Coefficient Of Normal Restitution Research Articles

Moderately dense granular gas of inelastic rough spheres

A kinetic theory for moderately dense gases of inelastic and rough spherical molecules is developed from the Enskog equation where a macroscopic state is characterised by 29 scalar fields which correspond to the moments of the distribution function: mass density, hydrodynamic velocity, pressure tensor, absolute temperature, translational and rotational heat fluxes, hydrodynamic angular velocity and angular velocity flux. The balance equations for the 29 scalar fields are obtained from a transfer equation derived from the Enskog equation where the kinetic and potential parts of the new moments of the distribution function and production terms are calculated from Grad’s distribution function for the basic fields. The transition from the 29 field theory to an eight field theory—with mass density, hydrodynamic velocity, absolute temperature and hydrodynamic angular velocity—leads to the determination of the transport coefficients of the Navier–Stokes and Fourier laws. The transport coefficients are functions of the normal and tangential restitution coefficients and of the local equilibrium radial distribution function. The transport coefficients in the limiting case of elastic rough spheres is also determined.

The normal restitution coefficient and critical sticking velocity of disk-shaped adhesive particles

The particle-wall collision is an important phenomenon in gas-solid flows. For non-spherical particles like disk-shaped particles, the classic models based on quasi-spherical assumptions cannot be directly applied. It is still challenging to evaluate the loss of kinetic energy by resolving the adhesion and viscoelastic damping because of their strong coupling. To address this issue, we employ the finite element method (FEM) to conduct a numerical investigation of the particle-wall collision dynamics of adhesive, disk-shaped particles. The viscoelastic model and adhesion are coupled to naturally reproduce the energy dissipation. Our findings reveal that the non-spherical nature of the particles influences energy dissipation through two distinct mechanisms: viscoelastic dissipation by changing the collision time and adhesion by changing the contact area. Explicit expressions of the restitution and the critical sticking velocity of the disk-shaped particles are given, which can be directly used in the numerical simulation under the Euler-Lagrange framework.

Experimental Study on Coefficient of Restitution of Small-Sized Spherical Particles during Low-Speed Impact

This study presents experimental investigations on the normal restitution coefficients of a titanium bead (Ti), zirconia bead (ZrO2), and amorphous zirconium alloy sphere (Amor). The research explores the influence of particle diameter and collision velocity on the normal restitution coefficient between two independent, identical spherical particles of different materials. The experimental findings demonstrate that increasing the particle diameter results in more effective plastic deformation, leading to higher energy losses and, subsequently, smaller coefficients of restitution. Similarly, higher particle velocities cause more energy dissipation during collisions, resulting in smaller restitution coefficients. Comparing particles of different materials, those with larger yield strengths exhibit more elastic behavior, experience less initial energy loss due to deformation, and reach the maximum restitution coefficient (elastic state) with fewer collisions. This finding suggests that material properties significantly influence the overall energy dissipation and elastic response in the particles. To validate the experimental results, existing models are compared and discussed. Furthermore, potential physical mechanisms responsible for the observed behavior are explored, providing valuable insights into the collision dynamics in spherical particle interactions. Overall, this study contributes to a better understanding of the factors affecting the normal restitution coefficient in particle collisions, enabling the design and optimization of particle systems for diverse applications in condensed matter and related fields.

Measurement and analysis of impact dynamic parameters of micron-sized single particles using particle shadow velocimetry

The impact dynamics of micro-sized particles is still a fascinating topic of research, which can contribute to the understanding of the mechanisms of adhesion and rebound of these particles. A high-precision measurement system was developed that combines hardware, software, and numerical simulation methods to determine the impact dynamics parameters of micro-sized single particles under high temperature conditions by using Particle Shadow Velocimetry (PSV). The impact dynamic parameters of three standard particles (impact velocity, impact angle, coefficient of restitution, critical sticking velocity, etc.) were measured as examples to validate the reliability of this measurement system. The results demonstrated the reliability of the developed particle image processing algorithm. Corrections on the impact velocity and rebound velocity of small particles at low impact velocities were necessary. Particle size had a significant influence on the variation characteristics of the normal coefficient of restitution in particle impacts. A power function relationship between the critical sticking velocity and particle size was obtained. This paper provides a strong technical guidance for the future research on the impact dynamics of fly ash particles produced in the boiler field or other related fields.

