NONLINEAR INERTIA—GRAVITY INTERNAL WAVE IN STRATIFIED FLUID

Based on the fundamental equations of geophysical fluid dynamics and on the consequence of verticaldensity stratification,travelling wave coordinates are used in this work to study the geometric topologicalstructures of nonlinear permanent wave in phase plane.Rigorous mathematical mechanics demonstratethat the solution of permanent solitary wave does not exist.Hamilton functions and 'action-angle' varia-bles are used to express the travelling wave system in the simplest form and the analytic solution of the nonlinear inertia-gravity internal wave is obtained.

INTERNAL WAVE EIGENMODES AND THEIR WEAK NONLINEAR RESONANT INTERACTION IN A NONLINEARLY STRATIFIED FLUID- A PRELIMINARY THEORETICAL AND EXPERIMENTAL STUDY

A preliminary theoretical and experimental study was conducted on internal wave modes and their weak nonlinear resonant interaction in a nonlinearly stratified fluid . An asymptotical solution of the modes and a dispersion relation of internal waves in a stratified fluid with density profile similar to that in our experiment were obtained theoretically . The resonant-interaction mechanism to 2nd order approximation is also discussed . The resonant interaction of the 3rd and 4th mode internal waves excited by the unstable 1st mode wave is analyzed on the basis of data obtained by conductivity probes. The resonant-interaction condition, , is examined . It is shown that the resonant instability increases with pycnocline thickness and wave maker driving frequency .

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.

The Evolution Equation for Second-order Internal Solitary Waves in Stratified Fluids of Great Depth

By using perturbation methods, the evolution equation is derived for the second-order internal solitarywaves in stratified fluids of great depth, which is a kind of inhomogeneous linearized Belljamin-Ono equation.

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.

Analysis of current induced by long internal solitary waves in stratified ocean

An approximate theoretical expression for the current induced by long internal solitary waves is presented when the ocean is continuously or two-layer stratified. Particular attention is paid to characterizing velocity fields in terms of magnitude, flow components, and their temporal evolution/spatial distribution. For the two-layer case, the effects of the upper/lower layer depths and the relative layer density difference upon the induced current are further studied. The results show that the horizontal components are basically uniform in each layer with a shear at the interface. In contrast, the vertical counterparts vary monotonically in the direction of the water depth in each layer while they change sign across the interface or when the wave peak passes through. In addition, though the vertical components are generally one order of magnitude smaller than the horizontal ones, they can never be neglected in predicting the heave response of floating platforms in gravitationally neutral balance. Comparisons are made between the partial theoretical results and the observational field data. Future research directions regarding the internal wave induced flow field are also indicated.

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 Fr 2. The theoretical analysis is qualitatively consistent with the experimental results presented by Ma Hui_yang.

Interactions of internal solitary waves in deep stratified fluids (I)——Weak interactions

The weak interactions of internal solitary waves in deep stratified fluids are investigated interms of Lagrangian coordinates, which include the head-on and overtaking collisions of solitary waves with. different modes, and the head-on collision of waves with the same mode. The analysis shows that each waveis governed by the intermediately long wave (ILW) equation for finitely deep fluids and the Benjamin-Ono(BO) equation for infinitely deep fluids. The main effect of the interaction is the phase shifts of each wave.

WAVES GENERATED BY A SUBMERGED BODY MOVING IN STRATIFIED FLUIDS AND VERTICAL STRUCTURES OF INTERNAL WAVES

This dissertation deals with the internal waves generated by a submergedmoving body in stratified fluids by combining theoretical and experimental methods. Our purpose isto provide some scientific evidences for non-acoustic detection of underwater moving bodies based onthe principles of dynamics of the internal waves. An approach to velocity potentials obtained bysuperposing Green''s functions of sources and sinks was proposed for Kelvin waves at the free surfaceor interface in a two-layer fluid. The effects of interacting surface- and internal-wave modesinduced by a dipole on the surface divergence field were investigated. A new theoretical modelformulating the interaction of a two-dimensional submerged moving body with the conjugate flow in athree-layer fluid was established. An exact solution satisfying the two-dimensional Benjamin-Onoequation was obtained and the vertically propagating properties of the weakly nonlinear long waveswere studied by means of the ray theory and WKB method. The above theoretical results arequalitatively consistent with those obtained in the experiments conducted by the author.

