EVALUATING TURBULENCE MODELS FOR 3-D FLOWS IN ENCLOSURE BY TOPOLOGY

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1 Ninh Inernaional IBPSA Conference Monréal, Canada Augus 5-8, 5 EVALUATING TURBULENCE MODELS FOR -D FLOWS IN ENCLOSURE BY TOPOLOGY Lars Køllgaard Voig ANSYS Europe Ld. Wes Cenral 7, Milon Park Abingdon, Oxfordshire OX4 4SA Unied Kingdom Lars.Voig@ansys.com ABSTRACT Compuaional Fluid Dynamics (CFD) has evolved from an academic ool o an imporan commercial design ool over he pas decades. In he presen paper he accuracy of wo equaion urbulence models for -D inernal air flows is invesigaed in erms of mean quaniies and opological aspecs. Resuls for pseudo -D and full -D simulaions using he k-ε model and he Baseline (BSL) k-ω model of Mener, respecively, are compared wih LDA measuremens (Nielsen, 99) and PIV measuremens (Pedersen & Meyer, ). In order o explain similariies and differences beween simulaion and experimens, a opological apparaus is oulined. The main conclusion of his work is ha he BSL k-ω model of Mener is superior o he k-ε model when flow srucures are o be prediced. INTRODUCTION The scope of Compuaional Fluid Dynamics (CFD) has developed from academic o highly commercial during he pas four decades. Wih he increased use of CFD for commercial purposes, he developmen of pracical guidelines has urned ou o be an imporan issue. Pracical guidelines for HVAC applicaions ypically involve suggesions for he choice of urbulence model, choice of boundary condiions, grid dependency sudies, grid refinemen near walls, convergence crierion and difference scheme (CFX,4), he ERCOFTAC Guidelines, (Sørensen & Nielsen, ). In he auhor s opinion hese guidelines are imporan and should be addressed when publishing papers on CFD. However, i is remarkably ha none of he pracical guidelines addresses a qualiaive measure such as he flow paern. The flow paern mus be assumed o be exremely imporan, when predicing for example disribuion of paricles and chemical reacions. I can be difficul o se up qualiaive measures of he flow paern. In many pracical seups his may no be possible a all. Neverheless, he heory from nonlinear differenial equaions (Grimshaw, 99) can provide such a measure for cerain geomeries. I is one of he main obecives of his paper o demonsrae how he heory from non-linear differenial equaions can provide a qualiaive measure of he accuracy of a CFD simulaion. Moreover, he resuls indicae ha he choice of urbulence model for inernal flows is no a sraighforward maer, and herefore his issue should be considered carefully. The es case considered is he Annex room (Nielsen, 99). The flow in his room conains boh regions of fully urbulen and purely laminar flow, which is represenaive for he flows in venilaed spaces (Davidson e al, ). Moreover, boh Laser Doppler Anemomeer (LDA) measuremens (Nielsen, 99) and Paricle Image Velocimery (PIV) measuremens (Pedersen & Meyer, ), exis for his geomery. Previous invesigaions showed ha - D simulaions can never reproduce he experimenal -D flow paern (Voig, ). Moreover, i was found ha a k-ω urbulence model provided beer resuls han a k-ε model, when he locaion of criical poins was o be prediced. In he presen paper he simulaions (Voig, ) are exended o hree dimensions, leading o he appearance of hree dimensional flow opologies. The paper is organised as follows: Firs he es case is presened. Then he opological apparaus is oulined followed by a shor descripion of he numerics wih special emphasis on he urbulence models. Then he compuaional resuls for pseudo - D and full -D simulaions are discussed and finally he conclusions are drawn. TESTCASE The es case employed in his sudy is he well known Annex es case (Nielsen, 99). The es case wih dimensions is shown in figure. Figure : The isohermal Annex es case - 9 -

