Computation of Developing Turbulent Flow through a Straight Asymmetric Diffuser with Moderate Adverse Pressure Gradient

Transcription

1 Journal of Applied Fluid Mechanics, Vol. 0, No. 4, pp , 07. Available online a ISSN , EISSN DOI: /acadpub.afm Compuaion of Developing Turbulen Flow hrough a Sraigh Asymmeric Diffuser wih Moderae Adverse Pressure Gradien S. Salehi, M. Raisee and M. J. Cervanes,3 Hydraulic Machinery Research Insiue, School of Mechanical Engineering, College of Engineering, Universiy of Tehran, Tehran, 55/4563, Iran Division of Fluid and Experimenal Mechanics, Luleå Universiy of Technology, Luleå, 9787, Sweden 3 Waer Power Laboraory, Norwegian Universiy of Science and Technology, Trondheim, 749, Norway Corresponding Auhor s.salehi@u.ac.ir (Received March 5, 06; acceped March 6, 07) ABSTRACT In his paper, numerical invesigaion of hree-dimensional, developing urbulen flow, subeced o a moderae adverse pressure gradien, has been invesigaed using various urbulence models, namely: he low- Re, he SST, he v f and a varian of Reynolds sress model. The resuls are compared wih he deailed velociy and pressure measuremens. Since he inle condiion is uncerain, a sudy was firs performed o invesigae he sensiiviy of he resuls o he inle boundary condiion. The resuls showed he imporance of including he conracion effecs. I is seen ha he developing flow inside he sraigh duc, is highly sensiive o he inle boundary condiion. The comparisons indicae ha all urbulence models are able o predic a correc rend for he cenerline velociy and pressure recovery inside he sraigh duc and diffuser bu he low-re and RSM urbulence models yield more realisic resuls. The SST model largely overpredics he cenerline velociy and boundary layer hicness in he sraigh duc. The comparisons of he numerical resuls also revealed ha he RSM model, due o is anisoropic formulaion, is able o reproduce he secondary flows. As expeced, he RSM model demonsraes he bes performance in predicion of he flow field and pressure recovery in he asymmeric diffuser. Keywords: Moderae adverse pressure gradien; Asymmeric diffuser; Turbulen developing flow; Compuaional fluid dynamics; RANS models.. INTRODUCTION Developing urbulen flow under adverse pressure gradien (APG) occurs in various fluid flow relaed engineering applicaions. Turbulen flow around aircrafs, auomobiles and inside he draf ube of hydraulic urbines, are a few examples of such applicaions. Hence, accurae predicion of urbulen flow under adverse pressure gradien can conribue in he opimizaion of fluid machineries. Indeed, he presence of APG can grealy affec he performance and efficiency of hydraulic machines. If a urbulen boundary layer flow encouners a large APG, flow becomes unsable and if he APG is sufficienly large, flow separaes from he wall and forms a recirculaing region. Such recirculaion region ofen has negaive consequences such as: increase in he pressure drag, decrease in lif and increase in head losses. More specifically, flow separaion could also have desrucive effecs on performance of hydraulic machinery equipmen such as draf ube where he pressure recovery is resriced by aachmen of flow o he walls. Turbulen flow under APG has been he subec of a large number of experimenal and numerical invesigaions. Among all he previous sudies, he experimenal invesigaions conduced by Clauser (954) and Bradshaw (967) can be considered as he firs sudies on his opic. Boh sudies measured characerisics of equilibrium urbulen boundary layers where he pressure gradien parameer is ep consan. Nagano e al. (993) experimenally sudied he behavior of urbulen boundary layer under moderae and srong adverse pressure gradiens. They showed ha wih increase in adverse pressure gradiens, he near wall velociy profile shifs below he sandard log-law, indicaing a reducion in he viscous sub-layer hicness.