EXPERIMENTAL RESEARCH ON INTERNAL WAVES GENERATED BY A MOVING BODY IN A STRATIFIED FLUID

Two-dimensional internal waves generated by a moving flat body in a continuous- ly stratified fluid are investigated by using three kinds of flow-visualization methods:electrolytic precipitation,hydrogen bubbles,and dye streaks.Attentions are paid mainly to the generation and propagation process in the upstream region under low internal Froude number(1/10π<F_r<1/2π).The features of the upstream disturbances as well as their relationship with stratification number K (K=1/F_r) are illustrated.The corresponding theoretical analysis is briefly presented,and by comparison,the experimental and theoretical results agree well.

Experiment on surface wake of internal waves generated by underwater vehicle in stratified fluids

The surface flow field of internal waves generated by the underwater vehicle is very weak.In order to study the characteristics of the surface wake of internal waves,a surface particle image velocimetry(PIV)technique which can be used to measure the flow field in the order of mm/s is developed.Breakthrough is made with respect to the key technique measuring the micro-velocity flow field of internal waves on the water surface in stratified fluids.The wake generated by SUBOFF model is measured in stratified fluid tank,and the surface flow field of internal waves is successfully measured for the first time.The experimental results are compared with predicted results by the classical Tuck’s internal wave theory.The results show the characteristics of the surface wake signature of internal waves and the variation of the angle between internal wave beams and the surface flow velocity of internal wave with the towing speed of the model are in good agreement.It provides support for further research on wake signature remote sensing of internal waves in laboratory.

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.

Time-frequency analysis of internal waves generated by a towed and self-propelled submerged body model

The spectrogram,based on a short-time Fourier transform,can visualize the time-dependent frequency spectrum of waves and is easy to compute.This time-frequency analysis method provides crucial information about waves generated by moving vessels and has been utilized to analyze Kelvin ship waves and internal waves.To further study the internal waves induced by a submerged body,an experiment is conducted for the towed and self-propelled SUBOFF model in a stratified fluid.The internal wave elevation signals are captured using electronic conductivity probes.Comparing with the calculation results of theoretical model,the high-frequency component of internal waves is identified.The high-frequency component has the exact same characteristics in both the towed and self-propelled model experiment and is consistent with the theoretical results for all Froude numbers.Therefore,this component is composed mainly of lee waves.Through spectral characteristics identification,a low-frequency component is discovered in the spectrogram in addition to the lee wave component.The intensity of the low-frequency component is tightly related to the vortex structure behind the submerged body.The vortex structure depends on the net momentum imparted by the submerged body.Therefore,this component is composed mainly of wake waves induced by the vortex structure.

跃变分层流体中球体内波散度场特征实验研究

本文对跃变分层流体中匀速拖曳运动球体内波诱导流场PIV试验测量得到的水平速度散度场随时间的演化过程及内波时频、波谱、传播速度等特征进行了研究。球体处于跃层中,测试水平面位于球心下方一倍直径处。分析获得了速度散度场整体特征及其随内傅氏数F_(ri)=U/N_(max)D的变化规律,比较分析了体效应内波(Lee波)和尾流效应内波(WIW波)的时频、波谱、传播速度等波动特征。研究表明:(1)在低F_(ri)时Lee波占优,水平速度散度等值线呈内凹的“>”形,在最大浮力振荡频率f_(max)附近有非常明显的频率主峰,速度散度曲线的波谱主峰也非常明显,内波传播速度与球体运动速度基本一致;(2)在高F_(ri)时WIW波占优,水平速度散度等值线呈“■”形,频谱曲线上明显存在多个峰,且主频接近球体潜深位置当地浮力振荡频率,波谱曲线上也存在多个较明显的峰,对应WIW波波长在2.4D~5.6D之间,传播速度远低于球体运动速度;(3)在高F_(ri)时WIW波分界线的斜率与参数F_(ri)呈指数分布,在低F_(ri)下Lee波包络线的斜率也符合此分布规律。

