Generation of artificial inflow turbulence including scalar fluctuation for LES based on Cholesky decomposition

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1 Generton of rtfcl nflow turbulence ncludng sclr fluctuton for LES bsed on Cholesky decomposton Tsubs OKAZE, Aksh MOCHIDA Tohoku Unversty, # Aob, Armk Az, Aob-ku, Send, , Jpn, Tohoku Unversty, # 6-6- Aob, Armk Az, Aob-ku, Send, , Jpn, mochd@sbne.pln.rch.tohoku.c.p Tsubs OKAZE. Introducton Most ttempts to couple engneerng computtonl flud dynmcs (CFD) models wth the mesoscle meteorologcl model (MMM) hve used turbulence models bsed on eynolds-verged Nver-Stokes equtons. ecently, ncresed ccess to computng power hs led to severl ttempts to couple lrge-eddy smulton (LES) nd MMM. egrdng the couplng of LES wth MMM, one of the bggest unresolved ssues nvolves the generton of n nflow turbulence whch stsfes not only the turbulent sttstcs but lso the nstntneous turbulent structures. To ths end, severl technques hve been developed n the felds of wnd engneerng nd buldng scence. These poneerng technques for genertng nflow turbulence wthn neutrl boundry lyer cn be clssfed nto two ctegores. The frst, nd lso the smplest, nvolves storng the tme hstory of velocty fluctutons obtned from prelmnry or recyclng LES computton (Lund et l., 998; Ktok nd Mzuno, ). The second nvolves rtfclly genertng nflow turbulence whch prescrbes turbulent sttstcs wthout conductng n LES computton (Lee et l., 99; Izuk et l., 999; Klen et l., 3; Xe nd Cstro, 8; Xun nd Izuk, 4). In recent yers, non-sotherml LES computtons wthn urbn boundry lyers hve been crred out to nvestgte het slnd phenomen nd so on. When LES s ppled to non-sotherml feld, not only the nflow velocty fluctuton but lso the temperture fluctuton should be reproduced. egrdng the generton of nflow turbulence consderng temperture fluctuton, only few studes hve been conducted bsed on the method usng the recyclng LES mentoned bove (Tmur et l., ; Jng et l., ) n the wnd engneerng feld. Ths pper proposes new method of genertng turbulent fluctutons n wnd velocty nd sclr qunttes such s temperture nd contmnnt levels bsed on Cholesky decomposton of the tme-verged turbulent flux tensors of momentum nd sclr. The rtfcl turbulent fluctutons generted by ths method stsfy not only the prescrbed profles for the turbulent fluxes of momentum nd sclr but lso the prescrbed sptl nd tme correltons. Accordng to the method proposed by Xe nd Cstro (8), two-dmensonl rndom dt re fltered to generte set of two-dmensonl dt wth the prescrbed sptl correlton. Then, these dt re combned wth those from the prevous tme step by usng two weghtng fctors bsed on n exponentl functon. The method ws vldted by pplyng t to LES computton of contmnnt dsperson n hlf-chnnel flow.. New method of genertng nflow turbulence ncludng sclr fluctuton In ths study, we express the vlues of wnd velocty nd sclr s f, the tme-verged vlues of f s f, nd the devton from the tme-verged vlue s f ' f f f, () where the subscrpt, =,, 3, ndctes the wnd velocty components n the stremwse, lterl, nd vertcl drectons (u, v, w), respectvely, nd = 4 ndctes the sclr vlue. A regulr mtrx of the turbulent fluxes of momentum nd sclr,, s defned s uu uv uw u vu vv vw v. () w u w v w w w u v w If the Cholesky decomposton of s mplemented, we obtn lower trngulr mtrx,, s (3)

2 Wth the lower trngulr mtrx, nd vrble stsfyng nd be rewrtten s, the fluctutons, f, cn f f f f. (4) Here, s the Kronecker delt. We extended the trnsformton orgnlly proposed by Lund et l. (998) usng eynolds stress tensors to consder the turbulent fluxes of sclr. The sttstcs obtned by usng Eq. (4) mthemtclly stsfy the prescrbed turbulent fluxes of momentum nd sclr n Eq. (). To mpose tme nd spce correltons for ech component of the fluctutons, the two-dmensonl dgtl-flter method proposed by Xe nd Cstro (8) nd then revsed by Kondo nd Izuk () ws employed. The prescrbed tme nd spce correltons re ssumed usng exponentl functons wth n ntegrl tme scle, T, nd length scle, L: t t t t r r r r exp, (5) T exp, (6) L The tme dvnces of