DEVELOPMENT OF TURBULENT CIRCULATION FROM AN AREA HEAT SOURCE IN STABLE STRATIFIED ENVIRONMENT*

Transcription

1 48, Vol. 6, Pt, Specal Issue Proceedngs of Internatonal Conference RAMM VLOPMN OF URBULN CIRCULAION FROM AN ARA HA SOURC IN SABL SRAIFI NVIRONMN A.F. KURBASKII Insttute of heoretcal and Appled Mechancs SB RAS and Novosbrsk State Unversty, Novosbrsk, Russa e-mal: L.I. KURBASKAYA Insttute of Computatonal Mathematcs and Mathematcal Geophyscs SB RAS, Novosbrsk, Russa u A three-equaton model of the turbulent transport of momentum and heat for smulatng a crculaton structure over the heat sland n a stably stratfed envronment under nearly calm condtons s formulated. he turbulent knetc energy (K), = ( / ) < u u, ts spectral flux (dsspaton), and the dsperson of turbulent fluctuatons of temperature < are found from dfferental equatons, thus the correct modelng of transport processes n the nterface layer wth the counter-gradent heat flux s assured. urbulent fluxes of momentum < u and heat < u are determned from < turbulence model mnmes dffcultes n smulatng the turbulent transport n a stably stratfed envronment and reduces efforts needed for the numercal mplementaton of the model. Numercal smulaton of the turbulent structure of the penetratve convecton over the heat sland under condtons of stably stratfed atmosphere demonstrates that the three-equaton model s able to predct the crculaton nduced by the heat sland, temperature dstrbuton, root-mean-square fluctuatons of the turbulent velocty and temperature felds, and spectral turbulent knetc energy flux that are n good agreement wth the expermental data and results of LS. Introducton urbulence closure models are often used as tools to analye and predct atmospherc boundary layer characterstcs. urng the last years numerous artcles dealng wth varous types of flows usng dfferent models have been presented. For stratfed atmospherc flows the most frequently used models are models (uynkerke 988), second-order closure models (Zeman and Lumley 979; Sun and Ogura 98) and thrd-order closure models (Andre et al. 978; Canuto et al. 994). ogether wth large eddy models (Moeng 984; Mason 989; F..M. Neuwstadt et al. 99) thrd-order closure models (Andre et al. 978) should be consdered as fundamental research tools because of ther large computer demands. A growng need for detaled smulatons of turbulent structures of stably stratfed flows motvates the development and verfcaton of computatonally less expensve closure models for appled research that should be kept as smple as possble n order to reduce computatonal demands to a mnmum. he deas underlyng the algebrac models represent an mprovement n buoyant flow modelng and could be used for appled modelng snce a full secondorder closure model s presently much too demandng. Indeed, the recent studes (Andren 99; Sommer and So 995) of the stable stratfed flows ndcate that a model wth a transport approxmaton ncludng buoyancy effects mght be the optmal way that combnes both computatonal effcency and predctve capablty. he algebrac modelng technques of prevous studes could be modfed to obtan an algebrac heat-flux model for buoyant flows. In order to avod usng the symbolc algebra software for nvertng a system of algebrac equatons for the turbulent heat fluxes < u and turbulent momentum fluxes < u u t s desrable to derve an explct algebrac heat-flux model here the heat fluxes are expressed explctly n terms of the mean gradents and the eddy dffusvtes. It should be ponted out that the use n hgher-order closure studes of the -equaton s now qute standard (Canuto et al. 994; Ilyushn and Kurbatsk 997). Results of computatonal modelng and smulaton of the atmospherc boundary layer (Andren 99; Ilyshn and Kurbatsk 996) showed the mportance