Plane Bottom Research Articles

A Nonlinear Schrеdinger Equation for Gravity-Capillary Waves on Deep Water with Constant Vorticit

The surface gravity-capillary waves on deep water with constant vorticity in the regionbounded by the free surface and the infinitely deep plane bottom are considered. A nonlinear Schrödinger equation is derived from a system of exact nonlinear integro-differential equations in conformal variables written in the implicit form taking into account surface tension. In deriving the nonlinear Schrödinger equation, the role of the mean flow is taken into account. The nonlinear Schrödinger equation is investigated for modulation instability. A soliton solution of the nonlinear Schrödinger equation that represents a soliton of the “ninth wave” type is obtained. bounded by the free surface and the infinitely deep plane bottom are considered. A nonlinear Schrödinger equation is derived from a system of exact nonlinear integro-differential equations in conformalvariables written in the implicit form taking into account surface tension. In deriving the nonlinear Schrödinger equation, the role of the mean flow is taken into account. The nonlinear Schrödinger equation is investigated for modulation instability. A soliton solution of the nonlinear Schrödinger equation that represents a soliton of the “ninth wave” type is obtained.

The combined refraction–diffraction effect on water wave scattering by a vertical flexible–porous structure

This article investigates the combined effect of refraction–diffraction due to bottom topography on water wave scattering by a vertical flexible–porous structure. This model problem assumes two partial configurations of the structure, such as surface-piercing and bottom-standing positions. Wave scattering is studied under the assumptions of small amplitude water wave theory and Darcy law for flow through the porous medium. The solution method uses Eigenfunction expansion of velocity potentials in the uniform bottom region and the Galerkin-Eigenfunction expansion in the variable bottom region. The structure deflection is obtained by Green’s function technique. From the porous boundary condition, a Fredholm integral equation with Green’s function as the kernel is derived, and its solution is approximated by Galerkin expansion. The associated boundary value problem is converted to a system of algebraic equations. The variations of reflection and transmission coefficients of incident waves are illustrated with parameters related to waves and flexible structure. The flexible structure effect on Bragg resonance is explored in the case of a rippled bottom. Furthermore, the effect of depth ratio on wave scattering is studied in the case of a sloping plane step-type bottom. Some striking results are derived from this model. Bragg resonance reduces with increasing thickness of flexible structure and friction of porous medium at Bragg primary frequency, whereas it increases at other frequencies. On the other hand, the resonance enhances when plate flexibility decreases and porosity increases. Further, Bragg resonance magnifies with a wide frequency band while shifting towards the left at higher incident angles. In this situation, structure deflection increases with the increased incident wave angle and diminishes for angles greater than a particular incident angle. Structure deflection increases with decreasing water-depth ratio in the presence of sloping plane bottom.

Numerical analysis and optimization of rectangular texture for gas foil thrust bearing

Accurate surface texture design is of great significance to improve the performances of gas bearings. In this article, the finite difference method and Newton’s method were combined to obtain the oil film pressure distribution, and the effect of rectangular groove on the lubrication performance was analyzed by changing representative texture parameters. The results show that the performances were more affected by the rectangular texture size compared with the distribution and the bottom shape of texture. The increasing of the surface texture number can only enhance the stability of the bearing. The bearing can provide 30% more bearing capacity by choosing larger size, radial arrangement and plane bottom. These results and analysis can provide technical reference for the bearing texture design.

Fluid Oscillations in Cylindrical Tanks with Longitudinal Damping Partitions

An asymptotic method for investigating small oscillations of fluids in cylindrical tanks with longitudinal damping partitions is presented. The effect of the number and width of partitions on the hydrodynamic parameters and the damping coefficients of fluid oscillations is studied in detail for the case of a circular cylindrical enclosure with a plane bottom. The numerical results are obtained using the finite element method. Basing on the asymptotic theory of vortex resistance a perturbation method applicable for determining the fluid oscillation damping in tanks of arbitrary shape with partitions of small relative width is developed.

Effect of electric field induced strain for double-well ferroelectrics

Based on the Devonshire-Laudon phenomenological theory, the effects of polarization due to dipole stretching and electric hysteresis loop and strain due to dipole’s 90° turning in tetrogonal phase by appling electric field are investigated with the statistic method. The numerical simulation method is used to exam the influences of material parameters, coupling coefficient of dipole, and temperature on hysteresis loop and electrostriction. The significative results are obtained: the lareger the ratio of the two material parameters, α 0/ β , the height of electric hysteresis loop is, and the broader the plane bottom of electrostriction is; the larger the coupling coefficient is, the wider the butterfly shaped strain loop is; when temperature is close to the Curie temperature, the separation of two electrostriction loops is farther at the intersection point. The expirical law is confirmed that the product of electrostrictive coefficient and the Curie-Weiss constant C is a constant. The above principles can explain explicitly the relevant experimental results of electrostriction.

