A review of heat transfer between concentric rotating cylinders with or without axial flow

ReviewA review of heat transfer between concentric rotating cylinders with or without axial flowM.Fénot *,Y.Bertin,E.Dorignac,lizelInstitut P,Cnrs,ENSMA e Universitéde Poitiers,UPR 3346,Département fluides,thermique,combustion,1Avenue Clément Ader,BP 40109,86961Futuroscope Chasseneuil Cedex,Francea r t i c l e i n f oArticle history:Received 29July 2010Received in revised form 11February 2011Accepted 12February 2011Available online xxx Keywords:Rotating cylinders Heat transfer Taylor e CouetteTaylor e Couette e Poiseuille Slotted gapa b s t r a c tHeat transfer in flow between concentric rotating cylinders,also known as Taylor e Couette flows,constitutes a long-existing academic and industrial subject (in particular for electric motors cooling).Heat transfer characteristics of those flows are reviewed.Investigations of previous works for different gap thickness,axial and radial ratio,rotational velocity are compared.Con figurations with axial flow and/or with slots on the cylinders are also considered.For each case,different correlations are presented.Finally,unresolved issues are mentioned for further research.Ó2011Elsevier Masson SAS.All rights reserved.1.IntroductionFlow dynamics between two concentric rotating cylinders constitutes an old academic subject since Couette [1]and Taylor [2],whose names are recalled in the term Taylor e Couette flow,which has become a reference in stability studies due to the gradual destabilizing of a flow lending itself to a rigorous mathematical approach.Moreover,this kind of flow has many industrial appli-cations,particularly in the fields of mechanical or chemical mixing equipment.It has consequently been the subject of several biblio-graphic reviews by Di Prima and Swinney [3],Cognet [4],and Maron and Cohen [5].The heat transfer in this flow and the impact of flow structures on heat transfer were more recently studied (Gazley [6]);there already existed numerous industrial applications of the rotating elements (rotation,outer wall of the rotating heat pipes,cooling of the lower extremities of the turbojet turbine .),especially in electric motors.Indeed,different studies [7,8]on the heat transfer of electric motors have demonstrated the importance of convective heat transfer within the cylindrical gap (area separating the rotor from the stator).In fact,the rotor is the locus for large-scale dissipations of electromagnetic origin,and its cooling is ensured principally by the air flow of the cylindrical gap.Two main families of rotating electric machines maybe distinguished;closed machines (rotor rotation without axial air flow:Taylor e Couette flow),and open machines (axial flow combined with rotor rotation:Taylor e Couette e Poiseuille flow).Moreover,as regards some motor technologies,the existence of grooves on the rotor (where copper threads,for example,may be coiled)is liable to signi ficantly modify the dynamic and the thermal flow behavior.Excepted a short part of the bibliographic review by Maron and Cohen [5],no very detailed analysis of heat transfer in a rotating annular gap has ever been carried out.This bibliographic review is focused on the heat transfer of Taylor e Couette flow patterns.The kind of thermal behavior to be studied is obviously linked to the dynamics of these kinds of flow.We shall present in detail the different forms of flow already encountered,but our survey is not exhaustive.2.The fundamentals of Taylor e Couette flowLet us first look at the different parameters of in fluence.We shall consider a basic system composed of two concentric cylinders (Fig.1).Its geometry is characterized by two radii,the outer radius of the inner cylinder R 1and the inner radius of the outer cylinder R 2,as well as their length L .The flow is then characterized by the following geometric parameters:hydraulic diameter:D h ¼ð4S p =P m Þ¼ð2½p ðR 22ÀR 21Þ =p ðR 2þR 1ÞÞ,annular gap thickness (alsoknown as cylindrical gap):e ¼R 2ÀR 1,radial ratio h ¼R 1/R 2,and axial ratio:G ¼L /R 2ÀR 1.