Aristeu Silveira-Neto

Elie Luis M. Padilla

Comptes Rendus Mécanique (CRM)

/orinsky’sf unstablesodèle sous-lysés. Onique.cesC. R. Mecanique 333 (2005) 599–604http://france.elsevier.com/direct/CRAS2BLarge-eddy simulation of transition to turbulencein a heated annular channelElie Luis M. Padilla, Aristeu Silveira-Neto∗School of Mechanical Engineering, Federal University of Uberlândia, Uberlândia-MG, Brazil 38400-902Received 9 December 2004; accepted 14 December 2004Available online 24 August 2005Presented by Marcel LesieurAbstractOne carries out three-dimensional large-eddy simulations of natural convection in a horizontal annulus using Smagdynamic subgrid model. The onset of transition to turbulence and turbulent regimes are analyzed. The characteristics oflows and their influence on the heat-transfer process are studied.To cite this article: E.L.M. Padilla, A. Silveira-Neto, C. R.Mecanique 333 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméSimulation des grandes échelles de la transition à la turbulence dans un canal annulaire chauffé. On développe desimulation des grandes échelles tridimensionnelles de la convection naturelle dans un anneau horizontal, grâce au mmaille de Smagorinsky dynamique. L’apparition de la transition à la turbulence et des régimes turbulents sont anaétudie aussi les charactéristiques des écoulements instables et leur influence sur les mécanismes d’échange thermPourciter cet article : E.L.M. Padilla, A. Silveira-Neto, C. R. Mecanique 333 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.Keywords: Heat transfer; Computational fluid dynamics; Natural convection; Transition to turbulence; Large-eddy simulation; Dynamisubgrid model; ChaosMots-clés : Transferts thermiques ; Dynamique des fluides numérique ; Convection naturelle ; Transition à la turbulence ; Simulation dgrandes échelles ; Modèle sous-maille dynamique ; Chaos* Corresponding author.E-mail addresses: epadilla@mecanica.ufu.br (E.L.M. Padilla), aristeus@mecanica.ufu.br (A. Silveira-Neto).1631-0721/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crme.2005.07.016600 E.L.M. Padilla, A. Silveira-Neto / C. R. Mecanique 333 (2005) 599–604l and ex-lications.e natural–4], theThe flowoldsteinayleighe,t naturalusingitted toulationoyancyizontale athich thed scalesflux. Then.e ex-umbers, withfractionaluated byd in thections.1. IntroductionThe study of natural convection between concentric cylinders has been the focus of several numericaperimental investigations since the 1930s, due to the enormous number of practical and technological appIn the recent years, many numerical works have been performed, some of them in three dimensions. Thconvection between concentric horizontal cylinders was studied by Beckmann [1]. In several later works [2effects of changes in the control parameters upon the local and global heat transfer were investigated.destabilization due to natural convection in annular geometries was studied by [3] and [4]. Kuehn and G[3] have looked at cylindrical concentric geometries filled with pressurized nitrogen over a wide range of Rnumber (Ra), corresponding to 2. × 102  Ra  7.7 × 107 and radii ratio of 2.6. Results have shown that thflow becomes first unstable in the plume region, close toRa = 2×105, fully-developing to turbulence asRa grows.McLeod and Bishop [4] studied the problem using helium at cryogenic temperatures, changing theRa from 8×106up to 2× 109. They have graphically represented the dramatic changes in the flow pattern whenRa was increasedand new flow structures at the upper part of the cavity have been found.From a numerical point of view, two-dimensional [5,6] and three-dimensional [7] studies using theκ–ε modelhave been performed assuming vertical symmetry. Information about the unstable behavior of turbulenconvection in periodic horizontal annuli can be found in [8,9]. Fukuda et al. [8] have tackled the problemdirect numerical simulations up toRa = 5× 105.In the present Note, unstationary numerical studies related to transition inside horizontal annuli submnatural convection are presented, assuming periodicity in the axial direction. We use the large-eddy simmethodology (LES) with a dynamic subgrid model.2. Physical problem and numerical methodAn incompressible flow of a Newtonian fluid (air) with constant physical properties is considered. The buterm is modeled through Boussinesq approximation. We work in an annular cavity formed by two horconcentric cylinders of radiiRi (inner) andRo. The cylinders are isothermic, with inner and outer surfactemperatureTi andTo respectively, withTi > To. The gap between cylinders