Two-layer Fluid Of Finite Depth Research Articles

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

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.

The Circulation Effect of Flow around Objects in Marine and Atmospheric Environments

Analytical expressions are obtained describing the hydrodynamic response to the circulation of flow of a two-layer fluid of finite depth around a point dipole. The dependences of the wave resistance and lifting force on the flow velocity, density jump, circulation, and depth of the sea are investigated. It is shown that the influence of the velocity circulation leads to a change in the lifting force applied to the dipole. Such changes are reversible in a relatively narrow range of the flow velocity around a pipeline. In addition to the flow around a pipeline, similar properties of the effect of circulation on the lifting force can also appear around self-propelled underwater objects and aircrafts.

Generation of internal waves by flow past a flat plate in a two-layer fluid of finite depth

This paper is concerned with an analytical study on the internal wave generated by a flat plate that is submerged and located at the interface of a two-layer fluid. The flow pattern can be constructed as it applies near the stern of a wide blunt ship or submarine. The Fourier transform and Wiener–Hopf techniques are applied to analytically solve the linearized problem for the Froude number F < 1 and different values of density ratio ρ. The main objective of this research paper is to investigate the impact of the density ratio on the downstream waves, as well as to minimize or eliminate the waves past the stern completely. It is shown that for the flat plate the wave amplitude was not eliminated, but it decreased for a small Froude number F as the density ratio decreased.

Transient flexural gravity waves in two-layer fluid

In the present manuscript, flexural gravity wave generation due to an initial disturbance in the case of two-layer fluid of finite depth in the presence of current is analyzed in three dimensions. From the plane progressive wave solution, effect of wave and current direction and the combined effect of current and compressive force on flexural gravity wave motion are discussed. Using Laplace and Fourier transform, the solution of the transient wave structure interaction problem is obtained and integral forms of velocity potentials, structural deflection and interface elevations are derived. Assuming the initial disturbances is on the plate covered surface asymptotic solutions for structural deflection and interface elevations are obtained using method of stationary phase for large time and space. The study reveals that the effect of opposing current and critical compressive force in the presence of current has significant impact on the plate deflection.

EFFECTS OF SURFACE TENSION ON TRAPPED MODES IN A TWO-LAYER FLUID

We consider a two-layer fluid of finite depth with a free surface and, in particular, the surface tension at the free surface and the interface. The usual assumptions of a linearized theory are considered. The objective of this work is to analyse the effect of surface tension on trapped modes, when a horizontal circular cylinder is submerged in either of the layers of a two-layer fluid. By setting up boundary value problems for both of the layers, we find the frequencies for which trapped waves exist. Then, we numerically analyse the effect of variation of surface tension parameters on the trapped modes, and conclude that realistic changes in surface tension do not have a significant effect on the frequencies of these.

Analysis of propagation of weakly nonlinear waves in a two-layer fluid with free surface

The study of wave motions in stratified fluids is one of the main tasks of hydrodynamics, which is caused by both theoretical and practical needs. Analytical analysis of propagation of weakly nonlinear wave packets in a two-layer fluid of finite depth in the presence of a free surface is performed. As a result, evolution equations of wave packets on the interface and the free surface in the form of the second-order nonlinear differential Schrodinger-type equations were derived. The form of internal and surface waves depending on the ratio of layer densities and the wave number considering the surface tension was analyzed. As a result, the effects of taking into account the second approximation in modeling wave motions in the two-layer system, which leads to blunting or sharpening of the wave crests and troughs were revealed. The analytical results are confirmed by field observations.

Hydroelastic response of a circular plate in waves on a two-layer fluid of finite depth

The hydroelastic response of a circular, very large floating structure (VLFS), idealized as a floating circular elastic thin plate, is investigated for the case of time-harmonic incident waves of the surface and interfacial wave modes, of a given wave frequency, on a two-layer fluid of finite and constant depth. In linear potential-flow theory, with the aid of angular eigenfunction expansions, the diffraction potentials can be expressed by the Bessel functions. A system of simultaneous equations is derived by matching the velocity and the pressure between the open-water and the plate-covered regions, while incorporating the edge conditions of the plate. Then the complex nested series are simplified by utilizing the orthogonality of the vertical eigenfunctions in the open-water region. Numerical computations are presented to investigate the effects of different physical quantities, such as the thickness of the plate, Young’s modulus, the ratios of the densities and of the layer depths, on the dispersion relations of the flexural-gravity waves for the two-layer fluid. Rapid convergence of the method is observed, but is slower at higher wave frequency. At high frequency, it is found that there is some energy transferred from the interfacial mode to the surface mode.

