Nomenclature. I. Introduction

Transcription

1 Effect of Intal Condton and Influence of Aspect Rato Change on Raylegh-Benard Convecton Samk Bhattacharya 1 Aerospace Engneerng Department, Auburn Unversty, Auburn, AL, Raylegh Benard convecton s an example of thermal nstablty where temperature dfference between the top and bottom caused by heatng the flud from below results n formaton of rolls. Raylegh Benard convecton s studed and the effect of change of aspect rato and change of ntal condton s nvestgated through numercal smulaton by the use of a commercal computatonal flud dynamcs package Fluent. A 2d model of Raylegh Benard convecton was created. The 2d doman was constructed wth fxed temperatures on the top and bottom wth dfferent aspect ratos. The crtcal Raylegh number was found n each case. The effect of ntal condton change was mposed n the form of an ntal velocty n the whole doman wth a fxed number of rolls. In the end a 3 dmensonal model was created to fnd out the change n spatal character of the roll structures n the doman. ρ U τ E P T j λ φ β d ν α R R ac Nu = densty = velocty n th drecton = stress tensor = total energy = pressure = temperature = coeffcent of bulk vscosty = dsspaton functon = coeffcent of thermal expanson = depth of the contaner = knematc vscosty = thermal dffusvty = Raylegh number = crtcal Raylegh number = Nusselt number Nomenclature I. Introducton Raylegh Benard convecton has been studed extensvely over the years. It occurs when a flud held n a contaner s heated from the bottom. A temperature gradent s thus set up between the top and the bottom of the flud. When ths gradent reaches certan value convecton rolls begn to form whch ndcates the onset of thermal nstablty of the system. Ths nstablty occurs due to the nterplay among the vscous force, thermal dsspaton, gravty and buoyancy force. Due to the heatng, flud n the bottom layer becomes lght n weght and tres to rse upward. Ths upward moton s prevented by gravty, vscous forces and thermal dsspaton. But after a certan temperature dfference s reached the buoyancy force overcome the opposton of the stablzng forces and the lght flud layer comes to the top. The heavy flud layer from the top start comng down to the bottom to occupy the vacant place, afterwards ths flud layer also gets heated and start rsng upwards. Thus a cycle of moton sets n whch expresses tself n the form of convecton rolls. Convecton rolls also form n surface tenson drven Benard- Marangon convecton. At the onset we can descrbe convecton by a sngle Fourer mode. When Raylegh number crosses R ac, more modes are generated through nonlnear nteracton. When there are only few modes (say 10 modes or so), one gets convectve patterns [1]. The patterns can be square, hexagonal, wavy etc. These patterns could be oscllatng as well. When Ra s ncreased even further, the ampltudes of these modes become large resultng n a chaotc 1 Graduate Teachng Assstant, Aerospace Engneerng Department, Auburn Unversty, and AIAA Student Member. 1

