Mixing In Stratified Fluids - WordPress
G%phy&Asrophya Fluid Dynamiu. 1979, Vol. 13, pp 3-230309-1929/99/1301-MY)3 S04.50/0O Gordan m d Bmsch scieoa Pnbiisbss hc., 1979Printcd in O m t MainMixing in Stratified FluidstP. F. LINDENDownloaded by [University of Cambridge] at 04:55 23 February 2012Department of Applied Mathematics & Theoretical Ph ysics, Silver Street.Cambridge. CB3 9EW. U.K.(Receiued October 26,1978; inftnal form Janwry 8,1979)Laboratory experiments on mixing in stratified fluids are examined in the light of somesimple energy arguments appiied to an anaiysis of rnixing at density interfaces. It is shownincreases,that as the stability of the flow, as measured by an overall Richardson number Ri,,,the velocity interface becomes considerably thicka than the density interface. This mismatchin interface thickness provida the extra kmetic energy rcquired for mixing in these stronglystratifíed flows. The behaviour of the flux Richardson number R f as a function of Ri,, is ais0discussed. It is found that the results from a number of different experiments are quitesimilar, and that Rfincreases frorn zero as Ri, does, reaches a maxirnum value (0.2I0.05)and then decreases again with further increase in Ri,,. Finally. some recent idem ofPosmentier (1977) on the formation of interfaoes in a stratified, turbulent flow are comparedwith observations.1. INTRODUCTJONThe fact that stable density stratification can have a substantial effect onmixing will be obvious to most of us. It is only necessary to visit a largeindustrial or urban complex when there is an atmospheric inversion to seeits influence. The stable stratification in the inversion layer inhibits thevertical mixing of pollutants keeping them concentrated near the ground,often producing unpleasant conditions. Indeed, a large part of theatmosphere is stably stratified and so the way in which mixing takes placein such a environment is an important determining feature of the globaldistributions of heat and water vapour. The oceans, too, are stablystratified with the warmest water at the top, and many phenomena dependupon the rate of vertical transpoiits of heat and salt due to turbuientmixing. A large number of industrial processes also involve mixing instably stratified environments. In the home stable density stratification is a*Invited Review Paper at the Geophysical Fluid Dynamics Symposium of the FifthEuropean Geophysicai Society Meeting-Strasbourg, 29 August-1 September, 1978.
Downloaded by [University of Cambridge] at 04:55 23 February 20124P. F. LINDENcommon feature with the warmest air near the ceiling of a room. The wayin which the air mixes in such a room is very important for our comfort.The dynamical effects of the stratification manifest themselves as abuoyancy force which acts as a restoring force whenever fiuid is displacedvertically from its equilibrium position. In order to mix fluid vertically it isnecessary to do work against this buoyancy force, and this work isextracted from the agency doing the mixing. If the fluid is being mixed bya turbulent flow, more energy may be required to work against thebuoyancy force than is available and the character of the flow may bealtered dramatically.In order to estimate the effects of the buoyancy force we need to relateit to the forces available to mix the fiuid. This is conveniently done byconsidering the flow between two horizontal planes separated by a verticaldistance D. We suppose that the lower plane is moving horizontally witha velocity AU relative to the upper, and that fluid at the lower plane ismaintained at a density difference Ap relative to that at the upperboundary. As we are interested in stable stratifícation, A p O. If g is thegravitational acceleration and p the mean density, the stratification ischaracterised by the non-dimensional variable Rio E gApD/p(AU)’. Wewill cal1 this an overall Richardson number as it relates to the grossparameters of the system.Suppose a flow is set up so that the vertical gradients of density andvelocity (i.e. Ap/D, AUfD) are constant. If we move a fluid parcel a smalldistance, d, then the change in potential energy is gApd’f2D and theavailable kinetic energy is p(AUd)’/2DZ. The ratio of potential energychange to kinetic energy change is then Rio. Consequently, if Rio 1 moreenergy is required to mix the fluid in this manner than is available in thevelocity chear. This result is independent of the vertical scale of theproposed motion.But we know, both from laboratory and field measurements, thatmixing does take place when Rio is large. In the ocean, for example, Rio 0(10) or larger, and yet turbulence and mixing have often beenobserved. We are led in such circumstances to relax the one constraint wehave imposed, namely, that the mean gradients of density and velocity areconstant. Hence, there must be non-uniformities in the velocity anddensity profiles. So we see that the presence of layers and interfaces isintimately bound up with turbulence in regions of high Rio. Then, Turner(1973, p. 320) has shown, we can match any local gradient Richardsonnumber with any Rio essentially because (ôU/ôz)*s (AU/D)’.There are many processes associated with density interfaces which havebeen identified in recent years, and a large fraction of the reasearch effortin this subject has been (and still is) the study of these individual
