A three-dimensional (3D) direct numerical simulation is combined with a
laboratory study to describe the turbulent flow in an enclosed annular
rotor-stator cavity characterized by a large aspect ratio G=(b-a)/h=18.32 and a
small radius ratio a/b=0.152, where a and b are the inner and outer radii of
the rotating disk and h is the interdisk spacing. The rotation rate Omega under
consideration is equivalent to the rotational Reynolds number Re=Omegab2/nu=9.5
x 104, where nu is the kinematic viscosity of the fluid. This corresponds to a
value at which an experiment carried out at the laboratory has shown that the
stator boundary layer is turbulent, whereas the rotor boundary layer is still
laminar. Comparisons of the 3D computed solution with velocity measurements
have given good agreement for the mean and turbulent fields. The results
enhance evidence of weak turbulence at this Reynolds number, by comparing the
turbulence properties with available data in the literature. An approximately
self-similar boundary layer behavior is observed along the stator side. The
reduction of the structural parameter a1 under the typical value 0.15 and the
variation in the wall-normal direction of the different characteristic angles
show that this boundary layer is three-dimensional. A quadrant analysis of
conditionally averaged velocities is performed to identify the contributions of
different events (ejections and sweeps) on the Reynolds shear stress producing
vortical structures. The asymmetries observed in the conditionally averaged
quadrant analysis are dominated by Reynolds stress-producing events in this
B\"{o}dewadt layer. Moreover, case 1 vortices (with a positive wall induced
velocity) are found to be the major source of generation of special strong
events, in agreement with the conclusions of Lygren and Andersson.Comment: 16 page

Anthony Randriamampianina

Canuto C.

Itoh M.

Owen J. M.

Sébastien Poncet

Vanel J. M.

English

arXiv

16 pagesInternational audienceA three-dimensional (3D) direct numerical simulation is combined with a laboratory study to describe the turbulent flow in an enclosed annular rotor-stator cavity characterized by a large aspect ratio G=(b-a)/h=18.32 and a small radius ratio a/b=0.152, where a and b are the inner and outer radii of the rotating disk and h is the interdisk spacing. The rotation rate Omega under consideration is equivalent to the rotational Reynolds number Re=Omegab2/nu=9.5 x 104, where nu is the kinematic viscosity of the fluid. This corresponds to a value at which an experiment carried out at the laboratory has shown that the stator boundary layer is turbulent, whereas the rotor boundary layer is still laminar. Comparisons of the 3D computed solution with velocity measurements have given good agreement for the mean and turbulent fields. The results enhance evidence of weak turbulence at this Reynolds number, by comparing the turbulence properties with available data in the literature. An approximately self-similar boundary layer behavior is observed along the stator side. The reduction of the structural parameter a1 under the typical value 0.15 and the variation in the wall-normal direction of the different characteristic angles show that this boundary layer is three-dimensional. A quadrant analysis of conditionally averaged velocities is performed to identify the contributions of different events (ejections and sweeps) on the Reynolds shear stress producing vortical structures. The asymmetries observed in the conditionally averaged quadrant analysis are dominated by Reynolds stress-producing events in this Bödewadt layer. Moreover, case 1 vortices (with a positive wall induced velocity) are found to be the major source of generation of special strong events, in agreement with the conclusions of Lygren and Andersson

Randriamampianina, Anthony

Poncet, Sébastien

HAL: Hyper Article en Ligne

Turbulence characteristics of the Bödewadt layer in a large enclosed rotor-stator system

4 pagesA three-dimensional direct numerical simulation (3D DNS) is performed to describe the turbulent flow in an enclosed rotor-stator cavity characterized by a large aspect ratio $G=(b-a)/h=18.32$ and a small radius ratio $a/b=0.15$ ($a$ and $b$ the inner and outer radii of the rotating disk and $h$ the interdisk spacing). Recent comparisons with velocity measurements have shown that, for the rotational Reynolds number $Re=\Omega b^2/\nu=95000$ ($\Omega$ the rate of rotation of the rotating disk and $\nu$ the kinematic viscosity of water) under consideration, the stator boundary layer is 3D turbulent and the rotor one is still laminar. Budgets for the turbulence kinetic energy are here presented and highlight some characteristic features of the effect of rotation on turbulence. A quadrant analysis of conditionally averaged velocities is also performed to identify the contributions of different events (ejections and sweeps) on the Reynolds shear stress

