Third-order Stokes wave solutions for interfacial internal waves in a three-layer density-stratified fluid

Interracial internal waves in a three-layer density-stratified fluid are investigated using a singular perturbation method, and third-order asymptotic solutions of the velocity potentials and third-order Stokes wave solutions of the associated elevations of the interfacial waves are presented based on the small amplitude wave theory. As expected, the third-order solutions describe the third-order nonlinear modification and the third-order nonlinear interactions between the interracial waves. The wave velocity depends on not only the wave number and the depth of each layer but also on the wave amplitude.

Second-order random wave solutions for interfacial internal waves in N-layer density-stratified fluid

This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The density interface displacements and the velocity potentials were solved to the second-order by an expansion approach used by Longuet-Higgins （1963） and Dean （1979） in the study of random surface waves and by Song （2004） in the study of second- order random wave solutions for internal waves in a two-layer fluid. The obtained results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies, and that the amplitudes of first-order wave components with the same wave numbers and frequencies between the adjacent density interfaces are modulated by each other. They also show that the second-order solutions consist of two parts： the first one is the first-order solutions, and the second one is the solutions of the second-order asymptotic equations, which describe the second-order nonlinear modification and the second-order wave-wave interactions not only among the wave components on same density interfaces but also among the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. It is also deduced that the results of the present work include those derived by Song （2004） for second-order random wave solutions for internal waves in a two-layer fluid as a particular case.

Investigation of the Behavior of a Jet Issued into Two-Layer Density-Stratified Fluid

This study experimentally investigates a jet flow issued into a two-layer density-stratified fluid in a tank and the resultant mixing phenomena. The upper and lower fluids are water and a NaCl- water solution, respectively, with the lower fluid issued vertically upward from a circular nozzle mounted on the tank bottom. Experimental highlights of the jet behavior and mixing phenomena are classified into three patterns according to the jet Reynolds number and mass concentration of the NaCl-water solution. The internal density current clearly occurs along the density interface, and the maximum jet height is predicted by the Froude number defined by the density difference between the upper and lower fluids. The effect of fluid thickness on the maximum jet height is also clarified.

Numerical Simulation of Jet Issuing Diagonally Upward into Density-Stratified Fluid in Cylindrical Tank

This study simulates the behavior of a jet issuing into a two-layer density-stratified fluid in a cylindrical tank and the resulting mixing phenomena. The upper and lower fluids are water and an aqueous solution of sodium chloride (NaCl), respectively, with the lower fluid issuing diagonally upward from a nozzle on the bottom of the tank. The angle between the centerline of the jet and the tank bottom is 60°. The phenomena when the Reynolds number Re of the jet is 475, 1426, and 2614 are simulated. The mass concentration of the aqueous solution of NaCl is 0.02. The simulation successfully grasps the jet behavior and the resulting mixing, which agree with the authors’ experimental results at the corresponding Re value. The secondary flows that appear in the horizontal cross-sections consist of a pair of vortices and flows along the tank wall. The secondary flow at the density interface represents the intrusion of an internal density current, which gives rise to mixing along the interface.

Behavior of a Jet Issuing Diagonally Upward into Two-Layer Density-Stratified Fluid in a Cylindrical Tank

This study is concerned with the experimental investigation of a jet issuing diagonally upward into a two-layer density-stratified fluid in a cylindrical tank and the resulting mixing phenomena. The upper and lower fluids are water and an aqueous solution of sodium chloride (NaCl), respectively, and the lower fluid issues from a nozzle on the bottom of the tank. The angle between the centerline of the jet and the bottom of the tank is 60o, and the mass concentration of the NaCl solution is 0.02. The investigation reveals that secondary flow is caused by the jet in the horizontal cross-sections of the tank and that it is composed of a pair of vortices. It confirms that the secondary flow at the density interface corresponds to an internal density current. The investigation also clarifies the effect of the Reynolds number of the jet on mixing between the lower and upper fluids.

