Critical viscous surface waves over an incline

Surface waves moving at a speed near some critical value on a viscous fluid flow down an incline are studied. An inhomogeneous equation of the Burgers type is derived as a model equation for the long time evolution of the surface waves, when the shear stress on the free surface and the deviation of the uneven bottom from an inclined plane are prescribed. A soliton-like wave and a shock-like front generated ahead of or behind a moving source on the free surface are discovered.     

变深度浅水域中非定常船波

以Green—Naghdi(G—N)方程为基础，采用波动方程／有限元法计算船舶经过变深度浅水域时非定常波浪特性．把运动船舶对水面的扰动作为移动压强直接加在Green-Naghdi方程里，以描述运动船体和水面的相互作用．以Series60 CB＝0．6船为算例，给出自由面坡高，波浪阻力在船舶经过一个水下凸包时变化规律，并与浅水方程的结果进行了比较．计算结果表明，当船舶经过凸包时，波浪阻力先增加，后减少，并逐渐趋于正常．同时发现，当船速小于临界速度时(Fr＝√gh＜1．0)，G—N方程给出的船后尾波比浅水方程的结果明显，波浪阻力也比浅水方程的结果有所提高，频率散射必须考虑．当船速大于临界速度时(Fr＝√gh＞1．0)，G—N方程的计算结果与浅水方程差别不大，频率散射的影响可以忽略．     

A finite element method for two‐dimensional water‐wave problems

In this paper, the authors treat the free‐surface waves generated by a moving disturbance with a constant speed in water of finite and constant depth. Specifically, the case when the disturbance is moving with the critical speed is investigated. The water is assumed inviscid and its motion irrotational. The surface tension is neglected. It is well‐known that the linear theory breaks down when a disturbance is moving with the critical speed. As a remedy to overcome the invalid linear theory, approximate non‐linear theories have been applied with success in the past, i.e. Boussinesq and Korteweg de Vries equations, for example. In the present paper, the authors describe a finite element method applied to the non‐linear water‐wave problems in two dimensions. The present numerical method solves the exact non‐linear formulation in the scope of potential theory without any additional assumptions on the magnitude of the disturbances. The present numerical results are compared with those obtained by other approximate non‐linear theories. Also presented are the discussions on the validity of the existing approximate theories applied to two types of the disturbances, i.e. the bottom bump and the pressure patch on the free‐surface at the critical speed. Copyright © 1999 John Wiley & Sons, Ltd.     

Internal solitary waves moving over the low slopes of topographies

Internal solitary waves moving over uneven bottoms are analyzed based on the reductive perturbation method, in which the amplitude, slope and horizontal lengthscale of a topography on the bottom are of the orders of , ^5/2 and ^−3/2, respectively, where the small parameter is also a measure of the wave amplitude. A free surface condition is adopted at the top of the fluid layer. That condition contains two parameters, δ and Δ, the first of which concerns the discontinuity of the basic density between the outer layer and the inner one; the second concerns the discontinuity of the mean density between them. An amplitude equation for the disturbance of order decomposes into a Korteweg-de Vries (KdV) equation and a system of algebraic equations for a stationary disturbance around a topography on the bottom. Solitary waves moving over a localized hill are studied in a simple case where both the basic flow speed and the Brunt-Vaisalla frequency are constant over the fluid layer. For this case, the expression for the amplitude of the stationary disturbance contains singular points with respect to basic flow speed. These singularities correspond to the resonant conditions modified by the free surface condition. The advancing speeds of solitary waves are changed by the influence of bottom topography, in a case where the long internal waves propagate in the direction opposite to the basic flow, but their waveforms remain almost unchanged.     

