Numerical simulation of turbulent flow through a straight square duct using a near wall linear k ε model.

Transcription

1 It. Jl. of Multphyscs Volume Number Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model. Ahmed Recha, Hassa Na, *, Glmar Mompea, ad Abdelatf EI Mara 3 Faculté des Sceces et Techologes de Tager, B.P 46 Tager, Maroc ahmed.recha@polytech-llle.fr Uversté des Sceces et Techologes de Llle, Polytech-Llle, LML - UMR CNRS 87, Boulevard P. Lagev, Vlleeuve d Ascq, Frace hassa.a@polytech-llle.fr glmar.mompea@polytech-llle.fr 3 Ecole Mohammada d Igéeurs, Laboratore de TurboMaches Aveue Ib Sa B.P Agdal, Rabat, Maroc mara@em.ac.ma ABSTRACT The am of ths work s to predct umercally the turbulet flow through a straght square duct usg Reyolds Average Naver-Stokes equatos (RANS) by the wdely used k ε ad a ear wall turbulece k ε f µ models. To hadle wall proxmty ad o-equlbrum effects, the frst model s modfed by corporatg dampg fuctos f µ va the eddy vscosty relato. The predcted results for the streamwse, spawse veloctes ad the Reyolds stress compoets are compared to those gve by the k ε model ad by the drect umercal smulato () data of Gavrlaks (J. Flud Mech., 99). I lght of these results, the proposed k ε f µ model s foud to be geerally satsfactory for predctg the cosdered flow. Key words: Computatoal flud mechacs; Reyolds Average Naver- Stokes k-epslo turbulece models; fte-volume CFD algorthm; duct flow.. INTRODUCTION The predcto of turbulet flows volvg secodary motos ad usg umercal smulatos of the Reyolds Average Naver-Stokes (RANS) equatos has great practcal value flud mechacs ad may applcatos ca be foud cetrfugal machery desg. I these smulatos, the am s to obta accurate values of pressure ad velocty for the flow feld. Amog the varous RANS models for turbulece, the two-equato turbulece k ε models have foud ther broad applcatos of feasblty the maorty of egeerg practce for the predctos of the complex turbulet flows. By far the most popular turbulece model s the stadard k ε model whch s * Author to whom all correspodece should be addressed.

2 38 Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model based o the lear stress-stra relato tally proposed by Boussesq. Ths type of closure has bee revealed robust ad effcet wth respect to CPU tme tha more hghorder models [,]. I ths work, the umercal vestgato of a low Reyolds umber turbulet flow through a straght square duct s carred out order to descrbe ts turbulet characterstcs. I spte of appearaces, ths flow s dffcult to carry out umercally ad/or expermetally. Ths cofgurato has bee frequetly chose by may authors [3,4,5,6,7] sce t s a relatvely smple geometry whch provdes a good case test to mprove turbulece models. Note that ths flow s asotropc ad volves a secodary flow the cross-stream plae, whch s abset the case of a plae chael. Ths secodary flow covects mea flow mometum from the cetre of the duct to the corers. Ths causes a bulgg of the streamwse velocty cotours towards the corers. By studyg the org of the secodary flow for the square duct, Mompea [8] ad Huser et al. [9], usg data of ths problem, cocluded that the streamwse vortcty s resposble for the geerato of the secodary flow. The equato of mea vortcty the streamwse drecto, Ω= W / y V / z, reads as: V W = v w y z y z ( ) Ω Ω z y v w where V ad W are the mea spawse ad ormal veloctes, v ad w are the turbulet ormal stresses, vw s the cross-stream correlato ad v s the Kematc vscosty. From ths formulato, the three terms o the rght sde of the above equato ca be terpreted as the vortcty producto (whch s lked to the asotropy of the cross-stream ormal stresses), the producto term due to the cross-stream correlato vw ad the vortcty vscous dsspato. As the Reyolds umber of the flow creases, the vscous dsspato term teds to be eglgble ad other terms become relevat. Accordg to these authors, t seems therefore possble that the streamwse vortcty has ts org the equalty of the ormal Reyolds stresses combed wth a presece of lateral wall gradet. I other words, the vortcty producto term, ( v w )/ y z,plays a crucal role the geerato of secodary flows. It s well kow that tradtoal two-equato models are capable of descrbg such flows. Whle, tradtoal two-equato models, the stadard k ε model, fal to predct the ear wall behavour ad asotropy of the ormal Reyolds stresses, t s show ths paper that these stadard k ε eddy vscosty models ca be mproved va the troducto of dampg fuctos of Va Drest type to produce the asotropy. Prevous studes have bee carred out ad provded terestg results, see Gavrlaks [], Huser & Brge [] ad Xu ad Pollard [4] for a low Reyolds umber turbulet flow. I ths work, the predcted results are compared wth the data of Gavrlaks [], the LES predcto of Xu [4] ad the expermetal data of Nsho [7] at a Reyolds umber 48 based o the duct hydraulc dameter ad o the bulk velocty. As the turbulet characterstcs are good agreemet wth the ad expermet, ths provdes the motvato for the study of ths flow from further models as o-equlbrum models [] ad olear turbulece models [3,4]. The paper s orgased as follows: the turbulece model k ε ad the troducto of dampg fuctos are brefly reported wth secto ; the fte volume umercal method s preseted secto 3. I secto 4, the ma results of ths work are preseted, dscussed ad fally some coclusos are draw. ν Ω ()

