Numerical study of the effect of wedge angle of a wedge-shaped body in a channel using lattice Boltzmann method

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1 Avalable onlne at Proceda Engneerng 56 (2013 ) th BSME Internatonal Conference on Thermal Engneerng Numercal study of the effect of wedge angle of a wedge-shaped body n a channel usng lattce Boltzmann method M.A.Taher a *,Y.W.Lee b, H.D.Km c a Department of Mathematcs, Dhaka Unversty of Engneerng & Technology, Gazpur, Bangladesh b School of Mechancal Engneerng, Pukyong Natonal Unversty, Busan, Korea c School of Mechancal Engneerng, Andong Natonal Unversty, Andong, Korea Abstract The am of ths paper s to study the flud flow behavor around a wedge shaped body wth dfferent wedge angles placed n a channel usng Lattce-Boltzmann Method (LBM). The LBM has bult up on the D2Q9 model and the sngle relaxaton tme method called the Lattce-BGK (Bhatnagar-Gross-Krook) model. The nfluence of the gap rato G*=G/H, where H s the dstance between two parallel walls, G s the gap between body and wall, on the flow feld s llustrated. The gap rato, G*, depends on the angle of wedge-shaped body (0 0 < <180 0 ). Streamlnes, vortcty contours and pressure contours are provded to analyze the mportant characterstcs of the flow feld for a wde range of non dmensonal parameters namely the Reynolds number (Re), Strouhal number (St) and the gap rato(g*). However, t s seen that the flow can be characterzed by three regons: () large gap rato, 0.44 <G* < 0.50, wth 0 0 < <55 0, () ntermedate gap rato, 0.20 G* 0.44, wth and () small gap rato, 0 < G* < 0.20, wth < < The smulaton results are compared wth expermental data and other numercal models and found to be very reasonable and satsfactory The Authors. authors, Publshed by Elsever by Elsever Ltd. Open Ltd. access Selecton under CC and/or BY-NC-ND peer-revew lcense. under responsblty of the Bangladesh Socety of Selecton Mechancal and peer Engneers revew under responsblty of the Bangladesh Socety of Mechancal Engneers Keywords: Lattce-Boltzmann; Bhatnagar-Gross-Krook; Reynolds number; Strouhal number; wedge-shaped, gap rato. Nomenclature C p Specfc heat at constant pressure (J.kg -2 K -1 ) Cs Speed of sound (m.s -1 ) c CL number e Dscrete partcle velocty vector (m s -1 ) Dscrete partcle dstrbuton functon eq Dscrete partcle dstrbuton functon f Sheddng frequency G Gap between body and wall (m) G* Gap rato H Channel heght (m) L Channel length (m) Re Reynolds number St Strouhal number U Characterstc velocty (m.s -1 ) * Correspondng author. Tel.: E-mal address: tahermath@yahoo.com The Authors. Publshed by Elsever Ltd. Open access under CC BY-NC-ND lcense. Selecton and peer revew under responsblty of the Bangladesh Socety of Mechancal Engneers do: /j.proeng

2 M.A.Taher et al. / Proceda Engneerng 56 ( 2013 ) u, v lud velocty n x and y drecton respectvely(m.s -1 ) w Weghtng factor Greek symbols Relaxaton parameter for momentum Densty of the flud (kg m -3 ) Vscosty of the flud (kg m -1 s -1 ) Coeffcent of wedge angle Knematc vscosty (m 2 s -1 ) Relaxaton parameter Wedge angle Abbrevatons BGK Bhatnagar-Gross-Krook CD Computatonal flud dynamcs CL Courant-redrehs- Lewy D2Q9 2D-9 veloctes LBM Lattce-Boltzmann Method N-S Naver-Stokes PD Partcle dstrbuton functon 1. Introducton In the past years, the Lattce Boltzmann Method (LBM) has attracted much attenton as a novel alternatve to tradtonal computatonal flud dynamcs(cd) methods for numercally solvng the Naver Stokes (N-S) equatons. Actually the LBM orgnated from the Lattce Gas Automata (LGA) method, whch can be consdered as a fcttous molecular dynamcs (MD) n whch space, tme and partcle veloctes are all dscrete. Lattce gas models wth an approprate choce of the lattce symmetry n fact represent numercal solutons of the Naver-Stokes equatons and therefore able to descrbe the hydrodynamcs problems have been dscussed by McNamara et.al [1] and We et al. [2]. Due to the samplng of the partcle veloctes around zero velocty, LBM s lmted to the low Mach number (nearly ncompressble flow) flow smulaton. It s commonly recognzed that the LBM can fathfully be used to smulated the ncompressble Naver-Stokes (N-S) equatons wth hgh accuracy and ths lattce BGK (LBGK) model, the local equlbrum dstrbuton has been chosen to recover the N-S macroscopc equatons by dfferent authors [3-5]. It s found that the smulaton results from LBM are n good quanttatve agreement wth expermental results. However, He and Luo [6] shown that the lattce Boltzmann equaton s drectly