Numerical simulation of fully nonlinear irregular wave tank in three dimension

A fully nonlinear irregular wave tank has been developed using a three‐dimensional higher‐order boundary element method (HOBEM) in the time domain. The Laplace equation is solved at each time step by an integral equation method. Based on image theory, a new Green function is applied in the whole fluid domain so that only the incident surface and free surface are discretized for the integral equation. The fully nonlinear free surface boundary conditions are integrated with time to update the wave profile and boundary values on it by a semi‐mixed Eulerian–Lagrangian time marching scheme. The incident waves are generated by feeding analytic forms on the input boundary and a ramp function is introduced at the start of simulation to avoid the initial transient disturbance. The outgoing waves are sufficiently dissipated by using a spatially varying artificial damping on the free surface before they reach the downstream boundary. Numerous numerical simulations of linear and nonlinear waves are performed and the simulated results are compared with the theoretical input waves. Copyright © 2006 John Wiley & Sons, Ltd.     

EFFECTS OF VARYING BOTTOM ON NONLINEAR SURFACE WAVES

The resonant flow of an incompressible, inviscid fluid with surface tension on varying bottoms was researched. The effects of different bottoms on the nonlinear surface waves were analyzed. The waterfall plots of the wave were drawn with Matlab according to the numerical simulation of the fKdV equation with the pseudo-spectral method. Prom the waterfall plots, the results are obtained as follows: for the convex bottom, the waves system can be viewed as a combination of the effects of forward-step forcing and backward step forcing, and these two wave systems respectively radiate upstream and downstream without mutual interaction. Nevertheless, the result for the concave bottom is contrary to the convex one. For some combined bottoms, the wave systems can be considered as the combination of positive forcing and negative forcing.     

Study on the head-on collisions of two-dimensional trains of solitons

The head on collisions of trains of solitons induced by a two-dimensional submerged elliptical cylinder at critical speed in shallow water are studied based on velocity potential theory. The boundary value problems are solved through boundary element method (BEM). The nonlinear free surface boundary conditions are satisfied. The mixed Euler–Lagrangian method is adopted to track the free surface through a time stepping scheme. The effects of thickness and velocity of the elliptical cylinder on the evolution of solitary waves have been investigated. Two sets of solitons are truncated from these trains of solitary waves. The head-on collisions of these solitons have been simulated. The wave profiles and velocity fields during collision have been analysed. The propagation of solitary waves is the transmissions of kinetic energy and the collision processes are the results of the dynamic balance of potential energy and kinematic energy.     

A non-hydrostatic numerical scheme for dispersive waves generated by bottom motion

A numerical scheme based on the staggered finite volume method is presented at the aim of studying surface waves generated by a bottom motion. We address the 2D Euler equations in which the vertical domain is resolved only by one layer. The resulting non-hydrostatic scheme is used to simulate surface waves generated by bottom motion in a water tank. Here we mimic Hammack experiments numerically, in which a bed section is moved upwards or downwards, resulting in transient dispersive waves. For an impulsive downward bottom thrust, free surface responds in terms of a negative leading wave, followed with dispersive train of waves. For an upward bottom thrust, amplitude of the leading wave decays as the wave propagates, and no wave of permanent form evolves— instead, there appears a train of solitons. In this article, we show that our numerical scheme can produce the correct wave profiles, comparable with the analytical and experimental results of Hammack. Simulations using intermediate and slow bottom motions are also presented. In addition, we perform a simulation of a wave generated by submerged landslide, that compares well against previous numerical simulations. Via this simulation, we demonstrate that our scheme can incorporate a moving wet–dry boundary algorithm in the run-up simulation.     

Swept-Wing Boundary Layer Receptivity to a Steady Free-Stream Vortex Disturbance

A single trailing vortex developed behind a micro-wing immersed in a free stream was used to study the vortex receptivity of a swept-wing boundary layer. As a result of the interaction, longitudinal-velocity disturbances develop in the boundary layer. On the swept wing, disturbance transformation occurs near the leading edge and is accompanied by the formation of a wave packet consisting of waves typical of cross-flow instability. Disturbances with other characteristics are also detected. These disturbances may be attributable to distributed boundary-layer receptivity to the free-stream vortex disturbance considered.     

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.      