2 behavor. A popular model called Lorenz equaton [2] attempts to capture ths feature. In Lorenz equaton, the temporal behavor s chaotc, but the spatal structure s regular. In Lorenz model the horzontal rolls flp drecton of rotaton at random ntervals, but the rolls do not move n space. When Ra s ncreased further, many more modes are generated. In ths case spatal and temporal varatons can be observed. When Raylegh number ncreases more the flow becomes turbulent, where many modes nteract wth each other. The tme seres of temperature or velocty sgnal s random. Ths regme s called turbulent. The doman sze s determned by the aspect rato whch s equal to L /d, where L s the length and d s the heght of the doman. The effect of aspect rato on the chaotc dynamcs n a system s of great scentfc nterest. Ahlers and Behrnger [3] were one of the frst to expermentally study the effects of aspect rato on the dynamcs of Raylegh Benard convecton Ths partcular nstablty problem has many mportant applcatons n a number of engneerng problems, such as n ol extracton from porous meda, energy storage n molten salts, crystal growth n space and chemcal engneerng of pants, collods and detergents. Apart from that ths knd of convecton s a crucal factor n the dynamcs of ocean and atmosphere. Benard (1901) frst reported expermental fndngs of convecton rolls. Subsequently Raylegh [4] and Jeffrey [5] came up wth the mathematcal soluton of the nstablty problem. The lnear stablty analyss of Lord Raylegh or Sr Harald Jeffrey gves the magntude of the crtcal wave drector, but t cannot be decded whether the patterns above onset wll consst of rolls, hexagons, or squares. Indeed all three patterns occur n RBC. Malkus and Verons [6] predcted that the stable planform for the case of free boundares and Boussnesq condtons should be straght rolls rather than e.g. squares or hexagons. Afterwards there have been a number of attempts to solve the nstablty problem both expermentally and theoretcally. A major aspect of these works was to fnd out the selecton of patterns for varous boundary and ntal condtons. Most notable are the works of Chandrasekhar [7], Koschmeder [8], etc. The smplest geometrc case s the rectangular enclosure. Though attempts have been made n past to nvestgate other shapes most notably crcular and square contaners. It has been found that the spatal character of the rolls,.e. ther axs, sze depends manly on the boundary condton and the non unformty of the temperature dstrbuton on the top and bottom plate. In number of the cases steady crcular or hexagonal rolls (for rectangular contaner) and crcular concentrc rolls were found for a crcular contaner. In the work presented here a commercal software package Fluent [9] was employed to smulate the Benard convecton. Varous parameters lke Nusselt number were calculated and compared wth the theoretcal value. In the end a 3d model was constructed to fnd the characterstcs of 3d nstablty. The man am of ths work was to fnd the effect of aspect rato change and that of mplementaton of ntal perturbaton on the nstablty regme. II. Basc Parameters and Governng Equatons The basc parameter controllng the onset of nstablty s the Raylegh number whch s defned as Ra =β Tgd 3 /ν α. Ths rato s a dmensonless number, whch s a result of the competton between the destablzng mechansm and the stablzng process. Convecton develops when the buoyancy s more effectve than the dsspatve process and thus when Ra s large. The lmtng condton after whch convecton sets n s called crtcal Raylegh number. The governng equaton for the problem s the contnuty, momentum whch s famously known as Naver Stokes equaton and energy equaton. ρ t ρu Contnuty equaton: + = 0; Momentum equaton: DU ρ Dt p = τ j + j ρg Energy equaton: DE U T ρ = p + ( λ ) + φ Dt j j To these equatons of moton, Raylegh appled the Boussnesq approxmaton. The approxmaton states that there are flows n whch the densty varaton s neglgble due to small temperature varaton. But the flow may be drven 2

3 by buoyancy. So the varaton of densty s neglected everywhere except n the buoyancy. On the bass of ths approxmaton for small temperature dfference between the top and bottom layer of flud, we can wrte, ρ=ρ 0 {1-β (T-T 0 ), where T 0 s the temperature of the lower plate. Ths approxmaton works well for Benard convecton problem where the temperature dfference s small between the top and the bottom plate. III. Numercal Smulaton To smulate Benard convecton numercally a commercal computatonal flud dynamcs package Fluent was used. Two cases wth aspect rato 2 and 7 were nvestgated. The mesh was generated n the preprocessor Gambt. For the aspect rato 2 case, a 2dmensonal rectangular geometry was modeled wth length 20 mm and heght 10 mm. A lamnar model was appled snce the flow n the nstablty regme s essentally lamnar (unless the Raylegh number s above 10000). An nfnte horzontal span was assumed for ease of soluton and ths assumpton was mplemented through perodc boundary condton on the sde walls. The temperature of the bottom plate was fxed at 60 o C and wall boundary condton was appled to them. The temperature of the top plate was determned from the temperature dfference whch was calculated from the formula for Raylegh number. Ths process was repeated wth dfferent Raylegh numbers. The flud for the smulaton was always selected as water wth ts propertes at 60 o c. For the aspect rato 7 case the wdth of the rectangle was taken as 2 mm and length 14 mm. The followng table shows the temperature dfference appled between the top and the bottom plate for both aspect rato cases. For both the lamnar and turbulence case analyss SIMPLE algorthm was used n Fluent. Ra T Table 1: Temperature dfference between the top and bottom plate for dfferent Raylegh numbers A. Turbulence modelng: Turbulent flows are characterzed by fluctuatng velocty felds. These fluctuatons mx transported quanttes such as momentum, energy, and speces concentraton, and cause the transported quanttes to fluctuate as well. Snce these fluctuatons can be of small scale and hgh frequency, they are too computatonally expensve to smulate drectly n practcal engneerng calculatons. Instead, the nstantaneous (exact) governng equatons can be tmeaveraged, ensemble-averaged, or otherwse manpulated to remove the small scales, resultng n a modfed set of equatons that are computatonally less expensve to solve. However, the modfed equatons contan addtonal unknown varables, and turbulence models are needed to determne these varables n terms of known quanttes. In cfd a number of turbulence models are avalable. The selecton of a partcular turbulence model depends on the problem defnton and the flow characterstcs. The avalable optons n Fluent are K-epslon, K-omega, Spallart Allmaras model etc. For Ra=10000 and case the turbulent case were selected for ths problem. The K-epslon turbulence model was mplemented wth default coeffcents and standard wall functon approach. The K-epslon model s a two equatons, sem emprcal model n whch two separate transport equatons are solved to obtan the requred turbulent velocty and length scale. K-epslon model s very robust and t gves reasonable accuracy for a wde class of flow problems..for the 3d smulaton a 0.005x0.05x0.05 rectangular parallelepped was created n Gambt. The Ra was fxed at In ths case also the lamnar model was selected for vscosty. 3

