Action of periodic surface pressure on an ice cover in the vicinity of a vertical wall

This paper presents the solution of the linear hydroelastic problem for steady forced vibrations of a semi-infinite ice cover under the effect of localized external load. The ice cover is simulated by a viscoelastic thin plate, the thickness of the fluid layer is assumed to be small, and the shallow water theory is used. The fluid is limited by a solid vertical wall, and the rectilinear edge of the elastic plate adjacent to the wall can be both free and clamped. The solution is obtained with the help of the Fourier integral transform. The behavior of the ice cover is studied depending on the frequency of the external load and boundary conditions on the edge of the plate. It is shown that, in the case of a free edge of the plate, there are considerable deflections on the edge, which could be comparable with deflections at the center of the pressure impact region. It is established that, due to the existence of wave movements of the type of edge waves, the external load energy is transferred to larger distances along the free edge, and there are significant bending moments on the edge of the clamped plate, which can lead to fracture of the ice cover with sufficiently great intensity of the external load.     

Action of a Local Time-Periodic Load on an Ice Sheet with a Crack

The problem of vibrations of an ice sheet with a rectilinear crack on the surface of an ideal incompressible fluid of finite depth under the action of a time-periodic local load is solved analytically using the Wiener–Hopf technique. Ice cover is simulated by two thin elastic semi-infinite plates of constant thickness. The thickness of the plates may be different on the opposite sides of the crack. Various boundary conditions on the edges of the plates are considered. For the case of contact of plates of the same thickness, a solution in explicit form is obtained. The asymptotics of the deflection of the plates in the far field is studied. It is shown that in the case of contact of two plates of different thickness, predominant directions of wave propagation at an angle to the crack can be identified in the far field. In the case of contact of plates of the same thickness with free edges and with free overlap, an edge waveguide mode propagating along the crack is excited. It is shown that the edge mode propagates with maximum amplitude if the vertical wall is in contact with the plate. Examples of calculations are given.     

Behavior of a Semi-Infinite Ice Cover Under a Uniformly Moving Load

This paper consideres the behavior of a semi-infinite ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a load moving with constant velocity along the edge of the cover at some distance from it. The ice cover is modeled by a thin elastic plate of constant thickness. In a moving coordinate system, the deflection of the plate is assumed to be steady. An analytic solution of the problem is obtained using the Wiener–Hopf technique. The wave forces, the deflection of the plate, and the elevation of the free surface of the fluid at different velocities of the load are investigated.     

Wave generation by an oscillating submerged cylinder in the presence of a floating semi-infinite elastic plate

The results of solving the linear problem of steady-state oscillations of a horizontal cylinder submerged in fluid on whose upper boundary a semi-infinite elastic plate with free edge floats are given. The remaining part of the fluid surface is free. The contour-distributed mass source method is used. The corresponding Green’s function is constructed using eigenfunction expansions. The hydrodynamic load and the amplitudes of vertical displacements of the free surface and elastic plate are calculated. Equivalence relations which demonstrate both symmetry of the apparent mass and damping factors and connection of the damping factors with the wave amplitudes in the far field are derived.     

Heat and Mass Transfer in MHD Micropolar Flow Over a Vertical Moving Porous Plate in a Porous Medium

An analysis is presented for the problem of free convection with mass transfer flow for a micropolar fluid via a porous medium bounded by a semi-infinite vertical porous plate in the presence of a transverse magnetic field. The plate moves with constant velocity in the longitudinal direction, and the free stream velocity follows an exponentially small perturbation law. A uniform magnetic field acts perpendicularly to the porous surface in which absorbs the micropolar fluid with a suction velocity varying with time. Numerical results of velocity distribution of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Also, the results of the skin-friction coefficient, the couple stress coefficient, the rate of the heat and mass transfers at the wall are prepared with various values of fluid properties and flow conditions.     