Elimination method of sliding blockage effect in discrete element simulation of rolling stone based on discrete fracture network and damping joint model

The particle essence of the discrete element method could prevent the rock block from sliding on the slope surface due to the stress concentration between the convex particles of the sliding block and the slope. To eliminate this sliding blockage effect and simulate more realistic rockfall, slope surface was transformed into a series of successive joints by the discrete fracture network technique, and a damping joint model was developed by adding a normal dashpot and a shear dashpot on the smooth joint model that does not have damping properties. The motion processes of sliding, rolling, and collision were simulated by the proposed method, and the damping and movement performances were investigated. The macroscopic sliding and rolling friction coefficients significantly increase with the microscopic friction coefficient and the shear damping ratio. The normal restitution coefficient of collision decreases with the normal damping ratio, while the shear restitution coefficient of collision decreases with both the shear damping ratio and the friction coefficient. Furthermore, the simulation of a rockfall case demonstrates the proposed method represented a more realistic movement process and more plentiful characteristics of collision, sliding and rolling, compared with the simulated results of Rocfall software.

Numerical simulation of collision dynamics between a dry particle and a liquid-coated wet particle

A comprehensive understanding of wet particle collisions in the presence of a liquid film is essential for industrial processes. In this study, we conduct a numerical investigation into the dynamics of both normal and oblique collisions between a dry particle and a liquid-coated wet particle. In normal collisions, the gas flow velocity near the stretching liquid bridge increases significantly, while oblique collisions exhibit no such increase. The wetting area of particles reaches its maximum when solid particles come into contact, at which point the viscous and capillary forces acting on the particles also reach their peaks. The capillary force causes kinetic energy dissipation to be one magnitude greater than viscous energy dissipation. The normal wet restitution coefficient rises with the logarithm of the modified Stokes number, reflecting energy dissipation effects. As the collision angle increases, the normal wet restitution coefficient gradually increases and surpasses the tangential wet restitution coefficient.

Experimental study of particle collision and rebound with moving walls

Particle–wall collision always occurs in transportation systems with multi-phase flows containing particles in pneumatic conveying, coal liquefaction production, and wastewater treatment. Especially within rotating fluid machinery, collision and rebound between particles and moving walls occur frequently. It is necessary to conduct research, as it can provide theoretical support for controlling particle trajectories and reducing particle erosion. In this study, experimental investigations are carried out to analyze the trajectories and restitution coefficients of particle collision and rebound at different velocities of the moving wall, wall roughnesses, and collision angles. Particle trajectories and the influence of particle rotation on restitution coefficients in normal collisions are investigated, and particle motion behavior, tangential and normal restitution coefficients of the particle collision and rebound, and effective rebound angles in oblique collisions are analyzed and discussed at different operating conditions. The differences in collision and rebound characteristics between normal and oblique particle impacts are identified.

Occurrence conditions of the reverse rotation of rockfall and its influence on the restitution coefficient

When rockfall occurs along dense rock slopes, the rotation direction of rockfall is not always downhill. Specifically, the rockfall may obtain a reverse rotation speed (RRS) after impact under certain conditions, the effect of which on the restitution coefficient (RC) cannot be ignored. According to the statistical results of the reverse rotation (RR) phenomena of blocks obtained from previous experiments, the occurrence of RR is correlated to the block shape, incident angle, and contact attitude. In this study, considering a typically shaped cubic block, the critical condition for the RR is preliminarily deduced. Based on the results, the influence of the RRS on the RC for four typically shaped blocks is examined using a customized device. Results show that the tangential RC (Rt) values of each block are not sensitive to the change in the RRS, the distribution is relatively concentrated and the values are high. Moreover, the normal RC (Rn) values are not sensitive to the RRS, and the distribution is relatively discrete. The RRS influences Rn; however, it is difficult to directly establish the relationship between them. To this end, considering the contact attitude and shape of the block, an integral variable, the impact coefficient (Ic), is proposed to determine the influence of RRS on Rn. Moreover, the impact-bounce behaviours of the block are categorized and analysed. For the block rebound following a single impact, Ic and Rn are positively and negatively correlated when the mass centre of the block (MC) is in front and behind the contact point (CP), respectively. For the block rebound following two successive impacts, with the increase in Ic, the Rn increases. These conclusions help clarify the mechanism of the influence of the RRS on RC and provide vital information and ideas for the development and optimization of a program to accurately predict rockfall trajectories.