2 The es case is isohermal and air is supplied a he inle wih a velociy of.455 m/s in order o ensure Re=5 based on he inle heigh. From experimens (Nielsen, 99) he urbulence inensiy in he inle is specified o be 4 %, and he lengh scale is chosen o be h/=.68 m (Voig, ). TOPOLOGY Topology can be a valuable ool when a qualiaive descripion of he flow is desired. As menioned in he inroducion his will be imporan when considering paricle movemen, since hese can be srongly influenced by he flow paern. Previously, a discussion of he opologies relevan for wo dimensional flow simulaions and PIV experimens has been presened (Voig e al, ). Below he resuls will briefly be summarized. In his work, opology is inerpreed as a sysemaic ool ha provides a full qualiaive descripion of he flow (Voig e al, ). The approach involves wo seps: Firs he criical poins are locaed and hen he criical poins are classified in accordance wih heory from non-linear differenial equaions (Grimshaw, 99). In wo-dimensional flows he ypes of criical poins ha can occur are resriced by he Hamilonian propery of he sream funcion. Thus, he only nondegenerae possibiliies for criical poins are cenres and saddles. The orbis near such criical poins are shown in figure. Figure : Aachmen poin locaed on a verical wall (lef). Separaion poin locaed on a verical wall (righ). When hree dimensional flows are considered, several ypes of criical poins can exis. For example a spiral poin can replace a cenre if he flow is no exacly wo-dimensional. The spiral can be eiher sable or unsable, see he orbis in figure 4. Figure 4: The orbis close o a sable spiral poin (lef) and an unsable spiral poin (righ) This concludes he presenaion of he opological apparaus. Only he ypes of criical poins found in he experimens and simulaions presened in his work has been described. However, several oher ypes of criical poins exis (Grimshaw, 99) NUMERICS * Figure : The orbis close o a cenre poin (lef), and a saddle poin (righ). For he saddle poins i has been suggesed o disinguish beween aachmen poins and separaion poins (Harnack, 999). The flow paern near an aachmen poin and a separaion poin is shown in figure. Turbulence models The governing equaions are coninuiy and he Navier-Sokes equaions. The equaions are imeaveraged, and he se of equaions are closed using a wo equaion urbulence model. Since wo differen urbulence models were employed and he choice of urbulence model had a significan impac on he flow, he esed models are discussed briefly in he following. For he ime-averaged equaions, he purpose of he urbulence model is o model he Reynolds sresses. On he assumpion ha here exis an analogy beween viscous sresses and Reynolds sresses, he ask can be reduced o modelling he eddy viscosiy appearing in he Boussinesq approximaion. A relaive simple way of modelling he eddy viscosiy is o model he spaial variaion of his quaniy by a ranspor equaion. More sophisicaed models can be employed by expressing he eddy viscosiy in erms of urbulen kineic energy, k, and dissipaion of urbulen kineic energy, ε. The relaion beween he eddy viscosiy, k and ε is found from dimensional

3 analysis, and ranspor equaions are derived for k and ε, respecively. These ypes of models are referred o as he wo-equaion urbulence models and hey are he mos widely used for engineering applicaions. For his reason i was chosen o focus on hese models in he presen sudy. The models employed were he k-ε model and he Baseline (BSL) k-ω model (Mener, 994) For he k-ε model he eddy viscosiy is given by k µ Cµ ρ ε = () This inroduces he new variables k and ε, ino he sysem of equaions. The new variables are modelled by wo ranspor equaions, which ake he form ( ρk) ρu k + = and ( ρε) ρu ε + = µ µ + k + ρε σ Pk k µ ε ε µ + ( Cε Pk Cε ρε) x + σ ε k () () A problem which occurs when using he k-ε model is ha he fricion velociy will depend on he disance o he firs grid poin. This problem can be circumvened by employing scalable wall funcions in which a limier is applied o he dimensionless near wall disance. The limiing value is ypically se o.6, which is he inersecion poin beween he logarihmic layer and linear near wall profile (CFX, 4) For he k-ω models he eddy viscosiy is given by µ k ρ ω = (4) This inroduces he variables k and ω ino he sysem of equaions. The BSL k-ω based model blends he k- ε model and he k-ω model. For his models he ranspor equaions for k and ω ake he form and ( ρk) ρu k + = µ k µ + + Pk σ k β ρkω (5) ( ρω) ρu ω + = k ( F )ρ σ ω ω µ ω µ + σ + ω ω ω + α Pk βρω k (6) The blending funcion F is designed o be one close o surfaces, corresponding o a k-ω model, and zero away from