A NEW FILTERED DYNAMIC SUBGRID-SCALE MODEL FOR LARGE EDDY SIMULATION OF INDOOR AIRFLOW

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1 A NEW FILTERED DYNAMIC SUBGRID-SCALE MODEL FOR LARGE EDDY SIMULATION OF INDOOR AIRFLOW We Zhang and Qngyan Chen Buldng Technology Program Massachusetts Insttute of Technology 77 Mass. Ave., Cambrdge, MA , USA ABSTRACT Large Eddy Smulaton (LES) wth a Dynamc Subgrd-scale Model (DSM) s a powerful tool to predct ndoor arflow. However, the model needs to average the model coeffcent over a homogeneous drecton. Snce most ndoor arflow does not have a homogeneous drecton, ths study proposed a new Fltered Dynamc Subgrd-scale Model (FDSM) wthout the need of a homogeneous flow drecton. The predcted ar velocty, ar temperature and turbulence dstrbutons agree reasonably well wth the expermental data. The results show that the FDSM can be used to smulate ndoor arflow. INTRODUCTION To desgn a comfortable and healthy ndoor envronment, one requres nformaton about the dstrbutons of ar velocty, ar temperature, relatve humdty, contamnant concentratons, and turbulent quanttes. These nformaton can be obtaned numercally by usng a Computatonal-Flud- Dynamcs (CFD) program wth a eddy vscosty model. However, the power spectrum of arflow may play an mportant role n thermal comfort. The power spectrum can only be calculated by Large Eddy Smulaton (LES) at present. The LES model should be a next-generaton tool to study ndoor arflow n buldngs, because t s unversal, has few or no adustable model coeffcents, can provde more flow nformaton, and can calculate flow that s dffcult to be determned by other CFD models. For example, wth an eddy-vscosty model, the mean arflow through a wndow over tme would close to zero for natural ventlaton under sothermal condtons. However, an LES smulaton can correctly predct the nstantaneous arflow through the wndow. Typcal ndoor arflow ncludes natural convecton, such as wnter heatng by a baseboard heater; forced convecton, such as free coolng n shoulder seasons; and mxed convecton, such as summer coolng wth an ar condtoner. The ndoor arflow s complex and s drven by pressure gradents and thermal buoyancy. Very few LES studes on ndoor arflow have been reported (Davdson and Nelsen 996, Emmerch, and McGrattan 998, Murakam et al. 995). The results by LES do not agree very well wth the expermental data probably due to a constant model coeffcent used n the Smagornsky model. Dynamc Subgrd-scale Model (DSM, Germano et al. 99) s avalable to calculate the model coeffcent as a functon of tme, space, and flow type. The model requres to average the coeffcent over a homogeneous flow drecton because the coeffcent fluctuates sgnfcantly. It s possble to reduce the fluctuaton by averagng the coeffcent through a flter, whch wll be dscussed extensvely n ths paper. In addton, for ndoor arflow, t s dffcult to fnd a homogeneous flow drecton. It s necessary to fnd a smple method to determne the DSM coeffcent for ndoor arflow. Ths nvestgaton proposes a Fltered Dynamc Subgrd-scale Model (FDSM) to determne the model coeffcent for flow wthout a homogeneous drecton. The FDSM has been appled to natural and mxed convecton flows. The correspondng expermental data avalable from the lterature and the computed results wth the DSM have been compared to examne f the FDSM can correctly predct ndoor arflow. THE DYNAMIC SUBGRID-SCALE MODEL (DSM) The LES requres the separaton of small-eddes from large-eddes wth a flter. For smplcty, the followng secton uses one-dmensonal notaton. The fltered velocty s: u = G(x, x )u (x) dx () where G(x, x ) s a flter functon. The flter functon s large only when G (x, x ) s less than the flter wdth, a length scale over whch averagng s performed. The flow eddes larger than the flter wdth are large-eddes and smaller than the wdth are small-eddes. In the physcal spaces, the paper uses a box flter,.e.: ( x ) G(x ) =, () 0 ( x > ) Wth the fnte volume method, t seems natural to defne the flter wdth,, as an average over a grd volume. Wth the flter, t s possble to derve the governng conservaton equatons for the momentum (Naver- Stokes equatons), mass contnuty, and energy. The fltered Naver-Stokes equatons for an ncompressble flow are:

2 u + t x u υ x x = u u u u P (u u ) = + ρ x x + g θ θ ) δ where the subgrd Reynolds stresses are 0 () = u u u u (4) LES also solves the fltered energy equaton wth subgrd heat fluxes wth heat transfer problem: h = u θ u θ (5) The terms, u u, and u θ, are unknown and need to be modeled. Dynamc Subgrd-Scale Model. The Reynolds stresses can be modeled usng the Smagornsky model n order to close the equatons. The Smagornsky model uses a constant model coeffcent that depends on the flow type. On the other hand, Germano et al. (99) proposed a dynamc subgrdscale model (DSM). The DSM calculates the model coeffcent by relatng the subgrd scale Reynolds stresses to two dfferent szes of flters. Snce the Reynolds stresses vary wth tme and locaton, the resultng model coeffcent s therefore a functon of tme and locaton. The DSM uses an explct test flter, G, wth a flter wdth of ( > ) to determne the turbulent stresses on the G flter: T u u u u = (6) The frst term on the rght sde of the equaton cannot be determned drectly, as that n Eq. (4). However, substtutng Eq. (6) from the Eq. (4) wth a test flter can elmnate the term: T = L (7) L where (8) The resolved turbulent stresses n Eq. (8), L, can be calculated explctly. Wth the defnton of the Smagornsky model, the stresses of the test flter, T, and that of the grd flter,, are: δ kk = C S S = C (9) δ T Tkk = C T S S = C T α (0) where s the grd flter wdth and the test flter wdth. The C and C T are the coeffcents of the grd and test flters, respectvely. Further, S = () u u ( + ), S = SS x x Substtuton of (9) and (0) nto (7) gves: L δ L kk = C T α C () where α = S S and β = S S. The C n () cannot be solved explctly because t s n the test flterng operaton. Germano et al. (99) extracted the C from the flterng operaton, and assumed: C C C T () C = C (4) Usng the least-square approach suggested by Llly (99), the C can be solved va < LM > C = (5) < MM > where M = ( α β ), and < > denotes a plane averagng over a homogeneous drecton. Wthout an average over the homogenous drecton, the C fluctuates a lot and that makes the soluton of the flow very unstable. The averagng procedure can dampen large fluctuatons of the C often encountered n a flow predcton. Ths procedure gves good results for smple flows wth at least one homogeneous drecton, such as a turbulent channel flow. However, the averagng procedure cannot be used for a flow wthout a homogeneous drecton. Smlarly, Llly (99) modeled the subgrd-scale heat fluxes ( h = u θ u θ ) wth a smplfed Boussnesq approxmaton to determne the subgrdscale Prandtl number (Pr SGS ), PR = (6) PrSGS C R where, P = u θ u θ (7) R = S θ x θ S x Fltered Dynamc Subgrd-Scale Model. Snce most ndoor arflow does not have a homogeneous drecton. Ths paper proposes a smple method to determne the model coeffcent n the DSM by usng a localzed flter technque. It s done by applyng a grd flter to Eq. (7). The use of the grd flter s to average the coeffcent and to smooth the large fluctuaton of the coeffcent. The technque wll lead to a stable numercal soluton. Then all the terms n Eq. (7) wll be related to the grd flter: (8)

3 T = L (9) Replacement of and T n Eq. (9) wth a model, such as the Smagornsky model or the mxed model (Zang et al, 99), gves rse to an error n satsfyng model Eq. (9). The error assocated wth a model s gven by model e L (T model = ) (0) For smplcty, the use of the Smagornsky model leads to the followng error equaton: e Usng the defntons of α = S S and β = S S, Eq. () can be re-wrtten as: e = L C T α + C () If we assume: C α C α, () T T C C (4) C C C T. (5) Then the equaton becomes e = L C T α + C = L CM (6) where M = α. The followng paragraphs dscuss the assumptons used n Eq.() to (5). Meneveau et al. (996) used DNS data to analyze the two hypotheses n Eqs.() and (4). They fltered DNS data at both a grd flter and test flter and compared the coeffcents obtaned wth and wthout the tme averagng procedure. Although the two coeffcents are not equal wthout averagng, they are smlar wth the tme averagng: DNS DNS C C T (7) DNS { C } { C T } DNS (8) where {} denotes the tme averagng. Meneveau et al. (996) also dscussed the mnmzaton error caused by approxmatng C = C wth and wthout usng the averagng technque. The study shows that the mnmzaton error wth the averagng s smaller than that wthout the averagng: { C } Errormn (, C ) (9) << Error mn (C, C = L C T S S + C S S () ) Snce the flterng s also a knd of averagng technque, the results obtaned by Meneveau et al. (996) may be extended to the grd flterng technque. Therefore, the assumptons used n Eqs. (), (4), and (5) should be vald. These assumptons should be better than those used n DSM (Eqs. () and (4)). Localzaton of the Coeffcent