derved form the Boltzmann equaton wth varous approxmatons by dscretzaton n both space and tme. In ther study, they demonstrated that smulaton results from LBM are n good quanttatve agreement wth expermental results. An overvew of LBM, a parallel and effcent algorthm for smulatng sngle-phase and multphase flud flows and also for ncorporatng addtonal physcal complextes have been dscussed by Chen and Doolen [7]. Taher and Lee [8] have nvestgated numercally the suppresson of flud forces actng on a bluff body wth dfferent control bodes. It s found that the flud forces actng on the man bluff body are effectvely suppressed f the control body (a thn plate or a small crcular cylnder) s placed at a sutable poston wth proper heght or dameter. Moreover,LBM has several advantages over other conventonal CD methods, especally n dealng wth complex boundares, ncorporatng of mcroscopc nteractons, and parallelzaton of the algorthm that are descrbed n the excellent books by authors [9-12]. The vscous flow past a bluff body and the resultng separated regon behnd t has been focus on numerous expermental and numercal nvestgatons. There s no doubt that an enormous corpus of lterature on the subject of bluff body wakes has developed snce the poneerng work of Strouhal and Von Karman. Ths flow stuaton s popular not only because of ts academc attractveness but also owng to ts related techncal problems assocated wth energy conservaton and structural desgn. Ths type of flow s of relevance for many practcal applcatons, e.g. vortex flow meter, buldngs, brdge, towers, masts and wres. A lamnar vortex sheddng regon s known to occur for the Reynolds number range extendng approxmately from and the unversal relatonshp between Reynolds and Strouhal numbers around a crcular cylnder have been studed by and Wllamson [13]. Actually many authors have been studed the vortex sheddng frequency behnd a crcular cylnder or square cylnder or two cylnders for dfferent cases both n numercally and expermentally However, n ths paper, the present authors want to study the flud flow behavors around wedge-shaped usng lattce Boltzmann method (LBM).As far we know, the problem has not been consdered before. The objectve of ths paper s to numercally study of flud flow behavor around wedgeshaped body usng LBM where flow can be drven wth the pressure (densty) gradents. Computatons are carred out for varous wedge angles rangng from 0 0 to and the Reynolds number rangng from 0 to 397, based on the characterstc length of the channel, the maxmum ncomng flow velocty (less than 0.1 lu) and also the nature of flud transport propertes. Here we have focused our attenton on the evoluton of streamlnes, vortcty contours, pressure contours as well as velocty profles and vortex sheddng frequency, to nvestgate the mportant characterstcs of the flow feld around wed-

3 234 M.A.Taher et al. / Proceda Engneerng 56 ( 2013 ) shaped body for a wde range of non-dmensonal parameters that present n our smulaton namely Reynolds number (Re), Strouhal number (St) and Gap rato (G*). Throughout our calculaton, we use Lattce-Boltzmann unts. 2. ormulaton of the problem The computatonal doman s to consder as a rectangular regon L H, where H s the heght and L= 4H s the length of the channel. A wedge-shaped body havng wedge angle placed symmetrcally between parallel walls as shown n g.1 g.1. Physcal model and coordnate systems. The wedge angle s defned by =, 0 1. It s noted that, n the case of equal to zero corresponds to flow over a horzontal flat plate whle equal to 1.0 corresponds to flow over a vertcal flat plate. So the angle measurement s very mportant n ths analyss. If G s the gap between wall and the body then G * = G/H s defned as a gap rato. It s noted that f the value of ncrease, then G * decreases. or convenent, t has been dscussed n the present study of the followng three dfferent cases. Table 1. The dfferent cases of wedge-shaped body: Case G * (=G/H) Case Case In an ncompressble flow, Reynolds number s the only parameter that controls the flow feld and s defne by Re=UD /, where U and D are the characterstc velocty and the length respectvely. In flud dynamcs, vortcty s the crculaton per unt area at a pont n the flow feld. Mathematcally, t s defned as, w u, where u s the flud velocty. The non dmensonal sheddng frequency, the Strouhal number, s defned as: St = fd/u, where f s the vortex sheddng frequency. Ths relaton s beleved to be accurate to ±1% n the Reynolds number. In ths secton, t s assumed that, x = 1lu = m, t = 1 ts = s. The flud propertes are taken to ar propertes. The knematc vscosty = m 2 /s, whch corresponds to lattce unt. All reported data are obtaned on our calculaton doman (lattce node). Thus the physcal doman of smulaton s 400 m 100 m. or accurate soluton, the Mach number, Ma, should be kept as small as possble. In general, the maxmum ncomng flud velocty U s consdered n the LBM n order of 0.2 or 0.1 or less. Therefore, the Reynolds number should be chosen very carefully. In order to smulate a fully developed lamnar channel flow upstream of the wedge-shaped body, a parabolc velocty profle can be used wth a maxmum velocty U at the mdpont of the channel. Ths velocty s chosen to be lower than 10% of the speed of sound for LBM smulatons to avod sgnfcant compressblty effects. In our smulaton, we use Zou-He Boundary condton to mplement Drchlet boundares on nlet/outlet. At the top and bottom wall, no slp boundary condtons were mposed by the standard bounce back treatment. In LBM, the movement of the flud partcles s modeled nstead of drectly solvng the macroscopc flud quanttes lke the velocty and the pressure. It s known as mesoscopc smulaton model, whch s based on the Boltzmann equaton. Neglectng external forces, the Boltzmann equaton (BE) wth BGK approxmaton can be wrtten as t e. x 1 ( eq ), = 0,1,2,3..., q-1 (1) Where, ( x, t) s the dscrete partcle dstrbuton functon and poston x and tme t defned by eq s the dscrete equlbrum dstrbuton functon at lattce

4 M.A.Taher et al. / Proceda Engneerng 56 ( 2013 ) eq (2) w[1 e. u ( e. u) u ] c 2c 2c The lattce weghtng factors, w, depend on the lattce model. or D2Q9 model, each node of the lattce has three knds of partcle: a rest partcle that resdes n the node, partcles that move n the co-ordnate drectons and the partcles that move n the dagonal drectons. So the total number of dscrete veloctes (e ) on each node n D2Q9 model s 9. Table 2. D2Q9 lattce veloctes. e 0 = (0, 0) e 1 = (c, 0) e 2 = (0, c) e 3 = (-c,0) e 4 = (0,-c) e 5 = (c, c) e 6 = (-c, c) e 7 = (-c,-c) e 8 = (-c, c) Here c s called the Courant-redrchs- Lewy (CL) number and s proportonal to x/ t, where x and t are the lattce space and the tme steps respectvely. Therefore the dscrete form of equaton (1) s called the Lattce- Boltzmann equaton (LBE) and can be defned as 1 eq ( x t e, t t) ( x, t) ( ), = 0,1,2, ,8 (3) The relaxaton parameter, =1/, depends on the local macroscopc varables and u.these varables should satsfy the followng laws of conservaton:, u e (4) The above expressons descrbe the relatonshps between the mcroscaled quanttes and the macro scaled physcal quanttes. Usng the Chapman-Enskog expanson,.e. mult-scale analyss, t s mathematcally provable that the LBM equaton (3) can recover the N-S equaton to the second order of accuracy n the lmt of low Mach number [5], f the pressure and the knetc vscosty are defned by P = Cs 2 and = ( -1/2) Cs 2 t. 3. Results and dscussons In ths problem, equaton (3) s an algebrac equaton. In conventonal CD methods for ncompressble N-S equatons, we need to solve the Posson equaton for the pressure, whle n LBM, solvng the equaton (3) we get all nformaton that we nterested to our study. In g.2, we compare our result wth analytcal soluton for Re =100, 200 n a channel n order to assess the accuracy of our method u Analytcal, Re=100 LBM, Re = 100 Analytcal, Re=200 LBM, Re = y/h g.2. Verfy LBM wth analytcal result for dfferent Reynolds numbers The sold lnes are the analytcal soluton and the dashed lnes are the data results obtaned from the smulaton. Ths fgure shows that the velocty profle n the channel s parabolc and the maxmum value at the mddle poston of the channel. It s obvously as we consder the fully developed lamnar parabolc flow and t s seen that our results are n excellent agreement wth analytcal soluton. Ths confrms the accuracy of our present smulaton. One mportant quantty taken nto account n the present study s the Strouhal number(st = fh/u)), computed from the heght (h) of the bluff body,