surfaces, corresponding o a k-ε model. The boundary layer is handled using an auomaic near-wall swich. This implies ha firs grid poin is no reaed as being ouside he viscous sub layer, bu is virually moved down hrough he viscous sub layer when he mesh is refined sufficien (CFX, 4) Numerical seup The compuaional grid was generaed using he commercial mesh generaor ICEM (ICEM, 4). The main room was modelled using xx59 nodes, and boh he inle and oule were meshed using 4xx59 nodes. The amoun of nodes in his mesh, which is referred o as he medium mesh, was approximaely 8,. The firs cell heigh was specified in order o ensure a y + value less han.7 everywhere, and he minimum angles were approximaely 8 degrees. For he medium mesh, a skech of he cenre-plane of he room is shown in figure 5. Figure 5: Compuaional mesh in he cenre plane (z= m) of he room. For he k-ε model a calculaion using a coarse mesh of approximaely 4,8 was carried ou. The resuls obained wih he coarse mesh and he medium mesh showed only insignifican differences. Thus, i is reasonable o assume, ha he presened resuls are no influenced by he grid size. The discreised equaions were solved using a coupled solver echnology (CFX, 4). Second order accuracy was employed using a blend facor of one. The convergence crierion was se o.5. For he k-ε model, a soluion converged o. was compared o he soluion converged o.5. No significan differences exised beween hese wo soluions, and he crierion of.5 was herefore assumed o be sufficien for having a converged soluion

4 Numerical locaion of criical poins Finding he locaion of no-slip criical poins in he cenre plane is a sraigh forward maer when assuming he w componen of he velociy is zero (Voig e al., ). Along he floor (y= m) he u componen of velociy was moniored a y=. m, and a criical poin was depiced when a change in sign was found. Similar for he verical wall below he inle (x= m), he v componen of velociy was moniored a x=. m. A criical poin was found when a change in sign occurs. The procedure for locaing in-flow criical poins was more complicaed. A conour plo of he u velociy and v velociy was creaed, using only he conour wih value zero. The locaion of he criical poin was found a he inersecion poin beween hese wo conours. When classifying he criical poins, several sreamlines were drawn in he region surrounding he criical poin. Based on he sreamlines, i was a sraighforward maer o deermine if a no-slip criical poin was an aachmen poin or a separaion poin. When classifying he in-flow criical poins he procedure of using sreamlines was more roublesome, since i could be difficul o disinguish beween a cenre and a spiral wih a slow spiralling moion. To ensure ha he used mehod does no predic a spiral poin when he poin is acually a cenre, he mehod was esed for a pseudo -D calculaion. For he pseudo -D calculaion i was known ha spiral poins canno occur. This can be ascribed o he Hamilonian propery of he sream funcion. This concludes he presenaion of he numerics, and he resuls are presened in he following secion. RESULTS Pseudo -D simulaion The flow solver always solves he governing equaions in hree dimensions, and herefore a wo dimensional flow was emulaed using a slip wall boundary condiion a he side walls (z= m and z= m) of he compuaional domain. The main reason for carrying ou pseudo -D simulaions was o ensure ha he compuaional mesh did no inroduce hree dimensional effecs in he flow. Moreover, he procedure for classifying criical poins was verified. The performance of urbulence models can be assessed by comparing velociies along wo verical lines a he cenral region in he cenre plane of he room, x= m and x=6 m (Voig, ). In figure 6 he numerical resuls using he k-ε model and he Mener model, respecively, are compared wih measuremens (Nielsen, 99). For x= m i is observed ha boh models predic higher velociies in e,.5 m < y < m. For example a y=.9 m, he simulaions gives a dimensionless velociy of approximaely.8, while measuremens gives a dimensional velociy of.78. A he floor, m < y < m, he k-ε model gives lower velociies, while he Mener model gives higher velociies han found in he LDA measuremens (Nielsen, 99). For example a y=. m he k-ε model predics a dimensionless velociy