Optical visualization of internal gravity waves in stratified fluid

The refractive index changes associated with flows in salt water systems allow such flows to be visualized by means of optical methods, e.g. schlieren and interferometry. Experiments that have been conducted by our group with internal gravity waves in stratified brine are reviewed. The experiments encompass visualization and quantitative measurement of internal gravity waves generated by a body oscillating around a fixed position and a test body moving with constant speed vertically through the stratified brine. It is also shown that the velocity field of an internal gravity wave can be measured by means of particle image velocimetry. References to respective wave theories are made.     

Over-reflection of travelling internal waves in a rotating stratified fluid

Summary It has been known for many years that waves of different kind propagating in a fluid and incident on a mean flow localized shear layer can exhibit, in the linear theory, a reflection coefficient larger than unity. When this phenomenon takes place the wave is said to be over-reflected at the shear layer. In this paper the over-reflection of inertial-gravity waves in a stratified rotating fluid in horizontal sheared motion is studied by making use of the constancy of the net vertical energy flux associated with the wave field. It is found that over-reflection can take place if some physical conditions, explicitly evaluated, are locally satisfied in the fluid. Such conditions are discussed and a comparison is carried out with the same phenomenon in a nonrotating system.

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Riassunto Si sa da molti anni che la teoria lineare di onde di diversa natura, propagantesi in un fluido ed interagenti con uno strato localizzato di shear, prevede, in alcuni casi, un coefficiente di riflessione maggiore dell'unità. Questo fenomeno prende il nome di super riflessione. In questo articolo si studia la super riflessione delle onde gravito-inerziali in un fluido stratificato in moto orizzontale con la velocità dipendente dalla sola coordinata verticale. Lo studio è portato a termine servendosi della condizione di continuità e costanza del flusso netto verticale di energia associata al campo d'onda. Si trova che la super riflessione avviene se sono soddisfatte alcune condizioni fisiche che sono esplicitamente determinate: tali condizioni sono inoltre discusse in dettaglio ed i risultati sono confrontati con quelli, classici, relativi alla propagazione di onde interne nei fluidi stratificati in assenza di rotazione.

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Резюме В течение многих лет известно, что линейная теория волн различной природы, которые распространяются в жидкости и взаимодействуют с локализованным пограничным слоем, предсказывает, что в некоторых случаях коэффициенты отражения могут превьщать единицу. Зто явление носит название супер-отражения (сверх-отражение) на пограничном слое. В этой статье исследуется супер-отражение инерциальной гравитационной волны в стратифицированной вращающейся жидкости в случаэ горизонтального движения со скоростью, зависящей только от вертикальной координаты. При рассмотрении учитывается постоянство результирующей вертикальной энергии, связанной с полем волны. Получено, что сверх-отражение может иметь место, если локально в жидкости выполняются некоторые физические условия. Проводится обсуждение этих условий и полученные результаты сравниваются с классическими результатами по распространению внутренних волн в стратифицированной жидкости в отсутствии вращения.

     

线性密度分层流体中半球体运动生成内波的实验研究

在密度线性分布的分层流试验水槽中,对贴近侧壁面的半球体模型运动生成的内波进行了实验研究. 采用多通道电导率阵列,测量分析了拖曳模型产生内波的波形结构、垂向位移场及其相关速度等. 实验结果表明:半球体模型产生的内波可以分为两类,一类是由定常源激发的体积效应内波,另一类是由非定常源激发的尾迹内波,由前者转换为后者的临界内Froude(Fr)数是Fr_s =1.6;与球体模型实验比较,Fr_s约为球体的2/3,转换更迅速,界线更清 关键词： 分层流 半球模型 内波 尾迹     

Nonlinear internal waves in a stratified fluid in a closed basin

Summary The problem of two layers of homogeneous inviscid incompressible fluid of density ϱ₁ and ϱ₂ (ϱ₁>ϱ₂) separated by a free surface in a closed basin is studied via a variational principle. A system of ordinary differential equations in the time variable, which approximate the original problem, is derived and discussed.

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Riassunto Si usa un principio variazionale per studiare il problema del moto di due fluidi omogenei incompressibili non viscosi di densità ϱ₁, ϱ₂ (ϱ₁>ϱ₂) separati da una superficie libera in un bacino chiuso. Dopo alcune approssimazioni si deriva e discute un sistema di equazioni differenziali ordinarie.

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Резюме С помоыью вариационного принципа исследуется проблема двух слоев однородной невязкой несжимаемой жидкости с плотностями ϱ₁ и ϱ₂ (ϱ₁>ϱ₂), раазделенных свободной поверхностью в замкнутом резервуаре. Выводится и обсуждается система обыкновенных дифференциальных уравнений по временной переменной, которая аппроксимирует исходную проблему.

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Work done under the auspices of C.N.R.-G.N.A.F.A.

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Partially supported by NSF under grant PHY77-18762.     