3 It. Jl. of Multphyscs Volume Number GOVERNING EQUATIONS To predct the turbulet flow, the statstcal approach s used applyg the Reyolds decomposto, whch cossts splttg velocty ad pressure to a average ad a fluctuatg part. As all studes of Reyolds stress modellg, ay flow varable φ ca be decomposed as follows: φ = φ ϕ () where φ s the mea turbulet value ad ϕ s the fluctuatg compoet. The equatos goverg the mea velocty U ad the mea pressure P are obtaed from the RANS equatos for a compressble flow: U = (3) U t P U ( U U ) = ( ν ρ u u ) (4) where ρ s the flud desty ad τ = u u s the Reyolds stress tesor. I order to acheve closure of ths system of equatos, a Reyolds stress model that les τ to the global hstory of the mea velocty feld must be supplemeted. A varety of expressos have all depedetly establshed that, to the lowest order, τ ca be represeted by the stadard eddy vscosty form [5,6]: τ = kδ ν S 3 t (5) where k s the turbulet ketc eergy, v t s the eddy vscosty, S = ( U / x U x / )/ s the mea stra rate tesor ad δ s the Kroecker tesor. Note that ths approach, the problem was reduced from dervg equatos descrbg the evoluto of τ to dervg equatos for k wth a approprate eddy vscosty v t. I ths study, the eddy vscosty s expressed as: νt = C k µ ε (6) where ε s the dsspato rate of the turbulet ketc eergy k ad C µ s a model costat. I the two-equato k ε cotext, the turbulet ketc eergy trasport equato s derved from the cotracto of the Reyolds stress trasport equato, ad s gve by: k (7) U k = k k t ε where the rght-had sde represets the trasport of k by the combed effects of turbulet U trasport ad vscous dffuso t k k = ν, the turbulet producto k = ν t S σ k ad the sotropc turbulet dsspato ε.

