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[en] Three different sets of shallow water equations, representing different levels of approximation are considered. The numerical solutions of these different equations for flow past bottom topography in several different flow regimes are compared. For several cases the full Euler solutions are computed as a reference, allowing the assessment of the relative accuracies of the different approximations. Further, the differences between the dispersive shallow water (DSW) solutions and those of the highly simplified, hyperbolic shallow water (SW) equations is studied as a guide to determining what level of approximation is required for a particular flow. First, the Green-Naghdi (GN) equations are derived as a vertically-integrated rational approximation of the Euler equations, and then the generalized Boussinesq (gB) equations are obtained under the further assumption of weak nonlinearity. A series of calculations, each assuming different values of a set of parameters emdash undisturbed upstream Froude number, and the height and width of the obstacle, are then presented and discussed. In almost all regions of the parameter space, the SW and DSW theories yield different results; it is only when the flows are entirely subcritical or entirely supercritical and when the obstacles are very wide compared to the depth of the fluid that the SW and DSW theories are in qualitative and quantitative agreement. It is also found that while the gB solutions are accurate only for small bottom topographies (less than 20% of the undisturbed fluid depth), the GN solutions are accurate for much larger topographies (up to 65% of the undisturbed fluid depth). The limitation of the gB approximation to small topographies is primarily due to the generation of large amplitude upstream propagating solitary waves at transcritical Froude numbers, and is consistent with previous analysis. (Abstract Truncated)
[en] A qualitative appraisal of the effects of floating debris on dam break flood waves is presented. Reference is made to the Johnstown flood of 1879 which killed 23,000 people, and which seems to have been aggravated by large amounts of debris in the flood wave. Water velocities and depths needed for debris generation are estimated by comparison with wind velocities needed to uproot trees, and these velocities are compared with the velocities and depths in typical dambreak flood waves, confirming the debris production potential of such waves. Preliminary experiments were carried out with a horizontal flume and simulated debris. It was found that debris accumulations of the magnitude to be expected following a dam break in a forested or urbanized valley can have a major influence on the nature of the dam break surge. The magnitude of the effect depends mainly on the debris generated or collected by the flood wave and on friction forces between the debris accumulation and valley floor and walls. The debris can decrease the speed of the surge and increase its height relative to the simple water surge, and constrictions can allow the surge to build up again. 5 refs., 2 figs., 1 tab
[en] We investigate the possibility of generating quantum-correlated quasi-particles utilizing analogue gravity systems. The quantumness of these correlations is a key aspect of analogue gravity effects and their presence allows for a clear separation between classical and quantum analogue gravity effects. However, experiments in analogue systems, such as Bose–Einstein condensates (BECs) and shallow water waves, are always conducted at non-ideal conditions, in particular, one is dealing with dispersive media at non-zero temperatures. We analyse the influence of the initial temperature on the entanglement generation in analogue gravity phenomena. We lay out all the necessary steps to calculate the entanglement generated between quasi-particle modes and we analytically derive an upper bound on the maximal temperature at which given modes can still be entangled. We further investigate a mechanism to enhance the quantum correlations. As a particular example, we analyse the robustness of the entanglement creation against thermal noise in a sudden quench of an ideally homogeneous BEC, taking into account the super-sonic dispersion relations. (paper)
[en] This paper introduces a new shallow water wave equation with nonlinear strength. By using peakon bifurcation equation, we show that as the nonlinear-strength parameter varies, this nonlinear-strength equation has many interesting new solutions, which are called tanh-peakon and hyperbolicon because they can be expressed as tanh and hyperbolic functions. Exact expression of these new solution are detailed derived
[en] A modified form of Green's integral theorem is employed to derive the energy identity in any water wave diffraction problem in a single-layer fluid for free-surface boundary condition with higher-order derivatives. For a two-layer fluid with free-surface boundary condition involving higher-order derivatives, two forms of energy identities involving transmission and reflection coefficients for any wave diffraction problem are also derived here by the same method. Based on this modified Green's theorem, hydrodynamic relations such as the energy-conservation principle and modified Haskind-Hanaoka relation are derived for radiation and diffraction problems in a single as well as two-layer fluid.
[en] This paper concerns the calculation of the water particle kinematics generated by the propagation of surface gravity waves. The motivation for this work lies in recent advances in the description of the water surface elevation associated with extreme waves that are highly nonlinear and involve a spread of wave energy in both frequency and direction. To provide these exact numerical descriptions the nonlinear free-surface boundary conditions are time-marched, with the most efficient solutions simply based upon the water surface elevation, η, and the velocity potential, phi, on that surface. In many broad-banded problems, computational efficiency is not merely desirable but absolutely essential to resolve the complex interactions between wave components with widely varying length-scales and different directions of propagation. Although such models have recently been developed, the calculation of the underlying water particle kinematics, based on the surface properties alone, remains a significant obstacle to their practical application. The present paper tackles this problem, outlining a new method based upon an adaptation of an existing approximation to the Dirichlet-Neumann operator. This solution, which is presently formulated for flow over a flat bed, is appropriate to the description of any kinematic quantity and has the over-riding advantage that it is both stable and computationally efficient. Indeed, its only limitation arises from the assumed Fourier series representations. As a result, both η and phi must be single-valued functions and are not therefore appropriate to the description of overturning waves. The proposed method is compared favourably to both existing analytical wave models and laboratory data providing a description of the kinematics beneath extreme, near-breaking, waves with varying directional spread. The paper concludes by investigating two important characteristics of the flow field beneath large waves arising in realistic ocean environments
[en] With a new projective equation, a series of solutions of the (2 + 1)-dimensional dispersive long-water wave system (LWW) is derived. Based on the derived solitary wave solution, we obtain some special fractal localized structures and chaotic patterns. (general)
[en] In this Letter, the band structures and band gaps of liquid surface waves propagating over two-dimensional periodic topography was investigated by plane-waves expansion method. The periodic topography drilled by square hollows with square lattice was considered. And the effects of the filling fraction and the orientation of bottom-hollows on the band gaps are investigated in detail
[en] In this paper, the entangled mapping approach (EMA) is applied to obtain variable separation solutions of (1+1)-dimensional and (3+1)-dimensional systems. By analysis, we firstly find that there also exists a common formula to describe suitable physical fields or potentials for these (1+1)-dimensional models such as coupled integrable dispersionless (CID) and shallow water wave equations. Moreover, we find that the variable separation solution of the (3+1)-dimensional Burgers system satisfies the completely same form as the universal quantity U1 in (2+1)-dimensional systems. The only difference is that the function q is a solution of a constraint equation and p is an arbitrary function of three independent variables.
[en] In this work, an experimental study of spreading of crude oil is carried out in a wave tank. The tests are performed by spilling different volumes and types of crude oil on the water surface. An experimental measurement technique was developed based on digital processing of video images. The acquisition and processing of such images is carried out by using a video camera and inexpensive microcomputer hardware and software. Processing is carried out by first performing a digital image filter, then edge detection is performed on the filtered image data. The final result is a file that contains the coordinates of a polygon that encloses the observed slick for each time step. Different types of filters are actually used in order to adequately separate the color intensifies corresponding to each of the elements in the image. Postprocessing of the vectorized images provides accurate measurements of the slick edge, thus obtaining a complete geometric representation, which is significantly different from simplified considerations of radially symmetric spreading. The spreading of the oil slick was recorded for each of the tests. Results of the experimental study are presented for each spreading regime, and analyzed in terms of the wave parameters such as period and wave height. (author)