Modeling Ellipsoidal Block Impacts by an Advanced Rheological Model

In this paper, an advanced rheological model for impacts of ellipsoidal blocks on deformable ground surfaces, introducing the effects of block eccentricity and orientation at impact, is presented. This allows us to assess impact penetration and force, restitution coefficients, and block trajectories. A parametric analysis was carried out by considering different block aspect ratios, impact angles and initial block orientations at impact. The results are presented in terms of restitution coefficients, penetration and force time histories, maximum penetration depth, maximum force and rotational/total kinetic ratios. Impacts along the major block axis, versus those along minor axis, are characterized by larger penetrations (ranging from 3.3 to 50%), shorter impact durations (ca 50%) and very slightly larger vertical forces (ranging from 0.3 to 60%) according to the model parameter used. In contrast, the impact angle is shown to strongly affect maximum penetration and force values, and markedly increase rotation at impact. Analogously, normal restitution coefficient is severely dependent on impact angle, with a variation of more than two orders of magnitude. A mathematical expression for computing the energetic restitution coefficient from the normal and tangential apparent restitution coefficients and the ratio between the rotation and total kinetic energy is proposed. This overcomes the drawback of classical restitution coefficients greater than one when a change in block rotation occurs allowing us to bracket the coefficient of restitutions values to support and improve classical rock fall simulations also highlighting their intrinsic limitations. Finally, the effects of block geometry and initial angular velocity on rockfall simulations were analyzed by implementing the approach in the HyStone simulation code. The simulated frequencies of the maximum height during each ballistic trajectory follow an exponential distribution, whereas those for normal and tangential apparent restitution coefficients follow normal distributions.

Independent friction-restitution description of billiard ball collisions

The oblique impact of a cue ball moving on a horizontal surface with an arbitrary spin on an object ball at rest is described using the independent friction-restitution modeling of impact. The model yields theoretical expressions for the post-collision linear and angular velocities in terms of ‘constant’ coefficients of normal restitution, tangential restitution, and friction for both stick and slip regimes of impact. The model also predicts the post-impact velocities reached by the balls when, as a result of friction with the supporting surface, the pure rolling motion is established. Theoretical predictions are in satisfactory agreement with experimental data for collisions between regulation billiard balls, steel, brass, and rubber spheres.

Particle Collision Study Based on a Rotational Boundary Condition

The main engineering machinery for the hydrodynamic lifting of seafloor mineral particles is rotor machinery with rotating impeller motion. It is important to study the rebound mechanism of collisions between particles and rotating walls to improve the accuracy of numerical simulation of rotor machinery. In this study, the law of motion change after collisions between particles and rotating walls is investigated using an experimental research method. The results show that the deflection angle of the particles after collision decreases with increases in the rotational speed of the wall, and the spin angular velocity increases with increases in the rotational speed of the wall. The normal velocity coefficient of restitution under the rotating wall is not affected by the rotational speed of the wall. The tangential coefficient of restitution under rotational boundary condition is smaller than the tangential coefficient of restitution under the stationary wall, and the higher the rotational speed, the closer it is to the coefficient of restitution under the stationary wall. During collision in the experiment, the main mode of contact between the particle and the rotating wall is sliding contact. Sliding friction between the particle and the rotating wall results in energy loss in the tangential velocity of the particle, and also provides energy for deflection of the particle’s trajectory and increased kinetic energy from the spin angular velocity; sliding friction loss is affected by the speed of the wall.

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].

Assessment of kinetic theories for moderately dense granular binary mixtures: Shear viscosity coefficient

Two different kinetic theories [J. Solsvik and E. Manger (SM), Phys. Fluids 33, 043321 (2021) and Garzó et al. (GDH), Phys. Rev. E 76, 031303 (2007)] are considered to determine the shear viscosity η for a moderately dense granular binary mixture of smooth hard spheres. The mixture is subjected to a simple shear flow and heated by the action of an external driving force (Gaussian thermostat) that exactly compensates the energy dissipated in collisions. The set of Enskog kinetic equations is the starting point to obtain the dependence of η on the control parameters of the mixture: solid fraction, concentration, mass and diameter ratios, and coefficients of normal restitution. While the expression of η found in the SM-theory is based on the assumption of Maxwellian distributions for the velocity distribution functions of each species, the GDH-theory solves the Enskog equation by means of the Chapman–Enskog method to first order in the shear rate. To assess the accuracy of both kinetic theories, the Enskog equation is numerically solved by means of the direct simulation Monte Carlo method. The simulation is carried out for a mixture under simple shear flow, using the thermostat to control the cooling effects. Given that the SM-theory predicts a vanishing kinetic contribution to the shear viscosity, the comparison between theory and simulations is essentially made at the level of the collisional contribution ηc to the shear viscosity. The results clearly show that the GDH-theory compares with simulations much better than the SM-theory over a wide range of values of the coefficients of restitution, the volume fraction, and the parameters of the mixture (masses, diameters, and concentration).

