Rigid Horizontal Surface Research Articles

Oblique collision of a soft rubber disk with a rigid surface

An oblique collision of an object with a rigid surface involves an initial sliding phase that persists throughout the collision at glancing angles of incidence but which involves a subsequent grip phase at higher angles of incidence. The grip phase itself terminates towards the end of the collision if the contact region starts to slide backwards. Experimental evidence of the three separate stages of the collision process is presented using a high speed video camera to film the impact of a rubber disk on a rigid horizontal surface. A simplified model of the process is presented, providing analytical solutions that are at least qualitatively consistent with the experimental observations.

Energy loss in oblique collisions

The decrease in kinetic energy of a ball incident vertically on a rigid horizontal surface depends on the normal coefficient of restitution, [Formula: see text]. It is shown in the present paper that if the ball is incident obliquely on the surface then the total decrease in kinetic energy has two independent contributions, one depending on [Formula: see text], the other depending on the tangential coefficient of restitution, [Formula: see text].

On the question of the smallest rolling resistance coefficient value of elastic wheel on a rigid horizontal surface

As is known, rolling resistance of an elastic wheel on a rigid support surface is determined by internal friction losses (hysteresis) in the wheel material and friction losses in the contact of wheel with support surface (friction losses in the hub bearings and aerodynamic losses are neglected due to their smallness compared to with the above). To assess rolling resistance, the rolling resistance coefficient is used, which is defined as ratio of rolling resistance force applied to the wheel axle to the normal reaction on the wheel. Its value can be determined experimentally or by analytical dependencies. In this article, an analytical derivation of equation for calculating rolling resistance coefficient is given, followed by determining conditions under which this coefficient will be minimal.

Impact of a double sphere dropped vertically onto a horizontal surface

A simple experiment is described concerning the vertical bounce of a nonspherical object. The object consisted of two golf balls joined together. The balls were dropped onto a heavy, rigid horizontal surface and were filmed with a video camera to measure the coefficient of restitution (COR). For spherical objects, the COR is usually defined in terms of the vertical speed of the centre of mass and can vary between zero and unity depending on the elastic properties of the object and the surface on which it bounces. For nonspherical objects, the COR is best defined in terms of the vertical speed of the contact point. For the two golf balls, the centre of mass COR was found to vary between −4 and +2 depending on the incident spin and the angle of inclination of the long axis. The contact point COR was found to be 0.8 ± 0.2, regardless of the incident spin and angle of inclination.

A note on the solution of water wave scattering problem involving small deformation on a porous channel-bed

The solution of water wave scattering problem involving small deformation on a porous bed in a channel, where the upper surface is bounded above by an infinitely extent rigid horizontal surface, is studied here within the framework of linearized water wave theory. In such a situation, there exists only one mode of waves propagating on the porous surface. A simplified perturbation analysis, involving a small parameter e ( ≪ 1), which measures the smallness of the deformation, is employed to reduce the governing Boundary Value Problem (BVP) to a simpler BVP for the first-order correction of the potential function. The first-order potential function and, hence, the first-order reflection and transmission coefficients are obtained by the method based on Fourier transform technique as well as Green’s integral theorem with the introduction of appropriate Green’s function. Two special examples of bottom deformation: the exponentially damped deformation and the sinusoidal ripple bed, are considered to validate the results. For the particular example of a patch of sinusoidal ripples, the resonant interaction between the bed and the upper surface of the fluid is attained in the neighborhood of a singularity, when the ripples wavenumbers of the bottom deformation become approximately twice the components of the incident field wavenumber along the positive x-direction. Also, the main advantage of the present study is that the results for the values of reflection and transmission coefficients are found to satisfy the energy-balance relation almost accurately.

Basal entrainment by Newtonian gravity-driven flows

Gravity-driven flows can erode the bed along which they descend and increase their mass by a factor of 10 or more. This process is called “basal entrainment.” Although documented by field observations and laboratory experiments, it remains poorly understood. This paper examines what happens when a viscous gravity-driven flow generated by releasing a fixed volume of incompressible Newtonian fluid encounters a stationary layer (composed of fluid with the same density and viscosity). Models based on depth-averaged mass and momentum balance equations deal with bed-flow interfaces as shock waves. In contrast, we use an approach involving the long-wave approximation of the Navier-Stokes equations (lubrication theory), and in this context, bed-flow interfaces are acceleration waves that move quickly across thin stationary layers. The incoming flow digs down into the bed, pushing up downstream material, thus advancing the flow front. Extending the method used by Huppert [“The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface,” J. Fluid Mech. 121, 43–58 (1982)] for modeling viscous dam-break waves, we end up with a nonlinear diffusion equation for the flow depth, which is solved numerically. Theory is compared with experimental results. Excellent agreement is found in the limit of low Reynolds numbers (i.e., for flow Reynolds numbers lower than 20) for the front position over time and flow depth profile.

