Corresponding author: Tsubasa Okaze,

Transcription

1 Academc Artcle Journal of Heat Island Insttute Internatonal Vol. - (7) Large-Eddy Smulaton of on-isothermal Flow around a Buldng Usng Artfcally Generated Inflow Turbulent Fluctuatons of Wnd Velocty and Ar Temperature Tsubasa OKAZE * Akash MOCHIDA * * School of Envronment and Socety, Tokyo Insttute of Technology * Graduate School of Engneerng, Tohoku Unversty Correspondng author: Tsubasa Okaze, okaze.t.aa@m.ttech.ac.p ABSTRACT Ths study ams to apply artfcally turbulent fluctuatons of wnd velocty and ar temperature as nflow boundary condtons for a large-eddy smulaton of a non-sothermal flow around a buldng. The buldng was an solated hgh-rse structure wth a length rato of ::. The stratfcaton condton of the flow feld was weakly unstable. The artfcal turbulent fluctuatons satsfed the prescrbed profles for the turbulent fluxes of momentum and temperature. Moreover, they fulflled the prescrbed spatal and tme correlatons expressed usng exponental functons. The mean wnd velocty and mean ar temperature predcted from the large-eddy smulaton agreed well wth the expermental results. The turbulent knetc energy and the fluctuaton of the ar temperature also corresponded farly well wth the experment except for the regon mmedately behnd the nflow boundary. Key Words : Inflow turbulence generaton, Temperature fluctuaton, Large-eddy smulaton, Unstable stratfcaton, Synthetc method, Cholesky decomposton. Introducton In recent years, large-eddy smulaton (LES) computatons that consder the atmospherc stablty n bult-up envronments have been reported. Generaton of fluctuatons of wnd velocty and ar temperature fluctuaton used as an nflow boundary condton has become an mportant ssue n mplementng LES for non-sothermal flow felds. Kong et al. () proposed a method for generatng the nflow fluctuatons of wnd velocty and ar temperature. Ths method s based on the rescalng and recyclng laws for the flow feld proposed by Lund et al. () and s extended to the temperature feld. Hattor et al. (3) conducted drect numercal smulatons for very weak unstable and stable boundary layers usng the method proposed by Kong et al. () to generate nflow turbulent fluctuatons of wnd velocty and ar temperature. Jang et al. (4) performed LESs for boundary layers, both weakly stable and weakly unstable stratfcatons, usng Kataoka s recyclng method (5) to generate nflow turbulence and ar temperature fluctuatons. Tamura et al. (6) conducted an LES for a non-sothermal flow feld n the Tokyo metropoltan area wth nflow turbulence and ar temperature fluctuatons. A drver regon was establshed consstng of two domans n the prelmnary calculaton for generatng nflow turbulence and ar temperature fluctuatons. Matsuda et al. (7) nvestgated the greenery effect n the Tokyo Bay area usng LES. They set a long approachng secton over the Tokyo Bay to generate the wnd velocty and ar temperature fluctuatons. However, the methods for generatng the velocty and temperature fluctuatons based on a prelmnary or addtonal LES need addtonal computatonal costs n the prelmnary or addtonal calculatons. Furthermore, n these methods, the dstrbutons of mean wnd velocty and ar temperature and ts fluctuatons are obtaned after the prelmnary computaton. Ths means t s not easy to mpose the prescrbed profles of wnd velocty, ar temperature and the turbulent fluxes of the momentum and ar temperature. Thus, the process of the tral and error s usually needed. Recently, the present authors proposed a method for generatng turbulent fluctuatons of wnd velocty and varous scalar quanttes, such as ar temperature and contamnant concentraton, based on Cholesky decomposton of the tme-averaged turbulent flux tensors of momentum and scalar quanttes (8). The artfcal turbulent fluctuatons usng - 9 -

