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.

     

Lateral squeezing of horizontally stratified viscous layers

A fluid mechanics model for the slow deformations of layered geophysical materials is adopted. The problem consists of the slow viscous flow of two horizantally stratified fluid layers of different viscosities as being laterally squeezed between two frictionless planes. For solution a perturbation scheme is applied and the resulting approximation equations are solved numerically. The interface and free surface shapes, and the streamline patterns within the fluid layers are given.     

Convective plumes associated with local sources of a heat releasing impurity in a stably stratified medium

A class of self-similar solutions is found for steady convective plumes (axisymmetric and flat, laminar and turbulent) in a stably stratified medium associated with local sources of a weightless heat releasing impurity. The boundary layer approximation is used. Characteristic of such convection is the regime with neutral buoyancy, in which the growth in the buoyancy associated with the heat release is compensated by a reduction due to the ascent to less dense layers of the medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 156–158, January–February, 1984.     

Stability of stratified shear flows in channels with variable cross sections

For the instability problem of density stratified shear flows in sea straits with variable cross sections, a new semielliptical instability region is found. Furthermore, the instability of the bounded shear layer is studied in two cases: (i) the density which takes two different constant values in two layers and (ii) the density which takes three different constant values in three layers. In both cases, the dispersion relation is found to be a quartic equation in the complex phase velocity. It is found that there are two unstable modes in a range of the wave numbers in the first case, whereas there is only one unstable mode in the second case.     

Conjugate shear flows of a weakly stratified fluid

This paper studies the problem of pairs of horizontal shear flows of weakly stratified fluids with identical mass, momentum, and energy fluxes. The initial problem is reduced to a system of two scalar equations for the main- and perturbed-flow parameters by using bifurcation methods. The existence conditions for nontrivial branches of conjugate flows close to the main flow are investigated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 79–88, March–April, 2009.     

Critical layer formation in a stratified medium with mean shear flows

The problem of the excitation of internal waves with a given wave number k and frequency in a stratified medium with shear flows is considered. The internal wave field of the form v(z)exp(–it+ikx) established as t in a medium without dissipation has a singular point at the level z=z₀ (critical level), at which the flow velocity U(z) coincides with the phase velocity /k. Dissipative effects (viscosity and heat conduction) smooth out this singularity. An exact solution of the model equation describing as t and zz₀ the field excited by oscillating sources activated at t=0 is constructed with allowance for dissipation. This makes it possible to describe the limiting steady-state field, determine the critical layer as the neighborhood of the critical level in which dissipation effects are important, and to estimate its width and the rate of convergence to the limiting steady-state regime. The asymptotic behavior of the fields is examined for Ri1, where Ri is the Richardson number. It is shown that when the well-known Miles stability condition Ri>1/4 is satisfied there are no natural oscillations with a critical level.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 82–93, May–June, 1990.     

Space correlation measurements in a forced shear layer with wall injection

Comprehension of wall-injection flow in a channel in the presence of different geometric discontinuities is necessary as part of the general investigations concerning combustion instabilities of solid propellant rocket motors. In order to characterise the aerodynamic flow field and to evaluate the influence of an obstacle inside a porous channel in such a case, experimental studies were conducted on a 1/40 scale model of the new ARIANE V motor. In fact, the flow is only induced by wall-injection and the presence of an obstacle creates a particular shear layer development in the obstacle wake. Particular attention was given to the unstable dynamic conditions of the shear layer. A thermal seeding of the shear layer was made in order to qualify the heat transfer therein, and especially to emphasise the turbulent structure development. Transverse and longitudinal spatial correlations were measured to characterise turbulence scales in the shear layer. At the origin of the shear layer, the decay of turbulence memory is found to be similar to that observed in a forced flow boundary layer, but the injecting wall modifies the change in structures. The wall flow is found to preserve the turbulent structures in such a way that the turbulence memory predominates in a longitudinal direction.List of Symbols f frequency (Hz) - h channel height (m) - h _v obstacle height (m) - L mean dimension of large structures (m) - P pressure (Pa) - m mass flow rate (kg/s) - r distance between probes in X direction (m) - r ^+(–) distance between probes in Y direction (m) - u longitudinal velocity in X direction (m/s) - v transversal velocity in Y direction (m/s) - T temperature (K) - x,y,z axis system (m) - t = T-T _amb (K) - v kinematic viscosity (mVs) - density (kg/m³) - characteristic porous size (urn) - _u longitudinal rms (m/s) - I _u dynamic turbulence intensity _u /u _max - I _T thermal turbulence intensity - M Mach number u/a - Re _w wall Reynolds number v _w h/v - Re _c longitudinal Reynolds number u _c h/v - R _uT thermal dynamic correlation coefficient - St Strouhal number fh/u - X,Y,Z axis system normalised by the channel height h - X _S longitudinal position of the obstacle - X X-X _S - amb ambient reference - c flowing cavity - fav head end - l lateral direction - g longitudinal direction max maximum at a longitudinal position - w wall The authors thank the CNES for its financial support, and in particular E. Robert and R. Bee.     

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.     