4 IV. Results and Dscusson A. Effect of Aspect Rato Change: Fgure 1 shows the statc temperature contours at Ra= 700. It can be clearly seen that the sotherms are perfectly horzontal wthout any dstorton whch sgnfes that no nstablty has set up. Ths s n accordance wth the theory as t can be shown from lnear stablty analyss that for small dsturbances the crtcal Raylegh number for an enclosure wth aspect rato 2 s approxmately Fgure 2 shows that at Ra=1700 the nstablty has set n whch s evdent n the perodc nature of the statc temperature plot. The patterns formed are square rolls as ndcated by fgure 3. Number of rolls n ths case s two whch corresponds to the mnmum stable case. The wavelength of the rolls can be found from fgure 4 whch shows that the wavelength of the varaton of velocty magntude s around 10 mm whch s same as the depth of the flud. In the turbulent case wth Ra=10000 the rolls appeared to be less clear (fgure 5) and n fgure 6 whch s for Ra=30000, the patterns are broken. Ths s due to the result of ncreased perturbaton n the turbulent case. For the aspect rato 7 case almost the same results were obtaned. The number of rolls formed were 7 and the wavelength of the rolls were almost 2 mm.e. same as the depth of the flud layer. But n ths case one mportant devaton was perodcty n temperature profle at Ra 700 (fgure 8) whch sgnfes formaton of rolls. So t can be concluded that the crtcal Ra number comes down wth the ncrease of aspect rato. A reason for ths s the ncrease n surface area whch allows more heatng area. As more heatng area s avalable, the same temperature dfference between the top and the bottom plate n the ncreased aspect rato case causes more heat flux to enter the system. So the flud gets heated more quckly and the heatng enables a flud lump to overcome the stablzng effect of dsspaton and vscous forces. As an effect the convecton rolls appear more quckly n the ncreased aspect rato case. One prme aspect of Benard convecton s the heat transfer between the top and bottom surface of the flud. Before reachng the crtcal Raylegh number, when the flud s undsturbed the heat transfer takes place by dffuson only. But after the system reaches nstablty heat transfer n the flud s aded by convecton also. So Nusselt number whch s actually the rato of heat transfer n the flud n the dsturbed state to that of the undsturbed state s always greater than one after crtcal Raylegh number. The onset of nstablty can also be dentfed by the break n the nstablty curve. The frst emprcal formulas for the heat transfer by convecton were gven by Slverstone (1958). Accordng to hm for Nu=0.24(Ra) ^0.25. [1]. Fgure 9 shows a plot of Nusselt number vs. Raylegh number n whch the aforesad emprcal relaton was compared wth the data obtaned from the numercal study for aspect rato 2 case. The data shows a good match n the lamnar regme. But there s a tendency of devaton n the turbulent case.e. as the Raylegh number s ncreased more and more. B. Effect of ntalzaton of the velocty n the problem doman: To fnd out the effect of ntal condtons on the Benard convecton, an ntal perturbaton was appled n the form of an ntal velocty to the whole doman. The functon selected to acheve the ntalzaton was of the form v=a*(sn (p*y/0.01))*(cos (N1*p*x/0.02) + B* cos (N2*p*x/0.02)), where N1 and N2 are the number of rolls n the x and y drecton respectvely. Ths partcular form of velocty s actually the result of a lnear stablty analyss for an nfntely spanned doman n the horzontal drecton. The ntalzaton of the velocty was done by hookng a user defned functon to fluent. Through ths mplementaton the no of rolls n the x drecton was fxed at 6 and n the y drecton t was 2.As the result of ntalzaton 6 rolls formed n the doman whch s vsble n the contour plot of x veloctes (fgure 10). The contour plot of the y velocty showed 2 rolls as before (fgure 11). The Raylegh number was arbtrarly selected to be The ntalzaton case can be dffcult to attan n practcal experment as t nvolves a perodc velocty profle. Fgure 12 shows the result from a 3dmensonal test case wth Ra = In case of a 3 d doman the wall boundary condtons were appled along the sx faces. So the velocty contour plot s actually from a cross secton of the doman. It can be observed that there s a perodcty n the number of rolls (5 and 4 consecutvely) along the z axs. One mportant fndng from the results s all the patterns were square n shape. The structure of pattern formaton s an actve research topc n Benard convecton. Square rolls are the result of the smplest boundary condton,.e. nfnte span n the horzontal drecton. As dscussed prevously the structure and shape of the roll depends on the boundary condton, shape of the contaner etc. Also the presence of thermal nose affects ts structure. 4