*Corresponding author.E-mail address:fenot@let.ensma.fr (M.Fénot).Contents lists available at ScienceDirectInternational Journal of Thermal Sciencesjou rn al homepage:/locate/ijts1290-0729/$e see front matter Ó2011Elsevier Masson SAS.All rights reserved.doi:10.1016/j.ijthermalsci.2011.02.013International Journal of Thermal Sciences xxx (2011)1e 18The main dynamic parameter characterizing cylindrical gapflow is the rotor rotation speed u (rad/s).One may consequently de fine the tangential Reynolds number Re t ¼u R 1D h /n .The Taylor number is often preferred to the Reynolds number,for it may be interpreted as the ratio between centrifugal force and viscous force [9].It may be expressed in the form Ta ¼ðu 2R 1ðR 2ÀR 1Þ3=n 2Þð1=F g Þwith F g as a geometrical factor that differs from one author tothe next and may take into account the aspect ratio of the smooth annular gap.This ratio approaches 1once the cylindrical gap has become narrow;this is the number used by most authors.Gardiner and Sabersky [10]and Becker and Kaye [11]propose:F g ¼ðp 4ððR 1ÀR 2Þ=ð2R 1ÞÞ=1697ð0:0571ð1À0:652ðe =R 1ÞÞÞþ0:00056ð1À0:652ðe =R 1ÞÞÀ1Þ.Some authors (Ref.[12]for example)prefer the formulation Ta ¼ðu 2R 1ðD h =2Þ3=n 2Þwhich may take into account possible geometrical variations by hydraulic diameter.It should be noted that several authors,particularly in the more ancient studies,de fine the Taylor number as the square root of the classical form.In addition,heat transfer is de fined by the Nusselt number,whose de finition may likewise vary from one author to the next.Today ’s most widely used de finition is:Nu ¼ðhD h =l Þ¼ðÀl grad T j p =T w ÀT ref ÞðD h =l Þ¼ðF =S ðT w ÀT ref ÞÞðD h =l Þ,where the reference temperature is often postulated as equal to the fluid temperature or else,more generally,to the temperature of the second wall.The Nusselt number thereby deduced serves as a translation of the ef fi-ciency of heat transfer from one wall to the other by means of the fluid.As for the heat transfer surface S ,it is often de fined as the surface of the heating wall (S ¼2$p LR 1or S ¼2$p LR 2[13])Sometimes this surface is considered as intermediate with regard to the walls,and its exact de finition differs from one author to the next (generally S ¼2$p L ðe =ln ð1=h ÞÞ[14]).The more narrow the cylindrical gap,the smaller the differences between the de finitions of S .Fig.1.Taylor e Couette flow geometry.M.Fénot et al./International Journal of Thermal Sciences xxx (2011)1e 182Some authors do not use the hydraulic diameter as the char-acteristic length,but rather prefer the thickness of the annular gap:e ¼R 2ÀR 1.In the following pages,we will use the most widely accepted de finitions:Ta ¼ðu 2R 1ðD h =2Þ3=n 2Þand Nu ¼ðhD h =l Þ¼ðÀl grad T j p =T w ÀT ref ÞðD h =l Þ.3.Smooth cylindrical gap,Taylor e Couette flowIn this part,we shall focus on flow and convective heat transfer in a smooth annular gap subjected to inner cylinder rotation alone.A description of the flow structure due to rotation alone (Tay-lor e Couette flow)will be given prior to a presentation of the results concerning heat transfer.3.1.Flow structureFlow in a smooth and closed annular gap has been the subject of numerous theoretical and experimental studies.The first experi-ments were conducted by Couette [1]and Mallock [15]in 1887and 1888.Most studies involve a single rotating cylinder (generally the inner one).The rotation of the second cylinder makes more complex the flow as recorded by Andereck et al.[16](Fig.2).It is important to note that in this study a strict procedure has been followed to reduce the problem range (first,the outer cylinder is slowly accelerated,and then the inner cylinder is slowly acceler-ated as well).The following flow description will involve only one rotating cylinder con figurations since no heat transfer studies have been conducted on two rotating cylinders.3.1.1.Couette flowCouette observed that the torque required for rotation of the central cylinder rises linearly to a critical speed,and then the torque rises still more rapidly.In 1923,Taylor [2]highlighted the existence of a critical rotation speed u C .Under this rotational speed,the flow,referred to as Couette flow,is steady and laminar.It is driven by viscous drag force acting on the fluid.The streamlines are annular and centered on the rotation axis.Natural convection induced by gravity can greatly affect this flow.It is generally recognized that for a Rayleigh number Ra ¼ðPr $b $D T $e 3g =n 2Þlower than 104,the effects of natural convection are negligible.For a higher Rayleigh number,flow depends on the ratio between the Grashof number and the tangential Reynolds number Re t ¼u R 1D h /n (Guo and