Water wave diffraction by a bottom-mounted circular cylinder clamped to an elastic plate floating on a two-layer fluid

The wave diffraction by a bottom-mounted circular cylinder, which is clamped to the center of a floating circular thin elastic plate, in the two-layer fluid of finite depth is investigated for the time-harmonic incident waves of the surface and interfacial wave modes. Each fluid layer is inviscid, incompressible and of constant density. The flexural–gravity waves are composed of the propagating, decaying propagating and evanescent wave modes. Within the framework of the linear potential flow theory, a closed system of simultaneous linear equations is derived to solve the undetermined expansion coefficients with the methods of the angular eigenfunction expansion and the inner product. Explicit numerical computations are employed to test the convergence of the two series for the angular expansions and the evanescent wave modes. The horizontal forces and the moments exerted on the circular cylinder due to different wave modes are discussed in the case of the incident waves of either the surface or interfacial wave mode. It is obtained that the evanescent wave modes are appreciable parts for a high frequency.

Flexural gravity waves trapped in a two-layer fluid of finite depth

Trapped waves are of considerable interest in providing examples of discrete wave frequencies in the presence of a continuous spectrum. Under the usual assumptions of linear water wave theory, the existence of trapped modes supported by a submerged horizontal circular cylinder in a two-layer fluid of finite depth bounded above by a thin ice-cover and below by an impermeable horizontal bottom is investigated. The effect of surface tension at the surface of separation is neglected. In this case, two trapped waves are developed: waves with the higher wavenumber at the interface and waves with the lower wavenumber at the ice-cover. In this problem, a fifth-order boundary condition is satisfied at the upper surface which makes the problem more complex. Using multipole expansion method, an infinite system of homogenous linear equations is obtained. For a fixed geometrical configuration and a specific arrangement of a set of other parameters, the frequencies for which the value of the truncated determinant is zero are numerically computed and the trapped wavenumbers corresponding to those frequencies are obtained by using the dispersion relation.These trapped modes are compared with those for which the ice-cover gets replaced by a free surface. We also look into the effect of the variation of ice-parameters on the existence of trapped modes. Further, trapped modes in a homogenous fluid of finite depth bounded above by a thin ice-cover are recovered. Trapped modes due to a cylinder placed in either of the layers are mainly confined to the vicinity of interface and ice-cover only. These modes, in our case, exist with a cut-off value though there are trapped modes which are embedded in the continuous spectrum. So, above that cut-off value and far from interface and ice-cover, it is possible to have a unique solution to the radiation problem for the cylinder placed in either of the layers.

Hydroelastic interaction between obliquely incident waves and a semi-infinite elastic plate on a two-layer fluid

The hydroelastic response of a semi-infinite thin elastic plate floating on a two-layer fluid of finite depth due to obliquely incident waves is investigated. The upper and lower fluids with different densities separated by a sharp and stable interface are assumed to be inviscid and incompressible and the motion to be irrotational. Simply time-harmonic incident waves of the surface and interfacial wave modes with a given angular frequency are considered within the framework of linear potential flow theory. With the aid of the methods of matched eigenfunction expansion and the inner product of the two-layer fluid, a closed system of simultaneous linear equations is derived for the reflection and transmission coefficients of the series solutions. Based on the dispersion relations for the gravity waves and the flexural–gravity waves in a two-layer fluid and Snell’s law for refraction, we obtain a critical angle for the incident waves of the surface wave mode and three critical angles for the incident waves of the interfacial wave mode, which are related to the existence of the propagating waves. Graphical representations of the series solutions show the interaction between the water waves and the plate. The effects of several physical parameters, including the density and depth ratios of the fluid and the thickness of the plate, on the wave scattering and the hydroelastic response of the plate are studied. It is found that the variation of the thickness of the plate may change the wave numbers and the critical angles. The density ratio is the main factor to influence the wave numbers of the interfacial wave modes. Finally, the stress state is considered.

