Near-critical Speed Research Articles

Dynamics of interfacial gravity–capillary waves in three-dimensional fluids of great depth

This paper proposes a new isotropic bidirectional model for weakly nonlinear gravity–capillary waves propagating between two incompressible, inviscid, and immiscible fluids of different densities. The newly developed equation is a generalization of the celebrated two-dimensional Benjamin equation. It is derived based on the nonlocal formulation of water wave (i.e., the Ablowitz–Fokas–Musslimani formulation) and computed with the modified exponential time-differencing method. It is found that horizontally two-dimensional, fully localized traveling waves (known as lumps) exist in the model equation, and plane solitary waves are unstable subject to long transverse perturbations, which evolve into groups of lumps in the long-term dynamics. When considering topographical disturbances on the rigid upper boundary, the nonlinear effect becomes important when a uniform flow passes beneath a locally confined topography with a near-critical speed, and the phenomenon of time-periodic shedding of lumps occurs. Unlike the near-critical situation, the subcritical flows usually generate steady elevation waves, while the supercritical ones produce a V-shaped pattern of wake lines. Furthermore, it is shown that a uniformly accelerated motion can also generate lumps provided that the flow stays in the transcritical regime for a considerably long time.

Can the Waves Generated by Fast Ferries be a Physical Model of Tsunami?

The potential of long ship-induced waves to serve as a physical model for tsunami waves (called simply tsunami below) is examined. Such waves (wavelengths more than 200 m at depths down to 10–20 m) are induced by high-speed ferries sailing at near-critical speeds in semisheltered, relatively shallow areas. It is shown based on experience from Tallinn Bay, Baltic Sea, that for many aspects these waves can model nearshore dynamics and runup of tsunami caused by landslides, including processes of wave refraction, diffraction, and sea-bottom interaction in bays and harbors. Many governing nondimensional parameters (such as the nonlinearity, dispersion, Reynolds and Ursell numbers, surf similarity parameter, breaking parameter, etc.) of the largest ship waves and landslide tsunamis have the same order of magnitude. It is especially important that use of ship waves for wave propagation and runup studies allows their spatial structure to be accounted for adequately. Near-critical ship waves can therefore be used as a natural substitute for tsunami, for study under controlled and safe conditions.

Vortex pattern and wave motion produced by a bottom blunt body moving at a critical speed

An underwater object moving at a near-critical speed in a shallow-water domain had been observed to generate a sequence of upstream propagating solitary waves with an elongated depression of water surface and a train of dispersive waves followed in the downstream. This study presents the development of a two-dimensional stream function–vorticity based viscous fluid model with satisfied nonlinear free-surface conditions to study the generation of solitary waves and the induced vortex motion under the forcing of a moving object. A combined finite analytic and finite difference method is adopted to solve the flow field equations and free-surface boundary conditions in a transient curvilinear coordinate system. The model is shown to produce free-surface elevations in fairly good agreements with published results for a test case of a moving smooth bump. Other tests for the generation of recirculation zone behind a body of square shape in a confined fluid domain are also conducted to further verify the model performance. The results showing the generation of upstream advancing solitary waves and downstream vortex pattern by a blunt rectangular body moving at a critical speed along the bottom in a domain with free surface are presented. Comparisons of results from potential flow and viscous flow conditions are made to demonstrate the importance of viscosity to the wave generation. Different from the relatively regular vortex pattern occurred under the case of Re = 3500, the transition of the vortex motion for a larger Reynolds number (e.g. Re = 35,000) evolves without a regular pattern throughout the generation process of the advancing solitons. The effects of the size and bluntness of a moving object on the generated flow field and free-surface elevations are also analyzed and discussed.

Far-field vessel wakes in Tallinn Bay

The properties of wave fields induced by high-speed ferries and recently introduced conventional ferries with increased cruise speeds are analysed for a site in Tallinn Bay, the Gulf of Finland, the Baltic Sea, located about 3 km from the sailing line and up to 8 km from the wave production area. The analysis is based on high-resolution profiling of the water surface for about 650 wakes from fast ferries, measured during 4 weeks in June–July 2008. The new large conventional ferries with cruise speeds of 25–30 knots (~ 45–55 km/h) sail at near-critical speeds along extensive sections of eastern Tallinn Bay, and excite wakes equivalent to those of high-speed ferries. The peak periods of these wakes are between 10 and 13 s. The typical daily highest ship wave is approximately 1.2 m, measured prior to wake breaking. The largest recorded ship wave in calm conditions had a height of 1.5 m and in the presence of some wind wave background 1.7 m. The cumulative impact of ship wakes results in a gradual increase in the suspended matter concentration in near-bottom water over the course of a day. The largest and longest ship waves produce considerable wave runup at the coast and prevent several coastal sections from achieving an equilibrium state. The largest ship waves have an asymmetric shape both in terms of the water surface elevation above and below the mean level and in terms of the shape of the wave front and back. The overall intensity of anthropogenic waves has remained at the same level as it was in the year 2002, although the ships that produced the highest waves in the past are no longer in service.