Force Balance Model for Bubble Rise, Impact, and Bounce from Solid Surfaces.

A force balance model for the rise and impact of air bubbles in a liquid against rigid horizontal surfaces that takes into account effects of buoyancy and hydrodynamic drag forces, bubble deformation, inertia of the fluid via an added mass force, and a film force between the bubble and the rigid surface is proposed. Numerical solution of the governing equations for the position and velocity of the center of mass of the bubbles is compared against experimental data taken with ultraclean water. The boundary condition at the air-water interface is taken to be stress free, which is consistent for bubbles in clean water systems. Features that are compared include bubble terminal velocity, bubbles accelerating from rest to terminal speed, and bubbles impacting and bouncing off different solid surfaces for bubbles that have already or are yet to attain terminal speed. Excellent agreement between theory and experiments indicates that the forces included in the model constitute the main physical ingredients to describe the bouncing phenomenon.

Lubricated viscous gravity currents

AbstractWe present a theoretical and experimental study of viscous gravity currents lubricated by another viscous fluid from below. We use lubrication theory to model both layers as Newtonian fluids spreading under their own weight in two-dimensional and axisymmetric settings over a smooth rigid horizontal surface and consider the limit in which vertical shear provides the dominant resistance to the flow in both layers. There are contributions from Poiseuille-like flow driven by buoyancy and Couette-like flow driven by viscous coupling between the layers. The flow is self-similar if both fluids are released simultaneously, and exhibits initial transient behaviour when there is a delay between the initiation of flow in the two layers. We solve for both situations and show that the latter converges towards self-similarity at late times. The flow depends on three key dimensionless parameters relating the relative dynamic viscosities, input fluxes and density differences between the two layers. Provided the density difference between the two layers is bounded away from zero, we find an asymptotic solution in which the front of the lubricant is driven by its own gravitational spreading. There is a singular limit of equal densities in which the lubricant no longer spreads under its own weight in the vicinity of its nose and ends abruptly with a non-zero thickness there. We explore various regimes, from thin lubricating layers underneath a more viscous current to thin surface films coating an underlying more viscous current and find that although a thin film does not greatly influence the more viscous current if it forms a surface coating, it begins to cause interesting dynamics if it lubricates the more viscous current from below. We find experimentally that a lubricated gravity current is prone to a fingering instability.

An experimental and theoretical study of the dynamics of grounding lines

AbstractWe present an experimental and theoretical study of a thin, viscous fluid layer that flows radially under gravity from a point source into a denser inviscid fluid layer of uniform depth above a rigid horizontal surface. Near the source, the viscous layer lies in full contact with the surface, forming a vertical-shear-dominated viscous gravity current. At a certain distance from the source, the layer detaches from the surface to form a floating current whose dynamics are controlled by the viscous stresses due to longitudinal extension. We describe the dynamics of the grounded and floating components using distinct thin-layer theories. Separating the grounded and floating regions is the freely moving line of detachment, or grounding line, whose evolution we model by balancing the horizontal forces between the two regions. Using numerical and asymptotic analysis, we calculate the evolution of the system from a self-similar form at early times towards a steady state at late times. We use our solutions to illustrate how three-dimensional stresses within marine ice sheets, such as that of West Antarctica, can lead to stabilization of the grounding line. To assess the validity of the assumptions underlying our model, we compare its predictions with data from a series of laboratory experiments.

Ground Effect in Hydrodynamics of a Strip in a Stratified Fluid

The paper is devoted to the numerical investigation of stratified flow structure and dynamical characteristics of an imperme- able obstacle moving at some distance from the rigid horizontal surface. At the simplest cases of a plate moving along under- lying plane or in free space the calculation results are compared with the visualizations of exact solution and the Schlieren images of stratified flows in the laboratory experiments. The calculations reveal that accounting for the finestructure effects, i.e. medium stratification and diffusion, which are always present at the natural conditions, can influence essentially upon flow structure, forces and momentum distributions. The obtained results, which have a practical application to ground-effect vehicles and ekranoplans, show the stratification effects may lead to a noticeable increase in drag and decrease in lift of a body moving near the rigid surface.