4 3 Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model As for the trasport equato of the dsspato rate ε, a rather geeral expresso from whch may of the forms ca be derved. The modellg of ths equato s based o aaloges wth the k equato ad o pheomeologcal cosderatos. I ts most commoly used form, ths equato ca be wrtte as [7]: = ε ε ν t ε σ ε ε U C k ε t x x ε x k k Cε k (8) The Eqs. (7)-(8) cota fve costats C µ, C ε, C ε, σ k ad σ ε whose values are usually obtaed from calbratos wth homogeeous shear flows ad from the decay rate of homogeeous ad sotropc turbulece. The most commoly used values of these costats are: C µ = 9. ; C ε = 44. ; C ε = 9. ; σ k =. ; σ ε = 3.. Ths k ε model s most commoly used turbulet shear flow computatos wth a great deal of success. I the two-equato k ε models, Eqs. (7) ad (8) ther hgh-reyolds umber form, do ot provde the correct asymptotc behavour the wall rego. I order to hadle ear wall effects, dampg fuctos f µ are assocated wth the eddy vscosty defto (see for example [8,9]): ν = C f t µ µ wth fµ = ( aexp( bz ))( aexp( by )) () where the costats a ad b ca take varous values for each elemet of the Reyolds stress tesor τ. Prevous works have bee doe usg a pror test ad terestg results have bee obtaed, see refereces [] ad []. It s ths assumpto that we adopted ths paper. These correctos allow to corporate vscous molecular dffuso the vcty of sold boudares. The varables y ad z are the wall co-ordates based o the frcto velocty u = τ ad the cematc vscosty v, y = u τ y/v ad z = u τ z/v. τ k ε ρ w (9) 3. NUMERICAL METHOD The method used to solve the set of equatos (3)-(4)-(7)-(8) s the classcal fte volume method (FVM). The coservato equatos are tegrated over a cotrol volume, ad the Gauss theorem s used to trasform the volume tegrals to surface tegrals. 3.. SPACE DISCRETIZATION To descrbe the umercal algorthm, the RANS equatos ca be wrtte the form: φ t ( U ) ( J ) = S φ ϕ ϕ ( ) where φ ca be ay of the above varables UVW,,, Pk,,ε ad the overbar represets the esemble average. Ths s the so-called trasport equato for property φ. It clearly hghlghts ()

5 It. Jl. of Multphyscs Volume Number Table Equato Φ J φ S φ Masse coservato Mometum Eq. x-axs U Pδ ν U / x uu ( ) ( ) y-axs V Pδ ν V / x vu 3 ( ) z-axs W Pδ ν W / x wu Turbulet ketc eergy k νt σ k k uu U / ε νt ε Dsspato ε C σ ε x ε uu k U ε Cε ε k Σ Ω Ω Κ γ Fgure Calculato Doma Ω ad cotrol volume Ω κ. the varous trasport processes. For a Cartesa system ( xyz,, ), the expressos of the varous terms of the equato () ca be gve by (see Table ). The key step of the fte volume method s the tegrato of the trasport equato () over a cotrol volume Ω k of boudary γ (see Fgure ). The, applyg the Gauss dvergece theorem, the resultg equato ca be wrtte as follows: ur r Ω φdω J U dγ J dγ = S t k γ ϕ. γ ϕ Ωk ϕ ()

6 3 Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model Y N Y Y W w θ e E Y s S X X X X Fgure Arragemet for a staggered grd. N Ω k u u W W U w p k u u θ ε U e E Ω kw Ω k Ω ke u u W s Ω ks S Fgure 3 Cotrol volume for a staggered ad otato used. The equato () gves the assessmet of ay scalar flow property φ the cotrol volume Ω k. Ths equato s the base of the space dscretzato of FVM. There s o eed for all varables to share the same grd; a staggered mesh may tur out to be advatageous. Ths arragemet s show Fgures ad 3. To solve the dfferetal equato, a fte volume

