Three-dimensional interaction between uniform current and a submerged horizontal cylinder in an ice-covered channel

The problem of interaction of a uniform current with a submerged horizontal circular cylinder in an ice-covered channel is considered. The fluid flow is described by linearized velocity potential theory and the ice sheet is treated as a thin elastic plate. The potential due to a source or the Green function satisfying all boundary conditions apart from that on the body surface is first derived. This can be used to derive the boundary integral equation for a body of arbitrary shape. It can also be used to obtain the solution due to multipoles by differentiating the Green function with its position directly. For a transverse circular cylinder, through distributing multipoles along its centre line, the velocity potential can be written in an infinite series with unknown coefficients, which can be determined from the impermeable condition on a body surface. A major feature here is that different from the free surface problem, or a channel without the ice sheet cover, this problem is fully three-dimensional because of the constraints along the intersection of the ice sheet with the channel wall. It has been also confirmed that there is an infinite number of critical speeds. Whenever the current speed passes a critical value, the force on the body and wave pattern change rapidly, and two more wave components are generated at the far-field. Extensive results are provided for hydroelastic waves and hydrodynamic forces when the ice sheet is under different edge conditions, and the insight of their physical features is discussed.

Hydroelastic waves propagating along a frozen channel with non-uniform thickness of ice

Hydroelastic waves propagating along a channel covered with ice of non-uniform thickness are considered. The channel has a rectangular cross section. The fluid in the channel is inviscid and incompressible. The ice is modeled as a thin elastic plate. The ice thickness changes linearly. The problem is reduced to the problem of the wave profile across the channel, which is solved using the normal modes of an elastic beam with non-uniform thickness. It is shown that with the decrease in the change in the ice thickness, the modes approach the normal modes of an elastic beam with a constant thickness. The behavior of the dispersion relations of the hydroelastic waves depending on the parameter describing the change in the ice thickness is studied.

Estimating the lithospheric flexure of a plate with non-uniform flexural rigidity: a quantitative modelling approach

Lithospheric deformation is a fundamental process in plate tectonics. It is, therefore, critical to determine how the lithosphere responds to geological loads to better understand tectonic processes. The lithosphere can be modelled as the flexure of a thin, elastic plate over long-term (> 105 yr) geological timescales. The partial differential equation for the flexure of an orthotropic plate is used indirectly to calculate theoretical admittance and coherence, which are then compared against the observed admittance and coherence to invert for the non-uniform flexural rigidity (or effective elastic thickness, Te) of the plate. However, the process for accurately recovering variable lithospheric flexure remains unresolved, as the classical lithospheric model may overestimate the deflection of the plate. Here we adopt the classic lithospheric model with applied external and internal loads at the surface and Moho, respectively, and assume that the compensation material is denser than the mantle material beneath the Moho. The lithospheric flexure errors are derived mainly from the Te and Moho recovery errors in this lithospheric model. Synthetic modelling is then performed to analyse the influence of the Te and Moho errors. The analysis of synthetic modelling shows that: (1) the Te error-induced flexure errors exhibit a rippling pattern, and the rippling pattern is broader in high Te regions; (2) the Moho error-induced flexure errors mainly occur in the low Te regions, and applying Airy isostasy theory in low Te regions may still greatly overestimate the lithospheric deformation amplitude; and (3) the lithospheric flexure errors are dominated by the Te and Moho errors in the high and low Te regions, respectively.

Wave dispersion and dissipation in landfast ice: comparison of observations against models

Observations of wave dissipation and dispersion in sea ice are a necessity for the development and validation of wave-ice interaction models. As the composition of the ice layer can be extremely complex, most models treat the ice layer as a continuum with effective, rather than independently measurable, properties. While this provides opportunities to fit the model to observations, it also obscures our understanding of the wave-ice interactive processes, particularly, it hinders our ability to identify under which environmental conditions these processes are of significance. Here, we aimed to reduce the number of free variables available by studying wave dissipation in landfast ice. That is, in continuous sea ice, such as landfast ice, the effective properties of the continuum ice layer should revert to the material properties of the ice. We present observations of wave dispersion and dissipation from a field experiment on landfast ice in the Arctic and Antarctic. Independent laboratory measurements were performed on sea ice cores from a neighbouring fjord in the Arctic to estimate the ice viscosity. Results show that the dispersion of waves in landfast ice is well described by theory of a thin elastic plate and such observations could provide an estimate of the elastic modulus of the ice. Observations of wave dissipation in landfast ice are about an order of magnitude larger than in ice floes and broken ice. Comparison of our observations against models suggests that wave dissipation is attributed to the viscous dissipation within the ice layer for short waves only, whereas turbulence generated through the interactions between the ice and waves is the most likely process for the dissipation of wave energy for long periods. The separation between short and long waves in this context is expected to be determined by the ice thickness through its influence on the lengthening of short waves. Further studies are required to measure turbulence underneath the ice independently of observations of wave attenuation to confirm our interpretation of the results.