Wave Scattering by a Submerged Sphere in Three-Layer Fluid

Using linear water wave theory,three-dimensional problems concerning the interaction of waves with spherical structures in a fluid which contains a three-layer fluid consisting of a layer of finite depth bounded above by freshwater of finite depth with free surface and below by an infinite layer of water of greater density are considered.In such a situation timeharmonic waves with a given frequency can propagate with three wavenumbers.The sphere is submerged in either of the three layers.Each problem is reduced to an infinite system of linear equations by employing the method of multipoles and the system of equations is solved numerically by standard technique.The hydrodynamic forces(vertical and horizontal forces)are obtained and depicted graphically against the wavenumber.When the density ratio of the upper and middle layer is made to approximately one,curves for vertical and horizontal forces almost coincide with the corresponding curves for the case of a two-layer fluid with a free surface.This means that in the limit,the density ratio of the upper and middle layer goes to approximately one,the solution agrees with the solution for the case of a two-layer fluid with a free surface.

Trapped Modes in a Three-Layer Fluid

In this work,trapped mode frequencies are computed for a submerged horizontal circular cylinder with the hydrodynamic set-up involving an infinite depth three-layer incompressible fluid with layer-wise different densities.The impermeable cylinder is fully immersed in either the bottom layer or the upper layer.The effect of surface tension at the surface of separation is neglected.In this set-up,there exist three wave numbers:the lowest one on the free surface and the other two on the internal interfaces.For each wave number,there exist two modes for which trapped waves exist.The existence of these trapped modes is shown by numerical evidence.We investigate the variation of these trapped modes subject to change in the depth of the middle layer as well as the submergence depth.We show numerically that two-layer and single-layer results cannot be recovered in the double and single limiting cases of the density ratios tending to unity.The existence of trapped modes shows that in general,a radiation condition for the waves at infinity is insufficient for the uniqueness of the solution of the scattering problem.

Second-order solutions for random interfacial waves in N-layer density-stratified fluid with steady uniform currents

In the present paper, the random interfacial waves in N-layer density-stratified fluids moving at different steady uniform speeds are researched by using an expansion technique, and the second-order asymptotic solutions of the random displacements of the density interfaces and the associated velocity potentials in N-layer fluid are presented based on the small amplitude wave theory. The obtained results indicate that the wave-wave second-order nonlinear interactions of the wave components and the second-order nonlinear interactions between the waves and currents are described. As expected, the solutions include those derived by Chen （2006） as a special case where the steady uniform currents of the N-layer fluids are taken as zero, and the solutions also reduce to those obtained by Song （2005） for second-order solutions for random interfacial waves with steady uniform currents if N = 2.

Numerical Study of the Mixing of Density-Stratified Fluid with a Jet

Density stratification of LNG （liquefied natural gas） is produced in a storage tank when one LNG is loaded on top of another LNG in the same tank. Mixing LNG by a jet issued from a nozzle on the tank wall is considered to a promising technique to prevent and eliminate stratification in LNG storage tanks. This study is concerned with the numerical simulation of a jet flow issued into a two-layer density-stratified fluid in a tank and the resultant mixing phenomena. The jet behavior was investigated with the laboratory-based experiment of the authors＇ previous study. A numerical method proposed by the authors is employed for the simulation. The upper and lower fluids are water and a NaCl-water solution, respectively, and the lower fluid is issued vertically upward from a nozzle on the bottom of the tank. The Reynolds number （Re） defined by the jet velocity and the nozzle diameter ranges from 95 to 2,378, and the mass concentration of the NaCl-water solution Co is set at 0.02 and 0.04. The simulation highlights the jet-induced mixing between the upper and lower fluids. It also clarifies the effects of Re and C0 on the height and horizontal spread of the jet.

Mixing of Density-Stratified Fluid in a Cylindrical Tank by a Diagonal Jet

This study experimentally investigates the mixing of two-layer density-stratified fluid in a cylindrical tank by a diagonal jet.The upper and lower fluids are water and an aqueous solution of sodium chloride(NaCl),respectively,and the lower fluid issues from a nozzle on the tank bottom.The angle between the jet centerline and the tank bottom is 60°,and the mass concentration of the NaCl solution is 0.02.The mixing in cases that the Reynolds numbers of the jets are 713,2319,and 3565 is investigated.The velocity fields in the central vertical cross-section are measured with a PIV(particle imaging velocimetry)system by tracing nylon particles with the diameter of 80μm.The concentration fields in the section are visualized using Rhodamine B as the fluorescent dye.They are also measured using PLIF(planer laser induced fluorescence)from visualized images and the progresses of the mixing are evaluated quantitatively.The investigation clarifies the relationship between the mixing phenomena and the Reynolds number of the jet.