wth Least-Square Approach. The present nvestgaton uses the leastsquare approach to obtan the localzed coeffcent, the C n Eq. (6), as suggested by Llly (99). At any gven pont n a space, x, the e s a functon of the C but depends on the x. In order to obtan an optmal C, the e must be ntegrated over the entre flow doman. On the other hand, the least-square approach requres the optmzaton over the entre flow doman. However, the square of the resdual, e e, may have a locally volent change. The e e should be ntegrated n the entre flow doman wth a smooth functon. Thus, the ntegrated square of the error functon, (C), s E E ( C) = G( x, x ) e ( x ) e ( x ) dx (0) Substtute Eq. () nto Eq.(7), and Eq. (7) reads: E (C) = G( x, x )(L CM ) dx () Snce the least square condton for the Eq.() s E (C) = 0, then the optmal model coeffcent C s C obtaned as: G( x, x ) L M dx C = () G( x, x ) M M dx The C s obvously a functon of tme and space and s nconsstent wth the defnton gven by Germano et al. (99). The C here seems superor to the one proposed by Germano et al. (99), because t can be appled to nhomogeneous flows. The smooth functon G (x, x ) should be chosen for the entre flow doman and may depend on the turbulent scales. Although the smooth functon can be n many forms, a box flter may be the most convenent. The flter can be ether a grd flter or a test flter: L M C = (wth the grd flter) () M M or M M L C = (wth the test flter) (4) M Eqs. () and (4) are now defned as the Fltered Dynamc Subgrd-scale Model (FDSM). The FDSM- G s wth grd flter for Eq. () and FDSM -T wth test flter for Eq.(4). The FDSM s much smpler than those proposed by Ghosal et al. (995) and Meneveau et al. (996).

4 The FDSM- G or FDSM- T can also be locally negatve. A negatve C ndcates a negatve eddy vscosty and mples an energy transfer from small scales to the resolved scales or backscatter, accordng to Pomell et al.(99). However, the negatve C can also lead to numercal nstablty. In order to avod the nstablty, the present nvestgaton uses C = max (0.0, Eq.() or (4)) (5) Smlarly, we can also calculate the Prandtl number of the dynamc subgrd heat fluxes by: Pr or SGS Pr SGS = = C C P R R R PR R R (wth the grd flter) (6) (wth the test flter) (7) APPLICATIONS TO INDOOR AIRFLOW To demonstrate the ablty of the new FDSM, the model s used to calculate several typcal arflows n rooms: F natural convecton n a room wth a heated wall and a cooled wall F mxed convecton n a room wth a heated floor and a cold ar et near the celng Natural Convecton. The nvestgaton has selected the natural convecton flow n a cavty as shown n Fg. (a). Cheesewrght (986) measured the ar velocty, temperature, turbulence energy, and heat transfer n the cavty. The flow characterstcs of the cavty are smlar to those n a room. The smple geometry elmnates many potental errors, such as those caused by the complex geometry of a baseboard heater. If there are dscrepances between the computed results and measured data, ths would allow us to dentfy the reasons. Snce natural convecton conssts of both turbulent and lamnar flows, t s very challengng to use LES to smulate the flows. Ths case enables us to use the DSM, whch can be averaged over the depth drecton. The depth drecton can be consdered homogeneous. The LES wth the DSM and FDSM (FDSM-G, FDSM-T) s used to predct the dstrbutons of ar velocty, temperature, and turbulence energy. Fg. (a) shows the cavty geometry (heght AC=.5 m, wdth AB=0.5 m, and the depth s 0.5 m) as well as the numercal grd dstrbuton used n the LES smulatons. The temperature dfference between the warm and cold walls, θ, s 45.8 K (left wall temperature, θ, s 68.0 o C, and rght wall temperature, θ, s. o C). All of the other walls were nsulated. The flow corresponds to a Raylegh number (Ra) of 5.0x0 0, smlar to that found n a typcal room. The Ra s defned as Fg. The predcted results of the natural convecton n a cavty wth the FDSM-G at depth =0.5 m. (a) The cavty geometry, (b) average ar velocty, (c) average ar temperature. 4

