Lattice Boltzmann simulation of turbulent natural convection in tall enclosures.

Transcription

1 Lattce Boltzmann smulaton of turbulent natural convecton n tall enclosures Hasan SAJJADI 1*, Reza KEFAYATI 1* Department of Engneerng, Shahd Bahonar Unversty of Kerman, Kerman, I. R. Iran School of Computer Scence, Engneerng and Mathematcs, Flnders Unversty, Adelade, Australa E-mal: hasansajad@gmal.com Abstract In ths paper Lattce Boltzmann smulaton of turbulent natural convecton wth large-eddy smulatons (LES) n tall enclosures whch s flled by ar wth Pr=0.71 has been studed. Calculatons were performed for hgh Raylegh numbers (Ra= ) and aspect ratos change between 0.5 to (0.5<AR<). The present results are valdated by fnds of an expermental research at Ra= Effects of the aspect ratos n dfferent Raylegh numbers are dsplayed on streamlnes, sotherm counters, vertcal velocty and temperature at the mddle of the cavty, local Nusselt number and average Nusselt number. The average Nusselt number ncreases wth the augmentaton of Raylegh numbers. The ncrement of the aspect rato causes heat transfer to declne n dfferent Raylegh numbers. Keywords: Lattce Boltzmann Method, Large Eddy Smulaton, Aspect rato, Natural convecton. Nomenclature c lattce speed Greek letters c dscrete partcle speeds ω Weghted factor ndrecton c p specfc heat at constant pressure τ c Relaxaton tme for temperature F external forces τ v Relaxaton tme for flow f densty dstrbuton functons ρ densty f eq equlbrum densty dstrbuton functons μ dynamc vscosty g nternal energy dstrbuton functons ν knematc vscosty g eq equlbrum nternal energy dstrbuton functons Δx lattce spacng g y gravty Δt tme ncrement H cavty heght Nu Nusselt number Pr Prandtl number R constant of the gases Subscrpts Ra Raylegh number avg average Stran rate tensor C cold S T temperature f flud x,y Cartesan coordnates H hot W cavty wdth t Turbulence 1. Introducton

2 Turbulence n fluds s ubqutous n nature and technologcal systems and represents one of the most challengng aspects n flud mechancs. The dffculty stems from the nherent presence of many scales that are generally nseparable among many other factors. Nevertheless, consderable progress has been made over the years towards more fundamental physcal understandng of turbulence phenomena through measurements, statstcal phenomenologcal theores, modelng and computaton [1-]. Also some expermental nvestgatons have been done for nstance Ampofo and Karayanns [3] studed low-level turbulence natural convecton n an ar flled vertcal square cavty whle the hot and cold walls of the cavty were sothermal at 50 and 10 C respectvely gvng a Raylegh number of Ra= Large Eddy smulatons (LES) provde a very promsng approach for the smulaton of turbulent flows because computaton tmes are sgnfcantly lower than those of Drect Numercal smulatons (DNS). Further, ther resoluton of turbulent structures s more accurate n comparson to Reynolds Averaged Naver_Stokes (RANS) smulatons [4-8]. The Lattce Boltzmann Method (LBM) s a computatonal alternatve for smulatng flud flows and s rapdly ganng attenton. It s an attractve method snce t s based on a smple core algorthm whch n turn makes t easy to adapt to complex applcaton scenaros. Moreover, the base algorthm of the LBM can easly be extended to capture addtonal physcal effects. Consequently, ths method s beng used as a unversal tool n a rapdly ncreasng number of research projects. However, the flexblty of the LBM comes at a hgh prce n terms of computatonal cost whch routnely requres the use of parallel supercomputers [9-15]. To model the flow large-eddy smulaton (LES) n a lattce- Boltzmann scheme for dscretzng the Naver Stokes equatons was used n prevous works. Lattce Boltzmann Method demonstrates that t can be a powerful method for smulatng of turbulence flows [16-0].The mplementaton of a LBM procedure s much easer for turbulence flows than that of tradtonal CFD methods. Meanwhle, t s more popular due to the balance between accuracy and effcency. Because of these advantages, t was appled at past works regularly. Yu et al. [1] consdered the applcaton of multple-relaxaton-tme (MRT) lattce Boltzmann equaton (LBE) for large-eddy smulaton (LES) of turbulent flows. They demonstrated that MRT-LBE s a potentally vable tool for LES of turbulence. Fernandno et al. [] nvestgated large eddy smulatons of turbulent open duct flow are performed usng the lattce Boltzmann method (LBM) n conjuncton wth the Smagornsky sub-grd scale (SGS) model. Whereas they found that the LBM smulaton results are n good qualtatve agreement wth the experments. Chen [3] proposed a novel and smple large-eddy-based lattce Boltzmann model to smulate two-dmensonal turbulence. He showed that the model s effcent, stable and smple for two-dmensonal turbulence smulaton. The Lattce Boltzmann Method (LBM) wth a forcng scheme s used to smulate homogeneous sotropc turbulence by Kareem et al. [4].They receved that the turbulence characterstcs of the flow are smlar to those obtaned n studes by the spectral method regardless of whch model s used. Recently Sajjad et al. [5] studed numercal analyss of turbulent natural convecton n square cavty usng Large-Eddy Smulaton n Lattce Boltzmann Method. They exhbted ths method s n acceptable agreement wth other verfcatons of such a flow. The man am of ths nvestgaton s to present Large-eddy turbulence model by Lattce Boltzmann Method (LBM) wth a clear and smple statement. Thus natural convecton turbulence flow n tall