Calculation of wind currents in the edge zone of a sea ice sheet

The steady wind currents in the ocean near the edge of an ice sheet simulated by a rigid cover are studied. The characteristic horizontal scales of the problem correspond to several tens of kilometers. In this case, depending on the depth and viscosity of the fluid, the effect of the Coriolis force on its motion may be comparable with the effect of the wind shear stresses applied to the ice-free surface. Taking the Coriolis force into account leads to the formation of a spiral flow structure in the vertical direction.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 132–141, November–December, 1992.     

Waves on the Surface of an Ideal Incompressible Heavy Fluid under Wind Loads

The gravity-forced motion of an ideal incompressible fluid of infinite depth is studied when a periodic pressure is applied to the surface of the fluid. This problem is solved on the basis of the small amplitude wave theory. The analytical solutions for the velocity potential, the velocity field, and the shape of the free surface are found. An expression for the horizontal force is obtained in the case of a traveling wave.     

Interaction of surface and flexural-gravity waves in ice cover with a vertical wall

The problem of the interaction of surface and flexural-gravity waves with a vertical barrier is solved in a two-dimensional formulation. It is assumed that the fluid is ideal and incompressible, has infinite depth, and is partially covered with ice. The ice cover is modeled by an elastic plate of constant thickness. The eigenfrequencies and eigenmodes of oscillation of the floating elastic ice plate, the deflection and deformation of ice, and the forces acting on the wall are determined.     

Note on the reflexion of water waves at a nearly vertical cliff in the presence of surface tension

In this note the two-dimensional problem of incoming surface waves against an approximately vertical wall or cliff in water of infinite depth is examined, using velocity potential formulation and linearized boundary-value problem theory for time-harmonic motion. The cliff has arbitrary profile but for simplicity is taken to be vertical at the free surface. The approximate first-order solution is determined subject to a dynamical edge condition apposite to the presence of surface tension, and contains partially reflected outgoing waves. The solution is obtained by perturbation theory in a form involving known unperturbed and first-order correction potentials that is applicable also to water of finite constant depth. The motivation for the note is to point out a correction in principle to the results of a recent investigation for a specific profile, in which reflexion is ignored and another error made in obtaining a first-order solution by a method that is restricted to infinite depth.     

Vibrations of an ice sheet with crack under a time-periodic load

The analytic solution of the problem of vibrations of an ice sheet with a rectilinear crack floating on the surface of an ideal incompressible fluid of shallow depth under the action of a local zone of the time-periodic pressure is obtained. The ice sheet is simulated by two thin viscoelastic semiinfinite plates of different thickness. Various conditions on the crack edges are considered. Far field asymptotics are investigated and it is revealed that the predominant directions of wave propagation at an angle to the crack can be distinguished in the far field in the case of contact of two plates of different thickness. In the case of contact of identical plates, a waveguide mode propagating along the crack is excited. It is shown that the waveguide mode is the same for the plates with the free edges and the free overlap since the part of the solution symmetric about the crack is the same while the difference between the solutions is caused by the antisymmetric part of the solution.     

Transient free convective flow in a vertical channel with constant temperature and constant heat flux on walls

This note presents transient motion of a viscous and incompressible fluid in a vertical channel due to free convective currents occuring as a result of application of constant heat flux at one wall and constant temperature on other wall. The method of Laplace transform is used to solve the problem. The transient behaviour of flow on velocity and temperature fields are shown on the graphs.     

Effects of thermal stratification on flow and heat transfer past a porous vertical stretching surface

The aim of this paper is to present the boundary layer flow of viscous incompressible fluid due to a porous vertical stretching surface with a power-law stretching velocity in a thermally stratified medium. Using a special form of Lie group transformations viz. scaling group of transformations, similarity solutions for this problem are obtained. The equations are then solved numerically. With increasing values of the stratification parameter, the velocity as well as temperature decreases. At a particular point of the porous stretching sheet, the velocity decreases with the increasing suction parameter. The dimensionless temperature at a point of the sheet decreases due to suction but increases due to injection. The findings of this study reveal that stratification and suction can be used as means of cooling the boundary layer flow region.     

