Fluid Of Finite Depth Research Articles

Modulational instability of a pair of collinear wave trains

A coupled system of three nonlinear evolution equations is derived for a pair of collinear wave packets over finite depth fluid. The wave packets are narrow banded having different carrier wave frequencies. The evolution equations are employed to perform stability analysis of a pair of collinear wave trains. It is observed that the region of instability as well as the growth rate of instability for counter-propagating waves is greater than that for co-propagating waves when everything else remains same. The region of instability increases with the increase in the depth of the medium. It is also found that the growth rate of instability of a wave train increases with the increase in wave-steepness of the second wave train.

Dynamical response of a floating ice sheet due to a forced oscillation in a running stream

This paper explores the response of a floating ice sheet to a forced, time-harmonic oscillatory pressure applied to the ice-covered surface of a uniform, finite depth fluid. The floating ice sheet is modeled as a thin elastic plate following the Euler–Bernoulli beam equation, with the additional consideration of in-plane compressive forces acting on the ice. Employing linear wave theory, the problem is presented as an initial boundary value problem and is solved using Laplace and Fourier transforms to obtain a mathematical expression for the ice sheet's deflection in terms of infinite integrals. These integrals are thereafter solved asymptotically for large time and distance using the stationary phase method. The asymptotic analysis indicates a growing response over time when the poles and stationary points of the phase functions coalesce. The deflection of the floating ice sheet is graphically presented, highlighting the effects of various non-dimensional parameters, such as flexural rigidity, compressive force, uniform current speed, and the angular frequency of the oscillatory pressure. Additionally, the group velocity and phase velocity of flexural gravity waves are derived from the dispersion relation and elucidated using diagrams. The results show that compressive force and current speed significantly influence wave amplitude, enhancing the oscillatory nature, while the flexural rigidity of the elastic plate and the angular frequency of the applied pressure have a substantial impact on the plate's deflection.

Load motion on an ice cover in the presence of a liquid layer with velocity shear

The behavior of an ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a pressure domain that moves rectilinearly at a constant velocity in the presence of a current with velocity shift in the upper layer is studied. It is assumed that the ice deflection is steady in the coordinate system moving with the load. The Fourier transform method is used within the framework of the linear wave theory. The critical velocities, the deflection of ice cover, and the wave forces are studied depending on the current velocity gradient, the shear layer thickness, the direction of motion, and the compression ratio.

Well-posedness of the stationary and slowly traveling wave problems for the free boundary incompressible Navier-Stokes equations

We establish that solitary stationary waves in three dimensional viscous incompressible fluids are a general phenomenon and that every such solution is a vanishing wave-speed limit along a one parameter family of traveling waves. The setting of our result is a horizontally-infinite fluid of finite depth with a flat, rigid bottom and a free boundary top. A constant gravitational field acts normal to bottom, and the free boundary experiences surface tension. In addition to these gravity-capillary effects, we allow for applied stress tensors to act on the free surface region and applied forces to act in the bulk. These are posited to be in either stationary or traveling form.In the absence of any applied stress or force, the system reverts to a quiescent equilibrium; in contrast, when such sources of stress or force are present, stationary or traveling waves are generated. We develop a small data well-posedness theory for this problem by proving that there exists a neighborhood of the origin in stress, force, and wave speed data-space in which we obtain the existence and uniqueness of stationary and traveling wave solutions that depend continuously on the stress-force data, wave speed, and other physical parameters. To the best of our knowledge, this is the first proof of well-posedness of the solitary stationary wave problem and the first continuous embedding of the stationary wave problem into the traveling wave problem. Our techniques are based on vector-valued harmonic analysis, a novel method of indirect symbol calculus, and the implicit function theorem.

H 2-solutions for an Ostrosky–Hunter type equation

The Ostrosky–Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth, for the evolution of nonlinear propagation of optical pulses of a few oscillations duration in dielectric media, and for the evolution of the propagation of ultra-short light pulses in silica optical fibers. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

Wave interaction with multiple thin flexible porous barriers in water of uniform finite depth

The interaction of water waves with multiple flexible porous barriers partially immersed in a uniform finite-depth fluid is investigated in the present study. Using linear theory, the physical problem is mathematically modeled as a boundary value problem. The eigenfunction expansion approach is used to generate the velocity potential function in terms of orthogonal eigenfunctions. The Havelock’s inversion formula is then used to convert the boundary value problem into a set of Fredholm-type integral equations. By choosing a suitable weight function, the multi-term Galerkin’s approximation technique is used to obtain the mathematical solutions of these integral equations in terms of the Chebyshev polynomial. Numerical solutions regarding the reflection and transmission coefficients, hydrodynamic forces, and dissipated wave energy at the barriers are obtained and graphically represented in several figures for various non-dimensional parameters. It is observed from the graphical representations that the permeability and flexibility of the barriers play a crucial role in the modeling of efficient breakwaters.