7 It. Jl. of Multphyscs Volume Number umercal method has bee employed. The pressure, the turbulet ketc eergy, k, the dsspato, ε, ad the ormal Reyolds stress compoets are treated the ceter of the cotrol volumes; the veloctes are computed the ceter of the faces ad the cross compoets of the Reyolds tesor are attached to odes located at the md-edges. The bggest advatage of the staggered arragemet s the strog couplg betwee the veloctes ad pressure. Ths helps the avod some types of the covergece problems ad oscllatos pressure ad velocty felds. The tegrals volume of the temporal ad source term of the equato () are approxmated by adoptg a Eulera scheme ad usg the theorem of the average. The advecto terms for the mometum equato arsg from the fte volume tegrato are approxmated by the quadratc upstream terpolato scheme for the covectve ketcs (QUICK) scheme of Leoard []. Ths scheme s secod order accurate space. More detals about the umercal procedure are gve [,3]. 3.. TEMPORAL DISCRETIZATION The statoary soluto was obtaed by a tme-marchg algorthm. For a frst order tme scheme, the system (3)-(4)-(7)-(8) s wrtte as: U = (3) U U t P ( UU ) = ( ν ρ U uu ) (4) K t K = U K U νt S ε ν t σ k K (5) ε ε ε ν ε = U Cε t t x K U ε t S C ν ε ε K σε (6) where the superscrpt s the prevous tme step ad s the ext oe. The covectve, dffusve, producto ad dsspato terms of the dfferet trasport equatos are treated by a explct Euler scheme. The advecto terms the k ε equato are dscretzed space usg the frst order upwd scheme. The dffuso terms are dscretzed wth a secod-order cell-cetred scheme. The pressure s treated by a mplct scheme, where the decouplg procedure for the pressure s derved from the Marker ad Cell (MAC) algorthm proposed by Harlow ad Welch [4]. The method of soluto cossts substtutg the equato for the velocty at the ew tme wth the dscretzed equato for mass coservato at the same tme level. Wth ths method, the pressure at the ext tme step s obtaed by the resoluto of the dscretzed Posso equato. The, the veloctes are calculated from the mometum equatos. Sce the lear system for the pressure equato s symmetrc ad postve defte, t ca be solved by the Cholesky procedure or by a precodtoed cougate method. Note that the prcple of mass flux cotuty s mposed drectly va the soluto of ths equato. Covergece was declared whe the maxmum ormalzed sum of absolute resdual ~ ε for each varable φ over all the

8 34 Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model z Mea flow y h w v x Wall bsector u Fgure 4 Flow geometry ad axes system. Symmetry boudary codtos z Wall y Wall Fgure 5 Quadrat of the square. computatoal doma was less tha 4. Ths maxmum error s for each varable determed as: ~ N ( φ φ ) ε = Max = φ where N s the umber of cotrol volumes.

9 It. Jl. of Multphyscs Volume Number COMPUTATIONAL DOMAIN AND BOUNDARY CONDITIONS The geometrcal cofgurato the square duct wth the referece axes s show fgure 4. The x-axs desgates the streamwse drecto. The ormal drecto s parallel to the y-axs ad the spawse drecto s parallel to the z-axs. The cross-secto s dvded to four quadrats. Because of symmetry, oly quadrat of the square eed to be calculated wth symmetry codtos at the ceter (see fgure 5). For the frst tme-step, at the let of the duct, a costat profle was gve to U, k ad ε. The secodary veloctes were talsed as l (V = ad W = ) all over the doma. The k ad ε let values were obtaed from the data usg the r.m.s veloctes ( u rms = ( τ / 3) / ) ad the eddy vscosty, that was about four tmes the molecular vscosty. At the outlet, a homogeeous Neuma boudary codto was used for all varables. I the ext tme-step, the calculated outlet values were used for the let codto. The same procedure s used for the followg tme-steps up to the covergece. The calculatos were carred out usg 4 4 grd pots, regularly spaced the crosssecto ad fve grd pots the streamwse drecto. The outcome wth ths grd was foud to be satsfactory. The grd covergece was checked usg grd pots the cross-secto, the maxmum dfferece observed betwee the two calculatos were less tha % the streamwse velocty ear the corer ( =.). A test case wth grd pots the streamwse drecto ad the cross-secto was made order to check the grd depedece the streamwse drecto, ad o relevat dfferece was observed because the maxmum dfferece was about % for the spawse velocty at =.6. The boudary codto values for k ad ε, at the frst grd pot ear the wall, was calculated takg to accout the fact that ths pot was the vscous sub-layer (always y p 5). Also, due to the use of a staggered grd, the value of k ad ε are ot defed at the wall. I ths paper, we cosder the followg boudary codtos for the equato of k ad ε, whch have bee used by Patel et al. [9]: k k or, ad ε = ν k or = = y z y ε = ν k z Whe ths s doe, t s ecessary to modfy the model tself ear the wall. It s argued that the effects that eed to be modelled are due to the low Reyolds umber ear the wall ad a umber of low Reyolds umber modfcatos of the k ε model have bee proposed; see Patel et al. [9] ad Wlcox [5] for a revew of these modfcatos. A codto for symmetry, homogeeous Neuma, was used for all the varables alog the wall bsectors of the square duct. 4. RESULTS Ths secto shows the results for a turbulet flow a square duct quadrat usg the lear k ε model wth ad wthout dampg fuctos f µ. Drect Numercal Smulato () of Gavrlaks [] s used to compare some turbulece characterstcs. The developed turbulet flow through a straght square duct has bee smulated. The has bee carred out for the Reyolds umber based o the hydraulc dameter of the duct (h) ad the mea flow velocty U m s 48. The Reyolds umber based o the frcto velocty u τ, s Re τ = u τ h/v = 3. The velocty rato U /U m for ths cofgurato was.33, U beg the mea cetrele velocty. The maxmum Kolmogorov scale s.5v/u τ. I the followg, the presetato of the characterstcs of the flow are cofed to oe duct quadrat.