2 ths method satsfy both the prescrbed profles for the turbulent fluxes of momentum and the consdered scalar quanttes. The authors then confrmed the flow feld and dsperson of a passve scalar n a boundary layer were well reproduced wth artfcally nflow boundary condtons, ncludng wnd velocty and passve scalar fluctuatons. evertheless, the performances of artfcally fluctuatons of wnd velocty and ar temperature n a non-sothermal flow feld have not been nvestgated. The obectve of ths study was to therefore apply artfcally turbulent fluctuatons of wnd velocty and ar temperature as nflow boundary condtons for a large-eddy smulaton of a non-sothermal flow feld around an solated hgh-rse buldng. The LES results were compared wth those of a wnd tunnel experment.. Overvew of nflow turbulence generaton and scalar fluctuaton We developed an artfcal generaton method of the fluctuatons of wnd velocty and scalar based on Cholesky decomposton of the Reynolds stress tensor and turbulent scalar flux (8). In ths study, the proposed method was employed to generate the wnd velocty and ar temperature fluctuatons for LES nflow boundary condtons. The proposed method s summarzed as follows. The wnd veloctes and scalars are expressed as f, the tme-averaged values of f are denoted as f, and the devatons from the tme-averaged value are expressed as f ' f f f, () where the subscrpts, =,, 3, represent the wnd velocty components n the streamwse, lateral, and vertcal drectons (u, v, w), respectvely, and = 4 represents the scalar value,. A regular matrx of the turbulent fluxes of the momentum and scalar, R, s defned as uu uv uw u v u v v v w v. R wu wv ww w u v w The Cholesky decomposton of R produces a lower trangular matrx, a : R R R a a a R. 3 R3 aa3 (3) R33 a3 a3 a a R 4 R4 aa4 R43 a3a4 a3a4 R44 a4 a4 a43 a a a33 Wth a and a varable that satsfes = and =, the fluctuatons, f, can be rewrtten as f f f f a. (4) The transformaton orgnally proposed by Le and Mon (9) uses () Reynolds stress tensors as a 3 3 matrx. Here, t s extended to account for the turbulent fluxes of a scalar wthn a 4 4 matrx. To mpose tme and space correlatons for each component of the fluctuaton, a two-dmensonal dgtal-flter method proposed by Xe and Castro () s employed. The prescrbed tme and space correlatons are assumed usng exponental functons wth an ntegral tme scale, T, and a length scale, L: and t t t t r r r r exp, (5) T exp. (6) L The update equaton for the artfcally fluctuatons on a grd pont ( n) s expressed as where t t, n t, nexp t T t t t, n exp, T y z t, n bmb nrm nn. m y n z Here, r s a random number satsfyng r = and r r =, y and z relate to the length scale of the flter n each drecton, and b k s a dgtal-flter coeffcent for the ntegral length scale n the plane n each drecton, as descrbed below: ~ ~ k bk / b b, (9) where ~ x k b k exp. () L By substtutng (as obtaned usng the new dataset of random numbers from Eqs. (8) to () nto Eq. (7) for each tme step, for the next tme step s obtaned. Then, the fluctuatons, f, are gven by substtutng nto Eq. (4). 3. LES of non-sothermal flow around a buldng 3.. Calculaton condtons For LES valdaton data, we selected a wnd tunnel experment for a non-sothermal flow feld around an solated hgh-rse buldng model wth dmensons DDD (H = D) n the streamwse (x), lateral (y), and vertcal (z) drecton, respectvely. It was conducted n a thermally stratfed wnd tunnel at Tokyo Polytechnc Unversty (). The stratfcaton of the approachng flow was weakly unstable. The total measurement ponts were 6. Of these ponts, 5 were n the vertcal center secton, y =. H. Another 8 ponts were located on the horzontal plane at z = /4 H. (7) (8) - 3 -

