Transient wave behaviour over an underwater sliding hump from experiments and analytical and numerical modelling

Flume measurements of a one-dimensional sliding hump starting from rest in quiescence fresh water indicate that when the hump travels at speed less than the shallow-water wave celerity, three waves emerge, travelling in two directions. One wave travels in the opposite direction to the sliding hump at approximately the shallow-water wave celerity (backward free wave). Another wave travels approximately in step with the hump (forced wave), and the remaining wave travels in the direction of the hump at approximately the shallow-water wave celerity (forward free wave). These experiments were completed for a range of sliding hump speed relative to the shallow-water wave celerity, up to unity of this ratio, to investigate possible derivation from solutions of the Euler equation with non-linear and non-hydrostatic terms being included or excluded. For the experimental arrangements tested, the forced waves were negative (depression or reduced water surface elevation) waves while the free waves were positive (bulges or increased water surface elevation). For experiments where the sliding hump travelled at less than 80% of the shallow-water wave celerity did not include transient behaviour measurements (i.e. when the three waves still overlapped). The three wave framework was partially supported by these measurements in that the separated forward and forced waves were compared to measurements. For the laboratory scale experiments, the forward free wave height was predicted reasonably by the long-wave equation (ignoring non-linear and non-hydrostatic terms) when the sliding hump speed was less than 80% of the shallow-water wave celerity. The forced wave depression magnitude required the Euler equations for all hump speed tested. The long-wave solution, while being valid in a limited parameter range, does predict the existence of the three waves as found in these experiments (forward travelling waves measured quantitatively while the backward travelling waves visually by video). Nevertheless, the forced wave transient development required non-linear and non-hydrostatic terms for higher sliding hump speeds. The forward free wave, controversially, does not need non-linear and non-hydrostatic terms until much higher hump speeds, suggesting that the forward free wave falls in the parameter space where long-wave equations apply whereas the forced wave more often falls into the parameter space requiring non-linear and non-hydrostatic terms. It does raise the question of why the forced wave transient dynamics does not impact on the initial transient dynamics where the forward free wave is in the long-wave theory, escaping the forced wave (at least for speeds less than 80% of the shallow-water wave celerity).     

Weakly non-linear waves in a fluid-filled elastic tube with variable prestretch

In the present work, by utilizing the non-linear equations of motion of an incompressible, isotropic thin elastic tube subjected to a variable prestretch both in the axial and the radial directions and the approximate equations of motion of an incompressible inviscid fluid, which is assumed to be a model for blood, we studied the propagation of weakly non-linear waves in such a medium, in the long wave approximation. Employing the reductive perturbation method we obtained the variable coefficient KdV equation as the evolution equation. By seeking a travelling wave solution to this evolution equation, we observed that the wave speed is variable in the axial coordinate and it decreases for increasing circumferential stretch (or radius). Such a result seems to be plausible from physical considerations.     

Head on collision of a planar shock wave with a dusty gas layer

The head-on collision of a planar shock wave with a dust-air suspension is studied numerically. In this study the suspension is placed inside a conduit adjacent to its rigid end-wall. It is shown that as a result of this collision two different types of transmitted shock waves are possible, depending on the strength of the incident shock wave and the dust loading ratio in the suspension. One possibility is a partially dispersed shock wave, the other is a compression wave. The flow fields resulting in these two options are investigated. It is shown that in both cases, at late times after the head-on reflection of the transmitted shock wave from the conduit end-wall a negative flow (away from the end-wall) is evident. The observed flow behavior may suggest a kind of dust particle lifting mechanism that could shed new light on the complex phenomenon of dust entrainment behind sliding shock waves.      