Zhang [17]).So,the effects of natural convection occasioned by gravity are preponderant when rotation speed is particularly low.Along with an increase in rotation speed (and in Re t ),the effects of natural convection seem to disappear and are replaced by classical cylin-drical gap flow.When none of the effects are dominant,structures with two or three cells are obtained [18].These effects are also dependant on the way the cylinder is oriented,that is to say vertically [18,19]or horizontally [20,21].3.1.2.Taylor vortex flowOnce the critical rotation speed u C noted by Taylor is exceeded,the flow presents instabilities structured in a toric (O ring)form and known as “Taylor vortices ”.They are counter-rotative and associ-ated by pairs,as is indicated in Fig.3.From both an experimental and a theoretical standpoint,Taylor [2]determined the critical value of rotor rotation speed for an in fi-nitely long and pronouncedly narrow annular gap Ta c ¼1708z 1700.As soon as they appeared,Taylor vortices are arranged periodically by pairs,which rendered it possible to de fine their axial wavelength as the axial space taken up by a doublet of vortices.TheaxialFig.2.Regimes observed in flow between two rotating cylinders [16].Fig.3.Taylor vortices [6].M.Fénot et al./International Journal of Thermal Sciences xxx (2011)1e 183wavelength of a pair of vortices for the critical Taylor number (or critical wavelength)is generally slightly less than the theoretical critical wavelength for an in finitely long and pronouncedly narrow cylindrical gap.Kirchner and Chen [22]showed that Taylor vortices originate,at the level of the inner cylinder,in the form of axially and radially propagated discs,which are transformed into classical vortices.Ghayoub et al.[23]provided a number-based demonstra-tion of the appearance of small and non-stationary Gortler-type vortices in the zone where Taylor vortices leave the stator.In cases where the rotation speed is still rising and the Taylor number exceeds a second critical threshold,(Ta/Ta c ¼1.2),Coles [24](for a narrow cylindrical gap:h ¼0.95)underscored the pres-ence of an azimuth wave regime (wavy mode);the Taylor cell boundaries are no longer perpendicular to the cylindrical axis,but rather present undulations or waves (Fig.4).The flow consequently becomes doubly periodic.As speed was gradually heightened,Cognet [4](once again with regard to a narrow cylindrical gap:h ¼0.908)observed a rise in the number of azimuth waves up to a maximum number that remains constant for 4.5 Ta/Ta c 25,and subsequently decreases until the azimuth waves disappear (Ta/Ta c z 96).The movement then becomes virtually periodic.Moreover,experiments by Coles [24]entailed the observation of up to 26different states (number of vortices .)for the same Taylor number.Each state depends not only on rotation speed,but also on previous flow history,which involves hysteresis effects marked by transitions that differ with regard to increasing and then decreasing rotation speeds.Each of these states corresponds to a given number of pairs of vortices (number of axial waves)and of azimuth waves (number of azimuth wave periods).Cylindrical gap geometry likewise assumes an important role with regard to the transitions.As concerns annular space thickness,Sparow et al.[25]and Roberts [26]have used linear stability theory when showing that the wider the cylindrical gap (small h ),the more frequent the appearance of vortices corresponding to high Taylor numbers.Cognet [4]provided experimental con firmation of these results (Fig.5).In addition,Snyder and Lambert [27]showed that in large annular gaps (small h ),Taylor cells are less rapidly affected by azimuth waves.Another parameter liable to exert in fluence on flow transitions is the axial length of the cylindrical gap.The results gathered by Cole [28]facilitate comprehension of the role assumed by annular gap length.In Fig.6and in accordance with axial cylindrical gap length,we may note:the rotation speed starting at which Taylorvortices appear (curve i),the rotation speed starting at which their waves appear (curve ii),and the corresponding theoretical numbers ( and *).Fig.6shows that regardless of cylinder length,vortices show up at essentially the same rotation speed.On the other hand,cylindrical gap length has a far greater impact on wave appearance.When cylinder length goes up,rotation speed