3 enclosures s nvestgated n a wde range of Raylegh numbers. Although natural convecton turbulence flow n confned convecton s not only a topc for analyss but s comparable for numercal and expermental nvestgatons. The frst LBM s consdered brefly and then large-eddy was appled n LBM whereas thers equatons s expressed completely. Fnally results of ths study are compared wth an expermental research.. Numercal Method -1. Problem statement The geometry of the present problem s shown n Fg.1. It dsplays a two-dmensonal enclosure wth heght of H and wdth of W. The temperatures of the two sdewalls of the cavty are mantaned at T H and T C, where T C has been consdered as the reference condton. The top and the bottom horzontal walls have been consdered to be adabatc.e., non-conductng and mpermeable to mass transfer. The densty varaton n the flud s approxmated by the standard Boussnesq model.the flud s assumed to be Newtonan, ncompressble and the lamnar whereas Prandtle number equals to Pr=0.71. Also t s assumed that Mach number s fxed at Ma= Lattce Boltzmann method Fg.1 Geometry of the present study In ths paper a square grd and DQ9 model s used for both flow and temperature functons. By detachment of Naver-Stocks equatons, governng equatons for flow and temperature functons are as follow: For the flow feld: 1 eq f x c t, t t f x, t f x, t f x, t tc F (1) v For the temperature feld: g x t g x t 1 g x t g x t (),,, eq, c Where the dscrete partcle velocty vectors defned c (Fg.), Δt denotes lattce tme step whch s set to unty. τ v, τ c are the relaxaton tme for the flow and temperature felds, respectvely.

4 Fg. The dscrete velocty vectors for DQ9 f eq,g eq are the local equlbrum dstrbuton functons that have an approprately prescrbed functonal dependence on the local hydrodynamc propertes whch are calculated wth Eqs. (3) and (4) for flow and temperature felds respectvely. Also F s an external force term. eq f,. 1 (. ) 1. 1 x t cu cu uu (3) 4 c c c g eq s s s cu. (4) 1 T cs For the -D case, applyng thrd-order Gauss-Hermte quadrature leads to the DQ9 model wth the followng dscrete veloctes c, where = : 0 0 C ccos[( 1) ],sn[( 1) ] 1 4 (5) c cos[( 5) ],sn[( 5) ] Where ω 0 =4/9, ω 1 4 =1/9, ω 5 9 =1/36 and c 3RT m (to mprove numercal stablty, T m s the mean value of temperature for the calculaton of c). Usng a Chapman-Enskog expanson, the Naver-Stokes equatons can be recovered wth the descrbed model. The knematc vscosty υ and the thermal dffusvty α are then related to the relaxaton tmes by: 1 v cs t and 1 c cs t (6) c Where c s s speed of sound and equal to. 3 In the smulaton the Boussnessq approxmaton s appled to the buoyancy force term.in that case, the external force F appearng n Eq. (1) s gven by: F 3 g y T (7) Where gy, β and ΔT are gravtatonal acceleraton, thermal expanson coeffcent and temperature dfference, respectvely. Fnally, the macroscopc varables ρ, u, and T can be calculated usng as follows: Flow densty: f (8)

5 Momentum: u j f c j (9) Temperature: T g (10) -3. Large Eddy Smulaton method t In ths model the man am s obtanng t and t ( ) where Pr t s turbulent Prandtle number whch Pr s assumed to be 0.5. In order to evaluate t we perform as follow: t Pr g t ( C ) S T. Prt g 1/ (11) C s consdered as Smagornsky Constant and n ths paper t s assumed as 0.1 [17] and s ganed from ( x) ( y), x and y are grd extents n X and Y drectons. For S we have: S S S (1) / S u u (13) -4. Lattce Boltzmann Method based on Large Eddy Smulaton model Large eddy model s easly appled n Lattce Boltzmann method the way t affects relaxaton tme [18-0]. total cs ( m 0.5) 0 t (14) Where total and 0 are total vscosty and ntal vscosty respectvely. m (15) 0 t 0 t t 0 cs cs cs cs To obtan t n Lattce Boltzmann method we have: 3 S Q (16) m If we put S n equaton (11): 8 eq ( ) 0 (17) Q e e f f 9 Pr g t ( C ) Q T. 4 m Prt g And f we substtute the above equaton n (15): 1/ (18)