A set of Boussinesq-type equations for interfacial internal waves in two-layer stratified fluid

Many new forms of Boussinesq-type equations have been developed to extend the range of applicability of the classical Boussinesq equations to deeper water in the study of the surface waves. One approach was used by Nwogu (1993. J. Wtrw. Port Coastal and Oc. Eng. 119, 618--638) to improve the linear dispersion characteristics of the classical Boussinesq equations by using the velocity at an arbitrary level as the velocity variable in derived equations and obtain a new form of Boussinesq-type equations, in which the dispersion property can be optimized by choosing the velocity variable at an adequate level. In this paper, a set of Boussinesq-type equations describing the motions of the interfacial waves propagating alone the interface between two homogeneous incompressible and inviscid fluids of different densities with a free surface and a variable water depth were derived using a method similar to that used by Nwogu (1993. J. Wtrw. Port Coastal and Oc. Eng. 119, 618--638) for surface waves. The equations were expressed in terms of the displacements of free surface and density-interface, and the velocity vectors at arbitrary vertical locations in the upper layer and the lower layer (or depth-averaged velocity vector across each layer) of a two-layer fluid. As expected, the equations derived in the present work include as special cases those obtained by Nwogu (1993, J. Wtrw. Port Coastal and Oc. Eng. 119, 618-638) and Peregrine (1967, J. Fluid Mech. 27, 815-827) for surface waves when the density of the upper fluid is taken as zero.     

Two-dimensional evolution equation of finite-amplitude internal gravity waves in a uniformly stratified fluid

We derive a fully nonlinear evolution equation that can describe the two-dimensional motion of finite-amplitude long internal waves in a uniformly stratified three-dimensional fluid of finite depth. The derived equation is the two-dimensional counterpart of the evolution equation obtained by Grimshaw and Yi [J. Fluid Mech. 229, 603 (1991)]. In the small-amplitude limit, our equation is reduced to the celebrated Kadomtsev-Petviashvili equation.     

具有密度跃层分层流体中回转体激发内波特性实验

在具有连续密度跃层的分层流体中,对长径比为7:1的回转体在迎水和背水运动下激发体积效应与尾迹效应内波特性开展了系列实验.结果表明,体积效应激发内波属于一种相对于回转体定常的多模态Lee波结构,而尾迹效应激发主控内波为相对于回转体非定常的拟Lee波结构,这是一类由湍流尾迹中大尺度相干结构作为移动源激发的内波结构,在Lee波与拟Lee波之间存在一个与长径比近似为线性关系的临界转捩Froude数Fr_c,当FrFr_c 时拟Lee波为主控内波,而且拟Lee波相关速度Froude数近似为一个常数0.8,其无因次峰-峰幅值随Fr的增大近似线性增大,其中Fr为回转体特征直径Froude数.结果还表明,回转体头部与尾部几何形式并不影响其激发内波临界转捩Froude数Fr_c、 拟Lee波相关速度Froude数及其峰-峰幅值变化特性.     

Turbulence of near-inertial waves in the continuously stratified fluid

By using the normal form of continuously stratified “primitive” equations of geophysical fluid dynamics with density (in the ocean), or potential temperature (in the atmosphere) playing the role of the vertical coordinate, we decouple vortex and wave motions in the system, introduce normal variables, and derive the effective Hamiltonian for waves with frequencies close to the inertial frequency (near-inertial waves, NIW). We then apply the weak turbulence approach to the random-phase ensembles of these waves. We show how the anisotropic scale-invariance of NIW may be exploited in order to obtain the stationary power-law spectra. The non-decay anisotropic scale-invariant dispersion laws of the NIW-type were not studied previously in the weak-turbulence literature.     

Hydraulic simulation of negatively buoyant plumes

Summary The results of a hydraulic simulation of denser-than-air plumes in a neutral atmosphere are presented and discussed. The behaviour of these amission is of particular practical interest in connection with problems of geothermal drilling. The experimental data, concerning the height and the horizontal distance from the source of the maximum plume rise, as well as the distance where the plume touches ground, were compared with simple analytical formulae available in the literature. It was found that plume rise is described with sufficient accuracy by Meroney's model; for touchdown distance an empirical formula, obtained fitting the data from the present experiment, is proposed. It was seen, moreover, that in the examined range of variability of the emission parameters the best value to be adopted for the entrainment coefficient is β=0.4.

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Riassunto Si presentano e discutono i risultati di uno studio in vasca idraulica del comportamento in atmosfera neutra di getti piú densi dell'aria, di particolare interesse in relazione a problemi di perforazione di pozzi geotermici. I dati sperimentali, riguardanti il massimo sovrainnalzamento dei pennacchi, la distanza orizzontale dalla ciminiera a cui tale massimo viene raggiunto, nonché la distanza alla quale il pennacchio ricade al suolo, sono stati confrontati con semplici formule analitiche reperibili in letteratura. Si è trovato che il ?plume rise? è descritto con sufficiente accuratezza dal modello di Meroney, mentre per la distanza di ricaduta si propone una formula empirica ottenuta per interpolazione dei dati di questo esperimento. Si è visto inoltre che nell'intervallo qui esaminato di variabilità dei parametri di emissione il valore piú opportuno da assegnare al coefficiente di ?entrainment? à β=0.4.