Trapped modes in a two-layer fluid of finite depth bounded above by a rigid lid

Based on linear water wave theory, we investigate the existence of trapped modes supported by a submerged horizontal circular cylinder in a two-layer fluid of finite depth bounded above by a rigid lid and below by an impermeable horizontal bottom. The effect of surface tension at the surface of separation is neglected. Under such a situation time-harmonic waves propagate with one wavenumber only, unlike the case where the upper layer has a free surface and the waves propagate with two wavenumbers. Using a multipole expansion method, we obtain a homogeneous infinite system of linear equations. For a fixed geometrical configuration, we numerically compute the values of those frequencies for which the values of the truncated determinant become approximately zero, which confirms the existence of trapped modes. We plot the dispersion curves and observe that trapped mode frequencies always exist below a cut-off value. Trapped mode frequencies are plotted against various values of the density ratios for different depths of either of the layers. We also observe the effect of submergence depth on trapped mode frequencies. We find that only a single mode exists for a depth of submergence of the cylinder greater than about 1.05 times its radius. As the depth of submergence decreases, further trapped modes appear. The existence of trapped modes shows that, in general, a radiation condition for the waves at infinity is insufficient for the uniqueness of the scattering problem.

Wave-making experiments and theoretical models for internal solitary waves in a two-layer fluid of finite depth

A laboratory wave-making method is developed for the internal solitary wave under the condition of giving its amplitude produced by oppositely and horizontally pushing two vertical plates placed separately in the upper- and lower-layer fluids of a large-scale density stratified tank where based on the Miyata-Choi-Camassa (MCC) theoretical model, the layer-mean velocities of the upper- and lower-layer fluids induced by the internal solitary wave are used as the velocities of the two plates. On this basis, a series of experiments is conducted to explore the applicability conditions for internal solitary wave theories with stationary solutions which are Korteweg-de Vries (KdV), extended KdV (eKdV), MCC and modified KdV (mKdV) models in a two-layer fluid of finite depth respectively. It is shown that for the nonlinear parameter ε and the dispersion parameter μ defined by the total water depth, there exists a critical dispersion parameter μ0, in the case of μ μ0, the KdV model is applicable for ε ≤μ, the eKdV model is applicable for μ ε ≤√μ, as well as the MCC model is applicable for ε &gt; √μ. However, in the case of μ ≥ μ0, the MCC model is still applicable for a wide range of ε. Furthermore, for the case where the ratio of depth between the upper- and lower-layer fluids is not close to its critical value, the mKdV model is mainly applicable for the case where the amplitude of the internal solitary wave is close to its theoretical limiting amplitude, however, the MCC model is also applicable for such a case. The investigation quantitatively characterizes the applicability conditions for four classes of internal solitary wave theories, and provides an important theoretical foundation for what kinds of theories can be chosen to model internal solitary waves in the ocean.

Internal Wave Propagation from Pulsating Sources in a Two-Layer Fluid of Finite Depth

The influence of internal waves is very important in the Engineering Analysis, Design and Optimization. To study the internal wave properties, we model a two-layer fluid and generalize to multiple layers. In a two-layer fluid with the upper layer having a free surface, there exist two modes of waves propagating due to the free surface and the interface. This is due to the density difference in the vertical direction of the water, due to the variation in salinity and temperature where waves from underwater structures are of importance. In this case the fluid is assumed to be non viscous, incompressible and the flow is non rotational. On the other hand, there is need for appropriate Green functions to analyze these properties. In this paper, we use the three dimensional Green functions for a stationary oscillating source to study the internal wave characteristics. Some of the behavior studied in this work includes effects of internal waves on the surface and internal wave amplitudes. Further, an investigation of the influence of internal waves on the wave length, frequency and period is made.

A Time-Domain Boundary Element Method for Wave Diffraction in a Two-Layer Fluid

A time-domain numerical model is established based on the higher-order boundary element method (HOBEM) to simulate wave diffraction problem in a two-layer fluid of finite depth. There are two possible incident wave modes (surface-wave mode and internal-wave mode) exist in the incident wave for a prescribed frequency in a two-layer fluid. For surface-wave mode, the hydrodynamic characters of fluid particles are similar to single-layer fluid. For the internal-wave mode, through the definition of a new function respected to velocity potentials of upper and lower fluid on the interface by using matching condition, a single set of linear equations is set up to compute the time histories of wave forces and wave profiles by using a fourth-order Runge-Kutta method. An artificial damping layer is adopted both on the free surface and interface to avoid the wave reflection. Examinations of the accuracy of this time-domain algorithm are carried out for a truncated cylinder and a rectangular barge, and the results demonstrate the effectiveness of this method.