2 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. Furhermore, he presence of an APG could significanly affec he disribuions of urbulence quaniies. Rai (986) conduced experimens in a wind unnel for hree arbirary adverse pressure gradien flows. The experimenal observaions clearly indicae ha wall-waes in adverse pressure gradien can be adequaely described by he wolayer model proposed. Direc numerical simulaion (DNS) has been employed by Lee and Sung (008) o invesigae he effecs of an adverse pressure gradien on a urbulen boundary layer. Their numerical resuls showed ha he mean flow quaniies are grealy affeced by an APG, and he coheren srucures in he ouer layer of he APG flows were more acivaed han hose in he zero pressure gradien flows. This was aribued o increased urbulence inensiies, shear sresses and pressure flucuaions in he APG sysems. More recenly, Inouea e al. (03) repored large eddy simulaion (LES) of a urbulen boundary layer a high Reynolds number subec o an adverse pressure gradien. The sreched-vorex model (Chung and Pullin, 009) was used for he subgrid-scale modeling. The resuls showed self-similariy in he velociy saisics over a wide range of Reynolds numbers. I was concluded ha he boundary layers under adverse pressure gradien are far from an equilibrium sae. Among numerous applicaions of adverse pressure gradien flows, diffusers are one of he mos commonly used flow devices in he indusry, especially in hydraulic machines. Therefore, invesigaion of urbulen flow hrough diffusers has been an imporan research opic for fluid mechanics researchers. In he following, some of he invesigaions relaed o urbulen flow hrough diffusers will be reviewed. Obi e al. (993) performed laser Doppler anemomery (LDA) measuremens in an asymmeric diffuser wih an expansion raio of 4.7, a single deflecion of 0 o and a urbulen fully developed inle condiion. Buice and Eaon (996) argued ha he experimenal daa from Obi e al. (993) had several deficiencies when comparing wih heir numerical resuls. The mos noiceable problem was ha he experimenal daa of Obi e al. (993) did no appear o saisfy he mass conservaion. Hence, hey performed new deailed measuremens in he reproducion of he Obi experimens using ho-wire and pulsed-wire measuremens. The experimenal wors by Obi e al. (993) and Buice and Eaon (996) have received much aenion because of he fullydeveloped inle condiion, presence of flow separaion and flow developmen downsream of he reaachmen poin. For example, Kalenbac e al. (999) conduced a numerical invesigaion on Obi diffuser using LES. They showed ha a deailed represenaion of he inflow velociy field is criical for accurae predicion of he flow inside he diffuser. They also found ha he sub-grid scale model plays a maor role for predicion of boh mean momenum and urbulen ineic energy. Schlüer e al. (005) also conduced LES compuaions of Obi diffuser flow. They used hree differen mesh sizes and differen modeling approaches o examine he influence of he mesh resoluion and four subgrid models, namely: no model (implici LES), he sandard Smagorinsy model, he dynamic Smagorinsy model and he dynamic localizaion model. The mesh refinemen sudy demonsraed improvemen in he predicions. Among he subgrid sraegies examined, he dynamic Smagorinsy model performed he bes. Iaccarino (00) used hree commercial CFD codes, namely: CFX, Fluen, and Sar-CD, and wo urbulence models (he low-re and he v f ) for he numerical simulaion of Obi diffuser. The numerical resuls showed ha he v f model produces more accurae resuls han he low-re model when comparisons were made wih he experimenal daa and LES predicions. The calculaions do no show any recirculaion region, while he v f model reproduces he lengh of he separaion bubble wihin 6 percen of he measured value. More recenly, El-Behery and Hamed (0) employed he commercial code Fluen o examine he capabiliies of several urbulence models in predicion of urbulen flow in he planar asymmeric diffuser of Obi diffuser. The comparisons showed ha he resuls of he v f urbulence model agree bes wih he experimenal daa while he RSM model was shown o give unexpeced poor resuls. Cherry e al. (008 and 009) performed experimens o measure he mean velociy field in wo separae 3D asymmeric diffusers using he magneic resonance velocimery mehod. The measured flow fields in boh diffusers showed 3D boundary layer separaion bu he srucure of he separaion bubble exhibied a high degree of sensiiviy o he diffuser geomerical dimensions. In heir experimen, a fully-developed flow condiion was esablished a he diffuser enry which is suiable for he urbulence modeling validaion proposes. Similar o Obi diffuser, he Cherry diffuser has been used as a benchmar for he invesigaion of performance of urbulence models and numerical approaches in recen years. A direc numerical simulaion of urbulen flow in he Cherry diffuser (Cherry e al., 008) was conduced by Ohlsson e al. (00) wih a massively parallel high-order specral elemen mehod and heir resuls were in good agreemen wih he experimenal daa. Jairlić e al. (00) applied LES and a zonal hybrid LES/RANS scheme o predic he urbulen flow hrough he 3D diffuser invesigaed by Cherry e al. (008). Boh modeling sraegies gave accepable resuls for he ime-averaged quaniies. Jeyapaul (0) also performed numerical simulaions on Cherry diffuser using RANS models. The resuls showed ha he linear eddyviscosiy models fail o predic separaion on he correc wall of he 3D diffuser. However, he explici algebraic RSM (EARSM) is able o predic separaion accuraely. The EARSM prediced 030