5 A. Aspect Rato 2 V. Fgures Fgure 1. Contour plot of statc temperature at Ra = 700. Fgure 2. Contour plot of statc temperature at Ra = Fgure 3. Contour plot of velocty n x drecton at Ra =

6 Fgure 4. Varaton of velocty magntude along the central lne of the rectangular doman along x axs at Ra=1700 Fgure 5. Contour plot of velocty n x drecton at Ra = Fgure 6. Contour plot of velocty n x drecton at Ra =

7 B. Aspect Rato 7 Fgure 7. Contour plot of velocty n x drecton at Ra = 1700 Fgure 8. Contour plot of statc temperature Ra = 700 Nu vs Ra Nu 3 smulaton result emprcal result Ra Fgure 9. Plot of Nusselt number vs. Raylegh number 7

8 C. Intalzaton of velocty Fgure 10. Contour plot of velocty n x drecton at Ra = 6000 Fgure 11. Contour plot of velocty n y drecton at Ra = 6000 Fgure 12. Contour plot of the velocty n the 3 d case showng the cross secton of the doman, Ra=6000 8

9 V. Concluson Ralegh Benard convecton was smulated numercally through the commercal package Fluent. The basc am was to fnd the effect of aspect rato change and ntal velocty condton on the onset of convecton. It was found that ncrease n the aspect rato brngs down the crtcal Raylegh number. But the wavelength of the roll was always consstent wth the depth of the flud layer. The three dmensonal smulaton showed that along wth the x and y component of wavelength t has got a perodcty n the z drecton also. The rolls formed n all the cases were square rolls whch are the deal soluton of the normal mode soluton. The plot of Nusselt number vs. Raylegh number showed good correspondence between the theoretcal and fluent result. For turbulent convecton t was found that the plot shows dvergence between the theoretcal and computatonal result. A proper choce of turbulence model wth approprate coeffcents may solve ths problem. References 1 Cross, M. C. and Hohenberg, P. C., Pattern Formaton Outsde Equlbrum, Rev. Mod. Phys, 65:851, Lorenz, E. N., Determnstc Non Perodc Flow, Journal of Atmospherc Scences, Vol. 20, March 1963, pp Ahlers, G. and Behrnger, R.P. Evoluton Of Turbulence From The Raylegh Benard Instablty, Phys. Rev. Lett., Vol. 40. pp , Lord Raylegh. On convecton currents n a horzontal layer of flud when the hgher temperature s on the undersde. Phlosophcal Magazne, 32: , Jeffrey, H. Some Cases of Instablty n Flud Moton, Proceedngs of the Royal Socety of London, vol 118, no.779.pp Malkus, W. V. R and Verons, G., Fnte Ampltude Cellular Convecton, J. Flud Mech. 38, 227, Chandrasekhar, S. Hydrodynamc and Hydro magnetc Stablty. Oxford Unversty Press, Koschmeder, E.L. Benard Cell and Taylor Vortces, Cambrdge Unversty Press, Fluent 6.2, Fluent Inc, NH, USA. 9