of -.6, he measuremens predics a dimensionless velociy of -., while he Mener model predics a dimensionless velociy of -.8. A x=6 m, boh models show good agreemen wih measuremens in he e, while he Mener model predics slighly lower velociies close o he floor. For example a y=. m he Mener model gives a dimensionless velociy of -.9 while measuremens gives -.. In he core region, m < y <.5 m, boh models show a deviaion from experimens which is difficul o explain. Despie he minor differences he overall conclusion is ha boh urbulence models provide resuls in good agreemen wih measuremens. Thus, he esed urbulence models are concluded o be equally good for predicing mean velociies in he cenral par of he room. H e ig h [m ].5 x= m u/u_ [-] Experimens k-epsilon Mener H e ig h [ m ].5 x=6 m u/u_ [-] Experimens k-epsilon Mener Figure 6: u componen of velociy a wo posiions in he cenre plane of he room. Experimens are LDA measuremens (Nielsen, 99) Nex he locaion and ype of sagnaion poins are evaluaed. Boh models prediced a separaion poin on he floor and an aachmen poin on he wall. The k-ε model gives significan lower values han experimens, see able, while he Mener model leads o values which are larger han in experimens

5 Table Disance from origin o no-slip criical poins. Experimens are based on PIV measuremens (Pedersen & Meyer, ) k-ε Mener Experimens Separaion [m] Aachmen [m] A similar rend is observed for he locaion of he inflow criical poin. The k-ε model predics he cenre poin a (x,y) (. m,.9 m), while his is locaed a (x,y) (.9 m,. m) using he Mener model. In he measuremens he in-flow criical poin is an unsable spiral poin, and i is locaed a (x,y) ( m,.7 m). The differen locaion of he criical poins leads o significan differen flow paern in he firs hird of he room. The k-ε model predics a recirculaion zone which is almos negligible in size, see figure 7, while he Mener model predics a recirculaion which covers almos one hird of he room lengh. everywhere in he compuaional domain. Therefore, he flow mus be wo dimensional for z consan, which implies ha only cenres and saddles can exis. Based on figure 7 and figure 8, he opology for a pseudo -D simulaion is shown in figure. Thus, i is concluded ha a pseudo -D simulaion could never capure he spiral poin found in he measuremens. Anoher imporan conclusion drawn from hese simulaions is ha he numerical seup is sufficien for disinguishing beween cenres and spiral poins. Figure 9: Sreamline for y= m. The flow is seen o be parallel wih respec o z. Figure 7: Sreamlines in he cenre plane of he room (z= m) based on pseudo -D simulaions using he k-ε model. Only half of he room is displayed. Figure 8: Sreamlines in he cenre plane of he room (z= m) based on pseudo -D simulaions using he Mener model. Only half of he room is displayed. From he sreamlines in figure 7 and figure 8, he numerical in-flow criical poin can be characerized as a cenre, while he measuremens predic a spiral poin (Voig, ). The flow paern a a plane hrough y= m is shown in figure 9. From he sreamlines i is obvious ha he flow is parallel which implies, ha he w velociy componen is zero Figure : Flow opology relaed o recirculaion zone below occurring below he inle. The k-ε and he Mener model prediced he same opology. This concludes he presenaion of he pseudo -D simulaions. I was concluded ha he k-ε model and Mener model appeared o be equally good for predicing he mean quaniies in he cenral par of he room. However, away from he cenral par of he room, he k-ε model was bad for predicing he size of he recirculaion zone occurring below he inle. Moreover, i was concluded ha he presen numerics are sufficien for classifying he criical poins. Finally i was concluded ha a pseudo -D simulaion can never capure he flow paern obained in experimens, since he flow is parallel for all values of z. -D simulaion Using he same grid as for he pseudo -D simulaions, he side walls of he compuaional domain were se o a no-slip wall. This inroduces shear a he side walls, which leads o simulaions in beer agreemen wih he physical seup