3 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. quaniaively he mean flow field, however he Reynolds sresses were incorrec and he wall pressure was under prediced. Cervanes and Engsröm (008) experimenally invesigaed he urbulen flow inside an asymmeric diffuser wih a moderae adverse pressure gradien using LDA echnique. The diffuser has a diverging upper wall, designed o yield an approximaely consan adverse pressure gradien, which opens up from an angle of.5 a he beginning o 7.5 a he end. The diffuser is a generic model of he recangular diffuser found a he end of mos hydropower urbines of he Francis and Kaplan ypes wih an elbow draf ube. The elbow draf ube is found immediaely afer he runner and, as shown in Fig., composed of a conical diffuser, an elbow and a sraigh recangular diffuser. Alhough he Reynolds number was small compared o full-scale urbines, he flow is sill fully urbulen and he large viscous lengh-scale allows deailed measuremens in he boundary layer up o y. They carried ou measuremens in seady regime and hree differen pulsaing regimes based on he dimensionless frequency ( / u ) namely: quasi-seady ( ), relaxaion ( ) and quasi-laminar ( ). This es case provides accurae experimenal daa for validaing CFD calculaions wih focus on adverse pressure gradien effecs and non-rivial boundary condiions: boh imporan for hydropower simulaions..0 m sraigh recangular duc wih a recangular cross secion of m is used upsream he diffuser. The flow is sill developing a he end of he sraigh duc as found in hydropower sysems. The hydraulic diameer of he sraigh duc calculaed by 4 Aduc / p is Dh 0. m. Hence, he duc lengh is 7.5 D h. The Reynolds number based on he bul velociy and duc hydraulic diameer is 4 0 (flow rae of m 3 /s). A conracion (9:) precedes he duc. In addiion, he flow is ripped a he exi of he conracion by mm plaes, exruding 9. mm from he walls, corresponding o abou 8% of he duc heigh and % of he duc widh. The ripping was necessary o achieve repeaable condiions in he es secion, as also found by Durs e al. (998). Following he duc is an asymmeric diffuser wih a diverging upper wall designed o yield an approximaely consan adverse pressure gradien. The main par of his experimenal seup, from he duc inle o he diffuser oule, is numerically modeled here. Fig 3(a) shows a 3D view of he geomery. In addiion, Fig. 3(b) represens a D schemaic of he diffuser wih he variaion of he diffuser angle. The diffuser opens up o a cross secion of m a he oule (x=.77 m). The diffuser angle increases from.5 a he beginning o 7.5 a x=.77 m. The flow field inside he diffuser has been invesigaed a hree saions, namely: x=.08 m,.357 m and.63 m. As shown in Fig 3. (b), he firs saion is 0 mm before he sar of he diffuser and he second and hird saions are a abou 40% and 80% of he diffuser lengh downsream, respecively. Fig.. Schemaic of a draf ube. In he presen sudy, numerical invesigaions are performed on he diffuser sudied experimenally by Cervanes and Engsröm in saisically saionary condiion. The main obecive of he presen sudy is o examine he effeciveness of various urbulence models in predicion of developing urbulen flow subeced o he adverse pressure gradien. To he bes of our nowledge auhors, he presen paper is he firs aemp o invesigae he 3D asymmeric diffusers wih he developing inflow condiion using differen urbulence models. (a) Fig.. Experimenal seup used by Cervanes and Engsröm (008) (dimensions in mm).. PROBLEM DESCRIPTION The geomery invesigaed in his paper has been sudied experimenally by Cervanes and Engsröm (008). A schemaic of he experimenal seup used for heir measuremens is shown in he Fig.. A (b) Fig. 3. (a) 3D l view of he domain. (b) D Schemaic of examined diffuser. 03

4 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , MATHEMATICAL FORMULATION 3. Mean Flow Equaions For an incompressible urbulen flow, he ime averaged equaions of coninuiy and momenum are wrien as: U 0 () x ( UU i ) P U i uu i x xi x x () where and are fluid densiy and inemaic viscosiy and uu i represens he Reynolds sress ensor. 3. Turbulence Modeling Equaions The urbulence models employed for compuaions are he low-re model (proposed by Launder and Sharma, 974), he SST model (Mener, 994), he v f model (Durbin, 995) and a varian of Reynolds sress model (Gibson and Launder, 978) Low-Re Model In his urbulence model, he unnown Reynolds sresses are obained from he Boussinesq hypohesis: U U i uu ( ) x x 3 i i where he urbulen viscosiy is calculaed via: c f (3) (4) To obain, ranspor equaions for he urbulence ineic energy, and is dissipaion rae are solved. The ranspor equaion for he urbulen ineic energy is wrien as: ( U ) P x x x x (5) where P, he generaion rae of urbulen ineic energy, is obained from: P uu i Ui x (6) The homogeneous dissipaion rae of urbulen ineic energy,, is obained by solving he following equaion: ( U ) c P x x x c f E (7) The empirical consans of he model are presened in Table. The homogeneous dissipaion rae can be relaed o he rue dissipaion rae hrough: ( ) x The damping funcions f and f are given by: (8) f exp 3.4 / ( 0.0 R ) (9) f 0.3exp( R ) where R / is he local urbulen Reynolds number. The erm E is firs inroduced by Jones and Launder (97) and is expressed as: Ui E x x (0) Table Empirical consan of linear model c c c SST Model The main idea of he SST model is o combine he robusness of he urbulence model near walls wih he capabiliies of he model away from he walls. The model uses he Boussinesq hypohesis (equaion (3)) o obain he Reynolds sresses. The urbulen viscosiy is calculaed via: a () max( a, SF ) where and ω are obained by solving he following ranspor equaions: * ( U ) P () x x x ( U ) S x x x (3) ( F) x x The blending funcion F is defined by: 03