A study of the effect of step excrescences and free‐stream disturbances on boundary layer stability

In this work, a study of the mechanism by which free‐stream acoustic and vorticity disturbances interact with a boundary layer flow developing over a flat plate featuring a step excrescence located at a certain distance from a blunt leading edge is included. The numerical tool is a high‐fidelity implicit numerical algorithm solving for the unsteady, compressible form of the Navier–Stokes equations in a body‐fitted curvilinear coordinates and employing high‐accurate compact differencing schemes with Pade‐type filters. Acoustic and vorticity waves are generated using a source term in the momentum and energy equations, as opposed to using inflow boundary conditions, to avoid spurious waves that may propagate from boundaries. The results show that the receptivity to surface step excrescences is largely the result of an overall adverse pressure gradient posed by the step, and that the free‐stream disturbances accelerate the generation of instabilities in the downstream. As expected, it is found that the acoustic disturbance interacting with the surface imperfection is more efficient in exciting the Tollmien–Schlichting waves than the vorticity disturbance. The latter generates Tollmien–Schlichting waves that are grouped in wave packets consistent with the wavelength of the free‐stream disturbance. Copyright © 2015 John Wiley & Sons, Ltd.     

Solitary waves on the surface of a viscoelastic fluid running down an inclined plane

In this paper, we study the existence and the role of solitary waves in the finite amplitude instability of a layer of a second-order fluid flowing down an inclined plane. The layer becomes unstable for disturbances of large wavelength for a critical value of Reynolds number which decreases with increase in the viscoelastic parameter M. The long-term evolution of a disturbance with an initial cosinusoidal profile as a result of this instability reveals the existence of a train of solitary waves propagating on the free surface. A novel result of this study is that the number of solitary waves decreases with in crease in M. When surface tension is large, we use dynamical system theory to describe solitary waves in a moving frame by homoclinic trajectories of an associated ordinary differential equation.     

Three-dimensional problem of exponentially stratified flow over bottom roughness in the case of finite depth

The three-dimensional problem of the flow of an exponentially stratified fluid of finite depth over bottom roughness is considered in the rigid roof approximation and in the presence of a free surface. In the rigid roof approximation the solution is obtained in the form of a Fourier series in the vertical Lagrangian coordinate, and the series coefficients are expressed in terms of single integrals outside a horizontal strip whose sides are parallel to the flow axis and tangential to the projection of the support of the function describing the bottom roughness. This makes it possible to investigate the near field in regions not considered in [1, 2]. The presence of a small parameter in the boundary condition at the free surface makes it possible to find, in the first approximation, the wave motions and nonwave disturbances at the free surface in the near and far fields. In the near field the width of the wave zone is of the order of the flow depth, expands with distance from the bottom and is broadest at the free surface. As distinct from the annular disturbances within the fluid, the pattern of the nonwave disturbances at the free surface depends on the polar angle. The law of similarity for the diverging waves at the free surface is also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 101–111, May–June, 1990.The authors are grateful to É. V. Teodorovich for discussing the formulation of the problem.     

Interaction between an unsteady pressure disturbance and a hypersonic boundary layer

The development of disturbances in a hypersonic boundary layer on a cooled surface is investigated in the case in which the characteristic velocity of disturbance propagation is small but greater than the flow velocity in the wall region of the three-layer disturbed zone with interaction. The nonlinear boundary value problem formulated involves a single similarity parameter that characterizes the contribution made by the main, on average either subsonic or supersonic, region of the boundary layer to the generation of the pressure disturbance. In the linear approximation, an analytical solution and an algebraic dispersion equation are derived. It is shown that only waves exponential in time and in the streamwise coordinate can propagate downstream when themain region of the undisturbed boundary layer is subsonic on average.     

Global volume conservation in unsteady free surface flows with energy absorbing far‐end boundaries

A wave absorption filter for the far‐end boundary of semi‐infinite large reservoirs is developed for numerical simulation of unsteady free surface flows. Mathematical model is based on finite volume solution of the Navier–Stokes equations and depth‐integrated continuity equation to track the free surface. The Sommerfeld boundary condition is applied at the far‐end of the truncated computational domain. A dissipation zone is formed by applying artificial pressure on water surface to dissipate the kinetic energy of the outgoing waves. The computational scheme is tested to verify the conservation of total fluid volume in the domain for long simulation durations. Combination of the Sommerfeld boundary and dissipation zone can effectively minimize reflections and prevent cumulative changes in total fluid volume in the domain. Solitary wave, nonlinear periodic waves and irregular waves are simulated to illustrate the numerical developments. Earthquake excited surface waves and nonlinear hydrodynamic pressures in a dam–reservoir are computed. Copyright © 2009 John Wiley & Sons, Ltd.     