MIXING IN STRATIFIED FLUIDS5processes. Some examples are(i) Kelvin-Helmholtz biHows(ii) Breaking internaí and interfacial wavesDownloaded by [University of Cambridge] at 04:55 23 February 2012(iii) Lateral intrusions(iv) Double-diffusive convectionThese, and other, processes have been discussed in a recent review bySherman, Imberger and Corcos (1978).In this review I intend to look at the subject from a different viewpoint.One difiiculty of treating each process individually is that it does notreadily yield any general principles about mixing in stratified fluids. It isoften hard to relate one process to another or to fit them into any generalframework. This is reflected in the larger number of different experimentsthat have been carried out in the last twenty years or so. Often seeminglyconflicting results are obtained from what appear to be similar situationsand at present it is often difiicult to obtain answers to apparently simplequestions.In order to arrive at a more unified view some simple energy argumentswill be given for mixing at a density interface. In the light of thesearguments a number of different experiments will be examined. In so doingattention will be restricted to the case where the driving force for thernixing is mechanical in origin; for example, by driving a shear flow orstirnng the fluid in some way. Mixing by convective processes is excluded.The restriction of considering mixing at an interface is made for convenience. It is not a very severe restriction, however, as it recognises thefact that interfaces are an inevitable consequence of mixing at high Ri,,. Inthis respect we are dealing with mixing and turbulence when the stratification is dominant: the situation is not a smail perturbation fromturbulence in a homogeneous fluid.This raises another interesting question. In their investigations of theentrainment into a steady gravity current flowing down a slope, Ellisonand Tumer (1959) found that the entrainment virtually ceased at highvalues of Ri,. Thus we have the possibility that the stratification can causea collapse of the turbulence when it is strong enough. The partition ofenergy during mixing in strongly stratified flows is examined in section 4.A comparison is made between a number of different rnixing processes,and it is found that there is considerable similarity between them. Thefinal part of the paper shows how the behaviour of the flux Richardsonnumber with stability influentes the formation of interfaces caused by mixing.2. ENERGETICS OF INTERFACIAL MIXINGIn any mixing event in a stably stratified fluid. work is done t o change
6P. F. LINDENthe potential energy of the fluid. This is recognised by the raising of thecentre mass of the fiuid column as mixing occurs. It is necessary, therefore,that there be enough energy available to accomplish this mixing. Thisplaces quite severe, and revealing, restrictions on the possible outcome ofmixing events. We will deal with two cases.Downloaded by [University of Cambridge] at 04:55 23 February 2012(a) The splitting of an interfaceIt has been suggested (Woods and Wiley, 1972) that Kelvin-Helmholtzinstability on an interface may cause it to mix to form a uniform layerwith sharp interfaces on either side. This case has been discussed byTurner, (1973, p. 322) and he shows that if initial linear gradientsu uz,p po-pzsplit into a well-mixed layer of thickness H , the change in potential energyis -gjlH3/12 and the change in kinetic energy is u2H3/24. These two areequal in magnitude when Ri, gp/a2 0.5. Even ignoring viscous dissipation, an additional supply of kinetic energy is required to perform thismixing when Rio 0.5.(b) Altering the thickness of an interfaceObservations by Thorpe (1973) indicate that mixing produced byKelvin-Helmholtz instability leads to an approximately linear variationin density and velocity across the interface, but that the interface haschanged in thickness. Similar behaviour has been observed in otherexperiments and this is the case discussed below.Suppose, as shown on Figure 1, that stratified fluid of total depth 2Dconsists of two layers with uniform density and velocity separated by aninterface in which the gradients of density and velocity are constant. Wedenote the density and velocity of the lower layer by p i and U , ,respectively, and those of the upper layer by p z and U z . As a result ofmixing it will be supposed that the interface changes its thicknesssymmetrically about the centre line z D , but that the two layers on eitherside remain uniform.Conserving mass and horizontal momentum we findInitial K.E.-Final K.E. & I ( A U ) -( d i,) ;Initial P.E.-Final P.E. Qg(Ap& -ApfS.; - 3D2(Api- A p f ) ) .(2.1)(2.2)