Analysis of Hydrodynamic Loads by a Vertical Hollow Cylindrical Structure in A Two-Layer Fluid of Finite Depth

This paper aims to study the effect of surge radiation by a vertically hollow cylinder floating in a two-layer fluid of finite depth. Since most of the important characteristics of the oscillating water column are preserved by the hollow cylindrical structure, so the proposed device can be considered as an oscillating water column (OWC) which is one of the important wave energy device. In this two-layer fluid system, the upper layer is bounded above by a free surface and the lower layer is bounded below by a solid impermeable bottom, and the submerged hollow cylindrical structure allows it to oscillate in a surge mode of motion in the upper layer. On the assumption that wave amplitudes are small in comparison to wavelengths, hence the linear water wave theory is applied to formulate boundary value problems by dividing whole domain into 2 sub-domains. The formulated boundary value problems for surge radiated potentials in each sub-domain; we execute to obtain solutions in each sub-domain by using variable separation and matched eigenfunction expansion methods. With the help of obtained surge radiated potentials, we derive the radiation forces by the device in terms of added mass and damping coefficients. A set of influences of different parameters of the device on the added mass and damping coefficients are demonstrated in both surface and internal wave modes of motion. Hydrodynamic coefficients oscillate highly near a particular frequency, which leads to the resonance phenomena. In addition to that, the 3D plots of the free surface elevation in surface and internal wave modes are presented. A density ratio γ = 0.97 has a higher oscillation than a density ratio γ = 0.93, 0.95. High oscillations occur around a particular frequency ω = 3.83 due to the resonance phenomenon in which the waves incoming match with the natural motion of the object. HIGHLIGHTS A floating hollow vertical cylindrical structure is considered in a two-layer fluid system having finite depth The separation of variables and matched eigenfunction expansion methods are utilized to obtain the solutions to the boundary-value problems The diffracted velocity potentials under the surge mode of motion are evaluated in the identified regions The added mass, damping coefficients, free surface and interface elevations in surface and internal wave modes are evaluated with different set of parameters of the device A number of results are presented regarding the effect of parameters of the device on the added mass, damping coefficients, free surface and interface elevations It is observed that the density ratio of the two-layer fluid has significant effect on the added mass, damping coefficient and the elevations GRAPHICAL ABSTRACT

A Heat Transfer Problem with a Velocity Slip in Porous Media: A Galerkin Approach

In this paper a steady flow of a viscous fluid of finite depth in a porous medium over a fixed horizontal, impermeable bottom with a velocity slip is studied. A Galerkin Method of solution is considered to solve Momentum and Energy equations as the truncation errors both in general and special cases may result in complex situations to get the analytical form of solutions. Using the obtained velocity profile, the Temperature, Mean velocity, Mean Temperature, and heat transfer rates (Nusselt Number) on the free surface as well as on the bottom plate are obtained. The effect of velocity slip in the obtained fields is studied wherever possible. A special case of low porosity is studied and the results are illustrated graphically.

Deformations and Wave Forces in the Motion of a Load on an Ice Cover in the Presence of a Current with Velocity Shear

The deformations of an ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a pressure domain that moves rectilinearly at a constant velocity in the presence of a current with velocity shift, as well as the wave forces exerted on a moving body, are studied. Fluid flow is not potential. The ice cover is modeled by a thin elastic plate with account for uniform compression. The motion of the load can occur at an arbitrary angle to the direction of current. It is assumed that the ice deflection is steady in the coordinate system moving with the load. The Fourier transform method is used within the framework of the linear wave theory. The maximum deformations of ice cover and the wave forces are studied depending on the current velocity gradient, the direction of motion, and the compression ratio.

