ENVIRONMENTAL FLUID MECHANICS

Transcription

1 ENVIONMENTAL FLUID MECHANICS Plumes & Thermals Why do hese srucures behave he way hey do? How much mixing do hey accomplish? enoi Cushman-oisin Thayer School of Engineering Darmouh College hp://hayer.darmouh.edu/~cushman/books/efm/chap1.pdf The ype of quesion ha we seek o answer: If a blob of conaminaed waer spills ino a sraified lake, how far down does i fall? Will i reach he level where ambien waer densiy equals he blob s densiy? During is fall, he negaively buoyan blob will enrain ambien fluid and ge progressively dilued. Is negaive buoyancy will be progressively reduced. Clearly, i will no reach as deep as he level where he ambien densiy is equal o he blob s iniial densiy bu will sop before hen. The deph where he blob sops is descen is a funcion of he mixing ha akes place along he way. 1

2 Plumes Plumes are common feaures in environmenal fluids. They occur whenever a persisen source of buoyancy creaes a rising moion of he buoyan fluid upward and away from he source. The cleares example is ha of hydrohermal vens a he boom of he ocean. hp:// Phoo by by Dalibor ibičić Anoher occurrence is he rising of freshwaer from he boom of he sea a submarine springs in karsic regions such as along he Dalmaian Coas of Croaia, where such feaures are called vrulje. The common urban smokesack plume is, however, somewha differen because he warm gas rises no only under is own buoyancy bu also under he propulsion of momenum (ineria). Such plume is more properly caegoried as a buoyan je or forced plume. uoyancy imporan a laer sages Je momenum imporan a he beginning Wha drives a plume is is hea flux, defined as he amoun of hea (expressed in joules) being discharged hrough he exi hole per uni ime. ecause i is more pracical in laer mahemaical developmens, his quaniy is divided by C p (he fluid's reference densiy imes is hea capaciy a consan pressure) and hen muliplied by (he fluid's hermal expansion coefficien) and he graviaional acceleraion g, giving rise o he buoyancy flux: g hea F C p ime Noe ha because hea per ime is expressed in J/s, he buoyancy flux is measured in m 4 /s.

3 Le us consider a hree-dimensional radially symmeric plume progressing verically from he boom hrough a homogeneous and resing fluid. If we denoe by T he emperaure of he ambien fluid, hen he emperaure inside he plume has he value T + T', in which T' denoes he emperaure anomaly (posiive in a rising plume, negaive in a sinking plume). To his emperaure anomaly corresponds a densiy anomaly ' = T'. I is convenien o define he local buoyancy, or reduced graviy, g' as: g gt The buoyancy flux F can be expressed as he inegral across he secion of he plume of he produc of he verical velociy w wih he buoyancy g'. In erms of he mean values across he plume, a reasonably good approximaion is: F w gt w 1) Hea budge along he plume reduces o F = consan = upsream value. We need now o esablish he mass and momenum budges. For his, consider an infiniesimal slice along he plume. ) Mass budge: Mass exiing a op = Mass enering a boom + Mass enrained hrough he side d ( w) u ) Verical-momenum budge: Verical momenum exiing a op = verical momenum enering a boom + Verical momenum enrained hrough side + Upward buoyancy force = d ( w ) 4) Closure assumpion: Enrainmen velociy proporional o verical velociy inside plume u a w wih coefficien a o be deermined empirically

4 The problem consiss of 4 equaions for 4 unknowns, each a funcion of he verical coordinae :, w, u and g' The soluion is self-similar:.157 F w.14 F u.8 F 6.8 1/ 1/ 1/ 1/ / 5/ Coefficien was fied o observaions (cone of x 8.9 o ) Comparison wih he urbulen je: angle = 11.8 o = x/5 =. x A plume is narrower. The fi of he cone angle ses he value of he coefficien a in he enrainmen assumpion: a.1 u. 1 w Comparing his coefficien o ha for a urbulen je (=.1), we deduce ha mixing is slighly more vigorous in a plume han in a je. Pu anoher way, buoyancy is a more effecive mixer han momenum. Plumes in a Sraified Environmen When a plume rises (or sinks) in a sraified environmen, i encouners a emperaure becoming closer o is own and progressively loses buoyancy. A some level, i will have los all buoyancy and will begin o spread horionally. Such is he case of a smokesack plume in a calm (no wind) and sraified amosphere ypical of he early morning. The obvious quesion o ask is how high does he plume reach? 4

5 The sraified ambien fluid is characeried by is sraificaion frequency N defined from dt N g To describe a plume in his ype of environmen, he same quaniies are needed as before, namely he plume's radius, averaged verical velociy w, enrainmen velociy u, and averaged buoyancy, each a funcion of he elevaion. The difference wih he previous secion is ha now he buoyancy flux F is no longer a consan along he axis of he plume. Four equaions are again a our disposal. 1) Mass budge: d ( w) u ) Momenum budge: d ( w ) ) Hea budge: 4) Enrainmen assumpion: d ( w) N w u a w The soluion o his se of equaions does no exhibi similariy, bu he equaions can easily be inegraed numerically. Saring wih iniial condiions (a level = ) such ha he momenum and volumeric flow are nil bu buoyancy flux finie, one finds he resuls shown in Figure below. In he figure, variables are made dimensionless by scaling as follows: w and by by F N by 5 F N F N where F is he saring buoyancy flux (a = ). The enrainmen parameer a is aken as.15 o fi observaions (growh of radius wih heigh). Noe ha his value is inermediae beween hose for he urbulen je (.1) and plume (.1). 5