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Turbulence characteristics of the Bo¨dewadt layer in alarge shrouded rotor-stator systemSe´bastien Poncet, Anthony RandriamampianinaTo cite this version:Se´bastien Poncet, Anthony Randriamampianina. Turbulence characteristics of the Bo¨dewadtlayer in a large shrouded rotor-stator system. Deville, Pr and Le, Dr and Sagault, Pr. Con-ference on Turbulence and Interactions TI2006, 2006, Porquerolles, France. 2006. <hal-00085153>HAL Id: hal-00085153https://hal.archives-ouvertes.fr/hal-00085153Submitted on 11 Jul 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00085153, version 1 - 11 Jul 2006Conference on Turbulence and Interactions TI2006, May 29 - June 2, 2006, Porquerolles, FranceTurbulence characteristics of the Bo¨dewadt layer in a largeshrouded rotor-stator systemS. Poncet†,∗, A. Randriamampianina††IRPHE UMR 6594 CNRS - Universite´s d’Aix-Marseille I et II, Technopoˆle Chaˆteau-Gombert, 49 rue F.Joliot-Curie BP 146, 13 384 Marseille ce´dex 13, FRANCE∗Email: poncet@irphe.univ-mrs.frABSTRACTA three-dimensional direct numerical simulation (3D DNS) is performed to describe the turbulent flow inan enclosed rotor-stator cavity characterized by a large aspect ratio G = (b − a)/h = 18.32 and a smallradius ratio a/b = 0.15 (a and b the inner and outer radii of the rotating disk and h the interdisk spacing).Recent comparisons with velocity measurements [1] have shown that, for the rotational Reynolds numberRe = Ωb2/ν = 95000 (Ω the rate of rotation of the rotating disk and ν the kinematic viscosity of water)under consideration, the stator boundary layer is 3D turbulent and the rotor one is still laminar. Budgets forthe turbulence kinetic energy are here presented and highlight some characteristic features of the effect ofrotation on turbulence. A quadrant analysis of conditionally averaged velocities is also performed to identifythe contributions of different events (ejections and sweeps) on the Reynolds shear stress.INTRODUCTIONBesides its primary concern to many industrialapplications, the rotor-stator problem has proveda fruitful means of studying turbulence in con-fined rotating flows. This specific configurationis indeed among the simplest ones where rota-tion brings significant modifications to the tur-bulent field. The present paper is devoted to thestudy of the turbulent flow in an enclosed high-speed rotor-stator system of large aspect ratio.The flow is of Batchelor type and belongs to theregime denoted IV by Daily and Nece [2]. Theseauthors provided an estimated value for the localReynolds number at which turbulence originateswith separated boundary layers, Rer = Ωr2/ν =1.5 × 105 (r the radial location) for G ≤ 25.However, experiments have revealed that transi-tion to turbulence can appear at a lower valueof the Reynolds number within the stator bound-ary layer (the Bo¨dewadt layer), even though theflow remains laminar in the rotor boundary layer(the Ekman layer). Recently, the 3D computedsolution presented here has been compared to ve-locity measurements performed at IRPHE [1]. Ithas been shown that, for the rotational Reynoldsnumber Re = 95000, the Bo¨dewadt layer is tur-bulent and the rotor one is still laminar. The pur-pose of this work is to provide detailed data ofthe turbulent boundary layer along the stator sidein a large enclosed system (G = 18.32).NUMERICAL APPROACHThe numerical approach is based on a pseudo-spectral technique using Chebyshev polynomialsin the radial and axial directions with Fourier se-ries in the azimuthal direction associated with asemi-implicit second order time scheme. An ef-ficient projection method is introduced to solvethe pressure-velocity coupling. A grid resolutioncomposed of N × M × K = 300 × 80 × 100respectively in radial, axial and azimuthal direc-tions has been used, with a dimensionless timestep δt = 2.75× 10−3. Numerical computationshave been carried out on NEC SX-5 (IDRIS, Or-say, France).RESULTS AND DISCUSSIONThe detailed description of the mean field and ofthe axial variations of the Reynolds stress tensoris given in [1]. Nevertheless, we recall the mainresults. A good agreement has been obtained be-tween