Simulation of surface gravity waves in the Dyachenko variables on the free boundary of flow with constant vorticity

Waves in deep water with constant vorticity in the region bounded by the free surface and the infinitely deep plane bottom are considered. Using the conformal variables and the conformal transform technique, a system of exact integro-differential equations solved relative to the derivatives with respect to time is derived and the equivalent system of equations is obtained in the Dyachenko variables. The efficiency of using the obtained system in the Dyachenko variables for investigating surface wave dynamics on the current of infinite depth with constant vorticity is demonstrated with reference to numerical experiments.

Laminar flow past the bottom with obstacles – a suspension approximation

Abstract From Albert Einstein’s study (1905) it is known that suspension introduced to a fluid modifies its viscosity. We propose to describe the influence of obstacles on the Stokesian flow as a such modification. Hence, we treat the fluid flow through small obstacles as a flow with suspension. The flow is developing past the plane bottom under the gravity force. The spatial distribution of suspension concentration is treated as given, and is regarded as an approximation of different obstacles which modify the fluid flow and change its viscosity. The different densities of suspension are considered, beginning of small suspension concentration until 40%. The influence of suspension concentration on fluid viscosity is analyzed, and Brinkman’s formula as fitting best to experimental data is applied.

Solitary waves in turbulent open-channel flow

AbstractTwo-dimensional turbulent free-surface flow is considered. The ensemble-averaged flow quantities may depend on time. The slope of the plane bottom of the channel is assumed to be small. The roughness of the bottom is allowed to vary with the space coordinate, leading to small variations in the bottom friction coefficient. An asymptotic analysis, which is free of turbulence modelling, is performed for large Reynolds numbers and Froude numbers close to the critical value 1. As a result, an extended Korteweg–deVries (KdV) equation for the surface elevation is obtained. Other flow quantities, such as pressure, flow velocity components, and bottom shear stress, are expressed in terms of the surface elevation. The steady-state version of the extended KdV equation has eigensolutions that describe stationary solitary waves. Time-dependent solutions of the extended KdV equation provide a means for discriminating between stable and unstable stationary solitary waves. Solutions of initial value problems show that there are transient solutions that approach asymptotically the stable stationary solitary wave, whereas other transient solutions decay asymptotically with increasing time.

Impact of a body with a plane bottom on a thin liquid layer at a small angle

The problem of the impact of a body with a plane bottom (of the type of a box) on a thin liquid layer at a small angle is solved in the two-dimensional formulation. The nonlinear shallow water equations are used, together with the method of matched asymptotic expansions. It is found that at certain values of the input parameters of the problem the liquid pressure diminishes near the lower end of the body and becomes smaller than the atmospheric pressure, which results in liquid separation from the box bottom. The numerical results show that all input parameters of the problem have a considerable effect on the nature of body motion. The liquid separation effect on body motion is analyzed.

Augmented Lagrangian for shallow viscoplastic flow with topography

In this paper we have developed a robust numerical algorithm for the visco-plastic Saint-Venant model with topography. For the time discretization an implicit (backward) Euler scheme was used. To solve the resulting nonlinear equations, a four steps iterative algorithm was proposed. To handle the non-differentiability of the plastic terms an iterative decomposition-coordination formulation coupled with the augmented Lagrangian method was adopted. The proposed algorithm is consistent, i.e. if the convergence is achieved then the iterative solution satisfies the nonlinear system at each time iteration. The equations for the velocity field are discretized using the finite element method, while a discontinuous Galerkin method, with an upwind choice of the flux, is adopted for solving the hyperbolic equations that describe the evolution of the thickness. The algorithm permits to solve alternatively, at each iteration, the equations for the velocity field and for the thickness.The iterative decomposition coordination formulation coupled with the augmented Lagrangian method works very well and no instabilities are present. The proposed algorithm has a very good convergence rate, with the exception of large Reynolds numbers (Re≫1000), not involved in the applications concerned by the shallow viscoplastic model. The discontinuous Galerkin technique assure the mass conservation of the shallow system. The model has the exact C-property for a plane bottom and an asymptotic C-property for a general topography.Some boundary value problems were selected to analyze the robustness of the numerical algorithm and the predictive capabilities of the mechanical model. The comparison with an exact rigid flow solution illustrates the accuracy of the numerical scheme in handling the non-differentiability of the plastic terms. The influence of the mesh and of the time step are investigated for the flow of a Bingham fluid in a talweg. The role of the material cohesion in stopping a viscoplastic avalanche on a talweg with barrier was analyzed. Finally, the capacities of the model to describe the flow of a Bingham fluid on a valley from the broken wall of a reservoir situated upstream were investigated.