Study of water waves with submerged obstacles using a vortex method with Helmholtz decomposition

The present study develops a 2‐D numerical scheme that combines the vortex method and the boundary integral method by a Helmholtz decomposition to investigate the interaction of water waves with submerged obstacles. Viscous effects and generation of vorticity on the free surface are neglected. The second kind of Fredholm integral equations that govern the strengths of vortex sheets along boundaries are solved iteratively. Vorticity is convected and diffused in the fluid via a Lagrangian vortex (blob) method with varying cores, using the particle strength exchange method for diffusion, with particle redistribution. A grid‐convergence study of the numerical method is reported. The inviscid part of the method and the simulation of the free‐surface motion are tested using two calculations: solitary wave propagation in a uniform channel and a moving line vortex in the fluid. Finally, the full model is verified by simulating periodic waves travelling over a submerged rectangular obstacle using nonuniform vortex blobs with a mapping of the redistribution lattice. Overall, the numerical model predicts the vortices' evolution and the free‐surface motion reasonably well. Copyright © 2008 John Wiley & Sons, Ltd.     

Modelling of dust lifting using the Lagrangian approach

The subject of this paper is dust lifting behind shock waves, a process that is important for the formation of explosive dust clouds in air. While Eulerian–Eulerian has been the standard numerical technique for such simulations, the Eulerian–Lagrangian technique has been used in this paper, making it possible to take into account more physical phenomena, such as particle–particle and particle–wall collisions. The results of the simulations are shown mainly graphically, as snapshots of particle positions at given times after the passing of the shock wave. The results show that the collisions, and the coefficient of restitution assumed for them, is important in determining the mobility and lifting of dust behind shock waves. The results also show that the idea of a horizontally travelling shock wave is an oversimplification: the strong pressure gradient at the surface results in a series of reflected waves generated at the surface and travelling into the gas phase.     

Wave propagation in linear viscoelastic plates of various thicknesses

Experiments were conducted to observe plate waves travelling in polymethylmethacrylate, polyethylene, and glass plates of various thicknesses at several distances of wave travel, when the plates have been subjected to a transverse surface impact by a steel ball. It has been shown that if Poisson's ratio is taken to be a real constant, the shapes of the plate waves can be calculated when the initial disturbance and the viscoelastic properties of the medium are known. It was found possible to predict quite well the shape of the plate waves observed in the experiments. The plate waves were shown, both theoretically and experimentally, to be of a different character than those observed on the free surfaces of large block materials. The effect of the height of ball drop and contacting ball radius on pulse shape was noted.     

Nonlinear motions of a flexible rotor with a drill bit: stick-slip and delay effects

In this article, a discrete model of a drill-string system is developed taking into account stick-slip and time-delay aspects, and this model is used to study the nonlinear motions of this system. The model has eight degrees-of-freedom and allows for axial, torsional, and lateral dynamics of both the drill pipes and the bottom-hole assembly. Nonlinearities that arise due to dry friction, loss of contact, and collisions are considered in the development. State variable dependent time delays associated with axial and lateral cutting actions of the drill bit are introduced in the model. Based on this original model, numerical studies are carried out for different drilling operations. The results show that the motions can be self-exited through stick-slip friction and time-delay effects. Parametric studies are carried out for different ranges of friction and simulations reveal that when the drill pipe undergoes relative sticking motion phases, the drill-bit motion is suppressed by absolute sticking. Furthermore, the sticking phases observed in this work are longer than those reported in previous studies and the whirling state of the drill pipe periodically alternates between the sticking and slipping phases. When the drive speed is used as a control parameter, it is observed that the system exhibits aperiodic dynamics. The system response stability is seen to be largely dependent upon the driving speed. The discretized model presented here along with the related studies on nonlinear motions of the system can serve as a basis for choosing operational parameters in practical drilling operations.     

Tests of a new numerical scheme on a “non-reflection and free-transmission” open-sea boundary for longwaves

A new numerical scheme of a “non-reflection and free-transmission” boundary for longwave equations proposed by Hino (1987) has been tested for a variety of cases. The test results verify the effectiveness of the method for (a) a single progressive wave train on a horizontal bottom, (b) two wave trains each propagating in opposite directions on a horizontal bottom, (c) a single wave train propagating on a sloping bottom with friction, (d) oscillatory flood waves in an open channel flow, (e) two-dimensional waves travelling obliquely to open boundaries and (f) water surface oscillation in a harbor by waves incident through an opening.     

