Two-layer Fluid Of Finite Depth Research Articles

Transverse instability of interfacial solitary waves

The linear stability of finite-amplitude interfacial solitary waves in a two-layer fluid of finite depth is examined analytically on the basis of the Euler equations. An asymptotic analysis is performed, which provides an explicit criterion of instability in the case of long-wavelength transverse disturbances. This result leads to the general statement that, when the amplitude of the solitary wave is increased, the solution becomes transversely unstable before an exchange of longitudinal stability occurs.

Scattering of Surface and Internal Waves by Rectangular Dikes

Abstract The scattering of surface and internal waves by a single dike or a pair of identical dikes in a two-layer fluid is analyzed in two dimensions within the context of linearized theory of water waves. The dikes are approximated as cylinders of rectangular geometry and are placed in a two-layer fluid of finite depth. In the study, both the cases of surface-piercing and bottom-standing dikes are considered. The solution of the associated boundary value problem is derived by a matched eigenfunction expansion method. Because of the flow discontinuity at the interface, the eigenfunctions involved have an integrable singularity at the interface and the orthonormal relation used in the present analysis is a generalization of the classical one corresponding to a single-layer fluid. The reflection coefficients and force amplitudes are computed and analyzed in various cases. The computed results in a two-layer fluid are compared with those existing in the literature for a single-layer fluid. The results obtained by the matched eigenfunction expansion method are compared with that of wide-spacing approximation method, and it is observed that the results from both the methods are in good agreement when the dikes are widely spaced. The general behavior of reflection coefficients for interface-piercing and non-interface-piercing obstacles is found to be different in both cases of surface-piercing and bottom-standing dikes. Moreover, for surface-piercing dikes, the results show the possibility of very large resonant motions between the dikes but with a very narrow bandwidth for the frequency of interest.

Wave drift force in a two-layer fluid of finite depth

Based on the momentum and energy conservation principles, a compact calculation formula is analytically derived for the wave-drift force on a 2-D body floating in a two-layer fluid of finite depth. In a two-layer fluid, two different wave modes (the surface-wave mode with longer wavelength and the internal-wave mode with shorter wavelength) exist not only in the incident wave but also in the body-scattered wave, and these wave characteristics are properly incorporated in the obtained formula. It is noted that, unlike the single-layer case, the wave-drift force can be negative in the incident wave of surface-wave mode, if the transmitted wave with internal-wave mode is large. Numerical computations are implemented for a Lewis-form body by means of the boundary-integral-equation method with Green’s function for the two-layer fluid problem. The effects of density ratio, interface position, and body motions on the wave-drift force are studied, and some important features are found for two-layer fluids.

Features of Wave-Packet Propagation in Two-Layer Fluid of Finite Depth

Features of Wave-Packet Propagation in Two-Layer Fluid of Finite Depth

Hydrodynamics of a body floating in a two-layer fluid of finite depth. Part 2. Diffraction problem and wave-induced motions

A linearized two-dimensional diffraction problem in a two-layer fluid of finite depth was solved for a general floating body and relevant wave-induced motions were studied. In a two-layer fluid, for a prescribed frequency, incident waves propagate with two different wave modes. Thus the wave-exciting forces and resulting motions must be computed separately for each mode of the incident wave. The boundary integral equation method developed by the authors in the Part-1 article was applied to directly obtain the diffraction potential (pressure) on the body surface. With the computed results, an investigation was carried out on the effects of the fluid density ratio and the interface position on the wave-exciting forces on the body and the motions of the body, including the case in which the body intersects the interface. By a systematic derivation using Green's theorem, all the possible reciprocity relations were derived theoretically in explicit forms for a system of finite depth; these relations were confirmed to be satisfied numerically with very good accuracy. Experiments were also carried out using water and isoparaffin oil as the two fluids and a Lewis-form body. Measured results for the sway- and heave-exciting forces and the heave motion were compared with the computed results, and a favorable agreement was found.

Hydrodynamic Load Associated with the Oscillations of a Cylinder at the Interface in a Two-Layer Fluid of Finite Depth

The results of numerically solving the linear problem of the small steady-state oscillations of a horizontal cylinder located at the interface between two fluids of different densities are presented. The hybrid element method is used. In this method the velocity potentials are represented by means of the finite element method in a narrow zone in the neighborhood of the body and by means of the boundary integral equations in the outer domain. The Green’s functions for an oscillating source in a two-layer fluid bounded from above by a free boundary and from below by an even horizontal bottom are derived. Numerical calculations of the apparent mass and damping coefficients are carried out for an elliptic cylinder beneath a free surface and for a cylinder with the cross-section in the form of a Lewis rib contour which floats on the free surface.