5 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , F anh min max,, * y y CD y (4) Helmholz ype. The Boussinesq approximaion is sill used for he evaluaion of he Reynolds sresses. The eddy viscosiy is given by: where CD y is he disance o he neares wall. 0 max,0 x x and The blending funcion F is zero away from surfaces ( model) and swiches o one inside he boundary layer ( model). The S erm in he urbulen viscosiy expression is he invarian srain rae and he second blending funcion F is defined by: 500 F anh max, * y y (5) The producion erm has a limier o preven he build-up of unrealisic high urbulence energy in he sagnaion regions: * P min P,0 (6) All consans are compued by a blend from he corresponding consans of he and he model via F F.The consans of he SST model are given in Table Table Empirical consan of v f Model * / / model The v f model (developed by Durbin, 995) is similar o he sandard model. In addiion, i incorporaes some near-wall urbulence anisoropy as well as non-local pressure-srain effecs. Insead of urbulen ineic energy,, he v f model uses a velociy scale v (velociy flucuaion normal o he sreamlines) for he evaluaion of he eddy viscosiy. The anisoropic wall effecs are modeled hrough he ellipic relaxaion funcion, f, by solving a separae ellipic equaion of he cvt (7) where he urbulen ime-scale T and he urbulen lengh-scale L are obained from he following expressions: T max,6 3/ 3/4 L cl max, c /4 The v ranspor equaion is expressed as: v ( Uv ) fv x x x v (8) (9) (0) An ellipic equaion is solved for he relaxaion funcion f: f v P x T L f c c 3 () The coefficiens of v f urbulence model are given in Table 3. Table 3 Coefficiens of v c v.3 f model 4 c / cd L / L c.9 c.4 c 0.3 c 0.3 L c RSM Model The Reynolds Sress Model (RSM) is he mos deailed and elaborae RANS urbulence model. In conras o isoropic Boussinesq hypohesis based models, which use algebraic expressions for he evaluaion of Reynolds sresses, he RSM models solve a parial differenial ranspor equaion for each Reynolds sress componen. The exac form of he Reynolds sress ranspor equaions, shown below, can be derived via a mahemaical manipulaion on momenum equaion: 033

6 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. ( Uuu i ) ( uuu i ) u p up i x x xi x convecion urbulendiffusion U U p u u uu i i i uu x x xi x producion pressuresrain uu i x x viscousdiffusion u u i x x viscousdissipaion () Clearly he diffusion, he pressure srain and he viscous dissipaion erms are no explicily defined and need o be modeled. The urbulen diffusion erm is modeled using (Gibson and Launder, 978): D Ti, uu i x x (3) Also he viscous dissipaion erm can be simply modeled hrough ( / 3). i i were discreized using he cenral differencing scheme. The pressure field is lined o he velociy field hrough he SIMPLE pressure correcion algorihm. The convergence crieria were se o 0-5 for all equaions. 4. Boundary Condiions Accurae represenaion of inle boundary condiion is criical for numerical resuls inside a diffuser (Kalenbac e al., 999). Since he conracion geomery is no fully nown and boundary condiions are no available from experimenal daa, simulaions are performed from secion x=00 mm which is placed righ afer he ripping par. A his secion, he velociy field for he sreamwise and normal componens is available from experimenal daa. Fig. 4 shows six differen verical and spanwise locaions where he velociy componens were measured using he LDA echnique. Here he varian of RSM proposed by Gibson and Launder (978) is employed where he pressure srain correlaion (or redisribuion erm) is expressed as: i i, i, i, w (4) where i, C uu i 3 i (5) Fig. 4. The locaion of available experimenal daa a secion x=00 mm. i, C Pi Ci i PC 3 (6) 3/ 3 3 C l i, w C u u mnnmi uu i nn uu nn i d 3/ 3 3 C l C m, n nmi i, nn,nn i d (7) The P i and Ci erms are producion and convecion erms respecively (defined in equaion ()); also P /P and C /C. The model coefficiens are given in Table 4. Re and a i are urbulen Reynolds number and Reynolds-sress anisoropy ensor, respecively. 4. NUMERICAL ASPECTS 4. Solver All equaions are discreized using finie-volume mehodology on a collocaed grid sysem. The open source C++ CFD code OpenFOAM (0) was used o perform he compuaions. The second order upwind differencing scheme was employed for discreizaion of he convecive erms in all ranspor equaions. Gradien and Laplacian erms Table 4 Coefficiens of RSM model C C C AA.58 exp Re 0.75 A /3C.67 C C C C max / 3 / 6 /,0 A 9/8 A A 3 A aa i i A aaa 3 i i C l C 3/4 / A code was developed o evaluae he inle boundary condiion using a 3D surface fi on he experimenal daa wih a polynomial inerpolaion scheme. Then, a moving average filer is applied on he inerpolaed daa. The sreamwise velociy is scaled by a facor o mach he experimenal flow 034

