plane rigid wall is presented. The bubble at its collapse can become toroidal. The liquid is assumed inviscid and incompressible, its flow being potential. The bubble surface movement is determined by the Euler method, the normal component of the liquid velocity on the bubble surface is found by the boundary element method. The technique also includes an algorithm for calculating the velocity and pressure fields in the liquid. The convergence of the numerical solution with refining the temporal and spatial steps is demonstrated. The results of the present technique are compared with some known numerical and experimental data by other authors, their satisfactory agreement is found. To illustrate the capabilities of the present technique the process of growth and collapse of a bubble in water near a wall is considered. The liquid pressure contours in the stage of the bubble collapse are given and the radial liquid pressure profiles on the wall and at a small distance from the wall where the liquid pressure is maximum are shown.

Velocity analysis for collapsing cavitation bubble near a rigid wall under an ultrasound field

Acoustic cavitation bubble and its production extreme physics such as shockwaves and micro-jets on a solid wall have attracted great interest in the application of ultrasound (e.g., ultrasonic medical, ultrasonic cleaning, and ultrasonic machining). However, the prediction and control of micro-jets induced by ultrasonic field have been a very challenging work, due to the complicated mechanisms of collapsing of cavitation bubbles. In order to determine the interaction of micro-jet with the key parameters that influence the acoustic cavitation, the dynamics of bubble growth and collapse near a rigid boundary in water is investigated. Using the method of mirror image, a revised bubble dynamics equation in radial oscillation for a bubble near a plane rigid wall is derived from the double-bubble equation (the Doinikov equation). In the present equation, the gas inside the bubble is assumed to be the van der Waals gas, and the weak compressibility of the liquid is also assumed. The revised equation is then employed to simulate numerically the dynamical behaviors of a bubble, using the fourth-order Runge-Kutta method with variable step size adaptive control. Numerical simulations of the motion characteristics and collapse velocities of a bubble near a rigid boundary or a free boundary have been performed, under various conditions of initial bubble radius, spacing between the center of the bubble and the wall, acoustic pressure and ultrasonic frequency, in order to explain the effects of these key parameters on the acoustic cavitation intensity. It is shown that, compared with free boundary, the effect of rigid boundary on the bubble plays a significant role in suppressing the bubble oscillation. The intensity of bubble collapsing is reduced as the increase of the initial bubble radius and ultrasonic frequency, and increased by enlarging the spacing between the center of the bubble and the wall. There exists an optimal acoustic pressure (almost 3.5 times bigger than the ambient pressure), at which the collapse of a bubble near a rigid wall can be the most violent. Furthermore, the relationship between the collapse velocity of a bubble near a rigid boundary and its micro-jet is described. Results demonstrate that the velocity of micro-jet is dependent on that of bubble collapse, and it can be controlled by adjusting the velocity of bubble collapse indirectly. Calculation results of the micro-jet in this paper are compared with some numerical and experimental results given in the literature and good apparent trends between them are obtained. These results will give important implications for further understanding the dynamics of cavitation bubble on a solid wall induced by the ultrasonic field and its different requirements in engineering applications.

Growth-and-collapse dynamics of small bubble clusters near a wall

The violent collapse of bubble clusters is thought to damage adjacent material in both engineering and biomedical applications. Yet the complexities of the root mechanisms have restricted theoretical descriptions to significantly simplified configurations. Reduced-physics models based upon either homogenization or arrays of idealized spherical bubbles do reproduce important gross cluster-scale features. However, these models neglect detailed local bubble–bubble interactions, which are expected to mediate damage mechanisms. To describe these bubble-scale interactions, we simulate the expansion and subsequent collapse of a hemispherical cluster of 50 bubbles adjacent to a plane rigid wall, explicitly representing both the asymmetric dynamics of each bubble within the cluster and the compressible-fluid mechanics of bubble–bubble interactions. Results show that the collapse propagates inward, as visualized in experiments, and that geometric focusing generates high impulsive pressures. This gross behaviour is nearly independent of the specific arrangement of bubbles within the cluster and matches predictions from the corresponding particle and homogenized models we consider. The peak pressure in the detailed simulations is associated with the centremost bubble, which causes a corresponding peak pressure on the nearby wall. However, the peak pressures in all cases are a small fraction – over a factor of ten times smaller in many cases – of those predicted in the corresponding reduced models. This is due to the enhanced focusing in the homogeneous model and the spherical constraint on each bubble in the particle models assessed. These would be important factors to consider in any subsequent predictions of wall damage based upon reduced models.

Stokes First Problem for an Unsteady MHD Third-Grade Fluid in a Non-Porous Half Space with Hall Currents

The well-known problem of unidirectional plane flow of a fluid in a non-porous half-space due to the impulsive motion of the rigid plane wall it rests upon is discussed in the context of an unsteady MHD third-grade fluid in presence of Hall currents. The governing non-linear partial differential equations describing the problem are converted to a system of non-linear ordinary differential equations by using the similarity transformations. The complex analytical solution is found by using the homotopy analysis method (HAM). The existing literature on the topic shows that it is the first study regarding the effects of Hall current on flow of an unsteady MHD third-grade fluid over an impulsively moving plane wall. The convergence of the obtained complex series solutions is carefully analyzed. The effects of dimensionless parameters on the velocity are illustrated through plots and the effects of the pertinent parameters on the local skin friction coefficient at the surface of the wall are presented numerically in tabular form.