6 total g C Q T g 9 Pr ( ). 4 m Prt 0 cs To obtan relaxaton tme n temperature functon equaton we have: t t / Prt h D 0 D 0 (0) cs cs 0 Where D c s Substtutng new relaxaton tme n equaton (1) and () yelds to Lattce Boltzmann equatons based on large eddy model. 1/ (19) 3. Boundary condtons: 3-1.flow Implementaton of boundary condtons s very mportant for the smulaton. The unknown dstrbuton functons pontng to the flud zone at the boundares nodes must be specfed. Concernng the no-slp boundary condton, bounce back boundary condton s used on the sold boundares. For nstance the unknown densty dstrbuton functons at the boundary east can be determned by the followng condtons: f 6, n f 8, n, f 7, n f 5, n, f 3, n f1, n (1) Where n s the lattce on the boundary. 3-.Temperature The north and south of the boundares are adabatc then bounce back boundary condton s used on them. Temperature at the west and east wall are known, n the west wall T H =1.0. Snce we are usng DQ9, the unknowns dstrbuton functon are g 1, g 5, g 8 at west wall whch are evaluated as follows; g1 T H ( 1 3) g3 (a) g5 T H ( 5 7) g7 (b) g8 T H ( 8 6) g6 (c) Nusselt number Nu s one of the most mportant dmensonless parameters n descrbng the convectve heat transport. The local Nusselt number and the average value at the hot and cold walls are calculated as: NU H T y (3) T x NU 1 H avg NU ydy H (4) 0 Fnally, the followng crteron to check for the steady-state soluton was used: n1 n 5 Error max T T 10 (5) 4. Code valdaton The tall enclosures were nvestgated at dfferent Raylegh numbers of 10 7, 10 8 and 10 9, wth three varous aspect ratos of A=0.5, 1and.Turbulent natural convecton wth LES method were conducted n ths paper. An extensve mesh testng procedure was conducted to guarantee a grd ndependent soluton.

7 Fnally numbers of the lattces for dfferent Raylegh numbers and average Nusselt number comparson of present results wth two prevous studes were dsplayed at Table.1. Table1. Comparson of mean Nu wth prevous works Ra Number Mesh Mean Nu(ths work) Mean Nu[] Mean Nu[16] * * * The present numercal method was valdated wth expermental researches of Ampofo and Karayanns [3]. A comparson wth velocty at the mddle secton of the cavty and local nusselt number on the hot wall were consdered wth expermental results of Ampofo and Karayanns [3] n Fg.3. Table 1 shows the comparson of the average Nusselt numbers for dfferent Raylegh numbers between present results and fnds of Barakoset al [], Dxt [16] as cavty was flled by ar wth Pr=0.71. Clearly t s seen that the results match prevous works. Furthermore ths table demonstrates needful varous meshes to utlze for dfferent Raylegh numbers. These comparsons show that the present study has a good agreement wth prevous studes. (a) (b) Fg.3 Comparson of the velocty on the axal mdlne (a) and local Nusselt number (b) between the present results and numercal results by Ampofo and Karayanns [3](Ra= ) 5. Results and dscussons Fgs.4 shows the contour maps for the sotherms at varous Raylegh numbers and dfferent enclosure aspect ratos. When Raylegh number ncreases, the convecton process n the sotherms for varous aspect ratos augments. The process s obvous whereas the sotherm of T=0.9 moves from the upper left corner of the enclosure at R=10 7 toward the cold wall of the enclosure at Ra=10 9. The sotherms are vbrant at Ra=10 7, but ths manner changes to a steady state at hgher Raylegh numbers. The ncrement of the aspect ratos causes the movement of the sotherms from the hot wall to the cold wall declnes. Ths phenomenon s qute obvous at Ra=10 7 whereas both sotherms of T=0.8 and 0. at A= set between two vertcal walls and move towards the horzontal walls. Fgs.5 dsplays the streamlnes at varous Raylegh numbers and dfferent enclosure aspect ratos. The streamlnes have a regular process untl Ra=10 8 and the changes are margnal nto the streamlnes, but at Ra=10 9 the form of the streamlnes change completely and they revolve around some dfferent vortexes at ths Raylegh number. The maxmum values of the streamlnes behave erratcally for dfferent aspect ratos and Raylegh numbers. The maxmum values of the streamlnes decrease from Ra=10 7 to 10 8 whle the value at Ra=10 9 jumps to a hgher amount of Ra=10 7 suddenly. Moreover, the maxmum values of the streamlnes have the most ther values at A=1 among dfferent aspect ratos and Raylegh numbers. It s