Investigation of acoustic radiation from a sphere vibrating on the free surface of a finite depth water using a boundary element method

In this study, a boundary element method (BEM) is applied to investigate acoustic radiation from a sphere vibrating in pulsating mode on the free surface of finite or infinite depth water. Effect of the free surface is introduced by employing a half-space Green’s function. A modified version of the Helmholtz integral equation (HIE) is used to calculate acoustic radiation from the sphere vibrating in pulsating mode on the free surface. Free-terms of the HIE are calculated using two different forms of integrals and “dummy” boundary elements. Moreover, to simulate finite depth fluid medium, a chain image-source method is used to derive a waveguide Green’s function. To demonstrate applicability of the method presented, calculated acoustic pressures are compared with those by finite element method (FEM) and analytical calculations. Additionally, the effects of submergence depth and vibration frequency on acoustic radiation are investigated for infinitely deep water together with those of water depth and field point distance on acoustic radiation for finite water depth medium. The calculations show that there is a good agreement between BEM, FEM and analytical solutions. Also, it is observed that field point distance significantly affects the convergence behavior of waveguide Green’s function. Furthermore, it is noted that submergence depth, domain depth and vibration frequency have pronounce influence on radiated pressure amplitude and pressure field pattern.

Fully Nonlinear Evolution of Free-Surface Waves with Constant Vorticity under Horizontal Electric Fields

This work presents the results of a direct numerical simulation of the nonlinear free surface evolution of a finite-depth fluid with a linear shear flow under the action of horizontal electric fields. The method of time-dependent conformal transformation for the description of the combined effects of the electric fields and constant vorticity is generalized for the first time. The simulation results show that strong shear flow co-directed in the wave propagation direction leads to the formation of large-amplitude surface waves, and, for some limiting vorticity value, a wave breaking process with the formation of an air bubble in the liquid is possible. The oppositely directed shear flow can cause the retrograde motion of a surface wave (wave propagation in the opposite direction to the linear wave speed). The simulations conducted taking into account the electro-hydrodynamic effects demonstrate that a high enough external horizontal electric field suppresses these strongly nonlinear processes, and the surface waves tend to preserve their shape.

Spatial Dynamics and Solitary Hydroelastic Surface Waves

This paper presents an existence theory for solitary waves at the interface between a thin ice sheet (modelled using the Cosserat theory of hyperelastic shells) and an ideal fluid (of finite depth and in irrotational motion). The theory takes the form of a review of the Kirchgässner reduction to a finite-dimensional Hamiltonian system, highlighting the refinements in the theory over the years and presenting some novel aspects including the use of a higher-order Legendre transformation to formulate the problem as a spatial Hamiltonian system, and a Riesz basis for the phase space to complete the analogy with a dynamical system. The reduced system is to leading order given by the focussing cubic nonlinear Schrödinger equation, agreeing with the result of formal weakly nonlinear theory (which is included for completeness). We give a precise proof of the persistence of two of its homoclinic solutions as solutions to the unapproximated reduced system which correspond to symmetric hydroeleastic solitary waves.

Unsteady free surface flow due to a line sink at an arbitrary location with surface tension

The unsteady flow generated when a line sink is located at an arbitrary location beneath the free surface of a fluid of finite depth is considered. Interest is in the evolution to a steady state flow, splashing flows and the drawdown of the surface. The location of the sink away from, but near to, a vertical barrier leads to some interesting new surface flows due to interactions and reflection, some of which have a profound impact on the nature of the critical drawdown flow parameters. Interesting new solutions with a delayed drawdown that are initiated by a reflected wave are presented. A linearized solution and a fully nonlinear solution are presented and the results discussed. Regions of steady, unsteady and supercritical flow are identified.

Comprehensive Perturbation Approach to Nonlinear Viscous Gravity–Capillary Surface Waves at Arbitrary Wavelengths in Finite Depth

This study presents a comprehensive analysis of the second-order perturbation theory applied to the Navier–Stokes equations governing free surface flows. We focus on gravity–capillary surface waves in incompressible viscous fluids of finite depth over a flat bottom. The amplitude of these waves is regarded as the perturbation parameter. A systematic derivation of a nonlinear-surface-wave equation is presented that fully takes into account dispersion, while nonlinearity is included in the leading order. However, the presence of infinitely many over-damped modes has been neglected and only the two least-damped modes are considered. The new surface-wave equation is formulated in wave-number space rather than real space and nonlinear terms contain convolutions making the equation an integro-differential equation. Some preliminary numerical results are compared with computational-modelling data obtained via open source CFD software OpenFOAM.