Viscous incompressible flow in a narrow channel with one free wall and fluid supply through a porous insert

The problem investigated relates the plane unsteady flow of a viscous incompressible fluid in a narrow channel one of whose walls is free and acted upon by a given load, while the other is rigidly fixed. The fluid enters the channel through a porous insert in the stationary wall. A model of the flow of a thin film of viscous incompressible fluid and Darcy's law for flow in a porous medium are used to find the distribution of fluid pressure and velocity in the channel and the porous insert in the two-dimensional formulation for fairly general boundary conditions in the case where the length of the porous insert exceeds the length of the free wall. In the particular case where the length of the porous insert is equal to the length of the free wall an exact stationary solution of the problem is obtained for a given value of the channel height. The stability of the equilibrium position of the free wall supported on a hydrodynamic fluid film is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–24, January–February, 1986.     

Apparent wall slip velocity measurements in free surface flow of concentrated suspensions

In this work we have experimentally measured the apparent wall slip velocity in open channel flow of neutrally buoyant suspension of non-colloidal particles. The free surface velocity profile was measured using the tool of particle imaging velocimetry (PIV) for two different channels made of plane and rough walls. The rough walled channel prevents wall slip, whereas the plane wall showed significant wall slip due to formation of slip layer. By comparing the velocity profiles from these two cases we were able to determine the apparent wall slip velocity. This method allows characterization of wall slip in suspension of large sized particles which cannot be performed in conventional rheometers. Experiments were carried out for concentrated suspensions of various particle volume concentrations and for two different sizes of particles. It was observed that wall slip velocity increases with particle size and concentration but decreases with increase in the viscosity of suspending fluid. The apparent wall slip velocity coefficients are in qualitative agreement with the earlier measurements. The effect of wall slip on free surface corrugation was also studied by analyzing the power spectral density (PSD) of the refracted light from the free surface. Our results indicate that free surface corrugation is a bulk flow response and it does not arise from boundary problem such as development of slip layer.     

Mixed convection row on vertical surface

Mixed convective heat transfer on a vertical surface has been investigated for uniform free stream velocity and for several values of Prandtl number. An exact similarity solution is obtained for the case of a wall temperature that is inversely proportional to the distance from the leading edge. The non-dimensional temperature distribution suggests that over large η the boundary layer temperature is less than the free stream temperature.     

Nonlinear interaction of surface waves in a basin covered with broken ice

The nonlinear interaction of periodic traveling waves of the first and second harmonics in a constant-depth uniform fluid covered with broken ice is considered. Uniform asymptotic expansions up to third-order values for the velocity potential of the fluid and the elevation of the basin surface are found by means of the multivariable expansion procedure. The dependence of the wave perturbations on the thickness of the ice and the interacting-harmonic characteristics is analyzed. Sevastopol. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 136–143, July–August, 1994.     

Generation of Transient Waves by Impulsive Disturbances in an Inviscid Fluid with an Ice-Cover

The dynamic responses of an ice-covered fluid to impulsive disturbances are analytically investigated for two- and three-dimensional cases. The initially quiescent fluid of infinite depth is assumed to be inviscid, incompressible and homogenous. The thin ice-cover is modelled as a homogenous elastic plate with negligible inertia. Four types of impulsive concentrated disturbances are considered, namely an instantaneous mass source immersed in the fluid, an instantaneously dynamic load on the plate, an initial impulse on the surface of the fluid, and an initial displacement of the ice plate. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the vertical deflexions at the ice-water interface are obtained by means of a joint Laplace-Fourier transform. The asymptotic representations of the wave motions for large time with a fixed distance-to-time ratio are derived by making use of the method of stationary phase. It is found that there exists a minimal group velocity and the wave system observed depends on the moving speed of the observer. For an observer moving with the speed larger than the minimal group velocity, there exist two trains of waves, namely the long gravity waves and the short flexural waves, the latter riding on the former. Moreover, the deflexions of the ice-plate for an observer moving with a speed near the minimal group velocity are expressed in terms of the Airy functions. The effects of the presence of an ice-cover on the resultant wave amplitudes, the wavelengths and periods are discussed in detail. The explicit expressions for the free-surface gravity waves can readily be recovered by the present results as the thickness of ice-plate tends to zero.     

