The phase configuration of the waves around an accelerating disturbance in a rotating stratified fluid

Ray theory is extended to consider the case of an accelerating disturbance which is producing waves in a rotating stratified fluid. Starting from the equations of motion, dispersion relations are derived for surface gravity waves, capillary waves, Rossby waves and internal-inertial waves. The wave system is studied in each case for the problem of a body starting impulsively from rest and for a body starting from rest and moving with constant acceleration.     

Disturbances to a rotating fluid with a sphere moving along the axis of rotation

I.IntroductionWhenabodyprojectedinthewaterisintranslationalmotionthroughwateritwillcertainlycausesdisturbances.Howtodetectthedisturbancesf'arawayfromthebodyandhowtodetermillebytheintbrlnationobtainedthepositionandvelocityofthebodyandthesiteofprojectionareimportant.Astheprojectedbodyisrotating,thefluidaroundisinauniformlyrotationrelativetothereferenceframerotatinginsynchronismwiththebody,andthisisaquestionofthemotionofabodyinarotatingfluid.SuchproblemswerefirststudiedbyProudman(1916)l']andTay…     

Far fields of internal gravitywaves generated by a perturbation source in a stratified rotating medium

The problem of constructing uniform asymptotics for the far fields of internal gravity waves generated by a moving source of perturbations in flow of a finite-depth stratified rotating medium is considered. The solutions obtained describe the wave perturbations both inside and outside the wave fronts and can be expressed in terms of the Airy function and its derivatives. Numerically calculated wave patterns of the excited wave fields are presented.     

Surface characters of internal waves generated by Rankine ovoid

A linear theory on the internal waves generated in the stratified fluid with a pycnocline is presented in this paper. The internal wave fields such as the velocity fields in the stratified fluid and velocity gradient fields at the free surface are also investigated by means of the theoretical and numerical method. From the numerical results, it is shown that the internal wave generated by horizontally moving Rankine ovoid is a sort of trapped wave which propagates in a wave guide, and its waveform is a kind of Mach front-type internal wave in the pycnocline. Influence of the internal wave on the flow fields at the free surface is represented by the velocity gradient fields resulted from the internal waves generated by motion of the Rankine ovoid. At the same time, it is also shown that under the hypothesis of inviscid fluid, the synchronism between the surface velocity gradient fields at the free surface and the internal wave fields in the fluid is retained. This theory opens a possibility to study further the modulated spectrum of the Bragg waves at the free surface.The project supported by the National Natural Science Foundation of China (40576010). The English text was polished by Keren Wang.     

An interfacial Gerstner-type trapped wave

Coastally trapped rotational interfacial waves are studied theoretically by using a Lagrangian formulation of fluid motion in a rotating ocean. The waves propagate along the interface between two immiscible inviscid incompressible fluid layers of finite depths and different densities, and are trapped at a straight wall due to the Coriolis force. For layers of finite depth, solutions are sought as series expansions after a small parameter. Comparison is made with the irrotational interfacial Kelvin wave. Both types of waves are identical to first order, having zero vorticity. The second order solution yields a relation between the vorticity and the velocity shear in the wave motion. Requiring that the mean motion in both layers is irrotational, then follows the well-known Stokes drift for interfacial Kelvin waves. On the other hand, if the mean forward drift is identically zero, we obtain the second order vorticity in the Gerstner-type wave. The solutions in both layers for the Gerstner-type interfacial wave are given analytically to second order. It is shown that small density differences and thin upper layers both act to yield a shape of the material interfacial with broader crests and sharper troughs. These effects also tend to make the particle trajectories at the interface in both layers become distorted ellipses which are flatter on the upper side. It is concluded that the effect of air excludes the possibility of observing the exact Gerstner wave in deep water.     

Experiments to study interactions between baroclinic lower flows and a stably stratified upper layer

Experiments were conducted on a rotating fluid annulus to study the basic interactions between baroclinic lower flows and a stably stratified upper layer. Sufficiently stable stratification is necessary for steady flows to emerge in the lower layer. Upward fluid motions make the baroclinic flows permeate into the upper layer. The stable stratification, however, suppresses upward motions so that zonal fluid velocities decrease with height. In fact, their maximum appears at the top level of the baroclinic lower layer and the sign of the radial temperature gradient changes there; namely, it is warmer on the inner side of the annulus in the upper layer. This temperature profile is reflected in a meridional fluid circulation mixing both layers. In the upper layer of the wave flow, there exists a critical level below and above which the zonal fluid velocities have opposite directions for the wave to have a phase shift of half a wavelength in appearance. The experimental results correspond to real atmospheric phenomena.     