rtfclly generted fluctutons on grd pont (m, n) re expressed s t t, m, n t, m, nexp t t, m, n N y N z t, m, n bmbn rm m, nn m n t T t exp, (7) T, (8) where r s rndom number stsfyng r = nd r r =, N y nd N z re the number of grd ponts ncluded n the generted plne n ech drecton, b k s dgtl-flter coeffcent for the ntegrl length scle n the generted plne n ech drecton. The ntervls of convoluton operton n Eq. (8) proposed by Kondo nd Izuk () were used n ths study. The flter coeffcent s defned s ~ N ~ b / k bk b, (9) ~ x k b k exp. () L By substtutng (s obtned usng new dtset of rndom numbers usng Eqs. (8) to ()) nto (7) for ech tme step, for the next tme step s obtned. Then, the fluctutons, f, re gven by substtutng nto Eq. (4). 3. Outlne of LES computtons A pror LES computtons for hlf-chnnel were crred out to vldte the reproducblty of the flow nd dsperson feld by pplyng the rtfclly generted wnd nd sclr fluctutons s nflow boundry condtons. Fg. s schemtc of the computtons. The computtonl condtons re summrzed n Tble. Frst, prelmnry LES computton ws conducted to obtn turbulent sttstcs. Ths requres the generton of wnd nd sclr fluctutons. A perodc boundry condton ws mposed n both the stremwse nd lterl drectons for the wnd feld. An verged pressure grdent ws mposed n the stremwse drecton s drvng Prelmnry LES Perodc condton only for wnd Smplng Generton of turbulent fluctutons of wnd nd sclr H H Lne source y z o x Mn LES Fg. Schemtc vew of LES computtons

3 force. The stndrd Smgornsky model ws ppled. For the sptl dscretzton n the governng equtons, second-order centrl dfference scheme ws used. A lne source ws plced on the ground cell mmedtely behnd the nflow boundry nd pssve sclr ws emtted. The emsson rte of the pssve sclr per unt tme nd unt re ws set to. (w c =.). The perodc boundry condton ws mposed only n the lterl drecton of the dsperson feld. The tme seres of the turbulent fluctutons of the wnd nd sclr vlues were stored on the y-z plne t x = 5.H. Then, the fluctutons n the wnd nd sclr vlues were rtfclly generted bsed on the Cholesky decomposton of the tme-verged turbulent flux tensor of momentum nd sclr whch were obtned from the dtbse collected t x = 5.H n the prelmnry smulton. Next, the mn LES computton ws crred out wth the rtfclly generted turbulent fluctutons s the nflow boundry condtons of the mn computton. By comprng the results for the flow nd the dsperson felds t x =.H, 3.H, nd 5.H n the mn clculton wth those obtned wth x = 6.H, 8.H, nd.h n the prelmnry clculton, the reproducblty of the flow nd dsperson felds when pplyng the rtfclly generted wnd nd sclr fluctutons s nflow boundry condtons ws vldted. The ntegrl length scles for prescrbng the spce correltons of the turbulent fluctutons of wnd velocty were ssumed to be L =.5H, bsed on the results of prevous study whch exmned the sme stuton of hlf-chnnel smulton for nvestgtng the effects of rtfclly generted fluctutons of the wnd velocty components on reproduced flow feld (Kondo nd Izuk, ). The ntegrl tme scles for prescrbng the tme correltons of the turbulent fluctutons were gven bsed on the frozen turbulence pproxmton known s Tylor s hypothess : T L/ U, () where U s the tme- nd spce-verged wnd velocty t the upper boundry. It s ssumed tht the ntegrl length nd tme scles for sclr dsperson re equl to those for the wnd velocty. The turbulent sttstcs for both smultons were collected nd verged over T* fter suffcent ntl clculton. The non-dmensonl tme, T*, s defned s T* =Tu*/H, where T s the clculton tme, u* s the frcton velocty on the surfce of the ground nd H s the domn heght. 