移动载荷作用下浮冰的非线性动力学响应

本文考虑非线性、惯性和阻尼的影响, 研究了任意深度二维理想流体顶部浮冰的振动. 对相关的拟微分算子进行展开并将非线性项保留至三阶后, 完全非线性问题被简化为仅与自由面上的变量相关的三阶截断模型. 为了验证简化模型的准确性, 重点关注了自由孤立波解. 在不考虑阻尼的情况下, 采用多重尺度方法推导了三阶非线性薛定谔方程(NLS), 利用该方程预测了任意水深下原始欧拉方程中自由波包型孤立波解的存在性及三阶截断模型的准确性. 相比于Dinvay等所提出的二阶模型, 三阶截断模型的优势在于其对应的三阶NLS具有准确的非线性项系数, 能够在最小相速度附近更好地模拟冰层的动力学响应. 进一步地对自由孤立波解进行数值计算, 数值结果表明三阶截断模型在分岔曲线和孤立波波形上均与完全欧拉方程吻合良好, 准确性高于二阶截断模型. 基于三阶截断模型, 探究了匀速局域化载荷作用下的浮冰非线性动力学响应并将时间依赖解与实验测量数据进行比较, 数值计算结果与实验记录吻合良好.      

航行体出水破冰的多场耦合效应与相似律

航行体出水破冰中的耦合效应及载荷特征, 是出水冰结构安全性评估的重要依据. 针对航行体出水破冰问题, 通过量纲分析, 获得了影响航行体动载荷及头部应力的主控参数和相似律. 基于LS-DYNA流固耦合计算方法, 得到了航行体在不同冲击速度、冰层厚度、冰层大小条件下的载荷特性. 计算结果表明, 航行体速度越大, 不同冰层对其过载和头部应力的影响差别越大, 这主要是因为航行体速度越大, 通过水介质对不同冰层的前期破坏程度不同. 对于无限大冰层, 当其厚度大于3倍航行体直径时, 航行体穿冰后期呈现稳定侵彻现象, 航行体的过载和头部应力只与航行体的速度和冰的动力学性能相关; 而对于薄冰, 航行体速度越大, 其头部应力反而越小, 这是因为航行体初速度越大, 其通过水的运动对冰的前期冲击破坏越严重, 冰层易开裂上鼓, 所以造成航行体头部应力较小. 对于径向尺寸为6倍航行体直径的碎冰, 当其厚度大于5倍航行体直径时, 碎冰对航行体运动特性的影响和无限大冰层几乎相同; 而当其厚度小于3倍的航行体直径时, 只有在初速度较低时, 碎冰的尺寸效应才可以忽略. 此外, 对比碎冰和无限冰层对航行体运动的影响可以看出, 越厚的冰受前期水的冲击破坏越小, 碎冰和无限冰层的影响规律基本一致; 而较薄的冰在前期水的冲击下破坏严重, 碎冰和无限冰层对航行体运动的影响都较小; 只有中等厚度的冰, 在较高冲击速度下碎冰和无限冰层才表现出径向尺寸效应相关的破坏程度, 如无量纲厚度为3的两种冰在航行体较高初速度40 m/s的条件下前期破坏差别较大, 导致后期对航行体运动特性的影响具有显著差异.      

Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet with prescribed surface heat flux

The similarity solution for the problem of mixed convection boundary layer flow adjacent to a stretching vertical sheet in an incompressible electrically conducting fluid in the presence of a transverse magnetic field is presented. It is assumed that the sheet is stretched with a power-law velocity and is subjected to a variable surface heat flux. The governing partial differential equations are first transformed into a system of non-linear ordinary differential equations, before being solved numerically by the Keller-box method. The numerical results obtained are then compared with previously reported cases available in the literature as well as the series solution for certain values of parameters, to support their validity. The effects of the governing parameters on the flow field and heat transfer characteristics are obtained and discussed.