7 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. rae afer inegraion. Finally, he velociy profiles are calculaed on he cell ceners a he inle of he compuaional grid and are imposed as inle boundary condiion. The 3D surfaces obained for boh he sreamwise and normal velociy componens afer smoohing are presened in Fig. 5. The spanwise componen is obained using he coninuiy equaion assuming a negligible sreamwise variaion of u-velociy ( u/ x 0), wih respec o v/ y and w/ z because of he duc lengh. The experimenal daa for he sreamwise velociy are no enirely symmeric. This may be induced by uncerainy in he conracion geomery and inflow condiion. (I) and urbulen lengh-scale ( l ). Turbulen inensiy is he raio of velociy flucuaions o he mean velociy (U0). Therefore, assuming an isoropic field, he urbulen ineic energy is obained by: 3 ( IU ) 0 (8) (a) (a) (b) (b) Fig. 5. 3D surfaces fied o he experimenal daa. (a) sreamwise and (b) normal velociy componens. Compuaions of he flow inside he described geomery are performed wih uniform and nonuniform velociy boundary condiions o sudy heir effecs on he resuls. Figs. 6(a)-(c) illusrae he developmen of he mean sreamwise velociy and flucuaing par of he sreamwise and normal velociies along he cenerline of he duc and diffuser for boh inle boundary condiions. I is observed ha he numerical resuls of Case (nonuniform inle velociy) are significanly closer o he experimenal daa han hose obained in Case (uniform). Hence, based on his comparison he firs approach is used for he predicions presened in he following secions. An isoropic urbulence field is assumed a he inle of all cases due o he lac of experimenal urbulen daa a he inle. The urbulen quaniies a he inle are calculaed using urbulen inensiy (c) Fig. 6. Comparison beween resuls of Case (non-uniform inle B.C.) and (uniform inle B.C.). (a) Mean sreamwise velociy and flucuaing par of (b) sreamwise and (c) normal velociies. The homogeneous dissipaion rae and specific dissipaion rae are also obained from: 035

8 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. (a) (b) (c) Fig.7. Resuls of he grid sudy; developmen of (a) mean sreamwise velociy and (b) is flucuaions along he duc and diffuser cenerline. Sreamwise (c) velociy and (d) flucuaions profiles a (c) secion x=08 mm. (d) c (9) 3/4 3/ /, / 4 l c l where l (urbulen lengh-scale) is se o 0.D h. Hence, by choosing a urbulen inensiy a he inle, all oher quaniies can be calculaed. Since here is no available experimenal daa on velociy flucuaions and urbulen inensiy a he duc inle, a sudy was performed on he effec of inle urbulen inensiy on he resuls of he mean flow and urbulen sresses in he sraigh duc and diffuser. The compuaions were performed using he RSM model and differen urbulence inensiies a he inle and inensiy of 3% was found o give accurae resuls. The pressure value a he inle is compued using he zero-gradien condiion normal o boundary. A he exi boundary, zero gradien boundary condiions along normal o he oule were imposed for all variables (flow and urbulence) excep pressure, which is considered as a consan value. Since he models used in he presen wor are low-reynolds number urbulence models, he near-wall region is resolved up o y +. Hence, no special reamen (such as wall funcions) is necessary for he wall boundary condiions of he mean flow and urbulence ranspor equaions, i.e., he velociy componens, urbulen ineic energy, homogeneous dissipaion rae, specific dissipaion rae. In addiion, for he RSM model, he wall boundary condiion of velociy componens, homogeneous dissipaion rae and all componens of Reynolds sress ensor are also se o zero. The pressure on wall is also obained using he assumpion of zero-gradien along normal o he wall. 4.3 Grid Sudy To sudy he sensiiviy of he compuaional resuls o he grid resoluion, a grid sudy was performed using he RSM urbulence model. For his purpose, four differen grids (from coarse grid of 470,000 cells o fine grid of 3,700,000 cells) have been generaed. All of hese grids are nonuniform and fully srucured. As i was menioned earlier all of urbulence models used in his invesigaion are low-reynolds number and hese grids are fine enough o resolve he near wall viscous sub-layer region. In addiion, he grid poins have been refined along he sream-wise direcion around he diffuser because of high sream-wise gradiens in hese regions. Table 5 presens he specificaions of he five compuaional grids generaed for he grid sudy; nx, ny and nz represen he number of nodes along sream-wise (x), normal (y) and cross-sream (z) direcions, respecively. The resuls obained from he grid sudy for four differen grids are presened in Fig. (7). The developmen of mean sreamwise velociy and is corresponding flucuaion along he cenerline of he duc and diffuser are illusraed in Fig. 7(a) and 7(b), respecively. Figs. 7(c) and 7(d) show 036