4. esults nd dscusson 4.. Turbulent sttstcs for generted fluctutons n wnd nd sclr Fg. shows comprson of the turbulent sttstcs between the trgeted vlues obtned from the prelmnry smulton nd the rtfclly generted vlues bsed on the method proposed n Chpter. The generted men wnd velocty nd concentrton re n good greement wth the trgeted vlues. The vrnce of the generted concentrton nd the turbulent sclr flux n the vertcl drecton s obtned from the rtfcl generton re lso n good greement wth the trgeted vlues. Thus, we confrmed tht the generted turbulent fluctutons n the wnd nd sclr vlues, s obtned by ths proposed method, stsfy the prescrbed tme-verged turbulent flux tensors of momentum nd sclr. 4.. eproducblty of flow nd dsperson felds wth rtfclly generted nflow boundry condtons Fg. 3 shows the stremwse chnge n the vertcl dstrbutons of the turbulent sttstcs for the flow feld. The men wnd velocty chnges very lttle n the downstrem regon nd s n good greement wth the trget vlue obtned from the prelmnry smulton. The turbulent knetc energy n the grd scle t x =.H, ust leewrd of the nflow boundry, s rpdly dmped by 4%, reltve to the trget vlue. As prevous reserchers hve ponted out, the cuses re suspected to be relted to the rtfclly generted fluctutons not generlly beng ble to stsfy the contnuty nd momentum equtons, whle the generted fluctutons dtbse does not nclude the fluctutons n pressure (Xe nd Cstro, 8; Kondo nd Izuk, ). The turbulence knetc energy ner the surfce more closely pproches the trget vlue s the flow moves downstrem, due to the reproducton of the turbulent knetc energy by the grdent of the men wnd velocty ner the surfce. In Fg. 3 (3), the vrnce n the stremwse component of the wnd velocty fluctuton decreses by 4% t x =.H nd ncreses s the flow moves downstrem. Ths stremwse chnge s smlr to tht of the turbulent knetc energy. On the other hnd, n Fg. 3 (4), the vrnce n the vertcl component of the wnd velocty fluctuton flls sgnfcntly t x =.H, especlly ner the surfce. One of the resons for ths decrese n the vertcl component of the wnd velocty fluctuton s thought to be tht both the ntegrl length scles n the stremwse nd vertcl drectons re set to the sme vlue. Fg. 4 ndctes the stremwse chnge n the vertcl dstrbuton of the men concentrton c. The result of the men concentrton obtned from the mn smulton wth rtfclly generted fluctutons s slghtly lrger thn the result of the prelmnry smulton ner the surfce. Ths dfference could be ttrbuted to the underestmton of the turbulent dffuson of the pssve sclr n the upwrd drecton due to the dmpng of the turbulent knetc energy ner the nflow boundry. The stremwse chnge n the vertcl dstrbutons of the turbulent flux of the pssve sclr for vertcl component w c s shown n Fg. 5. The pek vlues of w c for both smultons re observed t the sme heght n ech mesured lne. However, the turbulent flux t x =.H s somewht smller thn tht due to the underestmton of the turbulent knetc energy. The flux ner the ground surfce ncreses t x = 3.H nd 5.H wth n ncrese n the turbulent knetc energy ner the ground.

4 Tble Computtonl condtons Computtonl domn H(x) H(y) H(z) Grd rrngement 3(x) 3(y) 3(z) SGS model Stndrd Smgornsky model (C s =.) Tme dvnce Second-order Adms-Bshforth scheme Sptl dervtves Second-order centrl dfference scheme Couplng lgorthm SMAC method Prelmnry smulton Flow feld: Perodc boundry condton Inflow boundry Dsperson feld: c Mn smulton Flow nd dsperson feld: Artfclly generted fluctutons bsed on the method proposed n Chpter Prelmnry smulton Flow feld: Perodc boundry condton Outflow boundry Dsperson feld: c x Mn smulton u xc x Lterl boundres Perodc boundry condtons Upper boundry Slp boundry condton Ground surfce Werner nd Wengle s pproch (99) ws dopted. A lner /7 power lw dstrbuton of the nstntneous velocty ws ssumed. eynolds number.7 4 (u H /) Drvng force Averge pressure grdent (p x = -.5U /H) Turbulent Schmdt number Trget Generted.8.6 Trget Generted.8.6 Trget Generted.8.6 Trget Generted u / U...3 c U / q.5..5 c U / q...3 wc q () u/ U () cu / q (3) c U / q (4) wc/ q Fg. Comprson of turbulent sttstcs for the trgeted vlues obtned from the prelmnry smulton nd the rtfclly generted vlues bsed on the proposed method.8.6 Trget x=h.8.6 Trget x=h.8.6 Trget x=h.8.6 Trget x=h / U k / U.5. / U.. / U () u / U () k / U (3) u / U (4) w / U Fg. 3 Stremwse chnge n vertcl dstrbuton n turbulent sttstcs for flow feld