the 3D solution and the experimental data,whereas the axisymetric solution leads to a steadylaminar flow. The flow is of Batchelor type: thetwo boundary layers developed on each disk areseparated by a central rotating core characterizedby a quasi zero radial velocity and a constanttangential velocity. The Bo¨dewadt layer alongthe stator is centripetal, three-dimensional as theTownsend structural parameter is lower than thelimit value 0.15, and turbulent with turbulence in-tensities increasing from the axis to the peripheryof the cavity. On the contrary, the Ekman layeron the rotor is centrifugal and laminar. The tur-bulence is observed by the formation of turbulentspots developing along spiral arms towards theperiphery of the cavity, as seen in figure 1 fromthe instantaneous iso-values of the turbulence ki-netic energy within the stator boundary layer.Turbulence kinetic energy budgetsThe balance equation for the turbulence kineticenergy k can be written:A = P +DT +Dν +Π− ǫ (1)Fig. 1. Iso-contours of the instantaneous turbulencekinetic energy within the stator boundary layer.with the advection term A, the production termP , the turbulent diffusion DT , the viscous diffu-sion Dν , the velocity-pressure-gradient correla-tion Π and the dissipation term ǫ.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1−0.8−0.6−0.4−0.200.20.40.60.81z*APDTDνΠε(a)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1−0.8−0.6−0.4−0.200.20.40.60.81z*(b)Fig. 2. Budgets for the turbulence kinetic energyk/(Ωb)2 at: (a) r/b = 0.56, (b) r/b = 0.8.Figures 2a and 2b show the axial variations alongz∗ = z/h of the different terms involved in thetransport equation (1) at two radial locations. Itis clearly seen that all these terms vanish towardsthe rotor side (z∗ = 0), confirming the lami-nar nature of this zone up to the stator bound-ary layer (z∗ = 1). At the stator wall, the vis-cous diffusion balances the dissipation as wellknown in 3D turbulent boundary layer. Withinthe Bo¨dewadt layer, even though some interac-tion between the different terms involved is ob-served, the major contributions come from theproduction, the dissipation and the viscous dif-fusion terms. The production is balanced by thedissipation and the viscous diffusion, the level ofwhich increases in association with the thicken-ing of the boundary layer towards the periphery.The production increases with increasing radiusas already observed with the levels of the nor-mal Reynolds stresses [1]. The maximum of theproduction term is obtained at the wall coordi-nate z+ = zvτ/ν = 12 (vτ = ((ν∂Vθ/∂z)2 +(ν∂Vr/∂z)2)1/4 the total friction velocity and zthe axial coordinate) for r/b = 0.56 and at z+ =12.5 for r/b = 0.8, which confirms the approx-imately self-similar behavior of the Bo¨dewadtlayer. The levels of the viscous diffusion increasewhen moving towards the outer casing, where thehighest turbulence intensities prevail. It indicatesthat viscous effects still play an important role inthe turbulence towards these regions, which doesnot allow for a distinct delineation of the viscoussublayer. This indicates also the weak nature ofthe turbulence obtained at this Reynolds number.Conditional-averaged quadrant analysisTo gain a better insight on the near-wall struc-ture of the turbulent boundary layer along thestator side, a conditional-averaged quadrant anal-ysis is performed to identify the contributionsof intense intermittent bursting events (ejectionsand sweeps) on the Reynolds shear stress pro-ducing vortical structures. It corresponds to foursubdivisions of the fluctuation field according tothe combination of the tangential v′θ and axial-1012345-60 -40 -20 0 20 40 60totalQ1Q2Q3Q4(a)-1012345-60 -40 -20 0 20 40 60totalQ1Q2Q3Q4(b)Fig. 3. Variation with −∆r+ of the conditionallyaveraged Reynolds shear stress at z+ = 17 in thevicinity of (a) a strong ejection < v′θv′z | strongejection > / < v′θv′z > and (b) a strong sweep< v′θv′z | strong sweep > / < v′θv′z >.v′z velocity fluctuations [3,4]. Following the def-initions given in [5] in a fixed frame, a strongsweep is associated with −v′θv′z > βv′θ,rmsv′z,rmsand v′z < 0 (quadrant Q4) and a strong ejec-tion with −v′θv′z > βv′θ,rmsv′z,rms