6 The performance of he urbulence models were assessed by he mean quaniies in he cenral par of he room, see figure. The rends observed were similar o hose found for he pseudo -D simulaions, excep in he core region m < y <.5 m a x= m. In his region he k-ε model seems o be in beer agreemen wih experimens han he Mener model. Despie his difference i is he auhor s opinion ha he wo models are equally good for predicing he mean velociies in he cenral par of he room. negligible in size, while he Mener model predics a recirculaion which covers approximaely one hird of he room lengh. Comparing wih measuremens, figure 4, i is obvious ha he Mener model is superior o he k-ε model. x= m x=6 m.5.5 H e ig h [m ] Experimens k-epsilon Mener H e ig h [m ] Experimens k-epsilon Mener Figure : Sreamlines in he cenre plane of he room (z= m) based on -D simulaions using he k-ε model. Only half of he room is displayed u/u_ [-] u/u_ [-] Figure : u componen of velociy a wo posiions in he cenre plane of he room. Experimens are LDA measuremens (Nielsen, 99) Nex, he locaion and ype of criical poins were evaluaed. Boh models prediced a separaion poin on he floor and an aachmen poin on he wall. The locaions are summarised in able. By comparing he values in able wih he values in able, he noslip walls seem o suppress he recirculaion zone. Table Disance from origin o no-slip criical poins. Experimens are based on PIV measuremens (Pedersen & Meyer, ) k-ε Mener Experimens Separaion [m] Aachmen [m] For he k-ε model he in-flow criical poin is locaed a (x,y) (.6 m,. m), while for he Mener model he in-flow criical poin is locaed a (x,y) (.77 m,.9 m). This should be compared o he experimenally found in-flow criical poin a (x,y) ( m,.7 m). As for he pseudo -D simulaions, he wo urbulence models predic significan differen flow paerns when -D simulaions are carried ou. The resuls based on he -D simulaions are shown in figure and figure. The k-ε model predics a recirculaion zone which is almos Figure : Sreamlines in he cenre plane of he room (z= m) based on -D simulaions using he Mener model. Only half of he room is displayed. y[m].5.5 x[m] Figure 4: Sreamlines in he cenre plane of he room (z= m) based on experimens (Pedersen &Meyer, ). From he sreamlines in figure i is no possible o classify he in-flow criical poin for he simulaion using he k-ε model, while he sreamlines in figure indicaes ha he Mener model predics a sable spiral poin. The criical poin is concluded o be sable spiral poin since he sreamline shows a

7 spiralling moion owards he criical poin. The direcion of he spiralling moion is deermined by he vecors near he sreamline. In conras, he measuremens sugges an unsable spiral poin, see figure 4. In he measuremens he criical poin is concluded o be an unsable spiral poin since he sreamline is spiralling away from he criical poin. The direcion of he spiralling moion is again deermined by looking a he vecors near he sreamline. In figure 5 he flow paern in a plane, y= m, is shown. This indicaes ha he flow is highly hree dimensional and he sreamlines show how an unsable raecory emanaes from he spiral poin ino he flow. This explains why he in-flow criical poin has changed from a cenre in he pseudo -D simulaions o a sable spiral poin in he -D simulaions. If an unsable raecory emanaes from he spiral poin in he z direcion, he spiral poin canno be unsable according o he heory from non linear differenial equaions (Grimshaw, 99). Thus, he Mener model predics a flow paern in agreemen wih heory. Since no hree dimensional flow opology is available from he measuremens, i is no possible o draw furher conclusions on he differences beween experimens and simulaions. The opology based on he Mener model and experimens is shown in figure 6. Figure 5: Sreamline for y= m. The flow is seen o be symmeric around z= m. Figure 6: Flow opology relaed o he recirculaion zone occurring below he inle. The experimens prediced an unsable in-flow spiral poin, while he Mener model prediced a sable in-flow spiral poin. This concludes he presenaion of he -D simulaions. I was concluded, ha employing he no slip wall boundary condiion a he side walls ends o suppress he recirculaion zone below he inle. When predicing mean quaniies in he cenral region of he room, he urbulence models were almos equally good. However, away from he cenral par of he room, he k-ε model was bad