Internal waves around a disturbance in a fluid with arbitrary stratification and background shear flow

A three-dimensional ray theory is presented for calculating the phase configuration of internal waves around a moving disturbance in a flow with arbitrary stratification and background shear. The theory is applied to two-dimensional stratified shear flows which have been produced in the laboratory and good agreement is shown between the theoretical and experimental phase configurations. Good agreement is also shown when caustics and critical levels are present. This paper includes the wave systems arising from a combined translation and oscillatory motion of a source and it shows how distinct systems of waves arise from each, type of motion under different background conditions. This paper shows that for two-dimensional steady wave systems the critical level is at a well defined height which is independent of wavenumber but in three dimensions the critical height can in general vary with wavenumber.     

Internal gravity waves in critical generation modes and in the vicinity of trajectories of motion of perturbation sources

The field of internal gravity waves in a layer of an arbitrary stratified fluid is studied for critical generation modes and in the vicinity of trajectories of motion of the perturbation sources. The exact solutions describing the structure of a separate mode of the wave field in the vicinity of the perturbation source in the critical generation modes are investigated, and expressions for the total field representing the sum of all wave modes are obtained. In the vicinity of the trajectories of the perturbation sources, asymptotic representations of the eigenfunctions and eigenvalues of the basic vertical spectral problem of internal waves are constructed in the approximation of large wave numbers and asymptotic expressions for a separate mode of the wave field are obtained that describe the spatial structure and features of the fields of internal gravity waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 70–79, September–October, 2008.     

The phase configuration of trapped internal waves from a body moving in a thermocline

Some two-dimensional internal wave systems are generated experimentally by a cylinder moving with constant velocity in a thermocline modelled with stratified brine. The phase configuration of the waves is shown to compare quite well with ray theory, and the phase changes which result from energy reflections are well defined in some of the schlieren photographs.     

The generation of three-dimensional internal waves and attendant boundary layers in a viscous continuously stratified fluid. Construction of an analytical solution

An analytical solution of a linearized problem of the emission of periodic internal waves by part of a plane which oscillates with a small amplitude in an arbitrary direction in a viscous exponentially stratified fluid is constructed. Solutions of the dispersion equation are given for all positions of the emitting surface (arbitrary, vertical, horizontal, and critical when one of the beam propagation directions is collinear with the emitting surface). The possibility of transition to the case of a uniform fluid, which is important for applications, is analyzed.     

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     

Calculation and measurement of conical beams of three-dimensional periodic internal waves excited by a vertically oscillating piston

The structure of the wave field given by an exact solution of the linearized problem of radiation of three-dimensional periodic internal waves in a continuously stratified viscous fluid is analyzed numerically. The waves are generated by a piston, i.e., a disk lying on a fixed horizontal plane and oscillating in the vertical direction. The flow fields and the wave displacements are compared with the data of shadow visualization and measurements of the wave amplitudes made using a contact sensor. The calculated and observed wave patterns are in satisfactory agreement and the displacement distributions coincide correct to a fitting coefficient 0.7 < K < 1.1 characterizing the role of the nonlinear effects and other factors neglected in this model.     

Internal waves in a two‐layer system using fully nonlinear internal‐wave equations

In order to understand the nonlinear effect in a two‐layer system, fully nonlinear strongly dispersive internal‐wave equations, based on a variational principle, were proposed in this study. A simple iteration method was used to solve the internal‐wave equations in order to solve the equations stably. The applicability of the proposed numerical computation scheme was confirmed to agree with linear dispersion relation theoretically obtained from variational principle. The proposed computational scheme was also shown to reproduce internal waves including higher‐order nonlinear effect from the analysis of internal solitary waves in a two‐layer system. Furthermore, for the second‐order numerical analysis, the balance of nonlinearity and dispersion was found to be similar to the balance assumed in the KdV theory and the Boussinesq‐type equations. Copyright © 2009 John Wiley & Sons, Ltd.     