is labeledL and the axial lengthLax.The geometric characteristics are defined by the dimensionless parameters: radii ratioη = Ro/Ri and aspect ratioΓ = Lax/L.The problem presented in this Note is governed by the Navier–Stokes and the energy equations, in wfiltering process are applied according to the LES methodology [10], allowing us to separate the subgrifrom the large scales. This filtering process gives rise to subgrid-stress tensor and subgrid heat turbulentfiltered equations are non-dimensionalized [11] usingTi , To, L and viscosityν. They are written as:∇ · v = 0 (1)∂ v/∂t = −∇ · (vv) − ∇p − Gr T k + ∇ · [(1+ νt )(∇v + ∇vT )] (2)∂T /∂t = −∇ · (vT ) + ∇ · [(1/Pr + νt/Prt )∇T](3)whereGr = gβ(Ti − To)L3/ν2 is the Grashof number andk a unit vector in the direction of gravity acceleratioThe dimensionless eddy viscosityνt is evaluated using the dynamical subgrid scale model according to thpression presented by Lilly [12]. The turbulent thermal diffusivity is estimated using the turbulent Prandtl nPrt = 0.6 [13].A finite-volume method is employed on a staggered grid in order to discretize the differential equationsecond-order Adams–Bashforth and centered differences schemes for time and space respectively. Thestep method [14] has been used for the pressure-velocity coupling. The pressure correction field is evalsolving a Poisson equation, using the strongly implicit procedure method. A non-uniform mesh is employeradial direction, which is more refined near the walls. A uniform mesh is used in the angular and axial direE.L.M. Padilla, A. Silveira-Neto / C. R. Mecanique 333 (2005) 599–604 601ut in thersists ofk, it wasr. 1(b). Ine andtuencies,othersothermaldaround 0.3. ResultsSimulations corresponding to the stable and unstable flows in the horizontal annuli have been carried orange 4× 104  Ra  7.5× 105, with η = 2 andΓ = 2.8. The mesh has 16× 72× 24 cells in the radial, angulaand axial directions, respectively.According to Bishop et al. [15], the instability onset that appears at the upper part of the cavities conoscillations. The oscillations amplitude increases as the Rayleigh number increases. In the present worpossible to capture theses instabilities precisely with the aid of a numerical probe placed atr = 1.5, θ = 90◦ andz = 1.4, as illustrated by Fig. 1(a). Small periodic oscillations in the velocity and temperature are observed foRa 4.7×104. For higherRa values, these oscillations increase and change dramatically, as can be seen on Figthis figure, the radial velocity component is depicted for two differentRa. The fluctuations forRa = 5.0× 104 arevery small and periodic. AsRa increases, oscillations lose their periodic behavior, with increased amplitudtemporal frequency. ForRa > 1.5 × 105 oscillations display a chaotic evolution. AtRa = 7.5 × 105 they presena strongly irregular pattern of high amplitude. One observes also the presence of a wide range of freqcharacterizing a turbulent behavior.The effect of instability upon the temperature field manifests initially in the plume structure, and after inregions of the cavity, as can be seen in Figs. 2 and 3. In these figures, instantaneous dimensionless isurfaces for three differentRa are shown. One observes in Fig. 2 that, asR increases, instabilities multiply anintensify, the more affected region being the upper part of the cavity. In the flow corresponding toRa = 1 × 105,the thermal plume oscillates axially, with small amplitudes and, also, small movements along theθ direction. ForhigherRa, the oscillation is subject to an intense three-dimensionalization. ForRa = 5.8 × 105 one observeschaotic pattern with hot mass transport in both sides, showing turbulent characteristics. The oscillations a◦and 180◦ are very intense. In the lower part of the cavity the flow is unstable, but turbulence is less intenseFig. 1. (a) Geometric characteristics; (b) time distribution of radial velocity.Fig. 1. (a) Caractéristiques géométriques ; (b) distribution temporelle de la vitesse radiale.602 E.L.M. Padilla, A. Silveira-Neto / C. R. Mecanique 333 (2005) 599–604onalthe leftand [8].osociatedFig. 