Characteristics of an interfacial wave in a horizontal air-water stratified flow

Interfacial wave parameters, in this case the frequency, height, velocity, and slope, were investigated experimentally in a horizontal air-water stratified flow. Experiments were conducted with a parallel wire conductance sensor and PIV visualization in a rectangular channel, of which the width and height are 40 mm and 50 mm, respectively. In the experiments, the flow condition covered the liquid Reynolds number Re_l range of 450 to 3540 and the gas Reynolds number Re_g range of 14,000 to 70,000. The results revealed that the observed wave types according to the flow conditions in the rectangular channel are similar to those in a horizontal pipe. The frequency, height, and slope of the interfacial wave show complicated tendencies according to the combination of Re_g and Re_l, which affects the coalescence and breakup of the wave. Specifically, the wave height and wave slope have opposite tendencies regarding the criterion of Re_g = 34,000. For cases in which Re_g  ≥  34,000, the interfacial drag force significantly affects the height and slope of the disturbance wave. In contrast, for Re_g < 34,000, the growth of the wave has an important effect on the wave parameters. Finally, new empirical correlations for the frequency, height, and slope of the interfacial wave were proposed for application to the development of a droplet entrainment model in a horizontal stratified flow.     

The solitary waves in stratified shear flow

In this paper, the basic equation of internal long waves in stratified shear flow is derived under Boussinesq assumption, the first order approximation solution is given for solitary waves with the effects of slowly varying topograph at the sea bottom, weak stratification and basic shear flow. The Project Supported by the National Natural Science Foundation of China.     

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.     

On the propagation of a three-dimensional packet of weakly non-linear internal gravity waves

The evolution of a three-dimensional packet of weakly non-linear internal gravity waves propagating obliquely at an arbitrary angle to the vertical line is considered. Two coupled non-linear equations connecting variations of a packet amplitude and induced flows are derived. three-dimensionality of the packet having been found responsible for the non-linearity of the system. Explicit formulae for the induced flow vertical component and the mean density field variation caused by packet propagation have been obtained. The plane wave is shown to be unstable at any arbitrary slope of the wave vector. The non-linear equation describing the evolution of the two-dimensional packet is derived in the subsequent order of the asymptotic scheme.It has been found possible for the packet to collapse. The collapse of internal waves packets may be one of the possible mechanisms of “blini”-shaped regions of mixed waters formation in the ocean.     

Linear theory of the propagation of internal wave beams in an arbitrarily stratified liquid

Beams of harmonic internal waves in a liquid with smoothly changing stratification are calculated in the Boussinesq approximation taking into account the effects of diffusion and viscosity. A procedure of local reduction of the beam in a medium with an arbitrary smooth stratification to the case of an exponentially stratified liquid is constructed. The coefficient of energy losses in the case of beam reflection on the critical level is calculated. Parameters of internal boundary flows with split scales of velocity and density that are formed by a wave beam on discontinuities of the buoyancy frequency and its higher derivatives are determined. Institute of Problems of Mechanics, Russian Academy of Sciences, Moscow 117526. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 88–98, September–October, 1998.     

A numerical study of vorticity layers in a two-dimensional stratified shear flow

A numerical study is conducted to find out the conditions of occurrence of a secondary Kelvin-Helmholtz instability in the thin layers (referred to as baroclinic layers) that form in a stably-stratified shear layer. For this purpose, three high resolution calculations of a moderately stratified shear layer have been carried out, at a fixed Reynolds number. The wavelength of the initial perturbation is progressively increased, starting from the fundamental wavelength predicted by linear stability theory up to twice this fundamental wavelength. The baroclinic layer of the flow is shown to lengthen and destabilize progressively from one calculation to the other, eventually bearing a secondary Kelvin-Helmholtz instability. The structure and dynamics of the baroclinic layers of the three calculations are examined in the frame of a theoretical model proposed by Corcos and Sherman ([1]). An excellent agreement with the predictions of this model have been found. We next show that the stability of the layer is controlled by the large-scale Kelvin-Helmholtz vortex, via the strain field that it induces in the stagnation point region of the layer. A consequence of this study is that secondary Kelvin-Helmholtz instabilities are fostered by the pairing of primary Kelvin-Helmholtz vortices in a strongly-stratified shear layer.

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Sommario E stato condotto uno studio numerico per trovare le condizioni in cui insorge una instabilità secondaria di Kelvin-Helmholtz negli strati sottili che si formano in uno strato di scorrimento stabilmente stratificato. A questo scopo sono state effettuate tre simulazioni ad alta risoluzione a fissato numero di Reynolds e stratificazione bassa. La lunghezza d'onda della perturbazione iniziale è stata progressivamente aumentata dalla lunghezza fondamentale predetta dalla teoria lineare della stabilità fino a due volte questa stessa lunghezza. È stato osservato che da una simulazione all'altra lo strato baroclino del flusso si allunga e si destabilizza progressivamente, generando eventualmente un'instabilità di Kelvin-Helmholtz secondaria. Utilizzando il modello teorico proposto da Corcos e Sherman (1976), per le tre simulazioni sono state analizzate la struttura e la dinamica dello strato baroclino. È stato trovato un accordo eccellente con le predizioni di questo modello. È stato in seguito mostrato che la stabilità dello strato è controllato dai vortici di Kelvin-Helmholtz di larga scala attraverso il campo di deformazione che inducono nella regione del punto di ristagno dello strato. Una conseguenza di questo studio è che le instabilità secondarie di Kelvin-Helmholtz sono forzate dall'accoppiamento dei vortici primari in uno strato di scorrimento fortemente stratificato.