Exciting forces due to interaction of water waves with a submerged sphere in an ice-covered two-layer fluid of finite depth

Abstract Scattering of water waves by a sphere in a two-layer fluid, where the upper layer has an ice-cover modelled as an elastic plate of very small thickness, while the lower one has a rigid horizontal bottom surface, is investigated within the framework of linearized water wave theory. The effects of surface tension at the surface of separation is neglected. There exist two modes of time-harmonic waves – the one with lower wave number propagating along the ice-cover and the one with higher wave number along the interface. Method of multipole expansions is used to find the particular solution for the problem of wave scattering by a submerged sphere placed in either of the layers. The exciting forces for vertical and horizontal directions are derived and plotted against different values of the wave number for different submersion depths of the sphere and flexural rigidity of the ice-cover. When the flexural rigidity and the density of the ice-cover are taken to be zero, the numerical results for the exciting forces for the problem with free surface are recovered as particular cases.

Steady streaming confined between three-dimensional wavy surfaces

We present a theoretical and numerical study of three-dimensional pulsatile confined flow between two rigid horizontal surfaces separated by an average gap h, and having three-dimensional wavy shapes with arbitrary amplitude σh where σ ~ O(1), but long-wavelength variations λ, with h/λ ≪ 1. We are interested in pulsating flows with moderate inertial effect arising from the Reynolds stress due to the cavity non-parallelism. We analyse the inertial steady-streaming and the second harmonic flows in a lubrication approximation. The dependence of the three-dimensional velocity field in the transverse direction is analytically obtained for arbitrary Womersley numbers and possibly overlapping Stokes layers. The horizontal dependence of the flow is solved numerically by computing the first two pressure fields of an asymptotic expansion in the small inertial limit. We study the variations of the flow structure with the amplitude, the channel's wavelength and the Womersley number for various families of three-dimensional channels. The steady-streaming flow field in the horizontal plane exhibits a quadrupolar vortex, the size of which is adjusted to the cavity wavelength. When increasing the wall amplitude, the wavelengths characterizing the channel or the Womersley number, we find higher-order harmonic flow structures, the origin of which can either be inertially driven or geometrically induced. When some of the channel symmetries are broken, a steady-streaming current appears which has a quadratic dependence on the pressure drop, the amplitude of which is linked to the Womersley number.

The circular internal hydraulic jump

Circular hydraulic jumps are familiar in single layers. Here we report the discovery of similar jumps in two-layer flows. A thin jet of fluid impinging vertically onto a rigid horizontal plane surface submerged in a deep layer of less-dense miscible fluid spreads radially, and a near-circular internal jump forms within a few centimetres from the point of impact with the plane surface. A jump is similarly formed as a jet of relatively less-dense fluid rises to the surface of a deep layer of fluid, but it appears less stable or permanent in form. Several experiments are made to examine the case of a downward jet onto a horizontal plate, the base of a square or circular container. The inlet Reynolds numbers, Re, of the jet range from 112 to 1790. Initially jumps have an undular, laminar form with typically 2–4 stationary waves on the interface between the dense and less-dense layers but, as the depth of the dense layer beyond the jump increases, the transitions become more abrupt and turbulent, resulting in mixing between the two layers. During the transition to a turbulent regime, single and sometimes moving multiple cusps are observed around the periphery of jumps. A semi-empirical model is devised that relates the parameters of the laboratory experiment, i.e. flow rate, inlet nozzle radius, kinematic viscosity and reduced gravity, to the layer depth beyond the jump and the radius at which an undular jump occurs. The experiments imply that surface tension is not an essential ingredient in the formation of circular hydraulic jumps and demonstrate that stationary jumps can exist in stratified shear flows which can be represented as two discrete layers. No stationary circular undular jumps are found, however, in the case of a downward jet of dense fluid when the overlying, less-dense, fluid is stratified, but a stationary turbulent transition is observed. This has implications for the existence of stationary jumps in continuously stratified geophysical flows: results based on two-layer models may be misleading. It is shown that the Froude number at which a transition of finite width occurs in a radially diverging flow may be less than unity.