Solitary waves in stratified fluids and their interaction

A systematic procedure is proposed for obtaining solutions for solitary waves in stratified fluids. The stratification of the fluid is assumed to be exponential or linear. Its comparison with existing results for an exponentially stratified fluid shows agreement, and it is found that for the odd series of solutions the direction of displacement of the streamlines from their asymptotic levels is reversed when the stratification is changed from exponential to linear. Finally the interaction of solitary waves is considered, and the Korteweg-de Vries equation and the Boussinesq equation are derived. Thus the known solutions of these equations can be relied upon to provide the answers to the interaction problem.     

Numerical modeling of the generation of internal waves by uniform stratified flow over a thin vertical barrier

Numerical simulations of two‐dimensional stratified flow past an obstacle (thin vertical strip) were performed at relatively low Reynolds numbers. A finite differences solver was adopted to simultaneously solve Navier–Stokes equations together with transport equations for salinity (stratifying agent), and the standard Smagorinsky turbulent closure scheme was called in whenever necessary to account for turbulence. The emphases were on the evaluation of code for unsteady stratified flow applications as well as identification of transient and steady internal‐wave processes during flow past obstacles. Simulations were compared with laboratory experiments, where observations were made using a high resolution Schlieren technique and conductivity probes. Blocking was observed upstream of the obstacle, surrounded by near‐zero frequency internal waves, the phase lines of which joined those of lee waves through a transition zone in the proximity of the obstacle. This pattern was preceded by initial transients of the starting flow in which propagating internal waves played a dominant role. Confluence of isopycnals passing over/under the obstacle in the wake led to interesting flow phenomena, including the radiation of internal waves. The numerical simulations were in good agreement with observations, except that some phenomena could not be captured due to resolution issues of either numerical or experimental techniques. The efficacy of the code in point for stratified flow calculations with relevance to the atmosphere and oceans was confirmed. Copyright © 2011 John Wiley & Sons, Ltd.     

Explicit conditions for the existence of exceptional body waves and subsonic surface waves in anisotropic elastic solids

It is known that a subsonic surface (Rayleigh) wave exists in an anisotropic elastic half-space x₂  0 if the first transonic state is not of Type 1. If the first transonic state is of Type 1 but the limiting wave is not exceptional, a subsonic surface wave exists. If the first transonic state is of Type 1 and the limiting wave is exceptional, a subsonic surface wave exists when . It is shown that an exceptional body wave is necessarily an exceptional transonic wave, and could be an exceptional limiting wave. Only two wave speeds are possible for an exceptional body wave. We present explicit conditions in terms of the reduced elastic compliances for the existence of an exceptional body wave. If an exceptional body wave exists, conditions are given for identifying whether the transonic state is of Type 1. Hence, through the existence of an exceptional body wave we provide explicit conditions for the existence of a subsonic surface wave with the exception when needs to be computed.     

简化Holmboe分层流体密度剖面的形成及其内波实验

本研究在内波水槽中形成了具备Ｈｏｌｍｂｏｅ模式基本特征的密度剖面，在此流场中成功地进行了第一模态单板造波内波实验，并进而对半板造波的方案给予了分析和检验。     

Evolution of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid

The problem of the stabilization of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid is solved using both asymptotic and numerical methods. The analytical solution describes the structure of the main convective cells, including thin meridional jets flowing along the surface and plumes spreading from the flow convergence regions above the upper and lower poles of the sphere which gradually return the fluid particles to the neutral buoyancy horizon. The total width of the flows adjacent to the surface exceeds the thickness of the salinity deficit layer or the density boundary layer. The numerical solution of the complete problem in the nonlinear formulation describes the main convective cells and two systems of unsteady integral waves formed in the vicinity of the sphere poles. At large times, out of the entire system of internal waves only those nearest to the neighborhood of their horizon of formation remain clearly defined. The calculated flow patterns are in agreement with each other and the data of shadow visualization of the stratified fluid structure near a submerged obstacle at rest.     