10 36 Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model (a) (b) (c) Fgure 6 Streamwse flow cotours obtaed from (a), (b) k ε f µ, (c) k ε. Fgure 6 shows the cotours of streamwse velocty for oe quadrat, obtaed wth the stadard k ε model, the k ε f µ model ad the. By the examato of ths fgure, we remark that the flow s symmetrc accordg the corer bsector. The predctos are good agreemet wth data. The model k ε s uable to predct the effect of vortex duced by the secodary flow o the cotours. The profles of the mea streamwse velocty (U ) for several secto are show Fgure 7(a)-(c) respectvely, for =., ad.. I these fgures, we show the comparsos betwee the lear k ε model, the k ε f µ model, of Gavrlaks [] ad the expermetal data of Cheesewrght et al. [7]. At the secto =., Fgure 7(a) shows a strog dstorto o the mea velocty geerated by the secodary flow. O the other had, the lear k ε model used was uable to predct ths dstorto o the mea streamwse velocty

11 It. Jl. of Multphyscs Volume Number (a) =. preset smulato : Gavrlaks (99) preset smulato EXP : Cheesewrght (99) (b) = preset smulato : Gavrlaks (99) preset smulato EXP : Cheesewrght (99) U/U U/U (c) preset smulato : Gavrlaks (99) preset smulato EXP : Cheesewrght (99) =..75 U/U Fgure 7 The dmesoless mea streamwse velocty profles at dfferet sectos: (a) =., (b) =, (c) =. ths rego. After =, the agreemet betwee the results of the dfferet models ad expermets starts to mprove, ad s good at the wall bsector (see Fgure 7c, =.). The comparsos betwee the two models, of Gavrlaks ad expermetal data [6] for the spawse velocty ( V ) are preseted fgure 8(a) ad 8(b), respectvely, for two sectos, z /h =. ad =.8. There s a good qualtatve agreemet betwee the varous datasets of, measuremets ad k ε f µ. At z /h =.8, we ote that the spawse velocty ( V ) s overall well predcted by

12 38 Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model (a) preset smulato : Gavrlaks (99) preset smulato EXP : Cheesewrght (99) (b) preset smulato : Gavrlaks (99) preset smulato EXP : Cheesewrght (99) = = Fgure 8 The dmesoless mea spawse velocty profles alog two sectos: (a) =., (b) =.8. Table Values of costats a ad b used the f µ fucto uu vv ww uw a 4.6. b (a) (b) Fgure 9 Secodary velocty vectors. (a) square duct (left sde) ad k ε f µ model results (rght sde); (b) square duct (left sde) ad k ε model results (rght sde).