9 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. he variaion of sreamwise velociy and is flucuaion along a verical line placed on he symmery plane a secion x=08 mm. I can be noed from hese four figures ha alhough grid refinemen slighly affecs he velociy field inside he diffuser, i has more noiceable effecs on he mean and flucuaing velociy in he developing region (sraigh duc). The resuls obained on Grid 3 wih.40 6 nodes are almos grid independen and a finer mesh would no give very differen resuls. Thus, he subsequen compuaional resuls are obained using Grid 3. Figs. 8(a) and 8(b) show he chosen grid in symmery plane and inside he diffuser. Table 5 Specificaions of grids produced for he grid sudy Grid No. n x n y n z Disance from wall Grid y O() Grid y O() Grid y O() Grid y O() flow acceleraes immediaely and he shear layers induced by he ripping produce fas growing boundary layers. Figure 9 shows he mean sreamwise velociy (normalized wih U0 Q/ Aduc ) along he cenerline in he sraigh duc and he diffuser. In boh he numerical and he experimenal resuls, he sreamwise velociy sars wih an overshoo induced by he ripping and he sudden decremen of flow area. Afer he rip, velociy decreases, bu he rae of velociy decrease reduces along he cenerline due o he boundary layer growh. From x=500 mm o he sar of he diffuser, he velociy gradien becomes smaller, bu flow does no show clear signs of reaching he fully-developed condiion. As he flow eners he diffuser, he velociy drops rapidly due o he cross-secional area increase. I is seen ha all urbulence models have successfully prediced he general variaion of he sreamwise velociy along he cenerline. There are significan differences beween he resuls of hese models. The SST model largely overpredics he cenerline sreamwise velociy due o an overesimaion of he boundary layer hicness. I is observed ha he resuls of low-re and RSM urbulence models are closes o he experimenal daa boh inside he sraigh duc and he diffuser. (a) (b) Fig. 8. Main grid used for he compuaions. Cells disribuion on (a) Symmery plane and (b) diffuser. Fig. 9. Developmen of he sreamwise velociy along cenerline. 5. RESULTS AND DISCUSSION In his secion, he numerical resuls obained using four urbulence models are presened and compared wih he experimenal daa of Cervanes and Engsröm (008). The variaion of he velociy along he cenerline of he duc flows is generally sensiive o he developmen of he boundary layers in he duc. Non-ripped boundary layers will grow along he duc lengh, hereby forcing he bul flow o accelerae along he cenerline. However, in he presen geomery, he flow is ripped righ before he sraigh duc. Wih he curren ripping, he bul Fig. 0. Pressure coefficien disribuion along he cenerline of he upper wall. 037