Drift of an inclined plate counter oncoming waves

The dependence of the velocity of the motion of a tow with an inclined plate mounted in a wave water channel on the wave parameters, the submergence depth, and the angle of inclination and dimensions of the plate is experimentally investigated. The effect of tow motion counter to the waves is detected and theoretically justified. The free surface profiles for periodic waves above an inclined plate obtained using the elolutionary system of the Boussinesq approximation equations correspond to the measured ones. The pulse generated as a result of wave breakup is estimated.     

Numerical analysis of breaking waves using the moving particle semi-implicit method

The numerical method used in this study is the moving particle semi-implicit (MPS) method, which is based on particles and their interactions. The particle number density is implicitly required to be constant to satisfy incompressibility. A semi-implicit algorithm is used for two-dimensional incompressible non-viscous flow analysis. The particles whose particle number densities are below a set point are considered as on the free surface. Grids are not necessary in any calculation steps. It is estimated that most of computation time is used in generation of the list of neighboring particles in a large problem. An algorithm to enhance the computation speed is proposed. The MPS method is applied to numerical simulation of breaking waves on slopes. Two types of breaking waves, plunging and spilling breakers, are observed in the calculation results. The breaker types are classified by using the minimum angular momentum at the wave front. The surf similarity parameter which separates the types agrees well with references. Breaking waves are also calculated with a passively moving float which is modelled by particles. Artificial friction due to the disturbed motion of particles causes errors in the flow velocity distribution which is shown in comparison with the theoretical solution of a cnoidal wave. © 1998 John Wiley & Sons, Ltd.     

On the propagation of energy in linear conservative waves

Summary This paper is concerned with the question when and why the rate of energy propagation in a system of waves equals the group velocity. It is shown by the method of stationary phase that this equality holds, for travelling waves without dissipation, whenever this method applies. The reason why this result can be obtained by this kinematical method is investigated by a discussion of simple harmonic waves. It is shown that the choice of an expression for the energy density to be used in connection with a given wave equation is restricted by the conservation of energy in such a way that the average rate of work done divided by the average energy density always equals the group velocity. Finally some examples of wave motion are discussed to illustrate the derived formulae.     

Improvement of moving particle semi‐implicit method for simulation of progressive water waves

Precise simulation of the propagation of surface water waves, especially when involving breaking wave, takes a significant place in computational fluid dynamics. Because of the strong nonlinear properties, the treatment of large surface deformation of free surface flow has always been a challenging work in the development of numerical models. In this paper, the moving particle semi‐implicit (MPS) method, an entirely Lagrangian method, is modified to simulate wave motion in a 2‐D numerical wave flume preferably. In terms of consecutive pressure distribution, a new and simple free surface detection criterion is proposed to enhance the free surface recognition in the MPS method. In addition, a revised gradient model is deduced to diminish the effect of nonuniform particle distribution and then to reduce the numerical wave attenuation occurring in the original MPS model. The applicability and stability of the improved MPS method are firstly demonstrated by the calculation of hydrostatic problem. It is revealed that these modifications are effective to suppress the pressure oscillation, weaken the local particle clustering, and boost the stability of numerical algorithm. It is then applied to investigate the propagation of progressive waves on a flat bed and the wave breaking on a mild slope. Comparisons with the analytical solutions and experimental results indicate that the improved MPS model can give better results about the profiles and heights of surface waves in contrast with the previous MPS models. Copyright © 2017 John Wiley & Sons, Ltd.     

An invariant manifold approach for studying waves in a one-dimensional array of non-linear oscillators

We consider a one-dimensional linear spring-mass array coupled to a one-dimensional array of uncoupled pendula. The principal aim of this study is to investigate the non-linear dynamics of this large-scale system in the limit of weak non-linearities, i.e. when the (fast) non-linear pendulum effects are small compared to the underlying (slow) linear dynamics of the linear spring-mass chain. We approach the dynamics in the context of invariant manifolds of motion. In particular, we prove the existence of an invariant manifold containing the (predominantly) slow dynamics of the system, with the fast pendulum dynamics providing small perturbations to the motions on the invariant manifold. By restricting the motion on the slow invariant manifold and performing asymptotic analysis we prove that the non-linear large-scale system possesses propagation and attenuation zones (PZs and AZs) in the frequency domain, similarly to the corresponding zones of the linearized system. Inside PZs non-linear travelling wave solutions exist, whereas in AZs only attenuating waves are permissible.     