High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States

The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution function of each species when diffusion is absent (mass flux of each species vanishes). The corresponding associated eigenvalues as well as cross coefficients are obtained as functions of the coefficients of normal restitution and the parameters of the mixture (masses, diameters and composition). The results are applied to the analysis of the time evolution of the moments (scaled with a thermal speed) in two different nonequilibrium situations: the homogeneous cooling state (HCS) and the uniform (or simple) shear flow (USF) state. In the case of the HCS, in contrast to what happens for simple granular gases, it is demonstrated that the third and fourth degree moments could diverge in time for given values of the parameters of the system. An exhaustive study on the influence of the parameter space of the mixture on the time behavior of these moments is carried out. Then, the time evolution of the second- and third-degree velocity moments in the USF is studied in the tracer limit (namely, when the concentration of one of the species is negligible). As expected, while the second-degree moments are always convergent, the third-degree moments of the tracer species can be also divergent in the long time limit.

A comparison of pendulum experiments and discrete‐element simulations of oblique collisions of wet spheres

AbstractOblique collisions of two spherical particles coated with a thin layer of viscous liquid are considered. Experimental measurements are performed using particle tracking velocimetry and the results are compared to viscous force and capillary force models via numerical simulation using the discrete‐element method. Comprehensive experimental data for collisions with an impact angle between 0° and 60° are presented to ensure future models can be rigorously validated. Collisions are characterized by the normal Stokes' number (a dimensionless ratio of normal inertial forces to normal viscous forces), and the normal coefficient of restitution (a dimensionless ratio of postcollisional normal velocity to precollisional normal velocity). Good agreement was found between the models and the experiments at high Stokes' number, where the models are dominated by the normal components. As the tangential forces become more significant (i.e., at low to medium Stokes' number, and high collision angle), agreement between the simulations and experiments is poorer. Furthermore, the models predict a decreasing rotational velocity with increasing Stokes' number past some critical Stokes' number. A coefficient of friction term was implemented and found to improve agreement for the rotational velocity postcollision.

Study of the process of metal droplets with high surface tension impinging on wall

The behavior of aluminum droplets impacting a wall critically affects the operation of solid rocket motors. In the present study, the flow and rebound of aluminum droplets with a high surface tension are studied experimentally and numerically. In addition, the impact of aluminum droplets at different inclined angles is monitored experimentally. When the droplet is about to rebound away from the wall, it is stretched to its maximum length. An empirical correlation formula is proposed to predict the maximum length when a droplet is about to bounce off a flat plane. The velocity and pressure distributions of a droplet flowing over a flat plane and an inclined plane are compared by using the volume of fluid method. Furthermore, the restitution coefficient of the droplet is discussed in detail. When normal Weber number Wen ranges from 0 to 20, normal restitution coefficient en ranges from 0.3 to 0.6. When tangential Weber number Wet <10, tangential restitution coefficient et is subject to great uncertainty. When 10 < Wet < 80, et is maintained at 0.75. The formulas for the total restitution coefficient, normal restitution coefficient, and tangential restitution coefficient of the aluminum droplets are also presented. Based on the principle of conservation of energy, we calculate the remaining energy of the aluminum droplets impinging on a wall and provide the relationship between the restitution coefficient and the dissipated energy. The results elucidate the mechanisms at work when aluminum droplets collide with a wall.

Kinetic Theory and Memory Effects of Homogeneous Inelastic Granular Gases under Nonlinear Drag

We study a dilute granular gas immersed in a thermal bath made of smaller particles with masses not much smaller than the granular ones in this work. Granular particles are assumed to have inelastic and hard interactions, losing energy in collisions as accounted by a constant coefficient of normal restitution. The interaction with the thermal bath is modeled by a nonlinear drag force plus a white-noise stochastic force. The kinetic theory for this system is described by an Enskog–Fokker–Planck equation for the one-particle velocity distribution function. To get explicit results of the temperature aging and steady states, Maxwellian and first Sonine approximations are developed. The latter takes into account the coupling of the excess kurtosis with the temperature. Theoretical predictions are compared with direct simulation Monte Carlo and event-driven molecular dynamics simulations. While good results for the granular temperature are obtained from the Maxwellian approximation, a much better agreement, especially as inelasticity and drag nonlinearity increase, is found when using the first Sonine approximation. The latter approximation is, additionally, crucial to account for memory effects such as Mpemba and Kovacs-like ones.