5 ( θ θ ) gh Ra = (8) υα The computatons used no-slp velocty condtons for all the walls. The mesh employed were 6 6 for the heght (X), wdth (Y), and depth (Z) drectons, respectvely, and the tme step t =0.000 s. The ntal ar velocty was zero and ar temperature 45. o C for the whole flow doman. When the flow became statstcally steady, the averagng technque was used to obtan the mean value of the computed parameters, such as ar velocty and temperature. The averagng tme s about 0 s. Fgs. (b), (c) show the dstrbutons of average ar velocty, average ar temperature wth FDSM-G on the center secton (depth s 0.5 m), respectvely. The velocty feld s asymmetrc. The hot wall generates an upward flow near the wall and the cold wall a downward flow. The velocty n the center of the cavty s generally small. The flow s lamnar n the lower part of the hot wall and the upper part of the cold wall. Fg. compares the predcted and measured results n the span-wse of the cavty. The results wth the DSM (averagng n the depth drecton), the FDSM- G and the FDSM- T agree rather well wth the expermental data except near the wall regons. However, the FDSM- G and FDSM- T can predct flow wthout averagng along a homogeneous drecton. Ths s very mportant because there s no homogeneous drecton n most rooms wth natural convecton. As reported by Cheesewrght (986), the top and bottom walls of the cavty were not well nsulated. The heat loss n the lab envronment led to a lower mean ar temperature n the cavty. As a result, the predcted mean ar temperature n the cavty s hgher than the measured data as llustrated n Fg. (b). All models predcted a reasonably good temperature profle. The present nvestgaton calculates the turbulence energy as k =(u +v +w )/ (grd scale). Fg. (c) compares the computed k profles wth the expermental data. The DSM model under-predcts the turbulence energy near the walls. The performance of the FDSM- G and FDSM- T are smlar to that of the DSM, although they stll under-predct the turbulence energy, especally near the walls. The reason may be attrbuted to the ad hoc fx for the coeffcent C (C 0.0). Ths case shows that the FDSM- G and FDSM- T have the same performance as the DSM. Snce the FDSM- G and FDSM- T do not need averagng over a homogeneous drecton, they can be used to calculate more complex arflow. When comparng the FDSM- G and FDSM- T, the FDSM- G s slghtly better than the FDSM- T. Mxed Convecton. The present nvestgaton also appled the FDSM for mxed convecton flow n a room, as shown n Fg. (a). Blay et al. (99) measured the ar velocty, temperature, and turbulent energy dstrbutons for the case. The geometry of the test rg was H =.04 m long, L =.04 m wde, and D = 0.7 m deep. Ths s a scale-model of a room and has a homogeneous drecton (the depth drecton) so that the DSM can also be used. The nlet heght, h n, was 0.08 m, the supply ar velocty, U n, was 0.57 m/s, and the supply ar temperature, T n, was 5 C. The outlet heght was 0.04m. The test rg used a floor heatng system wth a floor temperature, T f, of 5 o C. All other wall temperatures, T w, were 5 o C. The correspondng gh n T Archmedes number ( Ar = ) s and U n the Reynolds number ( = U h n n Re ) s 678. ν The computatons used a non-slp velocty condton for all the walls. The meshes employed were 6 6 for the heght (X), wdth (Y), and depth (Z) drectons, respectvely. Fgs. (b) and (c) show the measured mean ar velocty dstrbuton and the averaged ar velocty dstrbuton usng the FDSM- G. The arflow patterns are almost the same between the measurements and computatons. The LES smulaton shows a small recrculaton n the left-bottom corner, but not n the experment. It s not clear f ths s due to nsuffcent fne measurng ponts or due to the numercal model used. Fgs. 4 and 5 further compares the predcted mean ar velocty, temperature, and turbulent energy dstrbutons usng the DSM, FDSM- G and FDSM- T wth the expermental data at two center sectons at X = 0.50 m and Y = 0.50 m. Fgs. 4(a) and 5 (a) show that the three subgrd-scale models gve very smlar ar velocty profles. The FDSM- G s slghtly better than the others. The predcted velocty profles agree reasonably well wth the expermental data. However, Fgs. 4(b) and 5(b) ndcate that the predcted ar temperature usng the three models s about.5 K hgher than the measured one, though the shape of the predcted temperature profles s the same as the measured one. The models may overpredct the heat transfer from the floor or underpredct the heat transfer to the other walls. Snce no detaled measurements on the heat transfer were avalable, t s dffcult to dentfy the actual cause of the dscrepances. Perhaps the subgrd-scale Prandtl number was not correctly modeled for the buoyancy effect. Fgs. 4(c) llustrates that the computed turbulent energy (k / ={(u +v +w )/} / ) profles at secton X = 0.50 m and the comparson wth the correspondng expermental data. The FDSM- G and DSM can predct the turbulence energy dstrbuton well, whle the performance of the FDSM- T was poor. At secton Y = 0.50 m, all three models overpredcted the turbulence energy as shown n Fg. 5(c). The models need to be developed further to predct the buoyancy effect correctly. 5