5 236 M.A.Taher et al. / Proceda Engneerng 56 ( 2013 ) the vortex sheddng frequency (f) and the velocty of the ncomng flud. The dmensonless sheddng frequency wth Reynolds number along the wake centerlne downstream of the wedge-shaped body s shown n g fh 2 / Re g.3.varaton of dmensonless sheddng frequency wth Reynolds number It s noted that the flow velocty profle, the poston, shape of the bluff (barrer) and the rato of the cross secton area of the bluff to the wall affect the Strouhal number(st) for the gven Reynolds number(re) of the flow regme. As the dmensonless vortex frequency ncreases when the gap rato decreases wth the Reynolds number n the range and consequently the Strouhal number also ncreases wthn ths range. The hgher frequency means that the process of vortex sheddng s faster. Therefore, the nature of the vortex sheddng s a strong functon of the Re. In order to gan further nsght nto the evoluton of vortex sheddng n the near-weak regon, the patterns of the vortcty contours for Re=100, 200 and 300 wth dfferent gap ratos are plotted n gs Re=100 Re=100 Re=200 Re=200 Re=300 Re=300 g.4.vortcty dstrbuton for = 45 0 wth Re g.5.vortcty dstrbuton for = 90 0 wth Re Re=100 Re=200 Re=300 g.6.vortcty dstrbuton for =135 0 wth Re or small wedge angle, hgher gap rato, t s seen that a par of vortces wth same strength and sze are formed behnd the body; a postve vortex (antclockwse) appears on the lower part of the body and a negatve vortex (clockwse) on the

6 M.A.Taher et al. / Proceda Engneerng 56 ( 2013 ) upper part of the body. urther, It s observed that for low wedge angles (< 55 0 ), the flow patterns are almost symmetrcal for all Reynolds number wthn the range up to 300. However, for large wedge angles, an unsymmetrcal flow patterns have been seen around the body as shown n the g.5-6. or, = 90 0 wth Re < 195, the flud flow behavors s almost symmetrc. Moreover, a sgnfcant changed n the flow s observed when Re 195. In g. 5-, the Von Karman Street s seen, whch s conssts of vortces n a regular arrangement. or Re=200, the wdth of the vortex street behnd the body s narrower compare to the Re = 300. The wdth s ncreasng wth the ncreasng of Reynolds number. When Re 230, the strong vortces are observed.e. the wdth of the vortex street and the numbers of vortces are remarkably changed. There are seven vortces are seen n g.5 wthn the consdered wake regon for Re=300. Ths knd of flow behavors are observed wth the wedge angles n the range of urther, the ncreases n the wedge angle (g.6) correspond to very low gap rato, a par of vortces are formed just behnd the body, and n addton, the wall proxmty effects are seen to gve rse to reverse Von Karman Street. t=20000 ts t=20500 ts t=21000 ts (d) t=21500 ts g.7. Streamlnes plot for dfferent tme steps wth Re = 260 and = A detaled vew of flow feld behnd the wedge-shaped body and changes n the vortex sheddng pattern wth dfferent tme steps are shown n g.7. The flow s developed untl 20,000 tme steps. In g.7, a new vortex s formng on the top of the body but the lower one s pulled away from the body. The formaton of the upper vortex s completed at 20,500 tme steps and consequently another new vortex on the bottom s formng and t s observed that the vortces are shed alternately wth dfferent tme steps. nally, the last plot, at tme steps t = 21500, s nearly dentcal wth the g.7. Ths evolutonary process s repeated approxmately every 1500 tme steps. Ths tme perod s strongly depends on the Reynolds number. If the Reynolds number ncreases, the tme perod becomes shorter. It s nvestgated that for Re=300, the tme perod s approxmately 1000 tme steps. The same phenomenon has been seen that for flow over an arfol at -90 degrees angles of attack documented by Rogers and Kwak [14]. t=5,000 ts t=10,000 ts t=15,000 ts g.8. Streamlnes plot for dfferent tme steps wth Re=300, =135 0 Typcal examples of nstantaneous flow felds are presented n terms of vortcty for varous tme steps wth wedge angle, =135 0, correspondng to the gap rato G * = Ths s easly understood by examnng streamlnes shown n gs.8 -. At t = 5,000 tme steps, two opposte vortces wth almost smlar sze and shape are seen just behnd the body and smultaneously three statonary vortces are seen on the top wall whereas two vortces are seen on the bottom wall of the channel. The recrculaton regons on the bottom wall are much larger than those of top wall. In g.8, the upper vortex just behnd the body a lttle extended and the two statonary vortces on the top wall far from the body are mergng but the one (near the body) s stll observed. However, the two vortces at the bottom wall have shown to tendency to become a large one. nally, at t = 15,000, the two vortces on the bottom wall converted to one bg sze vortex wth same center. After t =15,000 tmes steps, there s no changed of the flow feld. Thus t s concluded that, for hgher wedge angle, small gap ratos, the change of flow feld occurred untl approxmately 15,000 tme steps. However, the flow separaton on the wall s observed n ths case. Ths phenomenon s observed when Re 130.