goes down exponentially and is stabilized close to its theoretical number once the cylindrical gap has lengthened (G >40).Lastly,the presence of a temperature gradient can likewise affect flow stability.This is due to variable fluid properties (density and viscosity),to the natural convection occasioned by centrifugal forces,and to the natural convection occasioned by gravity,which is generally considered separately.Becker and Kaye [11]as well as Walowit [29]have shown that while a negative radial gradient stabilizes flow (the critical Taylor number increases),a positive gradient has the opposite effect.According to Walowit [29],the effects of the convection occa-sioned by centrifugal forces are negligible and the variations in fluid properties take on the main role.As regards the effects of natural convection occasioned by gravity,Ali and Weidman [30]have shown that the flow depends on the ratio between the Grashof numbers and the tangential Reynolds numbers Re t and that,as regards the Couette flow,the effects are negligible for a Rayleigh number Ra lower than 104.Gardiner and Sabersky [10]and Aoki et al.[31]have observed,in their strictly thermal studies,an unusually high critical Taylor number (104instead of 1700),which they attribute to the in fluence of natural convection in their experiments.Their conclusionsleadFig.4.Azimuth wave regime (wavymode).Fig.5.In fluence on the transition of annular gap thickness [4].Fig.6.Transitions and cylinder length [28].M.Fénot et al./International Journal of Thermal Sciences xxx (2011)1e 184one to believe that natural convection modifies e and stabilizes e cylindrical gapflow.3.1.3.TurbulentflowFinally,as the rotation speeds continue to rise after the disap-pearance of Azimuthal waves,randomfluctuations come to progressively dominate theflow.In the end(above Ta/Ta c z1300),flow becomes turbulent.The transition toward turbulence has beenconfirmed experimentally by Gollub and Swinney[32]and numerically by Alziary de Rocquefort and Grillaud[33].The Taylor numbers nonetheless differ,probably as a result of dissimilar geometries.In fact,Cognet[4]noticed that an infinite sequence of fluctuation appears progressively inserting a newflow frequency for each one of them until theflow becomes turbulent.3.2.Convective heat transferThe initial approaches to convective heat transfer were carried out using an analogy between heat transfer and momentum. Couette[1]and Taylor[2]both measured the resistant torque due to thefluid’s being viscously rubbed between the two cylinders. This type of measurement was likewise employed more recently by Yamada[34].On the basis of these studies,it is possible to deduce friction factors or coefficients that can be linked to heat transfer by means of Reynolds analogy.In1958,Gazley[6]was thefirst author to show a sustained interest in thermal study of Taylor e Couetteflow.Flow dynamics were found to create three different heat transfer situations as speed and consequently the Taylor number goes up.Thefirst corresponds to the laminar regime below the critical threshold,the second to the laminar regime with superimposed Taylor vortices occupying the annular gap,and the third to the turbulent regime with regard to the highest Taylor numbers presenting more or less fully formed structures.One should recall that the Nusselt number is calculated by these authors on the basis of the difference of temperature from one wall to the other.It should also be noted that all of these studies deal only with average Nusselt numbers for the cylindrical gap taken as a whole.3.2.1.Couetteflow heat transferBelow the critical Taylor number,most of the authors have concurrently found a constant value equal to1;this is particularly the case for Tachibana et al.[35].The value may vary in accordance with Nusselt;for instance,Becker and Kaye[11]found a value of2 by using the definition of a Nusselt number:Nu¼hD h/l(Nu¼h2e/ l)as regards a smooth cylindrical gap.This is the definition we have chosen in order to ensure that as thereby postulated,the Nusselt number remains identical,whatever be the configuration(smooth with or without clippingflow,slotted.).In every case,this Nusselt number corresponds to a conductive heat transfer between the two walls:only parallel to these walls does thefluid move,and the heat transfer is consequently carried out perpendicularly to theflow, which may be considered as conductive.Two phenomena may nonetheless arise and modify the above value.First,a radiative heat transfer involving the two cylinders probably explains the high Nusselt numbers observed by Gazley [6],who does not seem to have dealt specifically and at length with this kind of transfer.Second,there is the natural convection of which Gardiner and Sabersky[10],on the other hand,have taken particular note;since their results are in close agreement with those reported by Aoki et al.