Generation of beams of three-dimensional periodic internal waves by sources of various types

The energy and force characteristics of periodic internal wave beams in a viscous exponentially stratified fluid are analyzed. The exact solutions of linearized problems of generation obtained by integral transformations describe not only three-dimensional internal waves but also the associated boundary layers of two types. The solutions not containing empirical parameters are brought to a form that allows a direct comparison with experimental data for generators of various types (friction, piston, and combined) of rectangular or elliptic shape. The stress tensor and force components acting on the generator are given in quadratures. In the limiting cases, the solutions are uniformly transformed to the corresponding expressions for the problems in a two-dimensional formulation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 12–23, May–June, 2006     

Investigation of the effect of the Coriolis force on a thin fluid film on a rotating disk

The effect of the Coriolis force on the evolution of a thin film of Newtonian fluid on a rotating disk is investigated. The thin-film approximation is made in which inertia terms in the Navier–Stokes equation are neglected. This requires that the thickness of the thin film be less than the thickness of the Ekman boundary layer in a rotating fluid of the same kinematic viscosity. A new first-order quasi-linear partial differential equation for the thickness of the thin film, which describes viscous, centrifugal and Coriolis-force effects, is derived. It extends an equation due to Emslie et al. [J. Appl. Phys. 29, 858 (1958)] which was obtained neglecting the Coriolis force. The problem is formulated as a Cauchy initial-value problem. As time increases the surface profile flattens and, if the initial profile is sufficiently negative, it develops a breaking wave. Numerical solutions of the new equation, obtained by integrating along its characteristic curves, are compared with analytical solutions of the equation of Emslie et al. to determine the effect of the Coriolis force on the surface flattening, the wave breaking and the streamlines when inertia terms are neglected.     

Experiments to study baroclinic waves penetrating into a stratified layer by a quasi-geostrophic potential vorticity equation

The upward penetration of the mid-latitude baroclinic waves has been well known to influence the air motion in the stratosphere. To study this in the laboratory, we present a new type of rotating fluid annulus experiment where a radial temperature difference is imposed on water near the flat bottom to produce a baroclinic wave in a lower layer and the water surface is heated to create a stratified upper layer. To test this model, we carried out experiments to study baroclinic waves penetrating into the stratified layer by a quasi-geostrophic potential vorticity equation in the atmospheric dynamics. The baroclinic waves produced in the lower fluid layer were observed to penetrate into the stratified upper layer. As expected from the equation, the vorticities of the flows for wavenumbers 3, 4 and 5 decayed exponentially with height and showed a weaker decay for lower wavenumbers.     

分层流体中运动源生成的内波研究进展

针对两类密度分布模型------连续分层流体和间断分层流体, 综述了在运动潜体生成的Kelvin型和非Kelvin型内尾迹研究方面的现状, 内容侧重于运动源生成内波的解析理论和分层拖曳水槽中内尾迹实验方面的研究成果. 介绍了在连续分层流体中运动源生成的Kelvin型非线性内波的一般方程和在间断分层流体中Kelvin型内波的势流分析的一般方法; 概述了运动源诱生的先锋内孤立子、代数孤立子和平孤立波3类特殊非线性内波的研究进展, 其中运动潜体生成的平孤立内波被作者实验证实是一类极限孤立波, 并首次建立了共轭流动模型予以描述; 综合分析了在密度线性分布流体中潜体运动生成内波的动力学过程多样性特征, 其中包括内尾迹近场和远场的时空结构、不稳定结构、涡旋与湍流耦合结构以及湍流与内波相互作用结构等.      