2l-7MIXING IN STRATIFIED FLUIDSIiIIp2'"2IDDownloaded by [University of Cambridge] at 04:55 23 February 2012OLIp1IIIIIUu,IIIFIGURE 1. A diagrammatic represemtation of the vertical profiles of density (solid Iine) andvelocity (broken fine), before (a) and after (b) a mixing event. It is assumed that after mixing theprofiles have the same form, although the thicknesses of the transitionregions have been altered.Here kinetic energy is abbreviated to K.E. and potential energy to P.E.,AP P i -P z ,AU U1- Uz,d thickness of the velocity interface,6 thickness of the density interface.The subscripts i and f refer to the initial and final states, respectively. Ithas been assumed that AU remains unaltered; this is done simply forconvenience, and it turns out to be a good approximation in theexperiments to be discussed later.Before examining the experiments in detail we can draw two simpleconclusions from (2.1) and (2.2). From (2.1) we see that energy is onlyreleased from the velocity field if d, d,. Consequently, if an interface isobserved to get thinner as a result of mixing there must be someadditional source of energy. Indeed we need some turbulence in the layersto redistribute the mass from the interface through the layers.If, on the other hand, the thickness of the interface merely increaseswithout altering the densities in the iayers (as in Kelvin-Helmholtzinstability), the fact that the change in potential energy must be less than
8P.F. LINDENthe available kinetic energy impliesDownloaded by [University of Cambridge] at 04:55 23 February 2012If the velocity and density scales are equal thenThus we have a strict limit on the ultimate thickness an interface canreach by such a process.3. COMPARISON WITH LABORATORY EXPERIMENTSIn this section a number of different laboratory experiments will bediscussed in the light of the above energy argument. Only a very briefdescription of the experiments will be given ; for further details the readeris referred to the original papers.The first class of experiments concerns the mixing at an interfacebetween two horizontal layers of fiuid moving with different horizontalvelocities. These were performed by Koop (1976) who brought the twolayers together at the edge of a splitter plate, and observed the downstream evolution as mixing between the twg streams developed. As thePrandtl number (o V / K ; kinematic viscosity/molecular diffusivity) for hisflow was 0(103),the initial thickness of velocity interface was much largerthan the density interface. So to a good approximation di 0, and the flowis characterised by an overall (or initial) Richardson numberwhere AU is the velocity difference between the layers.From (2.3) we see that, with d i 0 ,Now x(l -x)s& for O x c 1 and so
Downloaded by [University of Cambridge] at 04:55 23 February 2012MIXING IN STRATIFIED FLUIDS9Thus, as Ri, increases, the final velocity interface becomes appreciablythicker than the final density interface. This is a result of the fact that athigh stabilities it is necessary to decelerate more of the fiow in order tomix the fluids.It is not possible to do much more than make a quaiitative comparisonwith Koop's results, due to ambiguities in determining d,/6, caused by thepresence of waves and molecular diffusion which continue to thicken theinterface after the turbulence has decayed. In any case (3.1) representsonly a strict lower bound on d f / 6 , as it ignores viscous dissipation.However, Koop finds that d f / S , increases from approximately 2 at thelowest initial Richardson number to about 7 at the largest, in qualitativeagreement with these ideas.Another experiment in this class was performed by Moore and Lqng(1971). They produced two layer, counter-current flow in an annular tank.They drove the flow by small directed jets of fiuid injected at the upperand lower boundaries, and were able to set up steady-state mixingexpenments by simultaneously withdrawing fluid at these boundaries.Experiments were run over a range of overall Richardson number 0.7 Ri, 61.5, where Ri, (gApD)/p(AU)2. Here D is the depth of thechannel, AU the velocity difference and Ap the density difference betweenthe layers. Vertical profiles of velocity and density across the channel weremeasured and a number of interesting results emerge. The first is shownon Figure 2 which is a plot of data contained in Table 1 of their paper.The figure shows the steady-state thickness of the velocity interface as afraction of the channel depth plotted against Ri,. At low values of Ri,,, theinterface occupies up to 80% of the channel width, but it becomes thinneras Ri, increases. For values or Ri, 10 the thickness becomes roughlyconstant with dlDNO.1.Suppose, for the moment, that the steady-state velocity and densityinterfaces have equal thickness. Then assuming di O and writing d d, in(2.4),we getAiternatively,D Riõ'dThe curve d / D R i õ l is shown on Figure 2. The fact that this curve andthe data points lie close together has some important consequences. For
Downloaded by [University of Cambridge] at 04:55 23 February 201210P. F.LINDENc9OI1OI'9I9IOoinINIoI
Downloaded by [University of Cambridge] at 04:55 23 February 2012MIXING IN STRATIFIED FLUIDSI1Rio lO, it seems that the upper bound on d/D given by (3.2) is almostattained. This means that the steady-state thickness is limited by theenergetics of the mixing which has been derived ignoring viscous dissipation. This is presumably because, although energy is continuously beinglost to dissipation it is continually supplied by forcing the flow. This is incontrast to mixing experiments in which the energy supply is limited (as inKoop’s case), where the viscous dissipation severely reduces the amount ofenergy for mixing and must be considered.For Rio 10 we see that d/D violates the criterion given in (3.2) and, aswe have already noted, the interface thickness remains roughly constant.Now (3.2) implies that d/D-O as Rio m. In an apparatus of finite sizethis implies that thickness of the velocity interface vanishes, and so thegradient Richardson number must vanish as Rio 