[31],it would appear that the same phenomenon of natural convection comes into play.Finally,Yoo [19]focused on the air driven into rotation between two horizontal cylinders for Rayleigh numbers ranging from1000to50000and a relatively large cylindrical gap(aspect ratio h¼0.5),and observed that heat transferfirst diminishes along with rotation speed(Fig.7) on account of the disturbance provoked by centrifugal forces with regard to natural convection.The higher the Rayleigh number,the more pronounced and belated the diminution.It may be noted that in Fig.6,the threshold value of the Nusselt number in the case of heightened rotation speed is1,which is tantamount to strictly conductive transfer.3.2.2.Taylorflow and turbulentflow heat transferTaylorflow and turbulentflow heat transfer are studies jointly here as transition from one to the other seems to have no effect at all on the heat transfer rate.None of the authors have noticed heat transfer change with turbulent transition(Ta z106).This is prob-ably due to the progressive appearance offluctuations bringing the flow to turbulent(see paragraph2.1.3Turbulentflow):as transition to turbulence is progressive,heat transfer variation iscontinuous.Fig.7.Evolution of the Nusselt number along with the tangential Reynolds number for different Rayleigh numbers and an aspect ratio h¼0.5with Re¼ðu R2e=nÞ.M.Fénot et al./International Journal of Thermal Sciences xxx(2011)1e185So,above the critical Taylor number,Nu rises appreciably in conjunction with the latter.And above this critical threshold,the transportation of matter from one wall to the other of the annular gap favors heat transfer.Becker and Kaye [36]observed a second,less pronounced transition (simple modi fication of slope)for Ta ¼104.The results were con firmed in the respective works of Nijaguna and Mathiprakasam [37]as well as Boua fia et al.[12,14].There would appear to exist another dynamic transition or differing flow behavior altering the evolution of convective heat transfer.It bears mentioning that thermal transition coincides for Peres [38]with the Taylor number corresponding to the shortest axial wave-length.In fact,wavelength diminishes,according to Peres [38]reaching a minimum corresponding to the second transition,and then it increases.Consequently,the transition seems to correspond to a situation in which the vortex number is maximum.Several reasons can explain the lack of this second transition for several authors.First of all,as flow state depends on previous flow history (Coles [24]),so evolution of axial wavelength can be different depending on the experimental protocol (rarely described)and so,there may be no shortest wavelength.Moreover,Refs.[31,39,40]have large Taylor ranges and their measurements are less numerous.So,it is possible that the authors have failed notice the first region (before the transition)which is relatively small.A third possible transition is shown by Tachibana et al.[35]and Tachibana and Fukui [41].Comparison between their two studies seems to show a transition for Ta ¼108.They are the only authors working at such a high Taylor number.The results obtained by Taylor [2]lead to the conclusion that following the appearance of vortices:Nu z Ta 1/4.Most authors have formulated the Nusselt number through experimental correlations of the aspect Nu ¼A Ta n with A and n of the constants depending a priori on the aspect ratio of the cylindrical gap.Several precise correlations have been put forward by different authors,among whom we wish to cite Becker and Kaye [36],Tachibana and Fukui [35]and Boua fia et al.[12,14].These correlations and the conditions of use are reported in Table 1.The curves in Fig.8show sensitivity identical to the Nusselt number of variations in the Taylor number recorded above Ta ¼104by researchers as different as Becker and Kaye [36]and Bjorklund and Kays [39],Aoki et al.[31]and Tachibana et al.