7 238 M.A.Taher et al. / Proceda Engneerng 56 ( 2013 ) g. 9. Pressure felds for dfferent wedge angles Apart from the velocty profle, pressure dstrbuton s mportant to understand the flow feld behavor around the bluff body. g.8 shows the pressure contours wth Reynolds number Re=300 for dfferent angles = 45 0, 90 0 and In all cases, the pressure contour at the frontal stagnaton pont has maxmum value where as just behnd the body has mnmum. g.9, for large gap rato, the varatons of the pressure contours are not sgnfcant because the recrculaton regons as well as the flow n the wake are fully developed. However, f the wedge angle ncreases, gs.9, the pressure contours becomes more complcated patterns and many recrculaton regons are observed. The pressure contours ndcate the locaton of vortex contours, where the pressure has a local mnmum value. urther ncreased the wedge angle, g.9, corresponds to very low gap rato, the wall effects are consderable. Therefore, there exst a lttle recrculaton regon and the varaton of pressure contours are not sgnfcant at the far from the body. The same characterstc has found n g Conclusons In ths study, the flow can be characterzed by three regons: () large gap rato, 0.44< G* <0.50, wth 0 0 < <55 0, () ntermedate gap rato, 0.20 G* 0.44, wth and () small gap rato, 0< G* <0.20, wth < < The nvestgatons of these regons are as follows: or case 1, = 45 0, the flow s almost symmetrcal. or case 2, = 90 0, the flow behnd the body s characterzed by a Karman vortex street when Re 195 and the vortces become stronger wth ncreasng the Reynolds number. The formaton and the sheddng of vortces are repeated durng a tme perod. Ths tme perod becomes shorter wth ncreasng the Reynolds numbers. or Re=260, the tme perod s approxmately 1500 tmes steps (lattce unt) where as for Re=300, t s seen approxmately 1000 tme steps. Ths knd of flow behavors are observed wth the wedge angles n the range of or case 3, =135 0, the wall proxmty effects are observed to gve rse to reverse Von Karman street and consequently a packet of vortces are created on the both channel walls and t s observed when Re 130. References [1] McNamara, GR., Zanett, G., Use of the Boltzmann equaton to smulate lattce-gas automata. Physcal Revew Letters 61, pp [2] Wee, X., Mueller, K., Kaufman, A E., The lattce Boltzmann method for smulatng gaseous Phenomena. IEEE Transactons on Vsualzaton and computer Graphcs10 (2), pp [3] rshch, U., Hasslacher, B., Pomeau, Y., Lattce-gas automata for the Naver-Stokes equaton. Physcal Revew Letters 56, pp [4] Qan, Y H., D Humeres, D., Lallemand, P., Lattce BGK models for Naver-Stokes equaton. Euro physcs Letter17 (6), pp [5] Chen, H., Chen, S., Matthaeus, WH., Recovery of the Naver-Stokes equatons usng a lattce-gas Boltzmann method. Physcal Revew A 45(8), pp [6] He, X., Luo, L S., Theory of the lattce Boltzmann method: rom the Boltzmann equaton to the lattce equaton. Physcal Revew E 56(8), pp [7] Chen, S., Doolen, G.D., Lattce Boltzmann method for flud flows. Annu. Rev. lud Mechancs 30, pp [8] Taher, M. A., Lee, Y.W., Numercal study on reducton of flud forces actng on a crcular cylnder wth dfferent control bodes usng Lattce-Boltzmann method. Int. J. Energy and Technology 4(18), pp.1-7. [9] Succ, S., The lattce Boltzmann equaton for flud dynamcs and beyond. Oxford Unversty press, UK. [10] Wolf-Gladrow, D.A., Lattce gas cellular automata and lattce Boltzmann models: An ntroducton, Sprnger, Berln. [11] Suukop, M.C., Thorne, D.T., Lattce Boltzmann modelng, An ntroducton for geoscentst and engneerng. Sprnger, Hedelberg [12] Taher, M.A., lud flow and heat transfer analyss of pure and nanofluds usng Lattce-Boltzmann method, Ph.D Thess, Pukyong Natonal Unversty, Busan, Korea. [13] Wllamson, C.H.K., Defnng a unversal and contnuous Strouhal-Reynolds number relatonshp for a lamnar vortex sheddng of a crcular cylnder. Physcs of lud 31(10), pp [14] Rogers, S.E., Kwak, D., Upwnd dfferencng scheme for the tme-accurate ncompressble Naver-Stokes equatons. AIAA Journal 28(2),pp