Scattering of flexural gravity waves by a pair of submerged vertical porous barriers located above a porous sea-bed

An analytical study is presented to investigate the scattering of oblique flexural gravity waves by a pair of totally submerged vertically placed porous barriers, located at some distance from each other, for a homogenous fluid flowing over a porous sea-bed. A thin ice-sheet, replacing the usual free surface, is considered as the upper surface where it is treated as a thin elastic plate. The complete analytical solution, under the assumption of small-amplitude theory and structural response, is acquired by employing eigenfunction expansion and least square method. Subsequently, computation for the reflection and transmission coefficients, energy loss and wave forces is carried out and discussed for different parameter values corresponding to the ice-sheet, porous sea-bed and porous barriers. This study establishes that the oscillatory behavior exhibited by the reflection of the waves. The vertical porous barriers are found to dissipate a significant portion of the wave energy when an increase in the inertial effect of the porous barriers takes place. The hydrodynamic force on the barriers also follows an oscillatory pattern and it increases when the length of the barrier is increased. It is demonstrated that, corresponding to various structural parameters, almost no reflection and full transmission take place for an impermeable sea-bed and also when only real porosity parameter of the porous sea-bed is considered. Further, variation in the elastic parameter of the floating ice-sheet is observed to command a considerable influence when the wave impinges upon the submerged vertical porous barriers.

The Primary Breaking Pattern of Main Roof Above the First Working Face Based on Main Bending Moment

Scientific mining is based on breaking regularity of roof above underground working face in coal mine. In order to explore the primary breaking pattern of main roof above the first working face, on account of theory of thin elastic plate, development of the breaking in each region of main roof is analyzed and the breaking sequence of each region is explored in virtue of main bending moment taken as the breaking parameter. The results indicate that the first broken point of main roof is midpoint of the long side. The breaking, which occurs on the top surface of main roof, is caused by the second main bending moment. The fracture in long side region starts from midpoint of the long side and develops along the length direction of the working face. The fracture in middle region starts from the center of main roof and develops along the length direction of the working face. The fracture in short side region starts from midpoint of the short side and develops along advance direction of the working face. There always is an extreme value order of control moment in each region, Mc > Mz > Md, when a single parameter is within a reasonable range. Due to this, the breaking sequence is the long side region, the middle region and the short side region although they end up with the same breaking pattern O-X. Mc, Mz and Md depend on the advance distance of working face and increase linearly with transverse loading. Besides, the short side of main roof becomes stable with the increase of the length of working face. Revealing the primary breaking pattern of main roof above the first working face contributes to learning breaking behavior of main roof and providing theoretical support for design of the working face and roof management.

Mechanical analysis of basic roof fracture mechanism and feature in coal mining with partial gangue backfilling

Coal mining with partial gangue backfilling (CMPGB) method has the advantages of both high filling efficiency and excellent workface capacity, which breaks through the technical bottleneck of full-section backfilling mining. In order to reveal the fracture mechanism and characteristics in CMPGB workface, this paper presents a comparative analysis of the filling ratio of different filling patterns in CMPGB. A local composite elastic foundation mechanical model of basic roof in CMPGB was established using thin elastic plate theory. Then, Galerkin’s semi-analytic solution process was designed according to local composite elastic foundation characteristics. A deflection equation of the basic roof was derived, and a critical condition of the basic roof breakage was given. Based on engineering calculation example of Ji15-31010 workface of Pingdingshan No. 12 Coal Mine, the following conclusions are drawn. (1) At the basic roof of caving section, tensile-shear failure occurred in workface, cutoff position, and transition section, while compressive-shear failure occurred in the central part of the goaf. The basic roof showed a typical local “C–X” failure characteristic. (2) The value of first caving span decreased from 32.7 to 31.4 m as the elastic foundation coefficient of backfilling body increased from 70 × 106 to 120 × 106 N/m3, with a decreasing amplitude of only 4.1%. The increase of density of backfilling body only changed the support structure of backfilling section and had an insignificant effect on the first caving span. (3) The value of the first caving span decreased from 59.1 to 21.68 m as the length of caving section increased from 40 to 140 m, indicating that the first caving span was mainly influenced by the length of caving section. The measured value of the first caving span of Ji15-31010 CMPGB workface was 29.8 m, which was close to the theoretical value of mechanical model.

On the Wiener–Hopf solution of water-wave interaction with a submerged elastic or poroelastic plate

A solution to the problem of water-wave scattering by a semi-infinite submerged thin elastic plate, which is either porous or non-porous, is presented using the Wiener–Hopf technique. The derivation of the Wiener–Hopf equation is rather different from that which is used traditionally in water-waves problems, and it leads to the required equations directly. It is also shown how the solution can be computed straightforwardly using Cauchy-type integrals, which avoids the need to find the roots of the highly non-trivial dispersion equations. We illustrate the method with some numerical computations, focusing on the evolution of an incident wave pulse which illustrates the existence of two transmitted waves in the submerged plate system. The effect of the porosity is studied, and it is shown to influence the shorter-wavelength pulse much more strongly than the longer-wavelength pulse.