of retanng the full prognostc equaton for the temperature varance, allowng a counter-gradent transport of heat n the upper half of the turbulent layer. he present paper proposes and evaluates a turbulence closure scheme that has been mplemented n order to make the model more useful for stable stratfed flows and ar polluton applcatons. In ths study the < model s appled. In the model the eddy-exchange coeffcents are evaluated from the turbulent knetc energy and the vscous dsspaton. he turbulent fluxes < u and < u u are cal he work was supported by the Russan Foundaton for Basc Research (grants No , No ) and Investgaton Grants RAS, Sberan Branch, No., No. 64. A.F. Kurbatsk, L.I. Kurbatskaya,

2 Труды Международной конференции RAMM Т. 6, Ч., Спец. выпуск 49 culated from fully explct algebrac models for the penetratve turbulent convecton from an area heat source (the urban heat sland) wth no ntal momentum under calm and stably stratfed envronment. he performance of the threeequaton model was tested by comparng the computed results wth the laboratory measurements of the low-aspect-rato plume (Lu et al. 997) and the LS data (Canuto et al. 994). Good agreements were found. he heat-sland mathematcal model A smple theoretcal model of the nocturnal urban heat sland cannot be appled to the case of near calm condtons when the ambent wnd speed approaches ero, because n low-aspect-rato plumes ( / <<, where s the mxng heght and s the heat sland dameter) the basc assumptons of classcal plume theory are volated ( Lu et al. 997). In most mathematcal models the thermal plume turbulence s parametered (cf. Byun and Araya 99). However, to analye and understand the turbulent structure of the urban-heat-sland phenomenon and ts assocated crculaton, the turbulence modelng s requred. he penetratve turbulent convecton s nduced by the constant heat flux H from the surface of a plate wth dameter (Fgure ). It smulates a prototype of an urban heat sland wth the low-aspect-rato plume ( / << ) under near calm condtons and stably stratfed atmosphere. he flow s assumed to be axsymmetrc. In the experment the mxng heght,, s defned as a heght where the maxmum negatve dfference between the temperature n the center of the plume and the ambent temperature a s acheved as shown n Fgure, and ncludes the nterface layer n the upper part of the plume. Analyss of the expermental data (Lu et al. 997) leads to the followng choce of the characterstc scale for the horontal (radal) velocty: / W = ( β gh / ρc ), β = ( / ρ) ( ρ / ) s the thermal expanson coeffcent, ρ mean densty, c specfc heat at constant pressure. For the heat sland wth a low-aspect-rato of the vertcal lnear scale to the p p p horontal Fgure a. Schematc dagram of the heat-sland crculaton ncludng horontal velocty dstrbuton and vertcal densty profles ( mxng heght, ρ densty of the reference atmosphere, ρ plume centerlne m densty). / [ g( / ) ] lnear scale the contnuty equaton yelds that the characterstc vertcal lnear scale s of the order ( Fr), and the characterstc scale for the vertcal velocty has the value ( W Fr) where Fr = W /( N ) s the Froude number, N = β s the Brunt-Vasala frequency. ( W N) /( g) a β s taken to be the characterstc temperature scale, and / s the tme scale. W Fgure b. Shadowgraph pcture of the heat sland. At t = 4 sec the full crculaton s establshed.. Governng equatons. Fundamental flud dynamcs equatons descrbng the crculaton over the low-aspectraton heat sland can be wrtten n the hydrostatc approxmaton (Pelke 984). In the absence of the Corols force and radaton, the governng equatons n non-dmensonal form for mean values of velocty and temperature n Boussnesq approxmaton are, V t + ( rv ) V + VW = H < v d