Study of tribological behaviour of fresh mortar against a rigid plane wall

The interaction between a fresh cement-based material and a solid wall represents a considerable consumption of energy during an extrusion proces. In this article, the tribological behaviour of extrudible mortar and rigid wall friction is studied using a tribometer especially developed in the laboratory. Dynamic friction coefficient is highlighted. For fixed wall roughness, the friction velocity effect is analysed. Then, for fixed friction velocity, the roughness effect is analysed. Finally, the material structure evolution near the wall and the material erosion are analysed. L,interaction de matériaux à base cimentaire frais et une paroi solide représente une consommation énergétique importante dans les procédés d,extrusion. Dans cet article, le comportement tribologique d,un mortier extrudable et d,une paroi rigide est étudié au moyen d,un tribomètre développé et mis au point au laboratoire. Dans les travaux, le coefficient de frottement dynamique est tout d,abord mis en évidence. Pour une paroi de rugosité fixée, l,effet de la vitesse de frottement est ensuite analysé. Dans une partie suivante, à vitesse de frottement fixée, l,effet de la rugosité de la paroi est analysé. Enfin, l,évolution de la structure du matériau au voisinage de la paroi est présentée et discutée ainsi que l,érosion.

Turbulent structure of air flow over wind-induced gravity waves

This is the second paper in a group of three that reports the systematic measurements of wind-generated water waves in a wind tunnel experiment. Here, the structure of the boundary layer on the air side of the water–air interface was analysed and compared with the boundary layer over a smooth plane rigid wall. The contribution of the wave-induced Reynolds stress was detected through filtering the spectrum of velocity fluctuations. Wave-induced Reynolds stresses became negligible for z > 5 H rms. The intermittency factor in the boundary layer over water waves was similar to that in a boundary layer over a rigid plane wall, with several differences near the interface. Here, the presence/absence of water damps out the turbulence. The quadrant analyses revealed that ejection and sweep events were dominant and more concentrated. At small fetches, the large-amplitude negative streamwise perturbations were preferentially lifted. Turbulence energy production peaked at z/δ = 0.2 and had a distribution similar to that observed for a self-preserving boundary layer with a strong adverse gradient pressure. The quadrant analysis contribution to the energy production revealed that ejections still dominated the balance and that the production was spatially modulated in the wind direction with a couple of cells and with a minimum in the area of the free surface wave height reduction.

On the contact behavior of a buckled Timoshenko beam constrained laterally by a plane wall

This paper discusses the effect of shear deformation of the cross-section on the contact behavior of a buckled Timoshenko beam in frictionless contact with a rigid plane wall. The deformation is assumed to be small, and only planar deformation is considered. The results are compared with those from the Euler beam model. In the case when a buckled Euler beam is in line contact with a plane wall, the plane wall exerts concentrated forces at the boundary points of the line-contact segment but no distributed force. In the case of a Timoshenko beam, on the other hand, the concentrated forces are replaced by a distributed contact force. The distribution of the contact force approaches the two concentrated forces at the boundary points when the shear effect is negligible. The line-contact deformation may become unstable when the edge thrust reaches a secondary buckling load. It is found that the normalized critical length of the contact segment is always smaller when the shear effect is more significant. On the other hand, the normalized secondary buckling load reaches a maximum when the slenderness ratio (beam length over thickness) is in the order of 45.

Deformation and stability of an elastica under a point force and constrained by a flat surface

This paper studies the deformation and stability of a pinned elastica under a point force moving quasi-statically from one end to the other. The elastica is constrained by a rigid plane wall containing the two ends. Three types of equilibrium configurations can be found; they are non-contact, one-point contact, and one-line contact on the side. A vibration method is adopted to determine the stability of the calculated deformations. In order to take into account the variation of the contact region between the elastica and the plane wall during vibration, an Eulerian version of the governing equations is adopted. It is found that all the point-contact deformations are unstable. On the other hand, there are two different mechanisms a line-contact deformation becomes unstable; one through a secondary buckling and the other through a limit-point bifurcation. In the secondary buckling, the length of the line-contact segment and the axial force satisfy the Euler buckling criteria for a pinned-clamped column. On the other hand, when a line-contact deformation becomes unstable via a limit-point bifurcation, the axial force does not exceed the Euler buckling load. The theoretical predictions are confirmed by experimental observations.