The effect of surfactants on the instability of a rotating liquid jet

It has long been known that the presence of surfactants on the free surface of a liquid jet can create surface tension gradients along the interface. The resulting formation of tangential stresses along the surface lead to Marangoni type flows and greatly affect the resulting dynamics of rupture. In this way surfactants can be used to manipulate the breakup of a liquid jet and control the size of droplets produced. In this paper we investigate the effects of insoluble surfactants on the breakup of rotating liquid jets with applications to industrial prilling. Using a long wavelength approximation we reduce the governing equations into a set of one-dimensional equations. We use an asymptotic theory to find steady solutions and then carry out a linear instability analysis on these solutions. We show that steady state centreline solutions are independent of viscosity to leading order and that the most unstable wavenumber and growth rate of disturbances decrease as the effectiveness of surfactants is increased. We also numerically solve these equations using a finite difference scheme to investigate the effects of changing the initial surfactant concentration and other fluid parameters. Our results show that differences in breakup lengths between rotating surfactant-laden jets and surfactant-free jets increase with the rate of rotation. Moreover, we find that satellite droplet sizes decrease as the rate of rotation is decreased with the effect of surfactants amplifying the reduction in sizes. Furthermore, the presence of surfactants at fixed rotation rates is shown to produce larger main droplets at low disturbance wavenumbers whilst satellite droplets are smaller for moderate disturbance wavenumbers κ≈0.7.     

B平面上斜压波热力结构特性的实验研究

首次用转环实验模拟方法研究了β效应对斜压波热力结构的影响,发现β效应有抑制流动的水平混合和垂直混合的作用,使流动趋于正压;β平面上急流随高度降低而减弱,在急流的内外两侧各有一个无量纲温度值分布的突跃区,它们的空间结构与大气环流中的极锋锋区和北极锋锋区的结构相似。     

Vibration analysis of a multi-span rotating ring with ray tracing method

A ray tracing method is applied to analyze both the free and forced responses in a rotating multi-span section ring. In this paper, element coordinates are established and coupled by dispersion and transmission matrices. Structure vibration displacements are expressed in a wave form with a combination of propagation, fast-attenuating and near-field waves. Meanwhile, an exciting force is considered as a point discontinues with different elements on both sides. The wave reflection and transmission matrices are introduced through coupling different elements by applying wave transmission coefficients and transfer matrices. For numeric computation, the reflection and transmission matrices are assembled, independent waveguide elements are integrated and the responses of rotating rings with non-uniform section area are derived. The structure modeling and a numeric computation with corresponding solutions illustrate the validity of the presented approach. The investigation result also shows that the presented approach can be extended to compute accurately on the dynamic characteristics of a rotating complex structure (high speed bearing cage).     

Quasi-periodic waves and quasi-solitary waves in stratified fluid of slowly varying depth

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper. The model considered here consists of a two-layer incompressible constantdensity inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall. The Korteweg-de Vries (KdV) equation with varying coefficients is derived with the aid of the reductive perturbation method. By using the method of multiple scales, the approximate solutions of this equation are obtained. It is found that the unevenness of bottom may lead to the generation of so-called quasi-periodic waves and quasisolitary waves, whose periods, propagation velocities and wave profiles vary slowly. The relations of the period of quasiperiodic waves and of the amplitude, propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented. The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.Project Supported by National Natural Science Foundation of China.     

Effect of baffles on sloshing modulated fluid force and torque fluctuations on the dewar driven by gravity gradient acceleration in microgravity

The dynamical behavior of fluids affected by the asymmetric gravity gradient acceleration is studied. In particular the effect of surface tension on partially liquid filled rotating fluids applicable to a full-scale Gravity Probe-B Spacecraft dewar tank with and without a baffle have been investigated. Results of slosh wave excitation along the liquid-vapor interface induced by gravity gradient acceleration indicate that the gravity gradient acceleration is equivalent to the combined effect of a twisting force and torsional moment acting on the spacecraft. As viscous force (between liquid and solid interface), and surface tension force (between liquid-vapor-solid interface) greatly contribute to the damping effect of slosh wave excitation, a rotating dewar with baffle provides more areas of liquid-solid and liquid-vapor-solid interfaces than that of a rotating dewar without baffle. Results show that the damping effect provided by the baffle reduces the amplitude of slosh wave excitation, lowers the fluid force, torque, and the moment arm of fluid torque fluctuations than that without baffle, and also lowers the degree of asymmetry in the liquid-vapor distribution.     