2. Instantaneous temperature isosurfaces, thresholds 0.25 (transparent) and 0.65.Fig. 2. Isosurfaces de température instantanée, seuils 0,25 (transparent) et 0,65.Fig. 3. Temporal evolution of temperature isosurfaces forRa = 1.5× 105.Fig. 3. Evolution temporelle de l’isosurfaces de température pourRa = 1,5× 105.In Fig. 3, the flow atRa = 1.5 × 105 is depicted for two different instants: 49.5 s and 50 s. The transitiflow presents three-dimensional oscillations. The plume oscillates with vigorous amplitude, moving from(49.5 s) to the right (50 s) and back, as experimentally observed in [15] and [4].We have compared satisfactorily some statistical properties with experimental results of Itoh et al. [16]Time-frequency energy spectra of the radial velocity component fluctuation for two values ofRa are presentedin Fig. 4. The velocity distribution was taken at the probe of Fig. 1(a). A line with a−5/3 slope corresponding tKolgomorov law has been drawn. One observes that the ‘inertial range’ (if our temporal spectrum may be asto a spatial spectrum thanks to some local Taylor hypothesis) increases withRa. T e spectrum atRa = 1.5 × 105is steeper than−5/3, demonstrating an energy concentration in the large scales.E.L.M. Padilla, A. Silveira-Neto / C. R. Mecanique 333 (2005) 599–604 603vectiond scalecreases,ture thental andanicalIng. 2 (5)lindrical. J. Heatnaturalorizontal. J. HeatFig. 4. Energy spectra of the radial velocity fluctuation.Fig. 4. Spectres d’énergie de la fluctuation de vitesse radiale.4. ConclusionThree-dimensional unstationary numerical simulations of the transitional flow submitted to natural conin an annular cavity were successful performed, employing an LES methodology with dynamic subgrimodel. We have been able to characterize the onset of the first instabilities. As the Rayleigh number inthe flow becomes fully irregular and chaotic when a turbulent regime is reached. It was possible to capflow dynamic characteristics, as well as the plume transition. The results compare very well with experimenumerical results from other authors.AcknowledgementsThe authors would like to tank CNPq-Brazil for the financial support and to the School of MechEngineering-Federal University of Uberlândia. We thanks also Marcel Lesieur for interesting comments.References[1] W. Beckmann, Die Wärmeübertragung in zylindrischen Gasschichten bei natürlicher Konvektion, Forschung auf dem Geb. des(1931) 165–178.[2] U. Grigull, W. Hauf, Natural convection in horizontal cylindrical annuli, in: Third Int. Heat Transfer Conf., 1966, 182–195.[3] T.H. Kuehn, R.J. Goldstein, An experimental study of natural convection heat transfer in concentric and eccentric horizontal cyannuli, ASME J. Heat Transfer 100 (1978) 635–640.[4] A.E. McLeod, E.H. Bishop, Turbulent natural convection of gases in horizontal cylindrical annuli at cryogenic temperatures, IntMass Transfer 32 (10) (1989) 1967–1978.[5] B. Farouk, S.I. Guçeri, Natural convection from a horizontal cylinder-laminar regime, J. Heat Transfer 103 (1982) 522–527.[6] M.I. Char, Y.H. Hsu, Comparative analysis of linear and nonlinear low-Reynolds-number eddy viscosity models to turbulentconvection in horizontal cylindrical annuli, Numer. Heat Transfer Part A 33 (1998) 191–206.[7] C.P. Desai, K. Vafai, An investigation and comparative analysis of two-and tree-dimensional turbulent natural convection in a hannulus, Int. J. Heat Mass Transfer 37 (16) (1994) 2475–2504.[8] K. Fukuda, Y. Miki, S. Hasegawa, Analytical and experimental study on turbulent natural convection in a horizontal annulus, IntMass Transfer 33 (4) (1990) 629–639.604 E.L.M. Padilla, A. Silveira-Neto / C. R. Mecanique 333 (2005) 599–604t FluidFederalckward-308–323.(1968)izontal[9] Y. Miki, K. Fukuda, N. Taniguchi, Large eddy simulation of turbulent natural convection in concentric horizontal annuli, Int. J. HeaFlow 14 (3) (1993) 210–216.[10] M. Lesieur, Turbulence in Fluids, Kluwer Academic, 1997.[11] E.L.M. Padilla, Large-eddy simulation of transition to turbulence in rotating system with heat transfer, Doctor Thesis, Universidadede Uberlândia, 2004.[12] D.K. Lilly, A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids A 4 (3) (1992) 633–635.[13] A. Silveira-Neto, D. Grand, O. Metais, M. Lesieur, A numerical investigation of the coherent structures of turbulence behind a bafacing step, J. Fluid Mech. 256 (1993) 1–25.[14] J. Kim, P. Moin, Application of a fractional step method to incompressible Navier–Stokes equations, J. Comput. Phys. 59 (1985)[15] E.H. Bishop, C.T. Carley, R.E. Powe, Natural convective oscillatory flow in cylindrical annuli, Int. J. Heat Mass Transfer 111741–1752.[16] M. Itoh, T. Fujita, N. Nishiwaki, M. Hirata, A new method of correlating heat-transfer coefficients for natural convection in horcylindrical annuli, Int. J. Heat Mass Transfer 13 (1970) 1364–1369.

Large-eddy simulation of transition to turbulence in a heated annular channel

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Large-eddy simulation of transition to turbulence in a heated annular channel

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