03. However, then theinterface would be unstable to Kelvin-Helmholtz billows, and this leads toa thickening of the interface. This is a contradiction and so it is necessaryto violate (3.2). The only way to do this is to relax the supposition thatthe velocity and density interfaces have the same length scales. This iswhat is observed. From the profiles published in their paper we see thatd/6 1.5, 2.0 and 3.0 for Rio 2.2, 4 and 15, respectively. Thus, as inKoop’s experiment, at high stabilities the velocity interface becomessignificantly thicker than the density interface, which allows the flow tobecome unstable at ali values of Rio.Another feature of Moore and Long’s experiment which deservescomment is the fact that they were able to set up and maintain a steadystate. It was noted in section 2 that mixing produced by the mean shearcontinually increases the thickness of the interface. So it is essential tohave some way of removing mixed fluid from the interface in order tokeep its thickness constant. In their experiment this is provided byturbulence in the layers produced by the jets of injected fluid at the topand bottom boundaries of the tank. Thus we see that forcing the flow inthis way both maintains the shear across the interface and also produces aflux of density across the layers on either side of the interface. This seemsto be the reason why the criterion (3.2) applies to this flow. It is envisagedthat from an initially sharp interface, once the forcing is begun thethickness of the interface increases until the limit described by (3.2) isreached. Further increase in d is not possible as there is insufficient energyin the mean shear, and the turbulence in the layers, by removing mixedfluid, is tending to reduce d rather than increase it.A consideration of the smail scaie turbulence induced by the jets inMoore & Long’s experiments leads us to the second class of experimentswhich can be discussed in these terms. These are flows in which the twolayers are stirred mechanically, but where there is no mean shear across
Downloaded by [University of Cambridge] at 04:55 23 February 201212P. F. LINDENthe interface. These experiments are usually conducted by stirring thelayers with two horizontal grids of bars which are oscillated verticallywith a small amplitude about their mean position (usually in the centre ofeach layer). This type of experiment has recently been reviewed by Turner(1973).For the moment we shall concentrate only on one aspect of theseexperiments. One of the first things to be noticed (Cromwell, 1960) wasthat the stirring tended to produce sharp density interfaces; a resultcontrary to one’s initial expectation. These experiments are examples ofwhat Turner (1973, p. 116) calls “external mixing processes”; i.e. theenergy for the mixing is supplied from a region (the centre of the layers)external to the region where the mixing occurs (the interface). We notedearlier that the observation that an interface became thinner with timeimplied that energy was being supplied in addition to that which isavailable locally in the mean shear. Hence, the question arises: doexternal mking processes always lead to sharp interfaces?In order to answer this question, consider the case of two layersseparated by a density interface of thickness 6, stirred continuously. Fluidis transported across the interface and so the density difference Apdecreases with time. The centre of mass of the system is being raised bythe mixing, and so the potential energy of the density stratificationincreases with time. From (2.2) we get that the time rate of change ofpotential energy is given byRearranging this equation we obtain an expression for the change of interface thickness with time: viz,asat - - (1 6 . t a(P.E.) g(302 - 6 2 f g ) .(3.3)Now a(P.E.)/òt O as exp
2l MIXING IN STRATIFIED FLUIDS 7 D I i Ip2 I '"2 I I I I u, p1 I I I IU OL - I I FIGURE 1.A diagrammatic represemtation of the vertical profiles of density (solid Iine) and velocity (broken fine), before (a) and after (b) a mixing event.It is assumed that after mixing the profiles have the same form, although the thicknesses of the transition regions have been altered.
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stratified fluids, with emphasis on the practical application of this work to pipe flows. The paper first of all discusses a fundamental parameter relevant to mixing in stratified fluids, the Richardson number, and then briefly outlines the basis of the theoretical calculations, stressing the assumptions that
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against gravity (Turner 1973). Thus, the problem of turbulence and mixing in a density- stratified fluid is complex : it is dependent on the interaction of two dynamic scales, one due to mechanical turbulence and the other from the buoyancy of the density field. Previous efforts to understand mixing processes in density-stratified fluids have
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Mixing Newtonian Fluids The ability of the modeling framework given by Eq. 4to capture the mixing of Brownian particles over more than five orders of Péclet number is clearly demonstrated in Fig. 3 A using three different Newtonian fluids (water μ 0.001 Pa·s, 20:80 wt % water:glycerol μ 0.054 Pa·s, and glycerol μ 1.2 Pa·s, Table S1).
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