     

Simultaneous visualization of density and velocity variations in a stratified shear flow

Two flow visualization techniques, the shadowgraph and the hydrogen bubbles, are combined to facilitate simultaneous observation of the vertical velocity profile and the density interfaces in a stratified mixing layer in an annulus. The experimental setup is described and some examples of the resulting photographs are presented and discussed.     

Mean flow generated by an internal wave packet impinging on the interface between two layers of fluid with continuous density

Internal waves propagating in an idealized two-layer atmosphere are studied numerically. The governing equations are the inviscid anelastic equations for a perfect gas atmosphere. The numerical formulation eliminates all variables in the linear terms except vertical velocity, which are then treated implicitly. Nonlinear terms are treated explicitly. The basic state is a two-layer flow with continuous density at the interface. Each layer has a unique constant for the Brunt–Väisälä frequency. Waves are forced at the bottom of the domain, are periodic in the horizontal direction, and form a finite wave packet in the vertical. The results show that the wave packet forms a mean flow that is confined to the interface region that persists long after the wave packet has moved away. Large-amplitude waves are forced to break beneath the interface.     

Effect of homogeneous shear flow on small long-lived perturbations in a stratified fluid

Conclusions In the transitional stage of the evolution of small perturbations in a flowing stratified fluid various structures may be formed. The form of these structures will depend on the initial conditions, the velocity shear, the stratification, and the dissipative characteristics of the fluid. In some cases in the intermediate stage the perturbations may increase.However, at large times in the final dissipative stage all the perturbations decay and the structure of the longest-lived of them is fairly simple. Owing to the shear effect the surviving perturbations are highly elongated in the direction of flow. However, under the influence of the stratification, from all the perturbations with _h L/g in the transverse plane perturbations of two types are gradually filtered out: internal waves and thin horizontal layered structures.In the presence of shear flow the layered structure in a stratified fluid is not only preserved but even becomes more sharply expressed owing to the suppression by the shear if many of the competing perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–102, March–April, 1988.     

Effect of molecular diffusion on stratified shear flow stability

The results of theoretical analysis, which show that molecular heat or salt diffusion can stabilize the flow, are experimentally confirmed with reference to the problem of the motion of a density-stratified fluid towards an opening in a vertical wall. An asymptotic regime in which neither stratification nor diffusion affects the integral characteristics of the flow is established.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 35–40, January–February, 1993.The authors are grateful to O. F. Vasil'ev for his valuable contribution to the organization of the research.     

Large amplitude internal gravity waves in stably stratified fluids: Linear model

Summary In this paper, we consider a linear model for internal waves in a stratified fluid. The equations are generalised by Fourier analysis after the manner of Scorer(1949) who considered an infinite atmosphere.

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Sommario Si considera un modello lineare per le onde interne in un fluido stratificato. Le equazioni sono generalizzate con un'analisi di Fourier secondo il metodo di Scorer(1949) che ha considerato un'atmosfera infinita.

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Deformations of an elastic pipe submitted to gravity and internal fluid flow

This paper describes the deformation of an elastic pipe submitted to gravity and to an internal fluid flow. The pipe is clamped horizontally at one end and free at the other end. As the fluid velocity increases, the shape changes from an elastic beam deflected by its own weight towards an horizontal position. The shape of the pipe is characterized experimentally and is compared with a theoretical model based on the Euler–Bernoulli approximation and the conservation of the fluid momentum. We study how the determination of the pipe deformation provides an estimation of the conveyed fluid flow. Finally, the vertical force produced by the conveyed fluid to lift off a mass is deduced.     

Velocity measurements in the field of an internal gravity wave by means of speckle photography

Speckle velocimetry with forward scattering has been applied to measure and visualize the two-dimensional velocity field in an internal gravity wave. The wave was produced by towing a cylinder in vertical direction, normal to its axis, through stratified salt water. Neutrally buoyant tracer particles whose density was matched with the density distribution of the stratification were uniformly distributed in the test fluid. The experimental results verify the results of a linear theory in the far field of the wave.Most part of this work was performed when the three authors were with Institut für Thermo- und Fluiddynamik, Ruhr-Universität Bochum.     

Stability of unstably stratified shear flow between parallel plates *

The linear stability of unstably stratified shear flows between two horizontal parallel plates has been investigated. The eigenvalue problem was solved numerically by making use of the expansion method in Chebyshev polynomials, and critical Rayleigh numbers were obtained accurately in the Reynolds number range of [0.01,100]. It was found that the critical Rayleigh number for two-dimensional disturbances increases with an increase of the Reynolds number. The result strongly supports previous stability analyses except for the analysis by Makino and Ishikawa (1985) in which a decrease of the critical Rayleigh number was obtained. For some cases, a discontinuity in the critical wavenumber occurs, due to the development of two extrema in the neutral stability boundary.     

Structure of stratified flow around a cylinder at low internal froude number

The flow pattern around a horizontal cylinder towed at constant velocity in a continuously stratified fluid is visualized by the shadow method. The velocities in the leading flow disturbance, i. e., in the flow-blocking region ahead of the cylinder, are presented. In the body wake, a new class of small-size structures in the density gradient field is revealed against the background of a smooth velocity profile. The evolution of the flow pattern with variation of the parameters of body motion is studied. Institute of Mechanics Problems, Moscow 117526. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 80–88, January–February, 1999.     

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.