Three-dimensional desingularized boundary integral methods for potential problems

The concept of desingularization in three-dimensional boundary integral computations is re-examined. The boundary integral equation is desingularized by moving the singular points away from the boundary and outside the problem domain. We show that the desingularization gives better solutions to several problems. As a result of desingularization, the surface integrals can be evaluated by simpler techniques, speeding up the computation. The effects of the desingularization distance on the solution and the condition of the resulting system of algebraic equations are studied for both direct and indirect versions of the boundary integral method. Computations show that a broad range of desingularization distances gives accurate solutions with significant savings in the computation time. The desingularization distance must be carefully linked to the mesh size to avoid problems with uniqueness and ill-conditioning. As an example, the desingularized indirect approach is tested on unsteady non-linear three-dimensional gravity waves generated by a moving submerged disturbance; minimal computational difficulties are encountered at the truncated boundary.     

Waves in microstructured solids and the Boussinesq paradigm

The emergence of soliton trains and interaction of solitons are analyzed by using a Boussinesq-type equation which describes the propagation of bi-directional deformation waves in microstructured solids. The governing equation in the one-dimensional setting is based on the Mindlin model. This model includes scale parameters which show explicitly the influence of the microstructure in wave motion. As a result the governing equation has a hierarchical structure. The analysis is based on numerical simulation using the pseudospectral method. It is shown how the number of solitons in emerging trains depends on the initial excitation. The head-on collision of emerged solitons is not fully elastic due to radiation but the solitons preserve their identity after collision and the speed of solitons is retained while the radiation keeps a certain mean value. That is why we have kept through this paper the notion of solitons.     

Regime diagram of the development of long-wave near-wall disturbances in a hypersonic boundary layer

A regime diagram of the development of slow near-wall disturbances induced by an unsteady self-induced pressure perturbation in a hypersonic boundary layer is constructed for a disturbance wavelength greater than the boundary layer thickness. It is shown that the main factors shaping the perturbed flow are the gas enthalpy near the body surface, the intensity of the viscous-inviscid interaction, and the nature (sub- or supersonic) of the main part of the boundary layer. Nonlinear boundary-value problems are formulated for regimes in which the near-wall boundary layer region plays a decisive role. Numerical and analytical solutions are obtained in the linear approximation. It is shown that intensification of the viscous-inviscid interaction or an increase in the role of the supersonic main region of the boundary layer impart generally supersonic properties to the main part of the boundary layer, i.e. the upstream propagation of the disturbances is damped and the disturbance growth downstream becomes more intense. Damping of the viscous-inviscid interaction and an increase in the role of the subsonic main part of the boundary layer have the opposite effect. Surface cooling increases the effect of the main part of the boundary layer on the formation of pressure disturbances and surface heating leads to an increase in the effect of the near-wall boundary layer region. It is also shown that for the regimes considered disturbances propagate in a direction opposite to that of the free stream from the turbulent flow region located downstream of the local disturbance development region.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 59–71. Original Russian Text Copyright © 2004 by Bogolepov and Neiland.     

Non-linear free surface flows and an application of the Orlanski boundary condition

The focus of this paper is the analysis of spatially two-dimensional non-linear free surface problems. The critical aspects of the problem concern the treatment of the non-linear free surface, the body boundary condition for large motions and the imposition of suitable radiation conditions. To address such complexities, time domain simulation was chosen as the method of analysis. With the use of a finite domain for simulation, a major concern is with the radiation condition to be applied at the open or truncation boundary. For the two-dimensional problem at hand, no theoretical radiation conditions are known to exist. An extension of the Orlanski open boundary condition, based on phase velocity determination at the free surface, is proposed. Three categories of problems were analysed using numerical simulation-namely, freely moving steep waves, waves over a submerged body and forced body motion. Simulation results have been compared with linear theory and experiments.