A generalized modified Kadomtsev-Petviashvili equation for interfacial wave propagation near the critical depth level

Propagation of interfacial waves near the critical depth level in a two-layer fluid system is investigated. We first present a generalized modified Kadomtsev-Petviashvili (gmKP) equation for weakly nonlinear and dispersive interfacial waves propagating predominantly in the longitudinal direction of a slowly rotating channel with gradually varying topography and sidewalls. For certain type of non-rotating channels, we find two families of periodic-wave solutions, which include solitarywave solutions and a shock-like solution as limiting cases, to the variable-coefficient gmKP equation. We also show that in this situation the gmKP equation has only unidirectional N-soliton solutions and does not allow soliton resonance to occur. In a rotating uniform channel, our small-time asymptotic analysis and numerical study of the gmKP equation show that, depending on the relative importance of the cubic nonlinearity to quadratic nonlinearity, the wavefront of a Kelvin solitary wave may curve either forward or backward, trailed by a small train of Poincaré waves. When these two nonlinearities almost balance each other, the wavefront becomes almost straight-crested across the channel, and the trailing Poincaré waves diminish.     

Formation of a tripolar vortex in a stratified fluid

This paper reports on an experimental study of vortices in a stratified fluid. The vortices were generated by two different stirring devices, viz. a rotating sphere and a rotating bent rod. It was found that the vortices created with the rotating sphere are mostly axisymmetric and stable, whereas the vortices produced with the bent rod generally show instabilities, under certain conditions leading to the formation of a tripolar vortex. This report concentrates on this tripolar structure and presents quantitative information about the flow obtained through streak photography of tracer particles.     

Simulation of non‐hydrostatic,density‐stratified flow in irregular domains

A numerical model has been developed for simulating density‐stratified flow in domains with irregular but simple topography. The model was designed for simulating strong interactions between internal gravity waves and topography, e.g. exchange flows in contracting channels, tidally or convectively driven flow over two‐dimensional sills or waves propagating onto a shoaling bed. The model is based on the non‐hydrostatic, Boussinesq equations of motion for a continuously stratified fluid in a rotating frame, subject to user‐configurable boundary conditions. An orthogonal boundary fitting co‐ordinate system is used for the numerical computations, which rely on a fourth‐order compact differentiation scheme, a third‐order explicit time stepping and a multi‐grid based pressure projection algorithm. The numerical techniques are described and a suite of validation studies are presented. The validation studies include a pointwise comparison of numerical simulations with both analytical solutions and laboratory measurements of non‐linear solitary wave propagation. Simulation results for flows lacking analytical or laboratory data are analysed a posteriori to demonstrate satisfaction of the potential energy balance. Computational results are compared with two‐layer hydraulic predictions in the case of exchange flow through a contracting channel. Finally, a simulation of circulation driven by spatially non‐uniform surface buoyancy flux in an irregular basin is discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

连续分层流体中垂直薄板的水动力特性

研究了在线性连续分层流体中水波与半潜式刚性垂直薄板相互作用的问题. 在 Boussinesq近似下，基于分离变量法，导出了具有自由面的平面前进波的色散关系，建立 了半潜式刚性垂直薄板的反射与透射能量、水平波浪力的计算方法. 对给定的频率，当它大 于浮力频率时，流场中只有一种模态的平面前进波，当它小于浮力频率时，流场 中有无数多个模态的平面前进波，并证明了对每一种模态的入射波，其它每个模态水波的反射与透射能量是 相等的. 对水面漂浮和座底半潜式薄板的反射与透射能量，以及作用在薄板上的水平波浪力 进行了数值计算分析，表明了流体的线性连续分层效应对这些水动力的影响是不可忽视 的. 特别地，在入射波频率小于浮力频率时，与第1模态入射波的能量转化量及其对薄板产 生的水平波浪力相比，其它模态入射波的能量转化量及其对薄板产生的水平波浪力都要大得 多.     

Dam-break release of a gravity current in a stratified ambient

The ‘dam-break’ initial behaviour of an inviscid gravity current which is released from a lock and then propagates over a horizontal boundary at the base of a stratified ambient fluid is considered. Analytical and finite-difference solutions of the one-layer shallow-water equations are developed and compared for the linear stratification in a rectangular channel case, and corroborated by numerical solutions of the full two-dimensional Navier–Stokes equations. Extensions of the shallow-water solution to non-linear stratification, release from an elliptical reservoir, and axisymmetric geometry are also presented. The results indicate that the shallow-water formulation captures well the essential features of the motion, which are qualitatively similar to the non-stratified case, but with details modified by the stratification; in particular, the forward propagation of the head and the backward spread of the depression wave are reduced when the stratification increases.