and v′z > 0(quadrant Q2). In the first quadrant Q1, v′θ > 0and v′z > 0, while in the third quadrant Q3,v′θ < 0 and v′z < 0. The quadrant analysis isapplied at z+ = 17 corresponding to the lo-cation of the maximum value of the turbulentshear stress. We have also considered the valueβ = 2 to determine strong events, as used in[3,4]. We display in figures 3a and 3b the varia-tions with ∆r+ = r±r+ (r+ the wall coordinatein the radial direction) of the conditionally aver-aged Reynolds shear stress normalized by the un-conditionally ensemble averaged Reynolds shearstress < v′θv′z > near a strong ejection (fig.3a)and a strong sweep (fig.3b). The contributionsof each quadrant are also presented. The figures3a and 3b clearly show that the ejection (Q2)and sweep (Q4) quadrants contribute much moreto the Reynolds shear stress production than thetwo other quadrants. On the other hand, it seemsthat the weakness of the turbulence in the presentsimulation accentuates the features observed inprevious works. The results obtained support theconclusions of Littell and Eaton [3] and Lygrenand Andersson [5], in contrast with the findingsof Kang et al. [4]: the asymmetries observedare dominated by Reynolds stress-producing co-herent structures (sweep and ejection). Lygrenand Andersson [5] concluded that clockwise vor-tices contribute much more to the Reynolds shearstress than counter-clockwise vortices. The samebehavior applies in the presence of a sweep event.In this case, the levels of the surrounding ejec-tions approach the strong sweep level and areeven slightly beyond the fixed criterion conditionβ = 2, as seen in figure 3b, while the levels ofsweeps around a strong ejection are less impor-tant (fig.3a), in agreement with the results of [5].Case 1 vortices, having induced near-wall veloc-ity in the direction of the crossflow, are foundto be the major source of generation of specialstrong events.CONCLUSIONDNS calculations have been used to describe theturbulent boundary layer along the stator sidein a large enclosed rotor-stator cavity. For therotational Reynolds number under considerationRe = 9.5 × 104, the Bo¨dewadt layer is 3D tur-bulent, whereas the Ekman layer on the rotor isstill laminar. The transition to turbulence is asso-ciated with the onset of localized turbulent spotsspiraling up along the stator side. The turbulencekinetic energy budgets have revealed that produc-tion is the major contribution with a maximumobtained at z+ ≃ 12 independently of the ra-dial location, confirming the self-similar behaviorof the Bo¨dewadt layer. The results of the quad-rant analysis support the conclusions proposedby [3,5]. The asymmetries observed by these au-thors have been clearly detected and the analysisof conditionally averaged streamwise and wall-normal velocity components confirms that theseasymmetries mainly arise from the contributionsof quadrants Q2 and Q4, responsible for the gen-eration of ejection and sweep events. Therefore,Case 1 vortices are found to be the major sourceof generation of special strong events.BIBLIOGRAPHY[1] S. Poncet & A. Randriamampianina ” ´Ecoulementturbulent dans une cavite´ rotor-stator ferme´e degrand rapport d’aspect,” C.R. Me´canique, vol.333, pp. 783-788, 2005.[2] J.W. Daily & R.E. Nece ”Chamber dimensioneffects on induced flow and frictional resistanceof enclosed rotating disks,” ASME J. Basic Eng.,vol. 82, pp. 217-232, 1960.[3] H.S. Littell & J.K. Eaton”Turbulence characteristics of the boundary layeron a rotating disk,” J. Fluid. Mech., vol. 266, pp.175-207, 1994.[4] H.S. Kang, H. Choi & J.Y. Yoo ”On themodification of the near-wall coherent structurein a three-dimensional turbulent boundary layeron a free rotating disk,” Phys. Fluids, vol. 10 (9),pp. 2315-2322, 1998.[5] M. Lygren & H.I. Andersson ”Turbulent flowbetween a rotating and a stationary disk,” J. FluidMech., vol. 426, pp. 297-326, 2001.

Turbulence characteristics of the Bödewadt layer in a large shrouded rotor-stator system

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Turbulence characteristics of the B\"{o}dewadt layer in a large enclosed rotor-stator system

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