3 The bulk Rchardson number, R, s. and s defned as R = (g / ) (/ ) / (U / ). Here, s the boundary layer heght, and U and denote the mean streamwse wnd velocty and ar temperature at the boundary layer heght, respectvely. f represents the mean temperature of the wnd tunnel floor, s the temperature dfference between and f : f, denotes the mean ar temperature n the boundary layer, and g s the gravty acceleraton. Fg. llustrates the computatonal doman. The computatonal doman was set to.5h(x)6.h(y)5.h(z), where H s the buldng heght. The cross secton of the computatonal doman corresponds to that of the wnd tunnel. An orthogonal grd system was appled. The calculaton condtons are summarzed n Table. LES was performed wth OpenFOAM ver..4. (). The standard Smagornsky model was used for the sub-grd scale turbulence model. The Smagornsky constant was set to.. The buldng wdth and heght were unformly dscretzed nto and 4 wth computatonal grds, respectvely. The grd sze was expanded at a dstance from the buldng wth an expanson rato of.8. A second-order lnear-nterpolaton scheme was used for the dffuson terms. In convecton terms, the frst-order upwnd scheme was automatcally mxed to 4% when the convecton boundedness crteron was volated to avod the appearance of unphyscal oscllatons n the result. Ths scheme was mplemented as flteredlnearv n OpenFOAM (). Fluctuatons of wnd velocty and ar temperature for the LES nflow boundary condton were artfcally at ntervals of /6 H on the nflow boundary (7.5 H (y) 6.5 H(z)) based on the method descrbed n Secton. The characterstc turbulent length, L, n each component of wnd velocty was assumed to be a constant value,.5 where s the boundary layer heght. The characterstc tme scale, T, was calculated from L and the mean streamwse wnd velocty at the boundary layer heght U based on Taylor s hypothess of frozen turbulence, T = L / U. The turbulent length and tme scales of the ar temperature were assumed to be same values of these of wnd velocty n ths study. A run-up calculaton was performed over t* = wth dmensonless tme, t*, whch s defned as t* = t u H / H. Here, t s the tme calculaton, and u H denotes the mean wnd velocty at the buldng heght of the nflow boundary. From that pont, the turbulent statstcs were collected and averaged over t* = 4 as the man smulaton. 3.. Results and dscusson The wnd velocty and turbulent statstcs were normalzed wth the mean wnd velocty at the buldng heght of the nflow boundary u H and buldng heght H. The ar temperature was normalzed wth the temperature dfference between the mean Fgure : Computatonal doman Table : Calculated condtons Computatonal.5 H(x) 7.5 H(y) 6.5 H(z) doman Grd arrangement 47(x) 4(y) 9(z) SGS model Standard Smagornsky model (C s =.) Tme advancement Frst-order backward Artfcally fluctuatons based on Inlet boundary the method proposed n Secton Outlet boundary Advectve outflow condton Upper and sde Flow feld: Slp wall boundares Temperature feld: Adabatc condton Flow feld: a two-layer model based on the lnear-power law expresson proposed by Werner and Wengle (3) was used to estmate wall shear stress. Ground and Temperature feld: The wall temperatures buldng were fxed wth the expermental result. A boundares two-layer model based on the lnear-power law expresson proposed by Tuchya et al. (4) was used to estmate heat flux on the walls. Reynolds number.5 4 (u H H ) z / H [] u / u H [] () u / u H () ( f ) / z / H [] u' / u H [] (3) u' / u H (4) ' / (θ θ f ) / [] Fgure : Comparson of ed values and artfcally z / H [] z / H [] θ' / θ [] values for nflow boundary condton - 3 -

4 () Scalar wnd velocty from the wnd tunnel experment shown n Fg. 3. The mean wnd velocty and ar temperature predcted from LES agreed very well wth those of the experment. The degree of agreement between the expermental and LES results was evaluated by the ht rate (5) and a factor of two of observatons (FAC) (6). Evaluated values of these two valdaton metrcs are compared n Table. The ht rate, q, s used as a metrc by the German VDI Gudelne (5) for the comparson of veloctes, and t s calculated from wth n q n for for else whle FAC s evaluated usng P O D () O, P O W () Ar temperature Fgure 3: Dstrbutons of nstantaneous scalar wnd velocty and ar temperature at x =.5 H (nflow boundary), y = H (center secton) and z =.5 H temperature of the floor and that of the buldng heght at the nflow boundary ( f H ). Fg. shows a comparson between the ed values and the artfcally values for the nflow boundary condton based on the method n Secton. The mean wnd velocty and ar temperature completely agree wth the ed values. The varance of the fluctuaton of streamwse component of wnd velocty and the fluctuaton of ar temperature are also n good agreement wth the ed values. Thus, we confrmed that the turbulent fluctuatons, as obtaned by ths proposed method, satsfy the prescrbed tme-averaged values. Fg. 4 llustrates the dstrbutons of scalar wnd velocty and ar temperature at x =.5 H (nflow boundary), y = H (center secton) and z =.5 H. The spatal heterogenetes of wnd velocty and ar temperature were observed at the planes y = H and z =.5 H. The fluctuatons on the nflow boundary were advected to the downstream regon. On the leeward areas of strong wnd regons on the buldng sde, the flow near the floor was elevated. Then, the hgh temperature ar was transported to the upper regon and the hgh temperature areas were scattered on the leeward of the strong wnd regon. The streamwse component of the mean wnd velocty and mean ar temperature obtaned from LES were compared wth those ormalzed u (LES) wth n FAC = n = P for.5 O. for O W and P W else () Here, s the total number of measurement ponts, O s the observaton result, and P s the predcted result. In ths study, D and W were set to.5 and.5, respectvely. Table shows the metrcs evaluated for the streamwse component of the mean wnd velocty and turbulent knetc energy (TKE). Accordng to the VDI gudelne (5), the lmt q.66 could be used as the qualty acceptance crteron. The metrcs are hgh for the streamwse component of the mean wnd velocty. However, the TKE has relatvely low metrcs y =.x R² = ormalzed u (Exp.) () Streamwse component of mean wnd velocty () Mean temperature Fgure 4: Comparson of expermental results wth LES results Table : Valdaton metrcs wth q and FAC Streamwse component of mean wnd velocty q y =.989x R² = ormalzed Θ (Exp.) FAC TKE ormalzed Θ (LES) - 3 -