10 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. SST v f RSM Fig.. Vecors of secondary flow (righ) and Conours of sreamwise velociy (lef) in secion x=08 m. The disribuion of pressure coefficien, defined by: C p p pam (30) U 0 along he cenreline of he upper wall is illusraed in Fig. 0. The pressure rises o a pea value and hen decreases due o he viscous-losses. As he flow eners he diffuser, he pressure sars o increase again. I is observed ha he numerical and experimenal resuls of pressure coefficien disribuions show he same rend. As menioned earlier, he diffuser was designed o yield an approximaely consan adverse pressure gradien, which is confirmed by he linear variaion of he pressure in he numerical and experimenal resuls. In addiion, as expeced he inviscid heory (which he diffuser is designed based on) over-esimaes he pressure coefficien along he diffuser upper wall. Figure shows he prediced secondary flows and he sreamwise velociy conours using differen urbulence models a secion x=08 mm, (i.e. 0 mm before he sar of diffuser). As expeced only he RSM model is able o predic fairly srong secondary flow paerns. Using alernaive voriciy form of he Reynolds-averaged Navier-Soes equaions, one can show ha he secondary flows are generaed by he axial mean voriciy, which becomes non-zero in a sraigh duc or pipe only if here are differences beween cross-sream normal Reynolds sress ( v w ) or Coriolis forces arising from a spanwise roaion (Speziale e al. 99). Therefore, he linear eddy viscosiy urbulence models, based on he Boussinesq hypohesis, are no able o predic secondary flows in a sraigh non-circular duc. Consequenly, he resuls presened in Fig. are reasonable. Secondary flow vecors in all numerical resuls seem o have upward direcions, which is obviously due o he geomery (he lower wall of diffuser is fixed, while he upper wall is diverging). In addiion, he RSM model has prediced wo roaing vorices (generaed by normal Reynolds sress anisoropy). To undersand he srucure of hese secondary flows i is bes o firs loo a such urbulence driven secondary flows generaed in he upsream sraigh duc. As shown in Fig., wih he RSM compuaions, wo couner-roaing vorices are observed in each corner a he secion x=500 mm ha force he flow o move from he cener of he duc o he corner and hen bac o he cener. By moving in he sreamwise direcion, i is seen ha he upper vorices become weaer in secion x=63 mm due o he geomery (diverging upper wall). The lower vorex is also merged o he main upward secondary flow forcing he fluid core displacemen o he upward direcion in a vorical paern. The sreamwise velociy conours in Fig. 3 indicae flaer velociy profiles for and v f models. Fig.. Vecors of secondary flow (righ) and Conours of sreamwise velociy (lef) in secion x=500 m prediced by RSM model. 038

11 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. SST v f RSM Fig. 3. Vecors of secondary flow (righ picures) and Conours of sreamwise velociy (lef picures) in secion x=63 m. The prediced velociy and he urbulence field inside he diffuser using differen urbulence models are furher presened and compared o he experimenal daa in he following. These quaniies are repored a hree differen secions of he diffuser symmery plane. The secions are placed a 08 (0 mm before he diffuser), 357 (55 mm ino he diffuser) and 63 mm (530 mm ino he diffuser), respecively. Numerical and experimenal profiles of he sreamwise componen of he velociy are compared in Fig. 4. I is observed ha he SST model largely over-predics he velociy field inside he diffuser, which is consisen wih he cenreline velociy predicions shown in Fig. 9. The oher urbulence models demonsrae beer sreamwise velociy predicions inside he diffuser. As menioned earlier, he flow enering he diffuser is no fully-developed and hus he significan differences seen in velociy predicions depends on he capabiliy of he urbulence models in reproducing he developing boundary layer in he sraigh duc. Fig. 4 reveals ha among he urbulence models, as expeced, he RSM urbulence model reurns he bes predicions for he sreamwise velociy. In addiion, a comparison of velociy profiles a he hree secions illusraes he rapid boundary layer growh along he diffuser iniiaed by he rip (i.e. he mean sreamwise velociy profile becomes less fla). The sreamwise velociy predicions of he RSM model are furher compared in log-scale in Fig. 5 in order o invesigae he effec of he adverse pressure gradien on he boundary layer developmen. I is observed ha he linear law of he wall, U y, holds in he viscous sublayer ( y 5) regardless of he adverse pressure gradien. However, wih he flow developmen, he wae region grows boh in srengh and size. Hence, he ouer par of he log region becomes smaller. The increase in he wae componen is due o he significan decrease in he sin fricion along he diffuser. This feaure of adverse pressure gradien on he fluid velociy developmen has been also repored in he lieraure (see for example Nagano e al., 993). The prediced normal velociy profiles inside he diffuser are compared wih he measured daa in Fig. 6. The normal velociy profiles inside he diffuser are affeced by boh he diffuser geomery and he secondary flows. Since he firs secion (x=08 mm) is us before he diffuser, he normal velociy disribuion is more affeced by he secondary flow moion. However, in he downsream saions he geomery mainly influences he normal velociy profile. As expeced, he normal velociies in he firs secion are lower han hose in he downsream saions. As shown earlier he eddy viscosiy urbulence models fail o predic correc urbulence driven secondary flows, and as a resul hey are no able o predic accurae normal velociy profiles in he firs secion. As expeced he RSM model produces a more accurae cross velociy profile in he firs saion due is abiliy in predicion of urbulence driven secondary moion as already shown in Fig.. In he wo downsream secions, he conribuion of he geomery on he normal velociy disribuion becomes dominan. Hence, he numerical resuls of differen urbulence models are more similar and are closer o he experimenal daa. Neverheless, 039