The effect of critically moving loads on the vibrations of soft soils and isolated railway tracks

The dynamic response of the railway track is strongly influenced by the underlying soil. For a soft soil and very high train speeds or for a very soft soil and regular train speeds, the train speed can be close to the speed of elastic waves in the soil. This paper presents a detailed study of the so-called “moving-load effect”, i.e. an amplification of the dynamic response due to the load movement, for the tracks on soft soil. The analysis is carried out by evaluating the related integrals in the wavenumber domain. The influence of the load speed is quantified for a large set of parameters, showing that the effect on the soil vibration is reduced with increase of the frequency, track width and inverse wave velocity. Therefore, the moving-load effect associated with vibratory train loads is negligible whereas the amplification associated with the moving dead weight of the train can be significant. The strong moving-load effect on a perfectly homogeneous soil, however, can be strongly diminished by a layered or randomly varying soil situation. This theoretical result is affirmed by measurements at a test site in Germany where the trains run on a very soft soil at a near-critical speed. The results for soft soils are compared with experimental and theoretical results for a stiff soil. It is found that the influence of the stiffness of the soil is much stronger than the moving-load effect. This holds for the soil vibration as well as for the track vibration which both show a minor dependence on the load speed but a considerable dependence on the soil stiffness in theory and experiment. Railway tracks can include soft isolation elements such as rail pads, sleeper shoes and ballast mats. For these types of isolation elements and normal soil conditions, the influence of the load speed is usually negligible. There is only one isolation measure for which the moving load may be effective: a track which is constructed as a heavy mass–spring system. The resonance of this track system is shifted to lower frequencies and amplitudes for increasing train speed. A critical train speed can be reached if the mass–spring system has a marginal bending stiffness along the track.

On a moving interface crack with a contact zone in a piezoelectric bimaterial

An inplane problem for a crack moving with constant subsonic speed along the interface of two piezoelectric materials is considered. A mechanically frictionless and electrically permeable contact zone is assumed at the right crack tip whilst for the open part of the crack both electrically permeable and electrically insulated conditions are considered. In the first case a moving concentrated loading is prescribed at the crack faces and in the second case an additional electrical charge at the crack faces is prescribed as well. The main attention is devoted to electrically permeable crack faces. Introducing a moving coordinate system at the leading crack tip the corresponding inhomogeneous combined Dirichlet–Riemann problem is formulated and solved exactly for this case. All electromechanical characteristics at the interface are presented in a closed form for arbitrary contact zone lengths, and further, the transcendental equation for the determination of the real contact zone length is derived. As a particular case of the obtained solution a semi-infinite crack with a contact zone is considered. The numerical analysis performed for a certain piezoelectric bimaterial showed an essential increase of the contact zone length and the associated stress intensity factor especially for the near-critical speed region. Similar investigations have been performed for an electrically insulated crack and the same behavior of the above mentioned parameters is observed.

Nonlinear mechanism of tsunami wave generation by atmospheric disturbances

Abstract. The problem of tsunami wave generation by variable meteo-conditions is discussed. The simplified linear and nonlinear shallow water models are derived, and their analytical solutions for a basin of constant depth are discussed. The shallow-water model describes well the properties of the generated tsunami waves for all regimes, except the resonance case. The nonlinear-dispersive model based on the forced Korteweg-de Vries equation is developed to describe the resonant mechanism of the tsunami wave generation by the atmospheric disturbances moving with near-critical speed (long wave speed). Some analytical solutions of the nonlinear dispersive model are obtained. They illustrate the different regimes of soliton generation and the focusing of frequency modulated wave packets.

Spatial wave dynamics of steady oblique wave interactions

In several models for waves on water of finite depth, a great variety of surface-elevation profiles for steadily traveling waves can be generated by a spatial evolution law in which the time-like variable is one of the horizontal space variables. This is the case for both linear and weakly nonlinear models, including the Kadomtsev–Petviashvili equation (KP-II). At the linear level, these waves correspond to steady oblique interactions of one-dimensional wave trains. The shape of the waves is determined in a similar way in all cases – by initial data that specifies the wave slope along a single line oriented either parallel or perpendicular to the direction of propagation. As one application we compute a steady two-dimensional wake due to a pressure disturbance moving at near-critical speed in a weakly nonlinear regime. We find that the disturbance can generate solitary waves traveling obliquely in the wake, a phenomenon recently realized experimentally and found to be relevant to the safe operation of high-speed ferries.