Waves induced by a submerged moving dipole in a two-layer fluid of finite depth

The waves induced by a moving dipole in a two-fluid system are analytically and experimentally investigated. The velocity potential of a dipole moving horizontally in the lower layer of a two-layer fluid with finite depth is derived by superposing Green’s functions of sources (or sinks). The far-field waves are studied by using the method of stationary phase. The effects of two resulting modes, i.e. the surface- and internal-wave modes, on both the surface divergence field and the interfacial elevation are analyzed. A laboratory study on the internal waves generated by a moving sphere in a two-layer fluid is conducted in a towing tank under the same conditions as in the theoretical approach. The qualitative consistency between the present theory and the laboratory study is examined and confirmed.

Hydrodynamics of a body floating in a two-layer fluid of finite depth. Part 1. Radiation problem

A linearized 2-D radiation problem was considered for a general floating body in a two-layer fluid of finite depth. A boundary integral-equation method was developed for directly computing the velocity potential on the wetted surface of a body which is immersed in both the upper and lower layers as a general case. To do this, appropriate time-harmonic Green’s functions were derived, and an efficient numerical method of evaluating those functions is proposed. Based on Green’s theorem, hydrodynamic relations such as the energy-conservation principle were derived theoretically for a case of finite depth, and we confirm that those relations are satisfied numerically with very good accuracy. Experiments were also carried out using water and isoparaffin oil as the two fluids and a Lewis-form body. Measured results for the added mass, the damping coefficient, and the amplitude of the generated waves are compared with the computed results, and a favorable agreement is found.

Surface effects of internal wave generated by a moving source in a two-layer fluid of finite depth

Based on the potential flow theory of water waves, the interaction mechanism between the free-surface and internal waves generated by a moving point source in the lower layer of a two-layer fluid was studied. By virtue of the method of Green's function, the properties of the divergence field at the free surface were obtained, which plays an important role in the SAR (Synthetic Aperture Radar) image. It is shown that the coupling interaction between the surface-wave mode and internal-wave mode must be taken into account for the cases of large density difference between two layers, the source approaching to the pynocline and the total Froude number Fr close to the critical number Fr2. The theoretical analysis is qualitatively consistent with the experimental results presented by Ma Hui-yang.

Trapped modes supported by submerged obstacles

In this paper, trapped modes are shown to exist for a pair of bodies that are submerged in deep water. An inverse procedure is used whereby a flow that represents a local oscillation is constructed and closed streamlines that surround the singularities of the flow are interpreted as body boundaries. The velocity potential for the flow is constructed from a symmetric combination of horizontal and vertical dipoles. This combination of singular solutions of Laplace's equation satisfies the free-surface boundary condition, and the spacing and relative amplitudes of the singularities are related in such a way that there is no radiation of waves to infinity. The existence of closed streamlines that surround the singularities is demonstrated by investigating how the structure of the streamlines varies as the parameters of the flow change.

Two-dimensional problem of a uniform flow of a two-layer fluid of finite depth past a circular cylinder

The linear steady problem of an irrotational uniform flow past a horizontal circular cylinder located in the upper or in the lower layer of a two-layer fluid is solved by the multipole-expansion method. The flow is perpendicular to the axis of the cylinder. The fluid is assumed to be inviscid and incompressible, and the flow in each layer is assumed to be potential. The upper layer can be bounded by a free surface or a solid lid, and the lower layer by a rigid horizontal bottom.

Forced Oscillations in a Two-Layer Fluid of Finite Depth

An initial value investigation is made of the development of surface and internal wave motions generated by an oscillatory pressure distribution on the surface of a fluid that is composed of two layers of limited depths and of different densities. The displacement functions both on the free surface and on the interface are obtained with the help of generalized Fourier transformation. The method for the asymptotic evolution of the wave integrals is based on Bleistein’s method. The behavior of the solutions is examined for large values of time and distance. It is found that there are two classes of waves—the first corresponds to the usual surface waves with a changed amplitude and the second arises entirely due to stratification. Some interesting features of the wave system have also been studied.

Multi-soliton solutions in a finite depth fluid

A systematic procedure for solving the Whitham equation in a two-layer fluid of finite depth is developed. An analytic solution which asymptotically evolves into exactly two solitons is exhibited. The characteristics of these solitons can be quite different from those resulting from the Korteweg-de Vries equation (the shallow water limit of the present theory).

Solitary waves in a finite depth fluid

It is shown that U( xi )=u0(cosh2 a xi +(ab)-2 sinh2 a xi )-1 is the exact stationary wave solution to the Whitham equation in a two-layer fluid of finite depth and represents the natural connection between the Benjamin-Ono deep water (a to 0) and Kortweg-de Vries shallow water (b to infinity ) theories.

Waves in an inhomogeneous fluid due to periodic pressures in the presence of a thin plate

The problem of motion of a two-layer fluid of finite depth subjected to pressure applied to its surface in the presence of a semiinfinite plate is considered in a linear formulation. Expressions governing the form of the originating surface and internal waves are obtained. Results are presented of a numerical computation of the elevation of the free surface and the interface caused by the application of the pressure.

Translational motion of a rigid body with two-layer fluid of finite depth

Translational motion of a rigid body with two-layer fluid of finite depth