12 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. he resuls of RSM model are sill in superior agreemen wih he measured daa. x=08 mm x=08 mm x=357 mm x=357 mm x=63 mm Fig. 4. Sreamwise velociy along a verical direcion inside he diffuser. Fig. 5. Law of he wall plos of RSM model. x=63 mm Fig. 6. Normal velociy along verical direcion inside he diffuser. Figure 7 shows comparisons for he sreamwise normal Reynolds sress ( u ) in he diffuser. The comparisons show ha he eddy-viscosiy based urbulence models, as expeced, significanly underesimae he urbulen normal levels. On he oher hand, he RSM model is able o produce accurae predicions for his quaniy. The comparison of resuls in he hree secions reveals ha he sreamwise normal Reynolds sress develops an ouer plaeau in he presence of an adverse pressure gradien. The inensiy of his plaeau increases in he sreamwise direcion. This propery of a urbulen boundary layer flow subeced o an adverse pressure gradien is more obvious in a semi-log plo as shown in Fig

13 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. x=08 mm x=08 mm x=357 mm x=357 mm x=63 mm Fig. 7. Sreamwise componen of normal Reynolds sress along verical direcion inside he diffuser. Fig. 8. Sreamwise componen of he normal Reynolds sress for he RSM model. x=63 mm Fig. 9. Cross-sream componen of normal Reynolds sress inside he diffuser. Comparisons v predicions using various urbulence models and experimenal daa are presened in Fig. 9. As can be observed he all eddy-viscosiy based urbulence models overpredic v levels. As expeced, he RSM model reurned more reliable v. Numerical and experimenal resuls of shear Reynolds sresses are illusraed in Fig. 0. I is seen ha he RSM model is able o produce accepable resuls for he urbulen shear Reynolds sress disribuions in all hree locaions. I is noed ha he and v f models are also yield accepable resuls for his quaniy. 04

14 S. Salehi e al. / JAFM, Vol. 0, No. 4, pp , 07. x=08 mm measuremens were employed for he numerical predicions. Four RANS urbulence models are employed for he numerical predicions. The resuls of cenerline velociy in he sraigh duc demonsraed he bes predicions can be obained using he low-re and RSM urbulence models. The SST largely overpredics he cenerline velociy and boundary layer hicness. On he oher hand, he RSM model produces mos precise resuls for he pressure recovery. I was observed ha he normal velociy disribuion inside he diffuser is affeced by geomery and secondary flows and he RSM model yields he bes predicions for his quaniy due o is anisoropy naure. I was furher shown ha sreamwise normal Reynolds sress is significanly underesimaed by he isoropic based urbulence models. These urbulence models overpredic he cross-sream normal Reynolds sress. In conras, he RSM model produces more precise resuls for he urbulen Reynolds sresses. ACKNOWLEDGEMENTS The auhors graefully acnowledge he financial suppor of Hydraulic Machinery Research Insiue of Universiy of Tehran and Swedish Hydropower Cenre ( REFERENCES x=357 mm x=63 mm Fig. 0. Shear Reynolds sress along verical direcion inside he diffuser. 6. CONCLUSION The 3D developing urbulen flow hrough a recangular asymmeric diffuser wih moderae adverse pressure gradien was numerically invesigaed and he numerical resuls of he predicions are compared wih he LDA daa. Such calculaions are imporan in hydraulic machinery flows, due o imporance of adverse pressure gradien in hese flows. I was observed ha he developing flow inside he diffuser is significanly sensiive o he inle boundary condiion. Therefore, he inle condiions exraced from experimenal Bradshaw, P. (967). The urbulence srucure of equilibrium boundary layers. J. Fluid Mech. 9, Buice, C. U. and J. K. Eaon (996). Experimenal invesigaion of flow hrough an asymmeric plane diffuser. CTR Annual research briefs- 996, Cervanes, M. J. and T. F. Engsröm (008). Pulsaing urbulen flow in a sraigh asymmeric diffuser. J. Hydraulic Res., 49 (), -8. Cherry, E. M., C. J. Elins and J. K. Eaon (008). Geomeric sensiiviy of hree-dimensional separaed flows. In. J. Hea Fluid Flow 9, Cherry, E. M., C. J.Elins and J. K. Eaon (009). Pressure measuremens in a hree-dimensional separaed diffuser. In. J. Hea Fluid Flow 30, -. Chung, D. and D. Pullin (009). Large-eddy simulaion and wall-modeling of urbulen channel flow. Journal of Fluid Mechanics 63, Clauser, F. H. (954). Turbulen boundary layers in adverse pressure gradiens. J. Aero. Sci. (), Durbin, P. A. (995). Separaed flow compuaions wih he v model. AIAA Journal 33 (4), Durs, F., M. Fischer, J. Jovanović and H. Kiura 04

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