分层流体中混合流体团运动生成内波的研究现状

本文综述了分层流体中混合流体团运动生成内波的研究进展状况,系统地介绍了它的应用背景、实验和理论结果以及有待进一步研究的问题。      

Hydrodynamics of natural oscillations of neutrally buoyant bodies in a layer of continuously stratified fluid

The flow patterns induced by floats of different shapes (sphere, short and long cylinders) freely sinking to the neutral-buoyancy horizon in a continuously stratified fluid are investigated using optical methods. General flow elements, both large-scale (waves, vortices, hydrodynamic wake) and fine-scale (boundary layers, extended autocumulative jets), are distinguished. For large times, the float oscillation frequencies are of the order of or greater than the buoyancy frequency of the medium. This indicates the significant effect of the induced flows on the motion of the float.     

Generation of Periodic Motions by a Disk Performing Torsional Oscillations in a Viscous, Continuously Stratified Fluid

The solution of the problem of nonlinear generation of periodic internal waves by a boundary flow on a vertical cylinder or a horizontal disk performing torsional oscillations in an exponentially stratified fluid is constructed. The calculations are in satisfactory agreement with the results of experiments in which both horizontal and inclined disks of various diameters and a model propeller performing periodic torsional oscillations, including oscillations against a background of uniform rotation, are used as perturbation sources. The experiments were carried out over a wide range of parameters including the laminar, transition, and turbulent flow regimes. The limits of applicability of the proposed analytic theory of wave radiation are determined.     

Direct Numerical Simulation of Turbulence in a Stably Stratified Fluid and Wave-Shear Interaction

Turbulence decay in a strongly stratified medium is simulated by a direct pseudo-spectral code solving the three-dimensional equations of motion under the Boussinesq approximation. The results are compared to non-stratified simulations results. We focus on the production of mean shear energy observed in the stratified case. We then simulate the decay of stratified turbulence when affected by an initial horizontal mean flow and show that this mean flow is the major component remaining at large t. Next, we give some analytical elements on wave-shear interaction by using a simple refraction calculation with WKB hypothesis. This calculation is illustrated by simulating the interaction between one monochromatic internal wave and a vertical shear profile. We conclude that the existence of singularities in the mean shear production term in the presence of internal gravity waves may be one of the possible mechanisms involved within stratified turbulent shear flows.     

Breaking and cascade of internal gravity waves in a continuously stratified fluid

The evolution of a few large scale high frequency standing internal waves confined to a vertical plane is studied numerically. The growth of nonlinear interactions leads to a transfer of energy toward small vertical scales and lower frequencies: the result is a steep energy decrease due to wave breaking. Induced mixing is evaluated. A parametric forcing is also introduced in order to compare with laboratory experiments. Wave breaking also occurs but as opposed to the unforced case different phases are next observed: internal wave growth due to constructive forcing alternate with energy decrease.     

Experimental Study of the Dynamic Effect of an Internal Solitary Wave on a Submerged Circular Cylinder

The hydrodynamic loads due to the interaction of a small-amplitude internal solitary wave with a submerged circular cylinder in a two-layer system of miscible fluids were studied experimentally. The dependence of the internal-wave transmission coefficient on the position of the center of the cylinder relative to the pycnocline and on the ratio of the cylinder diameter to the fluid-layer thickness was obtained. The effects of the pycnocline thickness and the depth of the center of the cylinder on the value of the hydrodynamic loads were studied. Visualization of the flow structure was performed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 36–44, November–December, 2005.     

Phase velocity spectrum of internal waves in a weakly-stratified two-layer fluid

The problem of steady-state internal waves in a weakly stratified two-layer fluid with a density that is constant in the lower layer and depends exponentially on the depth in the upper layer is considered. The spectral properties of the equations of small perturbations of a homogeneous piecewise-constant flow are described. A nonlinear ordinary differential equation describing solitary waves and smooth bores on the layer interface is obtained using the Boussinesq expansion in a small parameter.     

Numerical simulation of internal gravity waves using a lattice gas model

Internal waves are modelled in two different circumstances: in a continuously stratified fluid and at the interface between two immiscible fluids. This is done using the lattice gas approach. The standard single phase model and an immiscible two-phase model are both modified to incorporate gravitational interactions. Standing internal waves are set up in both models and are seen to oscillate under the action of the gravitational interaction. The results obtained suggest that the lattice gas approach can be a useful tool in the modelling of such phenomena. © 1998 John Wiley & Sons, Ltd.