Effects of bearing and shaft asymmetries on the instability of rotors operating at near-critical speeds

Information about the stability of vibratory motions becomes essential for ensuring safer designs of rotor-bearing systems and the operational safety. In particular, the influences of shaft and bearing parameters on the stability characteristics have to be quantified for design and diagnostic purposes. Such an analytical investigation is the objective of the present paper. Expressions for amplitude and phase modulation functions that quantify the effects of slowly varying the rotational speed on the motion characteristics are derived. Based on the modulation functions, stability regions in the parameter space are determined. The effects of bearing and shaft asymmetries on the stability of the rotor are illustrated.

Convection Effects for Rapidly Moving Mechanical Sources on a Half-Space Governed by Fully Coupled Thermoelasticity

The two-dimensional steady-state analysis of rapidly moving mechanical sources over the insulated surface of thermoelastic half-space is extended by allowing surface heat convection. An exact transform solution based on the fully coupled (dynamic) equations of thermoelasticity is obtained, and robust asymptotic expressions for the surface displacements and temperature change are extracted. Convection is manifest in these expressions in terms of a positive dimensionless parameter of small magnitude, and results in a surface response that is more complicated than that for the insulated surface. In particular, surface temperature changes decay less with distance from the source, and convection effects can dominate all surface responses at low and near-critical speeds.

A slender ship moving at a near-critical speed in a shallow channel

The problem solved concerns a slender ship moving at a near-critical steady speed in a shallow channel, not necessarily in symmetric configuration, involving the special phenomenon of generation of solitary waves. By using the technique of matched asymptotic expansions along with nonlinear shallow-water wave theory, the problem is reduced to a Kadomtsev–Petviashvili equation in the far field, matched with a nearfield solution obtained by an improved slender-body theory, taking the local wave elevation and longitudinal disturbance velocity into account. The ship can be either fixed or free to squat. Besides wave pattern and wave resistance, the hydrodynamic lift force and trim moment are calculated by pressure integration in the fixed-hull case; running sinkage and trim, by condition of hydrodynamic equilibrium in the free-hull case. The numerical procedure for solving the KP equation consists of a finite-difference method, namely, fractional step algorithm with Crank–Nicolson-like schemes in each half step. Calculated results are compared with several published shipmodel experiments and other theoretical predictions; satisfactory agreement is demonstrated.

An equation of motion for the emission of accelerating dislocations from a crack

Previous transient studies of dislocation emission from cracks have treated constant-speed motions which satisfy a dislocation force-based criterion that is, in effect, associated with micromechanical strain-hardening. This analysis requires, instead, that the force necessary to produce glide away from the crack be equal to, but not exceed, the yield stress level. As an illustration, an exact transient solution for the emission of screw dislocation from a mode III crack subjected to SH-wave diffraction is generated. Invoking the new criterion leads to an equation of motion for the dislocation. The equation shows that the dislocation initiates at an essentially zero speed, accelerates to a near-critical speed, and then decelerates to arrest at a finite distance from the crack edge. Expressions for parameters such as the instant of initiation and the arrest distance are also obtained, and studied as functions of material properties such as yield stress and Burgers vector magnitude.

Radiation of solitons by slender bodies advancing in a shallow channel

It is known from recent experiments that the disturbance due to a slender ship advancing in a shallow channel is essentially one-dimensional in the horizontal plane. In particular solitons can be radiated upstream in a transient manner. In this note we develop a theory for soliton radiation by slender bodies. It is shown that, when the ship speed is in the transcritical range, one-dimensional upstream influence can occur even when the channel width is nearly of the order of the ship length but much greater than the ship beam. The theory is also extended to one or more ships travelling in the same channel at near-critical speeds.

Stability of lateral oscillations of a railway vehicle

This paper, though self-contained, is an extension of previous work by one of the authors1). The lateral motions of a highly simplified model of a railway vehicle on a straight track are studied theoretically. By means of an analytic approach the asymptotic form of the periodic motion for high and for low (near-critical) speeds is developed. The stability of the periodic motion is investigated. It is found that any motion starting with small deviations from the periodic motion approaches the latter in the course of time. This approach is slow for very high speeds and for near-critical speeds, but fast for intermediate speeds.