Destabilization of long-wavelength Love and Stoneley waves in slow sliding

Love waves are dispersive interfacial waves that are a mode of response for anti-plane motions of an elastic layer bonded to an elastic half-space. Similarly, Stoneley waves are interfacial waves in bonded contact of dissimilar elastic half-spaces, when the displacements are in the plane of the solids. It is shown that in slow sliding, long-wavelength Love and Stoneley waves are destabilized by friction. Friction is assumed to have a positive instantaneous logarithmic dependence on slip rate and a logarithmic rate weakening behavior at steady-state.Long-wavelength instabilities occur generically in sliding with rate- and state-dependent friction, even when an interfacial wave does not exist. For slip at low rates, such instabilities are quasi-static in nature, i.e., the phase velocity is negligibly small in comparison to a shear wave speed. The existence of an interfacial wave in bonded contact permits an instability to propagate with a speed of the order of a shear wave speed even in slow sliding, indicating that the quasi-static approximation is not valid in such problems.     

Regimes of frictional sliding of a spring-block system

In the context of rate-and-state friction, we revisit the crossover between the creep and inertial regimes in the dynamics of a spring-block system as observed and described in the dry friction experiment of Heslot et al. (1994) and Baumberger et al. (1994). We show that the transition between the quasi-static motion of a spring-block and its dynamic motion occurs at a lower sliding velocity than that which minimises the steady-state friction coefficient. We perform a weakly nonlinear stability analysis combined with numerical studies with the continuation package Auto. In particular, attention is focused on the change of nature the Hopf bifurcation from supercritical to subcritical, as observed by Heslot et al. Comparing the results obtained for different friction laws, we conclude that the weakly nonlinear analysis provides a possible criterion for distinguishing which friction laws may be physically relevant.     

Acoustic waves emitted by a vortex ring passing near a wedge-like plate

Acoustic waves emitted by a vortex ring moving near a thin wedge-like plate of finite width have been studied. The experiments are performed for three configurations: the plate (A) is held edgeways to the direction of the vortex motion, (B) is held sideways to the direction, and (C) is held edgeways at an angle of 45° against the vortex motion. The observed sound wave is of dipole radiation type, and the magnitude of the pressure is large in the direction of the normal to the plate plane and small in parallel. The observed pressure is proportional to the third power of the vortex speed. The instantaneous force exerted on the plate by the vortex motion has also been examined. The force vector is mainly normal to the plate plane. The observed profiles agree within a reasonable degree of accuracy with the theoretical ones calculated for the vortex ring interacting with the flat plate of thickness zero.     

Wave action on a movable plate immersed into a fluid. Physical and numerical experiments

The effect of the motion of a rigid inclined plate immersed into a fluid counter incident waves is studied both experimentally and numerically. In the wave water channel experiments the velocity of a trolley with a plate running freely on rails is determined as a function of the wave parameters, the immersion depth, the angle of inclination, and the plate dimensions. The interaction between the traveling waves and the plate having a single translational degree of freedom along the horizontal axis is numerically calculated in the time-dependent, two-dimensional formulation. The dependence of the upwave motion effect on the parameters varied in the full-scale experiment is analyzed. In the numerical experiment a regime of the downwave plate motion at a constant high velocity is found to exist. The channel bottom effect is estimated and the behavior of the plate with a flap is studied.     

Shock-wave-initiated lifting of particles from a cavity

The dynamics of particles of the disperse phase in a turbulent gas flow in planar shock waves sliding along a solid surface with a trapezoid cavity is examined numerically. Lifting of particles from the cavity walls is calculated in the approximation of a rarefied gas suspension. It is shown that the intensity of the transient shock wave and the initial positions of particles have a significant effect on the particle-lifting properties. The height of particle lifting is found to nonmonotonically depend on the initial streamwise coordinate and shock-wave Mach number. It is shown that zones of aggregation and subtraction of particles may be formed at the cavity bottom. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 24–34, January–February, 2007.