6 Pracical summary of plume rise in a sraified environmen.9 level of res inerial overshoo.98 level of neural buoyancy We conclude ha a plume in a sraified environmen rises a disance of.9(f/n ) 1/4, in which F is he plume s iniial buoyancy flux and N he sraificaion frequency of he environmen. Verificaion for plumes over many differen lengh scales: 1/4 1/4 F F N N 6

7 Engineering quesion: If he plume is released in waer of limied deph, how can we assure ha he plume erminaes wihin he available deph? In mos cases, we don wan his. Insead, we wan he plume o sop rising before i reaches he surface. If discharge pipe is siuaed a deph H below he surface: 1/4 4 4 Plume rise.76 F H F NH NH 4 N.76 Thermals A hermal is a finie parcel of fluid consising of he same fluid as is surroundings bu a a differen emperaure. ecause of is buoyancy, a cold hermal sinks (negaive buoyancy), while a warm hermal rises (posiive buoyancy). The name was given by glider pilos o wha hey perceived as regions of warm air rising above a heaed ground in which hey could soar. Convecion in he amosphere does indeed proceed by means of rising hermals. The siuaion, however, can be quie chaoic, wih a collecion of hermals rising here and here a various imes, some of hem smaller and slower, and ohers larger and faser. Here, for he sake of undersanding he basic mechanism, we are be concerned wih a single hermal immersed in an infinie homogeneous fluid a res. 7

8 Experimens have been conduced in he laboraory, and i has been found ha all hermals roughly behave in similar ways. As hey rise (or sink), hey enrain surrounding fluid and become more dilue, hereby slowing down in heir ascen (or descen). The acual shape of a hermal, however, can vary considerably from one se of observaions o anoher. Here, basic dynamics supplemened by a few dimensionless numbers gleaned from experimens will be used o esablish a simple heory for he predicion of a hermal's behavior over ime. Descending hermals in a laboraory experimen. These hermals in waer are made visible by barium sulfae. The sem lef behind by each hermal is due o he manner a spherical cap was roaed o provoke he release. The second hermal (boom row) has a larger negaive buoyancy han he firs (op row). [From Scorer, 1997] 8

9 Define he hermal s average radius and volume V, hen relae he wo by: V m in which m is a shape facor (would be = 4/ = 4.19 if he hermal were a sphere). 1) Mass budge: dv dv Au a w in which A is he enclosing surface area, u he enrainmen velociy, and a a dimensionless consan ) Momenum budge: d hermal V w hermal added mass effec V d V w V in which g is he reduced graviy experienced by he hermal ) Hea budge: V consan in which is he oal buoyancy of he hermal Soluion: 4a 9m 64a V 79m 1/ 9m w 4a 1/ 79m 64a / 4 / / Then, obain elevaion by inegraing velociy over ime: w 6m a 1/ and find proporionaliy beween radius and heigh : a m Experimenal daa reveal an angle of 14 o. Thus, o an14.5 a.75 m 9

10 Oher experimenal evidence: 5.8 1/ 6m 5.8 a m a.94 Togeher wih a =.75 m, we obain: a = 1.9 and m = V w / / 4 / 1/ 1/ / Plo of he quaniy / (a ime consan during he life of hermal) versus he square roo of he hermal's buoyancy. Each numbered do refers o a differen laboraory experimen, and he solid line shows he bes linear fi. [From Scorer, 1997] Thermals in a Sraified Environmen A laboraory experimen of a hermal sinking in a sraified environmen. Noe he ulimae arres and spreading of he hermal once i loses is buoyancy. (From Scorer, 1997) Equaions: V m w dv a w d( Vw) V d( V) N Vw d a m d( w) w d( ) N a m m 4 N ( 4a = a end of journey ending radius erminal heigh end erminal 4 4 ) 4a m N m a 4 4a m N 1

11 Now, we can answer he quesion of how deep does a hermal sink or rise: Answer: Deph of sinking m 4 4a V a m N 1/4 V.17 N 1/4 in which V is he hermal s iniial volume, g is original reduced graviy g T iniial g and N he square of he sraificaion frequency of he ambien fluid dtambien g d N g iniial ambien Conclusions Jes, plumes and hermals are complex urbulen srucures, bu heir dynamical behavior can be modeled by a few relaively simple equaions, in which a dimensionless facor or wo can laer be adjused by laboraory experimenaion or field daa. Jes, plumes and hermals accomplish mixing, which is easily deermined from he soluion o he dynamical equaions. Mixing proceeds by enrainmen of ambien fluid ino he coheren srucure. The (laeral) enrainmen velociy is on he order of 1% of he average speed of he fluid wihin he srucure. No one has ye proposed heories ha predic he numerical values of - he angles relaing sie o disance, - he raio of enrainmen velociy o mean velociy. 11