In this paper, the propagation of water surface waves over one-dimensional periodic and random bottoms is investigated by the transfer matrix method. For the periodic bottoms, the band structure is calculated, and the results are compared to the transmission results. When the bottoms are randomized, the Anderson localization phenomenon is observed. The theory has been applied to an existing experiment (Belzons, et al., J. Fluid Mech. {\bf 186}, 530 (1988)). In general, the results are compared favorably with the experimental observation.
In this paper, the phenomenon of band gaps and Anderson localization of water waves over one-dimensional periodic and random bottoms is investigated by the transfer matrix method. The results indicate that the range of localization in random bottoms can be coincident with the band gaps for the corresponding periodic bottoms. Inside the gap or localization regime, a collective behavior of water waves appears. The results are also compared with acoustic and optical situations.
Here I present a general formulation of water wave propagation and scattering over topographical bottoms. A simple equation is found and is compared with existing theories. As an application, the theory is extended to the case of water waves in a column with many cylindrical steps.
The propagation of surface water waves over rough topographical bottoms is investigated by the multiple scattering theory. It is shown that the waves can be localized spatially through the process of multiple scattering and wave interference, a peculiar wave phenomenon which has been previously discussed for frozen light in optical systems (S. John, Nature {\bf 390}, 661, (1997)). We demonstrate that when frozen, the transmission of the waves falls off exponentially, and a cooperative behavior appears, fully supporting previous predictions. A phase diagram method is used to illustrate this dis…
The far field asymptotic of internal waves is constructed for the case when a point source of mass moves in a layer of arbitrarily stratified fluid with slowly varying bottom. The solutions obtained describe the far field both near the wave fronts of each individual mode and away from the wave fronts and are expansions in Airy or Fresnel waves with the argument determined from the solution of the corresponding eikonal equation. The amplitude of the wave field is determined from the energy conservation law along the ray tube. For model distributions of the bottom shape and the stratification de…
We consider steady gravity-driven flow of a thin layer of viscous fluid over a curved substrate. The substrate has topographical variations (`bumps') on a large scale compared to the layer thickness. Using lubrication theory, we find the velocity field in generalized curvilinear coordinates. We correct the velocity field so as to satisfy kinematic constraints, which is essential to avoid particles escaping the fluid when computing their trajectories. We then investigate the particle transport properties of flows over substrates with translational symmetry, where chaotic motion is precluded. Th…
Coupled mixed convective and stratified systems are common in natural flows. To study experimentally the associated dynamics, we use a singular property of water: its non-linear equation of state is characterised by a maximum density close to $4^{\circ}$C. By heating the top of a tank at $35^{\circ}$C and cooling the bottom at $0^{\circ}$C, a two-layer configuration spontaneously appears. The convective motion in the bottom layer consists mostly of a large-scale circulation and rising cold plumes. This turbulent flow generates propagating internal gravity waves in the upper stably-stratified l…
A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types taking into account the effect of the slowly varying bottom. The arising KdV equation with variable coefficients is studied numerically when the initial condition is in the form of the one soliton solution …
We present the results of hydrodynamic simulations using the method of multi-particle collision dynamics for a system of squirmer microswimmers moving under the influence of gravity at low Reynolds numbers. In addition, the squirmers are bottom-heavy so that they experience a torque which aligns them along the vertical. The squirmers interact hydrodynamically by the flow fields of a stokeslet and rotlet, which are initiated by the acting gravitational force and torque, respectively, and by their own flow fields. By varying the ratio of swimming to bulk sedimentation velocity and the torque, we…
In this paper we discuss a theoretical model for the interfacial profiles of progressive non-linear waves which result from introducing a triangular obstacle, of finite height, attached to the bottom below the flow of a stratified, ideal, two layer fluid, bounded from above by a rigid boundary. The derived equations are solved by using a nonlinear perturbation method. The dependence of the interfacial profile on the triangular obstacle size, as well as its dependence on some flow parameters, such as the ratios of depths and densities of the two fluids, have been studied.
Convection over a wavy heated bottom wall in the air flow has been studied in experiments with the Rayleigh number $\sim 10^8$. It is shown that the mean temperature gradient in the flow core inside a large-scale circulation is directed upward, that corresponds to the stably stratified flow. In the experiments with a wavy heated bottom wall, we detect large-scale standing internal gravity waves excited in the regions with the stably stratified flow. The wavelength and the period of these waves are much larger than the turbulent spatial and time scales, respectively. In particular, the frequenc…
A formulation of the shallow water equations adapted to general complex terrains is proposed. Its derivation starts from the observation that the typical approach of depth integrating the Navier-Stokes equations along the direction of gravity forces is not exact in the general case of a tilted curved bottom. We claim that an integration path that better adapts to the shallow water hypotheses follows the "cross-flow" surface, i.e., a surface that is normal to the velocity field at any point of the domain. Because of the implicitness of this definition, we approximate this "cross-flow" path by p…
Here we suggest an alternative understanding of the surface gravity wave propagation mechanism based on the baroclinic torque, which operates to translate the interfacial vorticity anomalies at the air-water interface. We demonstrate how the non-Boussinesq term of the baroclinic torque acts against the Boussinesq one to hinder wave propagation. By standard vorticity inversion and mirror imaging, we then show how the existence of the bottom boundary affects the two types of torque. Since the opposing non-Boussinesq torque results solely from the mirror image, it vanishes in the deep water limit…
We present explicit solutions for the ordinary differential equations system describing the motion of the particles beneath small-amplitude capillary-gravity waves which propagate on the surface of an irrotational water flow with a flat bottom. The required computations involve elliptic integrals of first kind, the Legendre normal form and a solvable Abel differential equation of the second kind.