Decay of shear layers and vortex sheets

An incompressible fluid is contained in the domain between two stationary infinited parallel rigid plates. It is assumed that for shear flows, the shear stress in an element of the fluid depends linearly on history of the velocity gradient in that element. It is supposed that initially two steady shear layers exist in the fluid and are symmetrically disposed with respect to the mid-plane. The time-dependent velocity field which results from the removal of the forces maintaining this steady flow is calculated in the cases when the fluid is Newtonian and when it is Maxwellian. The limiting cases when the shear layers reside in an unbounded space of the fluid and when they further become vortex sheets are discussed.     

Baroclinic generation of streamwise vorticity in a stratified shear layer

The existence of an organized streamwise vortical structure, which is superimposed on the well known coherent spanwise vorticity in nominally two-dimensional free shear layers, has been studied extensively. In the presence of stratification, however, buoyancy forces contribute to an additional mechanism for the generation of streamwise vorticity. The purpose of the current investigation is to force the three-dimensional instability in the stratified shear layer. In this manner, we experimentally observe the effect of buoyancy on the streamwise vortex tube evolution, the evolution of the buoyancy-induced instability, and the interaction between these two vortical structures. It is found that streamwise vortices resulting from vortex stretching are weakened in regions of locally stable stratification. Buoyancy-induced vortex structures are shown to form where the unstable part of the interface is tilted by the streamwise vortex tubes. These vortices strengthen initially, then weaken downstream, the time scale for this process depending upon the degree of stratification. For initial Richardson numbers larger than about 0.03, the baroclinically weakened vortex tubes eventually disappear as the flow evolves downstream and the baroclinically generated vortices dominate the three-dimensional flow structure.

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Sommario L'esistenza di una struttura vorticosa nella direzione della corrente, che è sovrapposta alla ben nota vorticità trasversale coerente in flussi di scorrimento (free shear layer) nominalmente bidimensionali, è stata ampiamente studiata. In presenza di stratificazione, però, forze di galleggiamento contribuiscono ad un meccanismo addizionale per la generazione di vorticità nella direzione della corrente. In questa indagine si forza l'instabilità tridimensionale dei flussi stratificati. In questa maniera si osserva sperimentalmente l'effetto del galleggiamento sull'evoluzione della vorticità nella direzione della corrente, l'evoluzione della instabilità indotta dal galleggiamento e l'interazione tra queste due strutture vorticose. Si trova che i vortici nella direzione della corrente, risultanti dall'allungamento del vortice, sono indeboliti in regioni con stratificazione localmente stabile. Si osserva, inoltre, che le strutture di vortici incotte dal galleggiamento si formano quando la parte stabile dell'interfaccia viene ruotata dal tubo di vortice nella direzione della corrente. Questi vortici si rafforzano inizialmente e s'indeboliscono a valle della corrente. Per questo processo la scala temporale dipende dal grado di stratificazione. Per numeri di Richardson inizialmente più grandi di circa 0,03, i tubi di vortici baroclinicamente indeboliti eventualmente spariscono non appena il flusso evolve lungo la corrente ed i vortici generati baroclinicamente dominano la struttura di flusso tridimensionale.

     

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.     

Laboratory experiments of a jet impinging on a stratified interface

 The entrainment rates of vertical and inclined jets impinging on a stratified interface are measured in water tank experiments. At moderate Richardson number, the entrainment rate of the vertical jet is proportional to Ri ^-1/2, independent of Reynolds number. The inclined jets are tilted at 15° from the vertical. In one case, the jet nozzle is rotated about a vertical axis, so that the inclined jet precesses, while in the other, it is stationary. The inclined jets entrain at a rate proportional to Ri ^-3/2, whether precessing or not. This behavior is consistent with a new model of stratified entrainment which accounts for vortex persistence. Received: 15 October 1996/Accepted: 19 December 1996     

Laboratory experiments on internal wave interactions with a pycnocline

Laboratory experiments have been performed to investigate the interaction of internal waves with a pycnocline. An oscillating cylinder generated internal wave beams, which were observed using the synthetic schlieren technique. Internal waves incident on the pycnocline layer excited higher-frequency modes. In the absence of shear, a discrete spectrum of harmonic modes was generated due to nonlinear effects. These harmonic modes might play a role in the formation of internal solitary waves which have been observed in ocean pycnoclines. With shear, a continuous spectrum of excited modes was found.     

Suppression of forced and convective flows of a stably stratified fluid in vertical channels

In order to improve certain technological processes, for example, single-crystal growing, it is desirable to be able to control the flow rate in order to influence the heat and mass transfer processes. For this purpose it is usual to employ rotation, an electromagnetic field or reduced gravity [1]. Here, with reference to simple solutions of the system of equations of free convection in infinite vertical channels, it is shown that the problem of reducing the intensity of the flow can be solved given a suitable relation between the degree of stable stratification (with respect to density) and the factors responsible for the flow. The possibility of using temperature stratification is considered, but all the conclusions are also fully applicable to concentration stratification.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 172–174, March–April, 1987.The authors wish to thank B. Ya. Martuzan for his useful comments and interest in the work.     

Observations of modal interactions in resonantly forced beam-mass structures

We present a collection of experimental results on the influence of modal interactions (i.e., internal or autoparametric resonances) on the nonlinear response of flexible metallic and composite structures subjected to a range of resonant excitations. The experimental results are provided in the form of frequency spectra, Poincaré sections, pseudo-phase planes, dimension calculations, and response curves. Experimental observations of transitions from periodic to chaotically modulated motions are also presented. We also discuss relevant analytical results. The current study is also relevant to other internally resonant structural systems.