13 It. Jl. of Multphyscs Volume Number (a) (b) (c) E-6.69E E E-6.63E E-6.69E E E E-6.63E E E E E E E E-6.69E-6.69E E E-6.63E E E E E E E-6.63E E E E E E-6 69E E Fgure The dmesoless spawse flow cotours obtaed wth the (a), (b) k ε f µ, (c) k ε. the k ε f µ model, but the level of (V ) s uderpredcted at the posto of strog velocty, e.g. at. (see Fgure 8(a)). It s also oted that the dfferece betwee the k ε f µ model ad the data, the rego of strog velocty, decreases as z /h creases. The lear k ε does ot gve ay secodary flow (V = ) for all the values of z /h. The fgure 9 depcts the secodary velocty vectors the quadrat obtaed from, k ε f µ ad k ε models. The results are show o the left sde of the dagoal, ad the k ε f µ or the k ε results model o the rght sde (cf. Fgure 9(a) ad 9(b) respectvely). As ca be see, the k ε f µ model s able to capture reasoably the vortces

14 33 Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model (a) uu (c) uu = z =.7 (b) uu z z z (d) uu =.3 = Fgure The ormal Reyolds stress uu profles alog the wall bsector at =.,.3,.7 ad.. Comparso of the predctos of the k ε ad k ε f µ models wth. structure. I other words, t s apparet that the cross-flow plae cotas a vortex alog the y ad z walls ad the secodary vector feld seems to be smlar to that obtaed by Gavrlaks [6]. I Fgure (c), we ca observed clearly that the spawse cotours obtaed by the k ε model are ot satsfactory. O the other had, the spawse cotours obtaed by the k ε f µ model are compared favourably wth those of the data (see Fgures (a) ad (b)). I order to well predct the profles of the Reyolds stresses, we have troduced the dampg fuctos f µ (see Eq. ()) whose the ma role s to descrbe better the patter of the flow the ear-wall regos. Accordg to our smulatos, the best values of costats a ad b whch allow the reproducto of the Reyolds stresses uu = uu uτ, ( 3) ad uw = uw u τ are gve Table.

15 It. Jl. of Multphyscs Volume Number (a) (b) vv.5.5 =. vv.5.5 = z z (c) (d) vv.5.5 =.7 vv.5.5 = z z Fgure The ormal Reyolds stress vv alog the wall bsector at =.,.3,.7 ad.. Comparso of the predctos of the k ε ad k ε f µ models wth. I Fgures,, 3 ad 4, comparsos betwee the two models ad drect umercal smulato data for the three ormal Reyolds stresses uu, vv, ww four dfferet sectos ( =.,.3,.7 ad.) ad for the shear stress uw two sectos ( =.7,. ) alog the wall bsector are show. The dstrbuto of the prmary ormal stress uu alog the wall bsector for sectos =.,.3,.7 ad. predcted by the k ε ad the k ε f µ models are preseted Fgure. We ote that the k ε f µ model gves a better overall agreemet wth the data at =.7 ad =.. Ths quatty s rather well predcted the rego z f 5 startg from the secto =.7. We clearly see that the maxmum of ths quatty gve by the k ε f µ s hghly mproved. The lear k ε uderpredcts ths compoet all the sectos. At the secto y/ h =., the maxmum of the ormal stress uu s uderestmated

16 33 Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model (a) ww.5 =. (b) ww.5 = z z (c) ww.5 =.7 (d) ww.5 = z z Fgure 3 The ormal Reyolds stress ww alog the wall bsector at =.,.3,.7 ad.. Comparso of the predctos of the k ε ad k ε f µ models wth. by about 7% by the k ε model. However, whe the k ε f µ model s used, ths dfferece ca be reduced up to 35% (cf. Fgure d). The profles of the ormal Reyolds stress compoets vv ad ww ad the tagetal stress uw alog the wall bsector gve by the k ε ad k ε f µ models are show Fgures, 3 ad 4 respectvely. It seems clear that the k ε model caot correctly reproduce all these stresses whereas the k ε f µ model gves very good results partcular for =.7 ad =.. Whe cosderg the secodary ormal stress vv (cf. Fgure ), the results from the k ε f µ model are overall agreemet wth the at =.,.3,.7, whereas at =., some substatal dffereces rema partcular for z f 5 (cf. Fgure d). As for the k ε model, t overpredcts ths quatty for z f startg from the secto =.. Cosderg the ormal stress ww (see Fgure 3), the k ε f µ model agrees well wth the, especally the rego z 4 at =.7 ad =.. However, at =.3 (see Fgure 3b), the two models gve completely dfferet