6 Fg. Comparson of the predcted and measured results at md heght of the cavty. (a) average ar velocty, (b) average ar temperature, (c) average turbulent energy. Fg. The predcted and measured mxed convecton flow n a room. (a) room geometry, (b) average velocty vectors obtaned from the experment (Baly et al,99), (c) average velocty vectors computed by the FDSM-G 6

7 Fg. 4 Comparson of the predcted and measured results on the center sectons (X=0.50m). (a) average velocty, (b) average temperature, (c) average turbulent energy (k / ). Fg. 5 Comparson of the predcted and measured results on the center sectons (Y=0.50m). (a) average velocty, (b) average temperature, (c) average turbulent energy (k / ). 7

8 The reason for developng the new model s to calculate ndoor arflow wthout a homogeneous flow drecton. However, we have not found sutable expermental or drect-numercal-smulaton data for ths type of flow from the lterature, although the flow exsts n most rooms. Valdaton of the model needs detaled nformaton of Reynolds stresses and heat fluxes. Further research n the drecton s needed. CONCLUSION Ths study has developed a new fltered dynamc subgrd-scale model (FDSM) for large eddy smulaton of complex flow wthout a homogeneous drecton. The model has been used to predct natural and mxed convecton flow n a room. The computed results are compared wth the expermental data avalable from the lterature and those wth the dynamc subgrd-scale model (DSM). The FDSM can correctly predct the average ar velocty and temperature dstrbutons n the room wthout a homogenous drecton. However, t s more dffcult to calculate the heat transfer near a wall and the turbulence energy dstrbuton n the room. The models need to be developed further to predct the buoyancy effect correctly. The performance of the FDSM s the same as that of the DSM, but can be used for nhomogeneous flows, such as more complcated arflow n a room. NOMENCLATURE Ar : Archmedes number C: DSM coeffcent (C=Cs ) Cs: Smagornsky model coeffcent D: Room depth G(x ): Flter functon g: Gravtatonal acceleraton H: Room heght L: Room wdth p : Grd fltered pressure Pr: Molecular prandtl number Pr SGS : Subgrd-scale prandtl number Ra: Raylegh number T: Temperature u : Veloctes u : Grd fltered veloctes u : Test fltered veloctes v: Velocty vector x : Cartesan space coordnate β : Thermal expanson coeffcent θ: Tempearture T : Dmensonless tme step t : Tme step : Flter sze ν: Knematc vscocty ACKNOWLEDGMENT The research was supported by the Center for Indoor Ar Research. REFERENCES Blay, D., S. Mergu, and C. Nculae Confned turbulent mxed convecton n the presence of a horzontal buoyant wall et ASME HTD- Vol., Fundamentals of Mxed Convecton, pp. 65-7, 99. Cheesewrght, R., K.J. Kng,and S. Za Expermental data for the valdaton of computer codes for the predcton of twodmensonal buoyant cavty flows, HTD-60, ASME Wnter Annual Meetng, Anahem, p.75, 986. Davdson, L. and P.V. Nelsen Large Eddy Smulaton of the flow n a three-dmensonal ventlated room. Precedng of 5 th Internatonal conference on ar dstrbuton n rooms ROOMVENT 96, pp. 6-68, 996. Emmerch, S.J. and K. B. McGrattan Applcaton of a Large Eddy Smulaton model to study room arflow ASHRAE Transactons, 04, 998. Germano, M. U. Pomell, P. Mon, and W. H. Cabot, A dynamc subgrd-scale eddy vscosty model, J. Physcs Fluds A, 760, 99. Ghosal, S., T. Lund, P. Mon, and K. Akselvoll, A dynamc localzaton model for large-eddy smulaton of turbulent flows, J. Flud Mechancs 86, 9, 995. Llly, D.K. A proposed modfcaton of the Germano subgrd-scale closure method, J. Physcs Fluds A, 4, 6, 99. Meneveau, C., T. Lund, and W. Cabot, A Lagrangan Dynamc Sub-grd scale Model of Turbulence, J. Flud Mechancs., 5, Murakam, S., A. Mochda, and K. Matsu Large Eddy Smulaton of non-sothermal room arflow, -comparson between standard and dynamc type of Smagornsky model- SEISAN-KENKYU, Journal of Insttute of Industral scence, Unversty of Tokyo, 47(); 7-, 995. Pomell, U., W. H. Cabot, P. Mon, and S. Lee, Subgrd-scale backscatter n turbulent & transtonal flows, J. Physcs Fluds A,, Zang, Y., R. L. Street, and J. R. Koseff, A dynamc mxed subgrd-scale model and ts applcaton to recalculatng flow, J. Physcs Fluds A, 5, 86, 99. 8