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On coupled waves in stratified CGL-plasma

Starting from the Chew-Goldbsrger-Low equations we derive wave-equations describing small amplitude disturbances in a horizontally stratified, continuously varying CGL-plasma. A set of equations of first-order matrix form is treated by the method of Clemmow and Heading.     

有限深密度分层流体中运动物体生成内波的一种等效质量源方法

采用一种等效移动质量源来模拟有限深密度分层流体中运动物体生成内波的体积效应和旋涡、湍流尾迹激发源,结合内波本征值问题及Fourier变换等方法,建立了计算运动物体生成内波垂向位移场的一种理论模型,提出了确定移动质量源速度、回转直径及长度的具体方法.利用该理论模型,对一类典型密度分层流体中运动球生成内波的波系与波形结构,以及波高特征等进行了数值模拟与分析,结果与Robey实验结果符合良好,表明了所建立理论模型的合理与有效性. 关键词： 分层流体 内波 等效质量源     

On the localization of 2D nonlinear internal waves in a two-layer fluid

A 2D generalized Gardner equation is used to describe 2D nonlinear internal waves in a two-layer fluid. Unlike the previous model based on the Kadomtsev-Petviashvili equation, the model considered here allows for the instability of a plane internal solitary wave. Such a possibility causes the wave to be localized in any direction. Relationships between the thicknesses and densities of the layers under the instability conditions are obtained.     

Far fields of internal gravity waves in stratified media of variable depth

The present paper is devoted to presenting the solution of the problem on the far field of internal gravity waves in a stratified medium whose depth linearly depends on the space variable. Uniform asymptotic representations of solutions are constructed; these representations enable us to describe the far fields of internal gravity waves excited by a moving source and characterize the main specific features of the wave fields near caustics and wave fronts.     

Soaring interfaces, vortices and vortex systems inside the internal waves wake past the horizontally moving cylinder in a continuously stratified fluid

The flow pattern around a horizontal cylinder towed at constant velocity along isopycnic plane in a continuously stratified liquid is visualized by conventional techniques of “Vertical slit-Foucault’s knife”, “Maksoutov’s slit-thread” and “horizontal slit-regular grating”. Using sensitive high-resolution methods allows detail studying such component of stratified flow structures as soaring interfaces, singular soaring vortices and vortex systems, which arise directly inside the internal waves field past the cylinder. These flow elements having high level of vorticity are separated from the downstream wake by a strip of fluid without any small-scale inhomogeneities. Formation of singular vortex dipoles on leading edges of soaring interfaces is investigated in details in a wide range of flow parameters.     

A large-eddy simulation of buoyant plumes in a convective boundary layer

Summary By means of a large-eddy simulation technique, the time-dependent large-scale flow field of a convective atmospheric boundary layer has been claculated. Using a conservation equation for the concentration dispersion we applied this simulation technique to dispersion from both passive and buoyant sources. The results were found to be in good agreement with laboratory and field experiments. It was possible to separate the mean plume motion into a part induced by convective turbulence and a part induced by plume buoyancy. We found that the part of the plume motion caused by convective turbulence was strongly influenced by plume buoyancy. On the other hand, it appears that the large-scale motions of convective turbulence have a strong influence on the entrainment processes governing the motion due to buoyancy. Paper presented at the GNFAO/EURASAP Meeting, Turin, September 1989. To speed up publication, proofs were not sent to the authors and were supervised by the Scientific Committee.     

Coherent X-rays excited by a relativistic electron crossing a periodic stratified structure in Bragg scattering geometry

The dynamic theory of coherent X-rays excited by a relativistic electron crossing an artificial periodic stratified structure in Bragg scattering geometry has been developed for the general case of asymmetric reflection. Expressions describing the spectral-angular characteristics of radiation have been derived and investigated.     

Shear surface acoustic waves in the stratified structure piezoelectric—High-temperature granular superconductor

The interaction of a Gulyaev-Bluestain surface wave with a granular high-temperature superconducting (HTSC) medium has been investigated. For piezoelectrics of symmetry 4mm and 6mm, dispersion equations have been derived that describe the characteristics of surface acoustic waves (SAWs). The temperature dependences of the SAW attenuation and phase have been calculated forZnO andBa ₂Si₂TiO₃ crystals. It is shown that at temperatures higher than the critical temperature an attenuation jump and a phase shift are observed. The effect intensifies with increase in the electromechanical coupling coefficient and with decrease in the thickness of the HTSC film. For theBa ₂Si₂TiO₃ crystal the attenuation jump and phase shift are11 dB/cm and38 deg/cm, respectively, at a frequency of820 MHz. The results obtained can also be generalized for periodic HTSC structures and can be used to design frequency-selective devices and fast-response bolometric photodetectors. Tomsk State University of Control Systems and Radioelectronics. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 49–52, April, 1999.