Visualization of In Situ Oxidation Process Between Plasma and Liquid Phase in Two Dielectric Barrier Discharge Plasma Reactors Using Planar Laser Induced Fluorescence Technique

Cold atmospheric plasma is considered to be a promising approach for decontamination purposes, e.g. dyeing water decoloration. In order to better understand the complex mechanism of the plasma physics coupled with the plasma chemistry involved in the interaction of the polluted water with the discharge plasma, a novel approach was proposed to study the in situ oxidation process between the plasma and liquid phase in two dielectric barrier discharge (DBD) plasma reactors with different bottom shape (concave vs. plane), by using the planar laser induced fluorescence technique to visualize the process dynamics. Rhodamine B was employed as the tracer dye, which was gradually decomposed by the combined effect of the chemically active radicals (OH, O, H2O2, etc.) as well as the intense UV radiation in the DBD plasma process. The results showed that the DBD plasma filaments induced certain fluctuation on the Rhodamine B liquid layer, which accordingly intensified the mass transfer to a large extent thus accelerated the oxidation process. The comparison of the measured concentration fields in the two DBD plasma reactors illustrated that the DBD reactor #1 with concave bottom showed higher oxidation efficiency than the DBD reactor #2 with plane bottom. Additionally, the experiments demonstrated that the oxidation efficiency in the DBD plasma water treatment was much better than that in the reactor with pure oxidation by ozone gas, which can be further improved by injecting the additional oxygen gas bubbles into the liquid phase in the plasma reactor.

On the Traveller Wave Propagation Phenomenon

In the first part of this paper we present the determination of the constant C of the traveller wave potential introduced by Franz Joseph von Gerstner, obtaining the calculation equation of this constant from the Bernoulli equation, under the condition of its real values, establishing the connection between the wave height h and length , as well as the water static deep H in the channel. Also, by writing the Bernoulli equation we determined the pressure variation function in a domain point by passing of the traveller wave, for the two real constant values corresponding to the radical signs. As follows of the traveller wave potential constant determination, we obtain also the trajectory of a liquid particle for the real value of the constant C, having the negative sign before the radical. We determined the same constant true value C from the calculus of the tide produced by the traveller wave, which rises up the water static level on the shore. Further, one introduces the numerical solving of the heavy and ideal liquid traveller wave unsteady motion on an inclined plane bottom at the positive angle , using the dimensionless equation system, in the case of a rectangular grid, studying the streamline stability conditions and presenting the error relaxation diagrams on the two calculus directions in a rectangular grid with two different steps.

A small resistive axisymmetric object in the presence of an underwater point-source field

A small, acoustically resistive and axisymmetric object is placed in a deep homogeneous sea environment with a hard plane bottom. The size of the object is small when compared to the depth of the water channel. The free surface of the sea is assumed to be soft. The source and the receiver are placed on the same vertical line and far away from the object. Given the positions of the source and the receiver, two problems are solved: The determination of the pressure field at the receiver from the position and the shape of the object and the determination of the position and the shape of the object from the pressure field at the receiver. The special case of smooth axisymmetric objects, i.e. objects generated by the rotation of differentiable curves, is studied. I provide results for the case of a floating object and for the case of an object or a “hillock” at the bottom of the sea.

On the motion of incompressible-medium particles in a domain with a periodically varying boundary

The motion of incompressible-medium particles in a cavity bounded by a plane bottom, vertical lateral walls, and a top boundary deformed in accordance with an arbitrary periodic law is investigated. The problem is reduced to solving the Hamilton equations with a time-periodic Hamiltonian. To study the system, the Hamilton system averaging method and the KAM (Kolmogorov, Arnold, Mozer) theory are used. A novel modification of the averaging procedure using the Poincare point map is proposed. Correct to an exponentially small cavity boundary deformation amplitude, the Poincare map points lie on the closed integral curves of an averaged autonomous Hamiltonian system. An asymptotic expansion of the averaged Hamiltonian in the amplitude is written down.

A Small Axisymmetric Obstacle in the Presence of an Underwater Point-Source Field

A small, acoustically hard and axisymmetric object is placed in a deep homogeneous sea environment with a hard plane bottom. The free surface of the sea is assumed to be soft. The source and the receiver are placed on the same vertical line, far away from the object. Given the positions of the source and the receiver, two problems are solved: the determination of the pressure field at the receiver from the position and the shape of the object, and the determination of the position and the shape of the object from the pressure field at the receiver. The special case of smooth objects generated by the rotation of differentiable curves is studied. We provide results for the case of a floating object and for the case of an object or a boss at the bottom of the sea.