[35];the exponents n presented by these authors are all sit-uated within the same range;while the multiplying factor may vary,it nevertheless remains close to 0.2.It should also be noted that the geometric studies are far from identical.As seen in Table 1,h ,e /R 1and G vary from one author to the next.Once the Taylor number exceeds 108,Tachibana and Fukui [41]have recorded an exponent n of 1/3rather than ¼,The deviation from their prior numbers (and from other authors)is explained by the non-overlaping of the measuring ranges utilized in the two studies:the studies of Tachi-bana and Fukui [41]is conducted for greater value of Taylor numbers.It likewise bears mentioning that Tzeng [42]observed much higher heat transfer than the other authors without any marked differences with regard to their respective con figurations.The results reported by Tzeng [42]were associated with Taylor numbers higher than those of the many other authors.But,the more signi ficant difference between Tzeng ’s experiment and other studies resides in the presence of high natural convection which seems to greatly affect heat transfer (see below).Aside from this particular case,the different results concerning to heat transfer are in close correspondence,with scattering occurring once the rotation speed goes up.Several parameters may exert some in fluence on the heat transfer depicted.Natural convection may continue to affect heat transfer e even after the appearance of Taylor vortices.Tzeng [42]has postulated a correlation taking into account the natural convection effects based on the tangential Reynolds number Re t ¼u R 1D h /n for a horizontal con figuration with h ¼0.895,e /R 1¼0.12and G ¼17.This correlation provides the ratio between the Nusselt number with rotation Nu U and the Nusselt number without rotation Nu 0:ðNu U =Nu 0Þ¼0:375ðGr $ðu R 1D h =n Þ$10À11Þ0:328with Gr ¼b $D T $e 3g =n 2.The correlation should not be used unreservedly,for there are differences between the thermal results of Ref.[42]and those of the other authors (Fig.8).The effect of cylindrical gap size on heat transfer has been studied by par Ball et al.[40].They observed that the more pronounced the rise of h ,the more pronounced the rise of the radial Reynolds number,and consequently of the Taylor number.This observation would appear logical,since a wide cylindrical gap tends to stabilize the flow and thereby delay the development of Taylor cells and diminish their effects on heat transfer.Most of the studies presented above involve global heat transfer,and only the numerical studies record local results.Thus,Fig.9presents the axial evolution of the Nusselt number numerically recorded by Ghayoub et al.[23]with a cooled stator and a heated rotor.The reference temperature chosen was the mixing temper-ature,of which the evolution is likewise depicted in Fig.8;the choice renders it possible to distinguish the Nusselt number Nu 2of the stator from that Nu 1of the rotor.The sinusoidal evolution corresponds to the alternation of the pairs of vortices presented in Fig.10.To be precise,when two vortices meet at the level of theTable 1Recapitulation of the different correlations for convective heat transfer in a smooth and open cylindrical gap.Note:The de finition of the Nusselt number has been modi fied so as to have it coincide with the other authors,and it corresponds to the de finition used in the present text.he /R 1GTaTa c C.L.thermal CorrelationsBecker and Kaye [36]0.8070.2381720à3,3Â1051994Cooled stator Heated rotorTa <Ta c :Nu ¼2Ta c <Ta <104:Nu ¼0,128Ta 0,367104<Ta <3.3Â105:Nu ¼0.409Ta 0.241Tachibanaand Fukui [41]0.75/0.9380.07/0.33 2.25/11.25108/5Â1012eCooled stator,Heated rotor or Heated stator,Cooled rotor Nu ¼0.092(TaPr)1/3Tachibana et al.[34]0.522/0.9710.13/0.92220/70000/1081730e 3000Cooled stator Nu ¼0.42(TaPr)0.25Heated rotor Bjorklundand Kays [35]0.8/0.9480.054/0.24632/1478000/4.1061770e 1994Cooled stator Nu ¼0.35(Ta)0.25Heated rotor Aoki et al.[31]eee5000/2Â105e eNu ¼0.44(Ta)0.25(Pr)0.3Ball et al.