11th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11) Southampton, UK, July 30 to August 2, 2019 LARGE-EDDY SIMULATION OF TAYLOR-COUETTE FLOW AT RELATIVELY HIGH REYNOLDS NUMBERS D. I. Pullin Graduate Aerospace Laboratories. California Institute of Technology Pasadena CA 91125, USA [email protected] Wan Cheng and Ravi Samtaney Mechanical Engineering, Physical Science and Engineering Division King Abdullah University of Science and Technology Thuwal, Saudi Arabia [email protected],[email protected] ABSTRACT Rei Nq Nr Ny Ta We discuss large-eddy simulations (LES) of the in- 5 10 compressible Navier-Stokes equations for Taylor-Couette 1 × 10 256 256 512 1:34 × 10 flow. The ratio of the two co-axial cylinder diameters is 3 × 105 512 512 768 1:21 × 1011 h = ri=ro = 0:909 with ri the inner cylinder radius and ro the outer radius. The outer cylinder is stationary while 6 × 105 512 512 768 4:83 × 1011 the inner cylinder rotates with constant angular velocity w . i 6 12 Subgrid stresses are represented using the stretched-vortex 1 × 10 1024 1024 1024 1:34 × 10 model (SVM) where the subgrid motion is modeled by sub- grid vortices undergoing stretching by the local resolved- Table 1. LES flows at varying Rei. For all cases, the do- scale velocity- gradient field. We report wall-resolved LES main size is a sector of p=10 in the azimuthal q direction at Re = dr = up to Re = 106 with the kinematic vis- i iwi n i n and is 2pd=3 in the spanwise y direction. cosity of the Newtonian fluid and d = ro − ri the cylinder gap. The present study focuses on the wall-turbulence be- havior at relatively high Rei. Comparisons are made with small azimuthal and spanwise sector of the two concentric direct numerical simulations (DNS) and with experimental cylinders containing two Taylor-rolls have been reported by results. Ostilla-Monico´ et al. (2016) with Ret ≈ 4200. Some turbu- lence features of TC flow at high Ret are similar to PC flow, and also to other canonical flows such as turbulent channel 1 INTRODUCTION flow. Presently we investigate Taylor-Couette flow at rela- Taylor-Couette (TC) flow of a viscous fluid in the an- tively large Reynolds numbers using the numerical tech- nular gap between two concentric cylinders, where one or nique of large eddy simulation (LES). Our aim in part is to both cylinders are rotating, is a classical turbulent flow that provide data at larger Rei than available from present DNS exhibits interesting shear-flow phenomena (Taylor, 1923; as a prelude to wall-modeled LES at even larger Ret . Grossmann et al., 2016). TC flow is rather more experimen- tally accessible than the related plane-Couette (PC) flow owing to the cylindrical geometry and the convenience of 2 Numerical method and physical models torque measurement. TC flow can be generated in the labo- ratory at relatively large Reynolds numbers and over a range 2.1 Numerical method of parameters that include h, Reynolds number and the ro- The governing equations for LES of incompressible tational speeds of both cylinders. viscous flow are derived by formally applying a spatial filter In PC flow the existence of extremely large struc- onto the Navier-Stokes equation. These are tures presents challenges for Direct numerical simulation (DNS). The DNS studies of Pirozzoli et al. (2014) achieved 2 ¶uei ¶ueiuej ¶ p˜ ¶ uei ¶Ti j ¶uei Re ≈ 1000 on a commonly accepted domain of (9p;1;4p) + = − + n 2 − ; = 0; (1) t ¶t ¶x j ¶xi ¶x ¶x j ¶xi in the unit of d which is the gap between two plates. For TC j flow, if Rei is sufficiently large, the wavelength of the Taylor rolls does not significantly influence the turbulent statistics where xi, i = 1;2;3 are Cartesian coordinates with uei the (Ostilla-Monico´ et al., 2015). DNS of TC flow utilizing a corresponding filtered velocity and pe the filtered pressure. 1 11th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11) Southampton, UK, July 30 to August 2, 2019 5 Figure 1. Visualization of an instantaneous flow field for Rei = 10 . Left: streamlines of the azimuth-averaged flow field (ur;uy); middle: instantaneous azimuthal velocity field at mid-span plane; right: instantaneous spanwise velocity field at mid- span plane. Ti j = ugiu j − ueiuej denotes the effect of unresolved scales “wall-resolved” meaning that the wall-normal grid size at on the resolved-scale motion, which is represented using a the wall is of order the local viscous wall scale ut =n where p subgrid-scale (SGS) model. We will use (x;y;z) as Carte- ut ≡ jtwj=r is the friction velocity with jtwj the magni- sian coordinates with (u;v;w) as the corresponding filtered tude of the wall