Experimental Investigations of Free Vibrations of Cylinder in Water Near a Plane Bottom

Free vibrations of a horizontal cylinder placed in a water tank close to the plane bottom were investigated. The cylinder had four degrees of freedom, namely displacements and rotations in the horizontal and vertical planes. The accelerations of both ends of the cylinder were measured in the vertical and horizontal direction. The experimental set-up had a very small damping. Several values of the gap between the cylinder and the bottom were considered, and such values of the initial displacements were taken that the interaction with the bottom could be detected without the cylinder hitting the bottom.Assuming potential flow of inviscid and incompressible fluid, with perfect boundary conditions, and small displacements and rotations in the vertical plane, a theoretical description of the undamped dynamics of the cylinder was established. The Kalman filter method was applied to decompose the vibrations into the components having dominant frequencies. The frequencies of these components and amplitudes were compared with the results of the theoretical investigation. The theoretical solution gives proper frequencies of the components of free vibrations, except for the initial stage of vibrations when the real mechanical system needs time to develop a full coupled vibration with the fluid. The damping cannot be described only by linear terms in the differential equations.

Nonlinear Interaction of Free Vibrations of Cylinder in Water Near a Plane Bottom

Free undamped vibrations in a fluid of an elastically supported cylinder, placed in the vicinity of a horizontal, plane and rigid bottom are considered. It is assumed that the fluid is incompressible and inviscid, and the boundaries are perfect. The two-dimensional problem is posed, that is the cylinder can oscillate in both the horizontal and vertical directions. The Lagrangian formulation is used, in which the kinetic energy of the cylinder and the fluid is calculated, as well as the potential energy of linear elastic vertical and horizontal springs. In the paper explicit expressions are established for the coefficients in the series describing the complex potential of the fluid. Also, explicit expressions are given for the kinetic energy of the mechanical system and the lift forces acting on the cylinder. The Euler equations of the variational formulation yield two nonlinear coupled equations for the unknown horizontal and vertical displacements of the centre of the cylinder. By substitution of finite differences in the variational formulation and differentiation with respect to the displacements at the assumed time steps, a stable numerical solution is obtained. The calculated displacements are irregular functions in time, but the trajectories of motion of the centre of the cylinder lie in a well-defined area delimited by an envelope. This envelope for small vertical displacements may be approximated by two parabolas and two vertical lines.

Waves akin to the Gerstner type in a rotating fluid

In rotating fluids the existence of exact solutions expressing a form of wave motions similar to that of Gerstner is not a priori questionable. Effectively explicit solutions in Lagrangian form, representing rotational waves of finite amplitude in deep water ( infinite depth ), are investigated briefly in what follows. A first solution extends that of Gerstner by including the rotation. A second solution with no transverse velocity describes non linear Kelvin type waves, and a third solution represents non linear edge waves on a sloping plane bottom. It is observed that the dispersion relations of these waves are just the same as for linear waves of infinitesimal amplitude.

Edge waves on a longshore shear flow

A theoretical analysis of the effect of a longshore mean shear flow on edge waves is performed in the framework of linear shallow water equations. A single equation describing edge waves as well as neutral shear waves is obtained. A numerical method of calculation is given for the dispersion relations and the wave pattern, accounting for any beach topography and any mean flow profile (remaining constant alongshore and with a straight shoreline). Numerical calculations are presented for a simple exponential flow profile and for a plane bottom. A Doppler shift in the frequencies and a variation in the offshore extension of the waves are found, depending on the maximum local Froude number of the current, F, defined as F = [V(x)/√gH(x)]max, where V(x) stands for the mean longshore current, H(x) for the depth, and g for the gravity. The maximum shift in frequency is for wave numbers of about Vx(0)2/gm and frequencies of about Vx(0), where Vx(0) is the shear at the shoreline and m is the beach slope. For instance, these maximum differences may reach about 40% for F=0.5. Waves of any wavelength can always propagate downstream, but they can propagate upstream only for F≤Fc∼0.7. For mean flows with F≥Fc only waves shorter or longer than some forbidden wavelengths can propagate against the current. An analytical dispersion relation of asymptotic general validity for short waves (corresponding to the gravity range in real beaches) is given. The numerical model as well as this analytical dispersion relation is tested by means of a nonplanar real topography.

Bifurcation and Stability of Capillary-Gravity Waves

Tlree-dimensional capillary-gravity waves of a fluid of finite depth are considered. The fluid moves stationary over a plane bottom. For smallest critical Froude numbers there is a two- dimensional wave bifurcating from the trivial surface configuration.