[40]0.437/0.6560.26/0.6431.4/77.24000/4Â105e Cooled stator Nu ¼0.069$h À2.9084$(R 1u e /n )0.4614l n (3.3361h )Heated rotor Tzeng [42]0.8950.12177962/2Â108eHeated rotorNu ¼8.854(R 1u e /n )0.262(Pr)0.4M.Fénot et al./International Journal of Thermal Sciences xxx (2011)1e 186rotor (A ),the hot flow is evacuated in the direction of the stator,thereby provoking maximal Nu 1and increased mixing tempera-ture.Conversely,when two vortices meet at the level of stator (B ),Nu 2reaches a maximum and the mixing temperature a minimum.One may also note that as regards Ta ¼2Â106,the secondary Nu 1maxima present at the bottom (B )might be in correspondence with the presence of the Gortler vortices.Moreover,the Nu peak grows more pronounced as the Taylor number goes up.This finding is most likely explained by a heightening of centrifugal forces and consequently of the speed of the vortices when they meet (in (A )or (B )).The impact of the Gortler vortices on total heat transfer remains relatively insigni ficant.4.Smooth cylindrical gap,Taylor e Couette e Poiseuille flow In the con figuration of the smooth and open cylindrical gap,axial flow is superimposed on the rotation effect provided by the rotor.As in the preceding part of this paper,a brief description of flow structure will be given prior to a section devoted to heat transfer.In the case of Taylor e Couette e Poiseuille flow,a new dynamic parameter is to be used in addition to rotation velocity u :average axial velocity of the fluid V a (m/s).The axial Reynolds number may now be de fined as Re a ¼V a D h /n .The most widely accepted de finition of the Nusselt number remains:Nu ¼ðhD h =l Þ¼ðÀl grad T j p =T w ÀT ref ÞðD h =l Þ¼ðF =S ðT w ÀT ref ÞÞðD h =l Þ,where the reference temperature is usually taken to be equal to a fluid temperature (mean temperature T mean ¼(T e þT 0)/2),even though,as is the case with Taylor e Couette flow,some authors take it as equal to the temperature of the second wall.4.1.Flow structureFlow results from the superimposing of two discrete mecha-nisms,one of them linked to the centrifugal effects of rotational flow,the other one axially driven.Two main transitions may be distinguished:1)a transition from laminar flow to turbulent flow when axial velocity rises;2)the appearance of «Taylor vortex »structures above critical rotation velocity.When these twophenomena are combined,we may discern the four types of flow described by Kaye and Elgar [43](Fig.11):-Laminar flow-Laminar flow with Taylor vortices -Turbulent flow-Turbulent flow with Taylor vorticesNevertheless,most of the authors have focused their studies on the two main transitions (Laminar to turbulent transition and Taylor vortex transition)rather than on the characterization of the flow encountered.So,we will analyze separately the two major transitions.4.1.1.In fluence of axial flow:turbulent flow transitionFew studies have focused on the transition from laminar to turbulent flow,while most of them have concentrated on the in fluence of axial flow on vortex appearance.It is generally consid-ered that,before the appearance of Taylor vortices,these flows are similar to those encountered in duct flow with a radial velocity as well.Some authors have analyzed the in fluence of parameters on turbulence transition.As is the case in Taylor e Couette flow,in flu-ence parameters are both geometrical and thermal.Cylindrical gap geometry assumes a prominent role in transitions.Kaye and Elgar [43]were the first researchers to experimentally determine the transitions with regard to two different numerical degrees of relative cylindrical gap thickness h for a wide range of Ta and Re a (see Fig.12).The transition to turbulence is more rapidly reached with a more narrow cylindrical gap for Re a <1500,Reynolds number above which the effects are likewise reversed.Concerning thermal effects on flow,Becker and Kaye [36]studied the in fluence of existing temperature gradient and heat transfer on flow structure in a wide range of axial flow rates and rotation velocities (see Fig.13).According to these authors:e For Ta ¼0,turbulence appears in both cases at Re a ¼1800e Transition from laminar flow to turbulent flow appears to be independent of the axial Reynolds number when heat transfer11010010001.00E+021.00E+031.00E+041.00E+051.00E+061.00E+071.00E+081.00E+091.00E+10N uTaTachibana et Fukui [41]Tachibana et al. [35]Bjorklund et Kays[39]Aoki et al.[31]BECKER et Kaye [36]Ball et al.[40]Tzeng [42]parison of the different correlations Note:the de finition of the Nusselt number has been modi fied so as to have it coincide with the other authors,and it corresponds to the de finition used in the present text.M.Fénot et al./International Journal of Thermal Sciences xxx (2011)1e 187。