for predicing he size of he recirculaion zone occurring below he inle. Experimens and he Mener model prediced a differen opology for he in-flow criical poin, bu since experimens were only available in he cenre plane, i was no possible o explain his difference. However, he Mener model prediced a opology in agreemen wih heory, and herefore i is reasonable o believe, ha he differences are caused by experimenal uncerainies. CONCLUSION In he presen work a opological approach was used o evaluae wo urbulence models for simulaing he flow in an enclosure. In paricular he locaion and classificaion of criical poins was examined. Moreover opology was used o explain he differences beween pseudo -D simulaions, -D simulaions and experimens. Pseudo -D simulaions using slip boundary condiions a he side walls were carried ou o usify, ha he used numerical seup were sufficien for classifying criical poins. When assessing he numerical seup he opological apparaus urned ou o be a valuable ool, since his provides heoreical informaion abou he possible flow paerns. Three dimensional effecs are inroduced when he boundary condiions a he side walls are changed o a no-slip condiion. Based on he full -D simulaions he following imporan conclusions were drawn: The k-ε model and he Mener model performed equally good when mean velociies in he cenral region of he room were prediced The k-ε model was bad for predicing he size of he recirculaion zone below he inle, while he Mener model was much beer. The Mener model and experimens prediced a differen opology for he inflow criical poin below he inle. Since experimens are only available in he cenre plane i is no possible o conclude furher on his difference. However, based on heory from non-linear differenial equaions he opology prediced by he Mener model is consisen wih he hree dimensional flow srucures. The opological apparaus has urned ou o be a valuable ool for assessing he accuracy of numerical echniques and for explaining similariies and differences beween experimens and simulaions. In general i is suggesed ha a Mener ype of model should be used for HVAC applicaions of his kind

8 NOMENCLATURE LATIN Cε, Cε, C µ : Empirical consans in he k-ε model f, f : Damping funcions in he k-ε model F: Blending funcion h: Heigh of inle k: Turbulen kineic energy L: Lengh of room Li: Lengh of inle Lo: Lengh of oule P k : Turbulence producion : ime or Heigh of oule U i : Mean velociy componens in urbulence ranspor equaions u_: Inle velociy u, v, w: Velociy componens in x,y,z direcion W: Widh of room x i : Spaial direcions in urbulence ranspor equaions GREEK α,: Consan in he BSL k-ω model of Mener β, β : Consan in he BSL k-ω model of Mener ε: Dissipaion of urbulen kineic energy ω: Specific dissipaion rae ρ: Densiy µ: Dynamic viscosiy µ: Eddy viscosiy σ k, σ ε: Consans in he k-ε model σ k,σ ω, σ ω: Consans in he BSL k-ω model of Mener Turbulence Models for Engineering Applicaions, AIAA Journal, Vol., No. 8, Augus, Pedersen, J.M. & Meyer, K.E.,. POD Analysis of Flow Srucures in a Scale Model of a Venilaed Room, Experimens in Fluids, Sørensen, D.N. & Nielsen, P.V.,. Qualiy conrol of compuaional fluid dynamics in indoor environmens. Indoor Air,, -7. Voig, L.K.,, Navier-Sokes Simulaion of Airflow in Rooms and Around a Human Body Ph.D. Thesis, Deparmen of Mechanical Engineering, Fluid Mechanics Secion, DTU. Voig, L.K.,. Validaion of Turbulence Models using Topological Aspecs, ROOMVENT, 7-76 Voig, L.K., Sørensen, J.N., Pedersen, J.M. & Brøns, M.. Review of four urbulence models using opology, In proceedings of he 8h Inernaional IBPSA Conference, Eindhoven, Neherlands, Augus -4,, pp. 5-. REFERENCES CFX Solver Manual, 4, Ansys INC. Davidson, L., Nielsen, P.V. & Topp, C.. Low-Reynolds Number Effecs in Venilaed Rooms: a Numerical Sudy, ROOMVENT, vol., 7-. Grimshaw, D., 99. Nonlinear Ordinary Differenial Equaions, Blackwell Scienific Publicaions, ICEM User Manual, 4, Ansys INC. Nielsen, P.V., 99. Specificaion of a Two dimensional Tescase, IEA Annex Research Iem.45, Technical Repor, Universiy of AAlborg, Denmark. Harnack, J.N., 999. Sreamline Topologies near a Fixed Wall using Normal Forms, Aca Mechanica, vol. 6, issue, Mener, F.R Two-equaion Eddy-Viscosiy - -