5 Fg. 6 shows the stremwse chnge n the vertcl dstrbuton of the vrnce of the pssve sclr fluctuton c. The results of the mn smulton re overestmted for ech lne lthough the dstrbuton profle nd the heght t whch the pek vlue of c ppers re smlr to the results obtned wth the prelmnry smulton. The underestmton of w c mentoned bove, leds to poor turbulent dffuson, nd the hgh wndwrd concentrtons would trend to the leewrd sde. As result, c s overestmted n the mn smulton. 系列 7 系列 8 系列 9 系列 Trget Generted wnd nd conc. Trget Generted wnd nd conc x =.H x = 3.H x = 5.H c U / q Fg. 4 Stremwse chnge n vertcl dstrbuton of men concentrtons men wnd nd conc. Trget Generted wnd nd conc..8.6 x =.H x = 3.H x = 5.H w'c' / U / q Fg. 5 Stremwse chnge of vertcl dstrbutons of turbulent flux of pssve sclr for vertcl component Trget Generted wnd nd conc. Trget Generted wnd nd conc x =.H x = 3.H x = 5.H c' U / q/ U Fg. 6 Stremwse chnge n vertcl dstrbutons of vrnce of pssve sclr fluctuton

6 4. Conclusons A new method for genertng the turbulent fluctutons n wnd veloctes nd sclr qunttes such s temperture nd contmnnts, bsed on the Cholesky decomposton of the tme-verged turbulent flux tensors of momentum nd sclr, ws developed. By employng 5 5 non-sngulr mtrx s turbulent flux tensor mtrx, the proposed method cn generte tme seres of wnd velocty, temperture, nd concentrton of contmnnts nd so on. Ths method cn be ppled to other rtfcl generton methods bsed on the Cholesky decomposton of the eynolds stresses, ncludng the synthetc eddy method proposed by Jrrn et l. (6). LES computtons for hlf-chnnel were crred out to vldte the reproducblty of the flow nd dsperson felds by pplyng the rtfclly generted wnd nd sclr fluctutons s nflow boundry condtons. The stremwse chnge n the men concentrton ws reproduced well by mens of perodc smulton. However, w c ws underestmted nd c ws overestmted due to the dmpng of the turbulent knetc energy cused by the contnuty nd momentum blnces for the rtfclly generted wnd veloctes not beng stsfed. Further nvestgtons nto the effect of the ntegrl tme nd length scles on rtfclly generted fluctutons, s well s the pplcblty of ths method to non-sotherml flow felds should be undertken. Acknowledgment The uthors re grteful for the support provded by the Grnt-n-Ad for Scentfc eserch (C) (Grnt No ). eferences Izuk S., Murkm S., Tsuchy N., Mochd A., 999: LES of flow pst D cylnder wth mposed nflow turbulence. Proceedngs of th Interntonl Conference on Wnd Engneerng, 9 98 Jrrn N., Benhmdouche S., Lurence D., Prosser., 6: A synthetc-eddy-method for genertng nflow condtons for lrge-eddy smultons. Interntonl Journl of Het nd Flud Flow, 7(4), Jng G., Yoshe., Shrsw T., Jn X. : Inflow turbulence generton for lrge eddy smulton n non-sotherml boundry lyers. J. Wnd Eng. Ind. Aerodyn., 4 6, Ktok H., Mzuno M., : Numercl flow computton round eroelstc 3D squre cylnder usng nflow turbulence. Wnd nd Structures, 5, Kondo A., Izuk S., : An nvestgton of rtfcl generton methods of nflow turbulence for LES bsed on the Cholesky decomposton of eynolds stresses: Development of semless mergng method of mesoscle/regonl meteorologcl models nd engneerng LES models (Prt ). Trnsctons of AIJ, Journl of Envronmentl Engneerng, 77(678), (n Jpnese wth Englsh bstrct). Klen M., Sdk A., Jnck J., 3: A dgtl flter bsed generton of nflow dt for sptlly developng drect numercl or lrge eddy smultons. Journl of Computtonl Physcs, 86, Lee S., Lele S. K., Mon P., 99: Smulton of sptlly evolvng turbulence nd the pplcblty of Tylor s hypothess n compressble flow, Physcs of Fluds, A4, 5-53 Lund T. S., Wu X., Squres K. D., 998: Generton of turbulent nflow dt for sptlly-developng boundry lyer smultons, Journl of Computtonl Physcs, 4(), Tmur T., Nozu T., Okud Y., Ohsh M., Umkw H., : Hybrd method of meteorologcl nd LES models for het envronment, USB proceedngs of 8 th Interntonl Conference on Urbn Clmtes Xe Z.T., Cstro I. P., 8: Effcent Generton of Inflow Condtons for Lrge Eddy Smulton of Street-Scle Flows. Flow, Turbulence nd Combuston, 8(3), Xun Y., Izuk S., 4: Effects of mposng the contnuty condton on rtfclly generted nflow turbulence for LES. USB proceedngs of 6 th nterntonl conference on Computtonl Wnd Engneerng