Study of the Jet-Induced Mixing of Two-Layer Density-Stratified Fluid in a Rectangular Tank

The mixing phenomena of a two-layer density-stratified fluid induced by a jet in a tank are experimentally investigated. The upper and lower fluids are water and a NaCl-water solution, respectively, with the lower fluid issued vertically upward from a nozzle at the bottom of the tank. The jet Reynolds number Re, defined by the jet velocity and the water kinematic viscosity, ranges from 90 to 4,200. The mass concentration of the NaCl-water solution Co is less than 0.08. The flow visualization makes clear the jet behavior relative to the density interface between the upper and lower fluids. The measurement of the concentration distribution of the water paint issued with the jet highlights the effects of Re and Co on the mixing between the jet and the ambient fluid. The measurement of the fluid velocity distribution with a PIV （particle image velocimetry） system successfully elucidates the relationship between the velocity field and the resultant mixing.

CONJUGATE FLOWS OVER A STEP IN A THREE-LAYER FLUID

The conjugate flows over a step with the height h_0, which might be positive or negative, was studied in a three-layer fluid and the coupled nonlinear equations were derived, with which the effects of varying height h_0 on the existence and evolution of conjugate flows were examined. It is concluded that the conjugate flow is sharply sensitive to the thickness of fluid layers and its characteristics alters remarkably due to the existence of the step. As the flow climbs up a step (h_0>0), the conjugate flow with a convex lower interface and a concave upper interface is allowed to appear, while the flow with the concave lower interface or the simultaneous concave interfaces will be depressed. As the flow goes down a step (i.e., h_0<0), on most occasions only one kind of conjugate flow could exist, which prossesses the form with the simultaneous convex interfaces and will disappear rapidly with the increase of the step depth.

Wave scattering by a horizontal circular cylinder in a three-layer fluid

Using linear water-wave theory,wave scattering by a horizontal circular cylinder submerged in a three-layer ocean consisting of a layer of finite depth bounded above by finite depth water with free surface and below by an infinite layer of fluid of greater density is considered.The cylinder is submerged in either of the three layers.In such a situation time-harmonic waves with given frequency can propagate with three different wave numbers.Employing the method of multipoles the problem is reduced to an infinite system of linear equations which are solved numerically by standard technique after truncation.The transmission and reflection coefficients are obtained and depicted graphically against the wave number for all cases.In a two-layer fluid there are energy identities that exist connecting the transmission and reflection coefficients that arise.These energy identities are systematically extended to the three-fluid cases which are obtained.

Numerical Simulation of Mixing by Interaction of a Vortex Ring with a Density Interface

This study numerically explores the mixing caused by a vortex ring launched upward into a two-layer density-stratified fluid in a rectangular tank so as to search for a highly efficient mixing method.The upper and lower fluid layers are an aqueous glycerol solution and an aqueous solution of potassium dihydrogen phosphate(KH2PO4),respectively.The phenomena of vortex rings with Reynolds numbers of 4000,6000,and 8000 were simulated.The vortex-in-cell method is used to calculate the dynamics of the upper and lower fluids,and the CIP method is used to trace the concentration of the lower fluid.The simulations reproduce the mixing process successfully and elucidate the relationship between mixing efficiency and the Reynolds number of the vortex ring.

Numerical simulation of the internal wave propagation in continuously densitystratified ocean

A numerical model is proposed to simulate the internal wave propagation in a continuously density-stratified ocean, and in the model, the momentum equations are derived from the Euler equations on the basis of the Boussinesq approximation. The governing equations, including the continuity equation and the momentum equations, are discretized with the finite volume method. The advection terms are treated with the total variation diminishing （TVD） scheme, and the SIMPLE algorithm is employed to solve the discretized governing equations. After the modeling test, the suitable TVD scheme is selected. The SIMPLE algorithm is modified to simplify the calculation process, and it is easily made to adapt to the TVD scheme. The Sommerfeld＇s radiation condition combined with a sponge layer is adopted at the outflow boundary. In the water flume with a constant water depth, the numerical results are compared to the analytical solutions with a good agreement. The numerical simulations are carried out for a wave flume with a submerged dike, and the model results are analyzed in detail. The results show that the present numerical model can effectively simulate the propagation of the internal wave.