On the mean square displacement of intruders in freely cooling granular gases

We compute the mean square displacement (MSD) of intruders immersed in a freely cooling granular gas made up of smooth inelastic hard spheres. In general, intruders and particles of the granular gas are assumed to have different mechanical properties, implying that non-equipartition of energy must be accounted for in the computation of the diffusion coefficient D. In the hydrodynamic regime, the time decay of the granular temperature T of the cooling granular gas is known to be dictated by Haff’s law; the corresponding decay of the intruder’s collision frequency entails a time decrease of the diffusion coefficient D. Explicit knowledge of this time dependence allows us to determine the MSD by integrating the corresponding diffusion equation. As in previous studies of self-diffusion (intruders mechanically equivalent to gas particles) and the Brownian limit (intruder’s mass much larger than the grain’s mass), we find a logarithmic time dependence of the MSD as a consequence of Haff’s law. This dependence extends well beyond the two aforementioned cases, as it holds in all spatial dimensions for arbitrary values of the mechanical parameters of the system (masses and diameters of intruders and grains, as well as their coefficients of normal restitution). Our result for self-diffusion in a three-dimensional granular gas agrees qualitatively, but not quantitatively, with that recently obtained by Blumenfeld [arXiv: 2111.06260] in the framework of a random walk model. Beyond the logarithmic time growth, we find that the MSD depends on the mechanical system parameters in a highly complex way. We carry out a comprehensive analysis from which interesting features emerge, such a non-monotonic dependence of the MSD on the coefficients of normal restitution and on the intruder-grain mass ratio. To explain the observed behaviour, we analyze in detail the intruder’s random walk, consisting of ballistic displacements interrupted by anisotropic deflections caused by the collisions with the hard spheres. We also show that the MSD can be thought of as arising from an equivalent random walk with isotropic, uncorrelated steps. Finally, we derive some results for the MSD of an intruder inmersed in a driven granular gas and compare them with those obtained for the freely cooling case. In general, we find significant quantitative differences in the dependence of the scaled diffusion coefficient on the coefficient of normal restitution for the grain-grain collisions.Graphic abstract

Granular Gas of Inelastic and Rough Maxwell Particles

The most widely used model for granular gases is perhaps the inelastic hard-sphere model (IHSM), where the grains are assumed to be perfectly smooth spheres colliding with a constant coefficient of normal restitution. A much more tractable model is the inelastic Maxwell model (IMM), in which the velocity-dependent collision rate is replaced by an effective mean-field constant. This simplification has been taken advantage of by many researchers to find a number of exact results within the IMM. On the other hand, both the IHSM and IMM neglect the impact of roughness—generally present in real grains—on the dynamic properties of a granular gas. This is remedied by the inelastic rough hard-sphere model (IRHSM), where, apart from the coefficient of normal restitution, a constant coefficient of tangential restitution is introduced. In parallel to the simplification carried out when going from the IHSM to the IMM, we propose in this paper an inelastic rough Maxwell model (IRMM) as a simplification of the IRHSM. The tractability of the proposed model is illustrated by the exact evaluation of the collisional moments of first and second degree, and the most relevant ones of third and fourth degree. The results are applied to the evaluation of the rotational-to-translational temperature ratio and the velocity cumulants in the homogeneous cooling state.

Probabilistic data-driven framework for performance assessment of retaining walls against rockfalls

Rockfall is a significant hazard to sites that are located at the foot of rock slopes. In such sites, there is a notable need to evaluate the potential for rockfall, estimate the extent of areas at risk, and design retaining structures to reduce the risk of rockfall-induced. This paper presents a probabilistic framework for predicting the formation and progression of rockfalls and for evaluating the performance of retaining walls under rockfalls. To this end, first a probabilistic model for the rock projectile motion on a slope is presented. The model accounts for prevailing uncertainties, i.e., the trigger points, rock shape, projectile path, and slope material properties, which include surface roughness, friction angle, and horizontal and normal coefficients of restitution. Next, Monte Carlo sampling is employed to propagate these uncertainties in the proposed model and generate a large dataset of rockfall realizations. The resulting dataset is subsequently utilized to develop an exceedance probability diagram of the rock endpoints, which is in turn used to estimate the location of the retaining wall for a target exceedance probability. Furthermore, the exceedance probability contours of the bounce height and the total kinetic energy of the rock in the projectile path are computed to produce the spatial variation of the exceedance probability at any desired location along the slope. Given the location of the retaining and the target exceedance probability, the probability contours are then employed to approximate the height and the structural capacity of the retaining wall to withstand the rock collision. Finally, a bivariate sensitivity analysis is performed to measure the performance of the retaining wall against the falling rocks and further evaluate the efficiency of the proposed probabilistic design.