shear stress and r the constant fluid density. velocity components. Two other coordinate systems are also utilized; the first is cylindrical coordinates (q;y;r) with 2.3 Cases implemented velocity components (uq ;uy;ur) which is useful for analyz- ing results while the second is general curvilinear coordi- Taylor-Couette flow is characterized by two concentric nates (x;y;h) used for the implementation of the numerical cylinders. Most generally, the inner cylinder has radius ri method. and rotates with angular velocity wi, while the outer cylin- The governing LES equations are discretized on der of radius ro rotates with angular velocity wo. For the a body-fitted computational domain (x;y;h) which is present study the outer cylinder is stationary; wo = 0. Two mapped from the physical domain in (x;y;z) co-ordinates. typical dimensionless parameters for TC flow with an sta- A fully staggered strategy is employed for velocity com- tionary outer cylinder are the radius ratio h = ri=ro and the ponents, with both physical velocity components u, w inner Reynolds number Rei = driwi=n. Here d = ro − ri is and their contravariant components uq , ur computed on the gap between two cylinders, and n is the kinetic viscosity x and h faces. Spatial discretization employs a fourth- of the fluid. We use fixed h = ri=ro = 0:909 and vary only order-accurate, central-difference scheme for all terms, ex- Rei. cept the convective term which uses a fourth-order energy- In LES implementation in the sense of cylindrical conservative scheme for the skew-symmetric form (Morin- (r;q;y) co-ordinates, the computational domain is a sec- ishi et al., 1998). For time integration, we utilize a tor of angle p=10 in the q-direction. The spanwise domain fully implicit scheme for viscosity terms and an Adam- length is Ly = 2pd=3. Periodic boundary conditions are im- Bashforth method for the convective term, combined with plemented in both q and y. Grid spacing is uniform in both the fractional-step method. The modified Helmholtz equa- the q and spanwise y-directions but is stretched in the r di- tion for velocity and the Poisson equation for pressure are rection. solved with a multigrid method with line-relaxed Gauss- The present study focuses on the wall turbulence be- Seidel smoothers. havior at relatively high Rei. For numerical verification 5 5 we utilize DNS Rei at 10 and 3 × 10 (Ostilla-Monico´ 5 6 et al., 2016). Higher Rei at 6 × 10 and 10 are also pre- 2.2 Physical model sented. For the four cases implemented, we list the mesh For LES we employ the stretched-vortex SGS model size (Nq ;Nr;Ny) in table 1. The Taylor number, which for (SVM) (Misra & Pullin, 1997; Voelkl et al., 2000; Chung & w0 = 0 is Pullin, 2009), where the subgrid flow is modeled by spiral vortices (Lundgren, 1982) stretched by the eddies compris- (1 + h)6 ing the local resolved-scale flow. Generally, the SGS terms Ta = Re2 ≈ 1:34Re2; (2) 64h6 i i are computed on an h face, which minimizes the require- ment of ghost values for implementation of wall bound- ary conditions. A consistent fourth-order central differ- is also listed in Table 1. ence scheme is used to derive SGS terms that incorporate For defining averages the flow is assumed to be sta- ghost points, via computing an additional set of SGS terms tistically stationary in time and spatially homogeneous in at the first above-wall center point. The present LES is the q direction only. The flow is non-homogenous in the 2 11th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11) Southampton, UK, July 30 to August 2, 2019 25 DNS 20 LES U+ 15 10 5 5 Rei = 10 100 101 102 103 r+ 5 5 Figure 2. Comparison of TC flow at Rei = 10 . Top; Figure 3. Comparison of TC flow at Rei = 3 × 10 . Top; mean flow velocity profiles U+; Bottom; turbulent inten- mean flow velocity profiles U+; Bottom; turbulent inten- 0 0 + 0 0 + 0 0 + 0 0 + 0 0 + 0 0 + sities (uq uq ) ;(uyuy) ;(urur) square symbols: DNS by sities (uq uq ) ;(uyuy) ;(urur) square symbols: DNS by Ostilla-Monico´ et al. (2016). Solid lines with filled squares: Ostilla-Monico´ et al. (2016). Solid lines with filled squares: present LES. present LES. spanwise direction owing the Taylor-roll structure. In what cylinder. The left sub-panel in figure 1 shows streamlines of follows “:: ˆ” will denote an average of a space-time de- the streamwise-averaged, instantaneous flow field. One pair pendent quantity in both time and the azimuthal (q) di- of Taylor rolls is observed. The center and right-hand panels rection, while “¯:: ” denotes an additional spanwise aver- show color coded images of the instantaneous stream-wise age.