3 5, Vol. 6, Pt, Specal Issue Proceedngs of Internatonal Conference RAMM < vw < u < v + + Re ( V V r ) + Fr Re V, () r r ( rv ) + ( rw ) =, () + ( rv ) + ( W ) = Re Pr ( r + + Fr ) + r < v < w. () In equatons () - () V s the mean horontal velocty, W mean vertcal velocty, v horontal turbulent velocty fluctuaton, w vertcal turbulent velocty fluctuaton, u amuthal turbulent velocty fluctuaton, mean temperature, turbulent temperature fluctuaton, Re = ( ) / ν Reynolds number, Pr = ν / k Prandtl number, k ther- W mal dffusvty coeffcent, ν knematc vscosty, H gven heght of the stratfed layer. In () - () and everywhere n the followng dscusson captal letters and brackets <... defne mean values of varables, and lower-case letters are reserved for turbulent fluctuatons.. he explct algebrac model of turbulent fluxes. A physcally correct descrpton of the effect of stable stratfcaton on the crculaton over the heat sland can be obtaned by usng a three-equaton turbulence transport model. he K, ts dsspaton and the dsperson of turbulent fluctuatons of temperature < are found from the dfferental transport equatons, and the turbulent fluxes of momentum < u u and heat < u are determned from fully explct algebrac gradent dffuson equatons. hs three-equaton turbulence model mnmes dffcultes n descrbng turbulence n stable stratfed flow and reduces efforts requred for ts numercal mplementaton. he explct algebrac model for the turbulent heat flux vector < u can be derved from exact transport equatons (Kurbatsk 988, Sommer and So 995) n the approxmaton of equlbrum turbulence τ τ U < u = {[ < u u + ( C ) < u ] + ( C ) g β < }, (4) C x x where τ = /, τ =< / are characterstc tme scales, k < / x / x = s the destructon of temperature fluctuatons. C,C are constant coeffcents, ther values are gven below. he expresson (4) consttutes a lnear system of algebrac equatons for < u. hs expresson turns out to be mplct for flux < u because < u s ncluded nto the rght-hand sde of (4). he easest way to obtan a fully explct model for < u s to use the gradent transport model for the fluxes < u u and < u n the rght-hand-sde of (4): u u S < = ν ( / ) δ, (5) < u = k ( ), (6) where S = (/ )( U / x + U / x ) s the mean stran tensor, ν C / = turbulent vscosty, µ k = C R / turbulent thermal dffusvty, τ R = τ / rato of the characterstc scales of temperature( τ ) and dynamc (τ ) turbulent felds. Coeffcents n (4) (6) have "standard" values calbrated by modelng the homogeneous turbulence n stable stratfed flows (e.g. Sommer and So, 995): C =.9, C.95,.,. 4 µ = C = C =. Substtuton of (5) and (6) nto (4) gves the fully explct algebrac model for the turbulent heat flux vector: R < u = C R [{ν + ( C ) x C

4 Труды Международной конференции RAMM Т. 6, Ч., Спец. выпуск 5 k } S + ( C ) k Ω ]( ) + + [( C ) / C ] ( / ) Rg β <, (7) where Ω = ( / )( U / x U ) s the mean rotatonal tensor, R =. 6. For normal turbulent stresses n the rght-sde of () the present work adopts a smple Boussnesq model that preserves certan ansotropy of the normal stresses, < v = ( / ) ν ( V / ), (8) < w = ( / ) ν ( W / ), (9) < u = ( / ) ν ( V / r). () For the shear stress the Boussnesq model (5) yelds: < vw = ( / ) ν ( V / + W / ). () Substtuton of (8) () nto () leads to the close form of the equaton for the mean radal velocty V. he vertcal mean velocty W s then found as a quadrature from the contnuty equaton (). Results of the modelng dscussed below n Secton 4 show that the proposed fully explct algebrac model for turbulent fluxes of momentum and heat (5) (7) provdes qute acceptable predctons of the structure of such a complex phenomenon as the heat sland n stratfed envronment. Quanttes, and < n the three-equaton model are found from the dfferental equatons of turbulent transport. quatons for and n non-dmensonal form are, ( ) ( ) { [Re ν rv W r ν + + = + ] } Fr + {[Re + ] } + P + G, () t σ σ + ( ) + ( ) { [Re ν rv W r + ν ] } {[Re Fr + ] } = Ψ, () t σ σ τ where P = ν [ Fr ( V / ) + Fr ( V / )( W / )] ( / ) ( V / + W / ) s the K producton due to shear, G =< w K producton due to the buoyancy force fluctuaton, σ and σ turbulent Prandtl numbers ( σ =., σ =.). Functon Ψ s wrtten n the form, Ψ = Ψ + Ψ b q ( U ) / + Ψ β g < u In (4) b < u u / q δ /, q = and coeffcents / + Ψ β g < u / q ( U ) /. (4) Ψ, Ψ, Ψ, Ψ have standard values (e.g. Andren 99) calbrated by solvng dfferent atmospherc boundary layer problems: 8 =. Ψ =. 4 Ψ =. 4 ; Ψ =. quaton for the dsperson of turbulent temperature fluctuatons < s wrtten n the followng close form, < + rv < + W < t r r = [ r( C < < v )] + [ C < w < ] < v < w < R. ( C =. ) (5)

5 5, Vol. 6, Pt, Specal Issue Proceedngs of Internatonal Conference RAMM. Boundary condtons. he problem of the development and evoluton of crculaton above the heat sland s assumed to be axsymmetrc. he doman of ntegraton s a cylnder of a gven heght H. he heated plate wth dameter s located at the center of the cylnder bottom (Fgure ). he outer boundary s located at the dstance R. 5 from the cylnder axs. At the ntal moment the medum s at rest and t s stably stratfed. Condtons V = ( / ) = ( / ) = ( / r) = < / = are prescrbed at the plume axs ( r = ) and at ts outer boundary ( r = R ). At the top boundary ( = H ) the ero-flux condton s enforced, V / = / = / = < / =. Boundary condton for the temperature at the top boundary s wrtten so that the vertcal temperature gradent s the same at two last mesh ponts, ( / ) = ( / ). = = J For the horontal mean velocty on the underlyng surface the no-slp condton s specfed, V. = = he surface heat source on the bottom boundary ( = ) has the se r /. 5. It prescrbes non-homogeneous boundary condtons for,, and <. Values of and for r /. 5 at the frst mesh pont ( = ) are taken to be (Panofsky et al. 977), / / = u [{7 +.5( / L)} +.85{ + ( / L)} ], = ( u / L) { +.5( / L) } ; u frcton velocty on the underlyng surface evaluated from the expermental data (Lu et al. 997), ts value was u / W. 45. he value p J / / L = u /( кβ gh / ρc ) s the Obukhov Monn scale ( к =.4 Karman constant). Outsde the surface heat source (.5 r / R / ) values of and at the frst mesh pont are chosen accordng to Andre et al. (978) as, = C µ, = ( u / к)( Fr + 4/ L). / u For.5 r / R / temperature s taken to be equal to the surface temperature, =. On the surface of = w the source r /. 5 the heat flux H s prescrbed. Boundary condton for the dsspaton < for r /. 5 at the frst mesh pont ( = ) s obtaned from the equaton (5) approxmated as locally balanced ( producton dsspaton ) and from the equaton (7) for the turbulent fluxes: < = / = [( H / ρc ) β g / W ] (R) Fr /( C ) p. ( + R( C ) /( C C )[ H / ρc ) β g / W ]) p For.5 r / R / a background value of dsperson (at = ) s specfed as a functon of the dsperson at the heated plate, < (, r /.5) < (, r /.5). scusson of results and concludng remarks Fgure shows streamlnes for postve values of the streamfuncton ( r / < counterclock-wse crculaton), and for negatve values of the streamfuncton ( r / clockwse crculaton). Streamlnes n Fgure a (experment) and Fgure b (computaton) are smlar. hey show the man upflow n the center generated by two vortces that reaches the nterface layer ( / ), and downflow n the outer regon. Common feature for both experment and computaton s the suppresson of the plume heght by the stable stratfcaton, the ncrease n the sdeway flow and turbulence of the plume. he flow above the vortex par (Fgure b) can be explaned by local crculaton over the real-lfe heat sland. In ths stuaton the heat s not transported from the urban surface nto the upper atmosphere, nstead t s accumulated n local crculatons.