Pedal wavemakers that generate surface gravity waves through bed orbital motion have been shown to produce particle-excursion patterns that mimic deep-water wave behaviour but in finite-depth channels. In this article, we report that gravity waves in a general viscous fluid can resonate through the action of pedal wavemakers. We analyse the linear response of waves in an infinite channel in terms of the displacement amplitude, frequency, and wavelength of the bottom action. We show that the system behaves as a long-pass filter in space and a high-pass filter in time with a sharp resonance affe…
We study the evolution of nonlinear surface gravity water-wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrödinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework …
A compact and efficient numerical method is described for studying plane flows of an ideal fluid with a smooth free boundary over a curved and nonuniformly moving bottom. Exact equations of motion in terms of the so-called conformal variables are used. In addition to the previously known applications for shear flows with constant (including zero) vorticity, here a generalization is made to the case of potential flows in uniformly rotating coordinate systems, where centrifugal and Coriolis forces are added to the gravity force. A brief review is given of previous results obtained by this method…
Active microswimmers are known to affect the macroscopic viscosity of suspensions in a more complex manner than passive particles. For puller-like microswimmers an increase in the viscosity has been observed. It has been suggested that the persistence of the orientation of the microswimmers hinders the rotation that is normally caused by the vorticity. It was previously shown that some sorts of algaes are bottom-heavy swimmers, i.e. their centre of mass is not located in the centre of the body. In this way, the algae affects the vorticity of the flow when it is perpendicular oriented to the ax…
We revisit the classical but as yet unresolved problem of predicting the breaking onset of 2D and 3D irrotational gravity water waves. This study focuses on domains with flat bottom topography and conditions ranging from deep to intermediate depth (depth to wavelength ratio from 1 to 0.2). Our calculations based on a fully nonlinear boundary element model investigated geometric, kinematic and energetic differences between maximally recurrent and marginally breaking waves in focusing wave groups. Maximally steep non-breaking (maximally recurrent) waves are clearly separated from marginally brea…
An initial value problem of the one-dimensional nonlinear Schrödinger (NLS) equation with constant dispersive and nonlinear coefficients can be solved using a compact finite difference scheme (Xie, Li, & Yi, 2009). A similar scheme is implemented in the signalling problem of the one-dimensional NLS equation with constant coefficient where it describes the propagation of surface gravity wave packet over a flat bottom. Various examples are illustrated. A similar compact finite difference scheme is modified and implemented further in the signalling problem of the one-dimensional NLS equation with…
We consider stability of an elastic membrane being on the bottom of a uniform horizontal flow of an inviscid and incompressible fluid of finite depth with free surface. The membrane is simply supported at the leading and the trailing edges which attach it to the two parts of the horizontal rigid floor. The membrane has an infinite span in the direction perpendicular to the direction of the flow and a finite length in the direction of the flow. For the membrane of infinite length we derive a full dispersion relation that is valid for arbitrary depth of the fluid layer and find conditions for th…
We consider the high-order nonlinear Schrödinger equation derived earlier by Sedletsky [Ukr. J. Phys. 48(1), 82 (2003)] for the first-harmonic envelope of slowly modulated gravity waves on the surface of finite-depth irrotational, inviscid, and incompressible fluid with flat bottom. This equation takes into account the third-order dispersion and cubic nonlinear dispersive terms. We rewrite this equation in dimensionless form featuring only one dimensionless parameter $kh$, where $k$ is the carrier wavenumber and $h$ is the undisturbed fluid depth. We show that one-soliton solutions of the clas…
It was predicted that the spin precession frequency of a stationary gyroscope shows various anomalies in the strong gravity regime if its orbit shrinks, and eventually its precession frequency becomes arbitrarily high very close to the horizon of a rotating black hole. Considering the gravity waves of a flowing fluid with vortex in a shallow basin, that acts as a rotating analogue black hole, one can observe the predicted strong gravity effect on the spin precession in the laboratory. Attaching a thread with the buoyant particles and anchored it to the bottom of the fluid container with a shor…