Green functions with pulsating sources in a two-layer fluid of finite depth

The derivation of Green function in a two-layer fluid model has been treated in different ways. In a two-layer fluid with the upper layer having a free surface, there exist two modes of waves propagating due to the free surface and the interface. This paper is concerned with the derivation of Green functions in the three dimensional case of a stationary source oscillating. The source point is located either in the upper or lower part of a two-layer fluid of finite depth. The derivation is carried out by the method of singularities. This method has an advantage in that it involves representing the potential as a sum of singularities or multipoles placed within any structures being present. Furthermore, experience shows that the systems of equations resulted from using a singularity method possess excellent convergence characteristics and only a few equations are needed to obtain accurate numerical results. Validation is done by showing that the derived two-layer Green function can be reduced to that of a single layer of finite depth or that the upper Green function coincides with that of the lower, for each case. The effect of the density on the internal waves is demonstrated. Also, it is shown how the surface and internal wave amplitudes are compared for both the wave modes. The fluid in this case is considered to be inviscid and incompressible and the flow is irrotational.

Exciting forces due to interaction of water waves with a submerged sphere in an ice-covered two-layer fluid of finite depth

Abstract Scattering of water waves by a sphere in a two-layer fluid, where the upper layer has an ice-cover modelled as an elastic plate of very small thickness, while the lower one has a rigid horizontal bottom surface, is investigated within the framework of linearized water wave theory. The effects of surface tension at the surface of separation is neglected. There exist two modes of time-harmonic waves – the one with lower wave number propagating along the ice-cover and the one with higher wave number along the interface. Method of multipole expansions is used to find the particular solution for the problem of wave scattering by a submerged sphere placed in either of the layers. The exciting forces for vertical and horizontal directions are derived and plotted against different values of the wave number for different submersion depths of the sphere and flexural rigidity of the ice-cover. When the flexural rigidity and the density of the ice-cover are taken to be zero, the numerical results for the exciting forces for the problem with free surface are recovered as particular cases.

Unsteady three-dimensional sources for a two-layer fluid of finite depth and their applications

To our friend Ernie Tuck, in celebration of his multi-faceted talents. The velocity potentials of various unsteady point sources are derived in this paper for a two-layer fluid of finite depth. Two-layer fluids are often used to study effects of density stratification on hydrodynamics of marine systems. The sources here are restricted to the upper fluid layer and the potentials of the induced flows are given for the whole fluid domain. The velocity potentials of a transient source of arbitrary strength and in arbitrary three-dimensional motion are derived first. The potentials of a time-harmonic source without forward speed, and then with forward speed, are obtained from the transient source by specifying the appropriate source strength and motion. These potentials are fundamental to the analyses of various types of body motion in finite water depths under the influence of surface and interfacial waves. As a sample application, a numerical solution of the radiation and diffraction problem for a floating rectangular barge is presented. The results indicate that internal waves can have a strong effect on the motions of the floating barge over a wide range of incident-wave frequencies.

Two-dimensional interactions of solitons in a two-layer fluid of finite depth

Two-dimensional (2D) interactions of two interfacial solitons in a two-layer fluid of finite depth are investigated under the assumption of a small but finite amplitude. When the angle ϑ between the wave normals of two solitons is not small, it is shown by a perturbation method that in the lowest order of approximation the solution is a superposition of two intermediate long wave (ILW) solitons and in the next order of approximation the effect of the interaction appears as position phase shifts and as an increase in amplitude at the interaction center of two solitons. When ϑ is small, it is shown that the interaction is described approximately by a nonlinear integro-partial differential equation that we call the two-dimensional ILW (2DILW) equation. By solving it numerically for a V-shaped initial wave that is an appropriate initial value for the oblique reflection of a soliton due to a rigid wall, it is shown that for a relatively large angle of incidence ϑi the reflection is regular, but for a relatively small ϑi the reflection is not regular and a new wave called stem is generated. The results are also compared with those of the Kadomtsev–Petviashvili (KP) equation and of the two-dimensional Benjamin–Ono (2DBO) equation.

Wave scattering by a thin elastic plate floating on a two-layer fluid

The hydroelastic interaction between an incident gravity wave and a thin elastic plate floating on a two-layer fluid of finite depth is analyzed with the aid of the method of matched eigenfunction expansions. The fluid is assumed to be inviscid and incompressible. A two-dimensional problem is formulated within the framework of linear potential theory for small-amplitude waves. The fluid domain is divided into two and three regions for semi-infinite and finite plates, respectively, with the matching relations representing the continuities of the pressure and velocity. A new inner product involving two single integrals is proposed, in which the vertical eigenfunctions in the open water region of the two-layer fluid are orthogonal. Then the orthogonality of the eigenfunctions with respect to the newly defined inner product is used to obtain a set of simultaneous equations for the expansion coefficients of the velocity potentials, and the edge conditions are included as a part of the equation system. The effects of the fluid density ratio and the position of interface on the wave reflection and transmission are discussed. Numerical analysis shows that the method proposed herein is effective with a higher rate of convergence.