6 Труды Международной конференции RAMM Т. 6, Ч., Спец. выпуск 5 a b Fgure. Streamlne contours for Fr =. 77, R e = 88. a) expermental data (Lu et al. 997). Sold lnes: counterclockwse moton. ashed lnes: clockwse moton. b) smulaton results. Fgure. Nondmensonal varances of velocty on / at center above the heat sland. he laboratory data (Lu et al. 997: Fr =. 77 ; R e = 88 ): H horontal velocty profle,? vertcal velocty profle. he computaton data: F horontal velocty varance, vertcal velocty varance; b computed ntensty of turbulence, q =< u. Fgure 4. urbulent knetc energy dsspaton normaled by W as a functon of /. computaton (at r / =.5) ; Moeng & Wyngaard s LS (redrawn from Canuto et al. 994). he turbulent plume structure s shown n Fgure as a dstrbuton of root-mean-square fluctuatons of vertcal and horontal turbulent veloctes as functons of / at the plume center. It should be mentoned that the smple gradent model (8) and (9) not only correctly predcts characterstc features of σ v / W and σ w / W dstrbutons, but also satsfactorly reflects ther ansotropc nature. Fgure 4 shows comparsons of the vertcal profle of the K dsspaton,, computed by the three-equaton model wth LS data of Moeng and Wyngaard (redrawn from Canuto et al. 994). Fgure 5 show that temperature profles nsde the plume have characterstc swellng : the temperature nsde the plume s lower than the temperature outsde at the same heght creatng an area of negatve buoyancy due to the overshootng of the plume at the center. he heght of the overshoot s maxmum at the plume center and t decreases away

7 54, Vol. 6, Pt, Specal Issue Proceedngs of Internatonal Conference RAMM Fgure 5. Computed mean temperature profle at varous locatons at t = 8 mn ( R e = 45, Fr =.88) ; F the ntal profle. from the center. he temperature anomaly at the plume center extends to a hgher heght than at the cross-secton r / =.. hs behavor of the vertcal temperature dstrbuton ndcates that the plume has a dome-shaped upper part n the form of a hat schematcally shown n Fgure a and also n fgure b (the shadowgraph of Lu et al. 997). References Andre, J. C., et al. Modelng the 4-hour evoluton of the mean and turbulent structures of the planetary boundary layer // J. Atmos. Sc Vol. 5. P Andren, A. A K dsspaton model for the atmospherc boundary layer // Bound.-Layer Meteor. 99. Vol. 56. P. 7-. Byun,.W., & S.Arya, S. P. A two-dmensonal mesoscale numercal model of an urban mxed Layer. I. Model formulaton,surface energy budget, and mxed Layer dynams // Atmos. nvron. 99. Vol. 4A. P Canuto, V. M., et al. Second-Order Closure PBL Model wth New hrd-order Moments: Comparson wth LS ata // J. Atmos. Sc Vol. 5. P uynkerke, P. G. Applcaton of the urbulence Closure Model to the Neutral and Stable Atmospherc Boundary Layer // J. Atmos. Sc Vol. 45. P Ilyushn, B. B., & Kurbatsk, A. F. Modelng of Contamnant sperson n the Atmospherc Convectve Boundary Layer // Ivestya, Atmospherc and Oceanc Physcs Vol., P. 7-. Kurbatsk, A. F. Modelng of non-local turbulent transport of momentum and heat. Novosbrsk: Nauka, p. Lu, J., et al. A Laboratory Study of the Urban Heat Island n a Calm and Stably Stratfed nvronment // J. Appl. Meteor Vol. 6. P Mason, P.J. Large-ddy smulaton of the convectve atmospherc boundary layer // J. Atmos. Sc Vol. 46. P Moeng, C.-H. A large-eddy smulaton for the study of planetary boundary layer turbulence // J. Atmos. Sc Vol. 4. P Neuwstadt, F.. M., et al. Large ddy Smulaton of the Convectve Boundary Layer: A Comparson of Four Computer Codes // urbulent Shear Flows 8 (Selected Paper from the ghth Int.Symp. on urbulent Shear Flows) / ds. F.urst et al. Sprnger- Verlag, 99. P Panofsky, H. A., et al. he characterstcs of turbulent velocty components n the surface layer under convectve condtons // Bound.-Layer Meteor Vol.. P Pelke, R. A. Mesoscale Meteorolgcal Modelng. Academc Press, 984. Sommer,. P., & C. So, R. M. On the modelng of homogeneous turbulence n a stably stratfed flow // Phys.Fluds Vol. 7. P Sun, W.-Y., & Ogura, Y. Modelng the evoluton of the convectve planetary boundary layer // J. Atmos. Sc. 98. Vol. 7. P Zeman, O. & Lumley, J. L. Buoyancy effects n entranng turbulent boundary layers: A second order closure study // urbulent Shear Flows I. Berln et al.: Sprnger-Verlag, 979. P