A classical experiment revisited: The bounce of balls and superballs in three dimensions

A description of the inelastic collision of a ball when it bounces on a rigid horizontal surface with arbitrary initial spin conditions is presented. I consider cases where rebound occurs with and without sliding, and when the ball grips the surface. Two rebound models are discussed in which friction is described in terms of the static and dynamic coefficients of friction and the friction is expressed in terms of horizontal coefficients of restitution. The azimuthal and vertical deviation angles of the ball after impact are predicted as a function of the incident angle. We present data from an experiment in which a ball is launched horizontally from the edge of the laboratory bench and then rebounds on a horizontal surface. Ordinary balls exhibit two rebound regimes, with and without sliding, and can be satisfactorily described by the first model. Superballs exhibit a grip behavior whose description requires the use of the second model.

Finite element simulation of a rolling automobile tyre to understand its transient macroscopic behaviour

This paper presents explicit finite element results for the transient macroscopic behaviour of a rolling automobile tyre. The aim of the research has been to develop the modelling methodology for an advanced LS-DYNA finite element simulation of a rolling tyre that can be used to provide internal transient stresses (and strains) to support the development of sensor systems technology. The methodology has been developed with an experimental 195 / 65 R15 tyre, which has provided physical test data to benchmark the performance of the numerical predictions for the free-rolling analysis on a rigid horizontal surface. Simulation results for a normal load of 3 kN and a speed of 20 km / h are presented to show the characteristics of stresses at the tyre / ground interface and, for the first time, internal stresses and strains at specific locations inside the tyre's structure. Such numerical results of the internal behaviour will be invaluable to tyre technologists who want to determine the optimal location for their sensor systems, and also to vehicle dynamists who want to establish the dynamic relationships between the contact patch stresses and the global tyre forces.

The relaxation of two-dimensional rolls in Rayleigh–Bénard convection

Large aspect ratio, two-dimensional, periodic convection layers containing a Boussinesq fluid of finite Prandtl number bounded by rigid or free horizontal surfaces are investigated numerically. The fluid equations are solved using both a standard pseudospectral and a Fourier integral method for the time evolution of finite initial perturbations, both random thermal perturbations and localized roll disturbances, into a final equilibrium state. The suggestion that a Fourier integral solution method is required to yield roll relaxation, the two-dimensional process increasing the convection wavelength to values larger than critical, is investigated. Roll relaxation is found for both free-slip and no-slip surfaces using either solution method as long as the initial state is chosen to be of the form of a localized roll disturbance. A wide variety of simulations are performed and roll relaxation is found to be independent of the periodic domain length, weakly dependent on the Rayleigh number and dependent upon the magnitude of the initial localized roll disturbances.

On the amphidromic structure of inertial waves in a rectangular parallelepiped

Enclosed rotating homogeneous fluids support waves that are restored by Coriolis forces. These waves propagate obliquely through the fluid under a fixed angle with respect to the rotation axis which is set by the ratio of the frequency of the wave and the inertial frequency (twice the angular frequency of the tank). In the particular case that boundaries are either parallel, or perpendicular to the rotation axis, eigenmodes may be expected. An example is Kelvin's (1880) axial can. However, the spatial structure of these waves can still be quite complicated, as shown here for waves in a ‘horizontal’ rectangular parallelepiped. The container's rigid horizontal top and bottom surfaces allow for modal expansion in the vertical, each mode being governed by the ‘inertial’ analogs of surface Poincaré and Kelvin waves. Combining these, as in the Taylor problem of reflecting surface Kelvin waves, one finds both the eigenfrequencies as well as the typical amphidromic structure (phase lines circling around nodal points) of the resulting inertial wave pattern in dependence of the remaining two parameters: the horizontal and vertical aspect ratios. This entails the approximative determination of eigenvalues of an infinite matrix equation, obtained with the Proudman–Rao method. It appears that certain specific frequencies persist as eigenfrequency for different aspect ratios, albeit belonging to different eigenmodes. Surprisingly, these correspond to the eigenfrequencies of an infinitely long channel of waves lacking along-channel structure. Inertial wave patterns are presented by means of the amplitude and phase of the internal elevation field (related to the axial velocity component) and of the two horizontal velocity components. These reveal predominant anticyclonic turning of phase and currents; maximum elevation and current amplitudes detached from the solid walls; and, occasionally, a nearly sloshing type of response (nodal lines, instead of points).