Modeling of the acoustic response from contrast agent microbubbles near a rigid wall

In ultrasonic targeted imaging, specially designed encapsulated microbubbles are used, which are capable of selectively adhering to the target site in the body. A challenging problem is to distinguish the echoes from such adherent agents from echoes produced by freely circulating agents. In the present paper, an equation of radial oscillation for an encapsulated bubble near a plane rigid wall is derived. The equation is then used to simulate the echo from a layer of contrast agents localized on a wall. The echo spectrum of adherent microbubbles is compared to that of free, randomly distributed microbubbles inside a vessel, in order to examine differences between the acoustic responses of free and adherent agents. It is shown that the fundamental spectral component of adherent bubbles is perceptibly stronger than that of free bubbles. This increase is accounted for by a more coherent summation of echoes from adherent agents and the acoustic interaction between the agents and the wall. For cases tested, the increase of the fundamental component caused by the above two effects is on the order of 8–9 dB. Bubble aggregates, which are observed experimentally to form near a wall due to secondary Bjerknes forces, increase the intensity of the fundamental component only if they are formed by bubbles whose radii are well below the resonant radius. If the formation of aggregates contributes to the growth of the fundamental component, the increase can exceed 17 dB. Statistical analysis for the comparison between adhering and free bubbles, performed over random space bubble distributions, gives p-values much smaller than 0.05.

Numerical Analysis of the Dynamics of Toroidal Bubbles Considering the Heat Transfer of Internal Gas

This paper describes a new numerical method for computing the motion of a toroidal bubble that is formed after a liquid microjet threads the bubble surface. We used the boundary element method combined with the finite volume method to calculate the toroidal bubble dynamics with consideration of the heat transfer inside the bubble. The unstructured triangular grids were used to analyze the internal field. We investigated the bubble collapse near a plane rigid wall. The results show that the initial bubble radius affects the motion of the toroidal bubble, the temperature fields inside the bubble and the pressure fields outside the bubble. When the initial radius is small, the internal thermal boundary layer becomes thick and the bubble collapse is accelerated. This violent collapse induces an extremely high-pressure region near the point of liquid microjet impact. The pressure at a rigid wall, therefore, becomes locally very high when the tiny bubble collapses near the wall.

Numerical Analysis of the Dynamics of Toroidal Bubbles Considering the Heat Transfer of Internal Gas.

This paper describes a new numerical method to compute the motion of a toroidal bubble that is formed after the liquid microjet threads the bubble. We use the boundary element method (BEM) combined with the finite volume method to calculate the toroidal bubble dynamics considering the heat transfer inside the bubble. The unstructured triangle grids are generated inside the bubble. We investigate the bubble collapse near a plane rigid wall. The results show that the initial radius affects the motion of the toroidal bubble, the temperature fields inside the bubble and the pressure fields outside the bubble. When the initial radius is small, the internal thermal boundary layer becomes thick and the bubble collapse is accelerated. This violent collapse induces the extremely high pressure region near the point of liquid-jet-impact. The wall pressure, therefore, becomes locally much high when the tiny bubble collapses near the wall.

1411 非構造格子を用いた非球形気泡の内部温度場に関する数値解析手法の改良(GS-5 キャビテーション)

We have improved the numerical scheme based on the boundary element method combined with the finite volume method to consider the heat transfer of internal gas in the final stage of bubble collapse. Unstructured adaptive grids are used inside the largely deformed bubble. We investigate the motion of the collapsing bubble near a plane rigid wall. The dynamics of a toroidal bubble, which is formed after the penetration of liquid microjet, are taken into account in the present analysis. The results show that the present scheme can handle the temperature fields for the toroidal bubble.

Model and experiments of a drop impinging on an immersed wall

We present a model describing the rebound of a drop impinging on a rigid plane wall immersed in water. This model is based on the resolution of the drop equation of motion in an unbounded fluid in which an additional pressure force is introduced accounting for the wall effect on the drop motion. This force is computed from a film drainage simulation model during the approach of the deformable particle to the wall. Results of the model have been compared with experimental trajectories of drops impinging vertically at terminal rise velocity against a horizontal wall immersed in water at rest. These trajectories have been obtained with the help of an image processing technique. A wide range of experimental conditions has been studied (drop diameter, interfacial tension, drop viscosity, and density). In most of the cases, the model predicts the experimental trajectories within a very good accuracy (height of bouncing, deformation, number of rebounds) even in the case of a significant deformation. The numerical results show that the rebound of a deformable inclusion against a wall in water is essentially governed by the balance between the added mass force and the film pressure force exerted on the drop during the impact. The model has also been successfully tested in the case of an impinging bubble at high particle Reynolds number, based on experimental data taken from Tsao and Koch [Phys. Fluids 9, 44 (1997)].

Thermal Effects of Internal Gas on the Oscillations of Nonspherical Bubbles Near a Plane Rigid Wall.

The oscillatory motion of a nonspherical gas bubble near a plane rigid wall is investigated using the boundary element method combined with the finite volume method. Heat transfer inside the bubble is taken into account in the analysis. We compare the present numerical results with both theoretical ones using a series expansion of spherical harmonics and numerical ones using an effective polytropic index. It is shown that when the bubble oscillation is moderate, the theoretical results are in good agreements with the present numerical ones. However, the results of polytropic analysis overestimate the amplitude of bubble oscillation. It is also shown that the temperature distribution inside the bubble is much dependent on the initial bubble radius. The thermal damping is important in dealing with the nonlinear oscillations of the bubbles. When the initial bubble radius is small and the driving frequency is out of resonant one, the thermal damping affects surface oscillations as well as radial oscillations more strongly.