5 Furthermore, Fg. 5 shows the dstrbutons of the streamwse component of the mean wnd velocty, mean ar temperature, TKE, and the varance of temperature fluctuaton n vertcal lnes at x =. H,. H,. H, and. H, and y =. The mean wnd velocty and ar temperature predcted from LES agree very well wth those of the experment. TKE corresponds relatvely well wth the experment, except for ust behnd the nflow boundary. TKE decreased by 4% of the mposed value as the nflow boundary condton mmedately behnd the nflow boundary. The cause of the above decrease of TKE lkely relates to the. u / u H [-] Exp. LES () Streamwse component of mean wnd velocty ( f / [-] Exp. LES () Mean ar temperature. k / u H [-] Exp. LES (3) Turbulent knetc energy.5 ' / [-] Exp. LES (4) Varance of ar temperature fluctuaton Fgure 5: Dstrbutons of statstcs n vertcal lnes at x =. H,. H,. H and. H fact that the artfcally velocty fluctuatons do not satsfy the contnuty and momentum equatons, as shown by Xe and Castro (). The varance of ar temperature fluctuaton agrees well wth the experment, except for behnd the nflow boundary. The dstrbuton of the varance of ar temperature fluctuaton shows a smlar trend as the dstrbuton of TKE behnd the nflow boundary. The varance rapdly decreased, smlar to the decrease of TKE. 4. Conclusons In ths study, a tme seres of wnd velocty and ar temperature fluctuatons was based on the Cholesky decomposton of the Reynolds stress tensor and turbulent heat flux. The artfcally fluctuatons were used as the nflow boundary condton of LES of flow around a buldng model under a weakly unstable condton. The mean wnd velocty and ar temperature predcted by LES usng the nflow turbulence were generally n good agreement wth the expermental results. Although TKE decreased by 4% of the mposed value as the nflow boundary condton mmedately behnd the nflow boundary, TKE and the varance of ar temperature fluctuaton agreed reasonably well wth the experment, except for the regon mmedately behnd the nflow boundary. Acknowledgement The authors are grateful for the support provded by the Grant-n-Ad for Scentfc Research (C) (Grant o ). References () Kong H., Cho H., Lee J.S., Drect numercal smulaton of turbulent thermal boundary layers. Phys. Fluds, (). () Lund T.S., Wu X., Squres K.D., Generaton of Turbulent Inflow Data for Spatally-Developng Boundary Layer Smulatons. J. Comput. Phys., 58 (998), pp (3) Hattor H., Houra T., agano Y., Drect numercal smulaton of stable and unstable turbulent thermal boundary layers. Int. J. Heat Flud Flow, 8 (7), pp.6-7. (4) Jang G., Yoshe R., Shrasawa T., Jn X., Inflow turbulence generaton for large eddy smulaton n non-sothermal boundary layers. J. Wnd Eng. Ind. Aerodyn., 4 (), pp (5) Kataoka H., Mzuno M., umercal flow computaton around aeroelastc 3D square cylnder usng nflow turbulence. Wnd Struct., 5 (), pp (6) Tamura T., ozu T., Okuda Y., Ohash M., Umakawa H. Hybrd method of meteorologcal and LES models for heat envronment. Proc. 8th Int. Conf. Urban Clm., (), Dubln, Ireland

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