17 It. Jl. of Multphyscs Volume Number uw uw =.7 = z z Fgure 4 The dmesoless Reyolds stress uw alog the wall bsector at =.7 ad.. Comparso wth results from those gve by the. At =.7 ad =., the k ε model overpredcts hghly the results. Fgure 4 shows the profle of the prmary shear stress uw obtaed by the k ε ad k ε f µ models. At =.7 ad =., the dffereces amog results of the k ε f µ models are ot substatal. The results of =.7 from the k ε model, dcate that ths model uderpredcts slghtly ths compoet for p z p6, whereas at y/ h =., they dcate that the model uderpredcts eve more ths quatty for p z p9. It s also oted that the k ε f µ model predcts the rght posto of the peak of the prmary shear stress uw at z 5 (cf. Fgure 4). Fgure 5 dsplays the ormal Reyolds stresses uu, vv, ww ad the prmary shear stress uw profles alog the wall bsector of the square duct at cetrele ( =.). These values are compared wth the of Gavrlaks [], the LES of Xu ad Pollard [4] ad the expermetal data from Nsho ad Kasag [7]. I global sese, the predcted results by the k ε ƒµ model dsclose that ths model ca gves a overall agreemet. It s see that the dscrepacy s rather weak the secodary ormal stresses vv, ww ad for the shear stress uw dcatg that our methodology may be cosdered as suffcet. The dstrbuto of the turbulet ketc eergy from the ad the k ε f µ model are compared Fgure 6. The dffereces amog our results ad the are ot substatal. The k ε f µ model yelds better results tha the results of the k ε model (see Fg. 6c). The fgure 6b shows the results of the k ε f µ model dcatg a qualtatvely good agreemet of k wth the profle obtaed by. 5. CONCLUSION The purpose of ths paper s to descrbe a study of the turbulet flow a straght square duct whch volves a secodary flow. The spatal dscretzato of the RANS equatos s performed by a fte volume method wth a staggered varable arragemet. The lear k ε model s employed for the predcto of the cosdered flow. Ths model has bee modfed by usg dfferet dampg fuctos. The preset correctos were valdated

18 334 Numercal smulato of turbulet flow through a straght square duct usg a ear wall lear k ε model (a) uu (b) vv.5 preset smulato preset smulato : Gavrlaks (99) LES : Hogy Xu () EXP : Nsho (989) = z preset smulato preset smulato : Gavrlaks (99) LES : Hogy Xu () EXP : Nsho (989) = z (c) ww.5.5 (d) uw preset smulato preset smulato.9 : Gavrlaks (99).8 LES : Hogy Xu () EXP : Nsho (989).7 = z =. preset smulato preset smulato : Gavrlaks (99) LES : Hogy Xu () EXP : Nsho (989) z ( ) Fgure 5 The dmesoless Reyolds stresses uu, 3 ad uw alog the wall bsector at =.. Comparso of the predctos of the k ε ad k ε f µ models wth, LES ad expermet. by comparg the calculated velocty profles ad the turbulet quattes wth the avalable drect umercal smulato data gve Ref. []. These comparsos yelded the followg coclusos: Mea velocty profles are, geeral, good agreemet wth the data partcular whe oe approaches the ceter of the duct ( =. or =.). Although some dscrepaces exst for the turbulet quattes, the k ε f µ model leads to smlar results. The evaluato of the cosdered model wth data reveals that some asotropy close to the wall ca be captured. The k ε f µ model tested ths work yelds cosderably better predctos tha those obtaed by the stadard k ε model.

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