Effect of work of adhesion on contact of an elastica with a flat surface

The JKR technique has enjoyed widespread usage in recent years for measuring the work of adhesion between an elastomeric cap and substrates of interest. Although some success has been seen in coating the cap with other materials, the requirement that one of the contacting members be elastomeric remains an important feature of the test. The use of a soft structure instead of a soft material for the flexible contact element is proposed here. A flexible strip is bent and then pushed onto a rigid horizontal surface. First a JKR type of analysis is carried out, in which the surface energy is assumed to be proportional to the contact area, and the effects of adhesion are represented by a bending couple at the two separation points of the strip with the surface. For a given work of adhesion, the magnitude of the unknown couple is varied until the total energy is minimized. Then a DMT type of model is considered, with forces acting in the cohesive zone outside of each separation point. The forces are either constant or are linear functions of the gap. For both analyses, the effect of adhesion on the contact length, displacements, and forces is investigated. The results can be applied to determine the work of adhesion, based on measurements taken from such a contact problem.

Finite-amplitude salt fingers in a vertically bounded layer

We compute numerically the amplitude of long thin fingers that form in a liquid stratified with sugar S* and salt T* (measured in buoyancy units), for which τ = kS/kT = 1/3 is the ratio of the two diffusivities and the Prandtl number is Pr = v/kT ∼ 103, where v is the viscosity. The finger layer in our model is bounded by rigid and slippery horizontal surfaces with constant T*, S* (the setup is similar to the classical Rayleigh convection problem). The numerically computed steady fluxes compare well with laboratory experiments in which the fingers are sandwiched between two deep (convectively mixed) reservoirs with given concentration differences ΔT*, ΔS*. The model results, discussed in terms of a combination of asymptotic analysis and numerical simulations over a range of density ratio R = ΔT*/ΔS*, are consistent with the (ΔS*)4/3 similarity law for the fluxes. The dimensional interfacial height (H*) in the reservoir experiments (unlike that in our rigid lid model) is not an independent parameter, but it adjusts to a statistically steady value proportional to (ΔS*)−1/3. This similarity law is also explained by our model when it is supplemented by a consideration of the stability of the very thin horizontal boundary layers with large gradients (∂S*/∂z) which form near the rigid surfaces. The preference for three-dimensional salt fingers is also explained by a combination of analytical and numerical considerations.

Axisymmetric particle-driven gravity currents

Axisymmetric gravity currents that result when a dense suspension intrudes under a lighter ambient fluid are studied theoretically and experimentally. The dynamics of and deposition from currents flowing over a rigid horizontal surface are determined for the release of either a fixed volume or a constant flux of a suspension. The dynamics of the current are assumed to be dominated by inertial and buoyancy forces, while viscous forces are assumed to be negligible. The fluid motion is modelled by the single-layer axisymmetric shallow-water equations, which neglect the effects of the overlying fluid. An advective transport equation models the distribution of particles in the current, and this distribution determines the local buoyancy force in the shallow-water equations. The transport equation is derived on the assumption that the particles are vertically well-mixed by the turbulence in the current, are advected by the mean flow and settle out through a viscous sublayer at the bottom of the current. No adjustable parameters are needed to specify the theoretical model. The coupled equations of the model are solved numerically, and it is predicted that after an early stage both constant-volume and constant-flux, particle-driven gravity currents develop an internal bore which separates a supercritical particle-free region upstream from a subcritical particle-rich region downstream near the head of the current. For the fixed-volume release, an earlier bore is also predicted to occur very shortly after the initial collapse of the current. This bore transports suspended particles away from the origin, which results in a maximum in the predicted deposition away from the centre.To test the model several laboratory experiments were performed to determine both the radius of an axisymmetric particle-driven gravity current as a function of time and its deposition pattern for a variety of initial particle concentrations, particle sizes, volumes and flow rates. For the release of a fixed volume and of a constant flux of suspension, the comparisons between the experimental results and the theoretical predictions are fairly good. However, for the current of fixed volume, we did not observe the bore predicted to occur shortly after the collapse of the current or the resulting maximum in deposition downstream of the origin. This is unlike the previous study of Bonnecaze et al. (1993) on two-dimensional currents, in which a strong bore was observed during the slumping phase. The radial extent R of the deposit from a fixed-volume current is accurately predicted by the model, and for currents whose particles settle sufficiently slowly, we find that R = 1.9(g′0V3 / v2s)1/8, where V is the volume of the current, vs is the settling velocity of a particle in the suspension and g’0 is the initial reduced gravity of the suspension.