拖曳体内波尾迹水下/水面波流特征试验研究

内波水面效应对于水下航行体探测具有重要意义,目前关于水下航行体内波尾迹水面特征的相关研究还较少。本文介绍了在大型分层流试验水池中同时开展的以研究内波尾迹特征及其诱导的水面流动特征的拖曳体内波水下/水面尾迹特征试验研究。研究采用电导率仪阵列测量拖曳体(SUBOFF)水下内波波高场及其随时间变化,采用基于光学粒子图像互相关分析的流场测试方法测量拖曳体激发内波诱导的水面流场,并定量分析拖曳体激发的内波波长及其扰动引起水面流动的波长。研究表明:所采用的水面流场测量方法可以测量水面mm/s低速流动。在低傅汝德数Fr_(i)下(Fr_(i)≤2.14),分层流体中拖曳体水面尾迹特征主要受水下体效应内波影响,其水面尾迹波长及传播速度与水下内波的波长、传播速度基本一致;随着Fr_(i)的增加(2.14≤Fr_(i)≤3.42),在水下尾流效应内波逐渐增强,但在水面流场未见明显尾流效应内波影响;随着Fr_(i)的进一步增加(3.42≤Fr_(i)≤10.27),水下呈现出明显的尾流效应内波特征,由于表面波效应增强,水面流动主要表现为Kelvin波特征。

Experimental investigation of the generation of large-amplitude internal solitary wave and its interaction with a submerged slender body

A principle of generating the nonlinear large-amplitude internal wave in a stratified fluid tank with large cross-section is pro- posed according to the ＇jalousie＇ control mode. A new wave-maker based on the principle was manufactured and the experi- ments on the generation and evolution of internal solitary wave were conducted. Both the validity of the new device and ap- plicability range of the KdV-type internal soliton theory were tested. Furthermore, a measurement technique of hydrodynamic load of internal waves was developed. By means of accurately measuring slight variations of internal wave forces exerted on a slender body in the tank, their interaction characteristics were determined. It is shown that through establishing the similarity between the model scale in the stratified fluid tank and the full scale in the numerical simulation the obtained measurement re- suits of internal wave forces are confirmed to be correct.

Characteristics of the Internal Waves Generated by a Towed Model with Rotating Propeller Under a Strong Halocline

Experiments are performed on the internal waves(IWs) generated by a towed model with rotating propeller in a density-stratified fluid with linear halocline; the Reynolds number ranges from 7 000 to 84 000, and the Froude number ranges from 0.7 to 8.1. The wave speed, amplitude and patterns are investigated on the basis of the multi-channel conductivity probe array technology and the cross correlation analysis method. It is shown that the propeller advances the transition from the body-generated IWs to the wake-generated IWs. Before the transition, the IWs are stationary to the translational model. An extra V-shaped wave with a narrow opening angle is generated by the propeller and the wave amplitude becomes larger with the increase of the thrust momentum,indicating that the propeller produces body and wake effects at the same time before the transition. After the transition, the Froude number associated with the wave speed drops down and fluctuates within 0.4—1.5, showing that the IWs are nonstationary to the model. The interaction of the drag momentum and the thrust momentum changes the characteristics of the wave amplitudes and patterns. The wave amplitude no longer simply grows with the Froude number but depends on the contrast of the drag momentum and the thrust momentum. Experimental results show that the most obvious contrast of the wave pattern contour maps appears when the drag momentum and the thrust momentum have the largest difference if other conditions are the same. When the ratio of the drag momentum to the thrust momentum is within 1—10, the wake can be considered as zero-momentum, meaning that the momentum difference is not enough to generate large scale structures in the wake.

海表外强迫效应对连续层化流体中先驱内孤立波生成的影响

移动的海面或海底强迫在共振条件下会向强迫上游周期性地生成内孤立波,这种内孤立波被称为先驱内孤立波。本文利用三维全非线性、全频散的非静力模型NHWAVE,模拟连续分层流体受到海面外强迫作用而产生的先驱内孤立波,研究了正负外强迫效应对先驱内孤立波生成过程和形态特征的影响。结果显示,负效应强迫作用下先驱内孤立波具有更长的生成周期,这与两层流体中的结论相同;在强迫区内部,正效应强迫使水体的等密度面维持较稳定的结构,而当强迫效应为负时,强迫区内水体的等密度面随时间起伏较大。在正负两种效应强迫作用下,流场均维持了逆时针的回流结构,不过在负效应强迫作用下,流场更为复杂。

Generation and propagation of internal wave and its interaction with ocean structures

Internal waves appear frequently at the interface of a density-stratified fluid and they can cause severe damage on ocean structures.In this paper an extensive review on the study of the generation and propagation of internal waves and the interactions between internal waves with ocean structures is given.Future research is discussed.Some comparative analyses are conducted for propagation of internal waves on slope topography in shallow water.The future developments regarding propagation and oceanographic characteristics of internal waves are also discussed.