8 clear that turbulence n the enclosures at A= s less than other two aspect ratos for varous Raylegh numbers and ther trend s smooth so the maxmum values of the streamlnes has the least ther values at A=. AR=0.5 AR=1 AR= Ra=10 7 Ra=10 8 Ra=10 9 Fg.4 Comparson of the sotherms at varous aspect ratos and Raylegh numbers AR=0.5 AR=1 AR= Ra=10 7 mn mn mn

9 Ra=10 8 mn mn mn Ra= mn mn mn Fg.5 Comparson of the streamlnes at varous aspect ratos and Raylegh numbers Fg.6 llustrates the dstrbuton of the local Nusselt number on the hot wall for dfferent aspect ratos and Raylegh numbers. The trend of the local Nusselt number s the same for varous Raylegh number and just ther values ncrease by the augmentaton of Raylegh number at varous aspect ratos. The effect of aspect ratos on the local Nusselt number s neglgble and the dfference s consderable at (Y/H>0.8 for A=,1 and Y/H>0.4 for A=0.5) whereas the sotherm of T=0.9 have an mportant nfluence on the sotherms. Fg.7 demonstrates the values of the vertcal velocty on the axal mdlne for dfferent Raylegh numbers and aspect ratos. When Raylegh enhances, the sudden changes near the vertcal walls happen n a closer place to the walls. The maxmum and mnmum of the vertcal velocty on the axal mdlne whch are close to vertcal walls augments wth the growth of Raylegh numbers. On the other hand, when the aspect rato ncreases, the vertcal velocty on the axal mdlne vbrates at Ra=10 9 and the ncrement of the aspect rato ncrease the ampltudes values of the vbratons. (a) (b)

10 (c) Fg.6 Values of the local Nusselt number for dfferent aspect ratos (a) A=0.5, (b) A=1, (c) A= (a) (b) (c) Fg.7 Values the vertcal velocty on the axal mdlne for dfferent Raylegh numbers and aspect ratos (a) A=0.5, (b) A=1, (c) A= Fg.8 exhbts the values of the temperature on the axal mdlne for dfferent Raylegh numbers and aspect ratos. It s clear that the value declnes wth the augmentaton of Raylegh numbers whereas ths fall s margnal at A=0.5 and has the greatest value at A=. When the aspect rato ncreases, the values change near the vertcal walls gets less and smooth.

11 (a) (b) (c) Fg.8 Values of the temperature on the axal mdlne for dfferent Raylegh numbers and aspect ratos (a) A=0.5, (b) A=1, (c) A= Fg.9 ndcates the value of average Nusselt number and the normalzed average Nusselt number. The fg shows that the average Nusselt number decreases wth the ncrement of the aspect ratos at varous Raylegh numbers. The values of the normalzed average Nusselt number demonstrates that the greatest effect of aspect ratos happens at Ra=10 7 whle the least effect observes at Ra=10 8. (a) (b) Ra=10 7

12 Ra=10 8 Ra=10 9 Fg.9 Average Nusselt number (a) and (b) non-dmensonal Nusselt number dstrbutons on the hot wall at dfferent aspect ratos and Raylegh numbers. 4. Conclusons Turbulent natural convecton n tall enclosures whch are flled wth ar wth Pr=0.71 has been conducted numercally by Lattce Boltmann Method (LBM). Ths study has been carred out for the pertnent parameters n the followng ranges: the Raylegh number of base flud (Ra= ), aspect ratos (AR) of the enclosure (AR=0.5-), some conclusons were summarzed as follows: a) A proper valdaton wth prevous numercal nvestgatons demonstrates that Lattce Boltzmann Method s an approprate method for turbulent flows problems. b) Generally, the decrease n the aspect ratos and the Raylegh numbers enhancement result n the augmentaton of heat transfer. c) The form of the streamlnes changes completely when Raylegh number ncreases from Ra=10 8 to The most effect of the aspect rato changes s observed at Ra=10 7 among consdered Raylegh numbers.. Reference [1] U. Frsch, Turbulence: The Legacy of A.N. Kolmogorov, Cambrdge Unversty Press, New York, [] G. Barakos, E. Mtsouls, D. Assmacopoulos, Natural convecton flow n a square cavty revsted: lamnar and turbulent models wth wall functon, Int. J. Numercal Method n Fluds, (1994), 18, pp

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