Bifurcation Analysis and Propagation Conditions of Free-Surface Waves in Incompressible Viscous Fluids of Finite Depth

Viscous linear surface waves are studied at arbitrary wavelength, layer thickness, viscosity, and surface tension. We find that in shallow enough fluids no surface waves can propagate. This layer thickness is determined for some fluids, water, glycerin, and mercury. Even in any thicker fluid layers, propagation of very short and very long waves is forbidden. When wave propagation is possible, only a single propagating mode exists for a given horizontal wave number. In contrast, there are two types of non-propagating modes. One kind of them exists at all wavelength and material parameters, and there are infinitely many such modes for a given wave number, distinguished by their decay rates. The other kind of non-propagating mode that is less attenuated may appear in zero, one, or two specimens. We notice the presence of two length scales as material parameters, one related to viscosity and the other to surface tension. We consider possible modes for a given material on the parameter plane layer thickness versus wave number and discuss bifurcations among different mode types. Motion of surface particles and time evolution of surface elevation is also studied at various parameters in glycerin, and a great variety of behaviour is found, including counterclockwise surface particle motion and negative group velocity in wave propagation.

Motion of a Load on an Ice Cover in the Presence of a Current with Velocity Shear

The behavior of an ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a pressure domain that moves rectilinearly at a constant velocity in the presence of a current with velocity shift is studied. Fluid flow is not potential. The ice cover is modeled by a thin elastic plate with account for uniform compression. The motion of the load can occur at an arbitrary angle to the direction of current. It is assumed that the ice deflection is steady in the coordinate system moving with the load. The Fourier transform method is used within the framework of the linear wave theory. The critical velocities and the deflection of ice cover are studied depending on the current velocity gradient, the direction of motion, and the compression ratio.

Generation of Two-Dimensional Unsteady Motion in a Viscous Incompressible Fluid of Finite Depth

Generation of Two-Dimensional Unsteady Motion in a Viscous Incompressible Fluid of Finite Depth

Investigation on Dynamic Pressure Characteristics Induced by an Internal Solitary Wave in a Two-Layer Fluid of Finite Depth

In this paper, the dynamic pressure characteristics induced by an internal solitary wave (ISW) in a two-layer fluid of finite depth are studied. A method for detecting ISWs by dynamic pressure is proposed. First, an accurate controllable ISW numerical flume is established based on the applicability conditions of three kinds of ISW theories. Then the relationship between the maximum dynamic pressure and the amplitude of the ISW under four density stratifications is studied. The results show that the dynamic pressure variation induced by the ISW is synchronous with that of interface displacement. Under the same density stratification, the maximum dynamic pressure has a strong linear relationship with the corresponding amplitude. Furthermore, the nonlinear error is less than 7 %, and the relationship is not affected by the ocean structure. According to the above characteristics of dynamic pressure, a series of dynamic pressure measurement experiments were carried out in a large internal wave flume (IWF) using a very low frequency (VLF) piezoelectric sensor. The experimental results show that the dynamic pressure measurement results are in good agreement with the numerical simulation results. Last, the Miyata-Choi-Camassa (MCC) theory is used to invert the measured maximum dynamic pressure to the amplitude of ISW, and compared with the measured amplitude, the maximum error is less than 6 %, which verifies the feasibility of the inversion method of ISW amplitude by using dynamic pressure measurement proposed in this paper, and provides a new method and idea for detecting ISW at sea in the future.

Evolution of Perturbations of a Free Surface From a Pulsating Underwater Source in a Fluid of Finite Depth

Evolution of Perturbations of a Free Surface From a Pulsating Underwater Source in a Fluid of Finite Depth

New solutions for periodic interfacial gravity waves

Two-dimensional periodic interfacial gravity waves travelling between two homogeneous fluids of finite depth are considered. A boundary-integral-equation method coupled with Fourier expansions of the unknown functions is used to obtain highly accurate solutions. Our numerical results show excellent agreement with those already obtained by Maklakov & Sharipov using a different scheme (J. Fluid Mech., vol. 856, 2018, pp. 673–708). We explore the global bifurcation mechanism of periodic interfacial waves and find three types of limiting wave profiles. The new families of solutions appear either as isolated branches or as secondary branches bifurcating from the primary branch of solutions.