Numerical Simulation of Flow Past a Rotating Cylinder Wei Zhang *, Rickard Bensow 1 Department of Shipping and Marine Technology, Chalmers University of Technology Gothenburg, Sweden * zhang.wei@chalmers.se ABSTRACT Flow past a rotating cylinder is investigated through primarily three-dimensional numerical simulation for Reynolds number and spin ratio ranging from 0.01 to 800 and from 0 and 15 respectively. It is found that the secondary and third stationary points found in the lift curve in two-dimensional simulation no longer exists in the three dimensional simulation. Centrifugal instabilities play a key role in the flow past the rotating circular cylinder, especially with respect to the maximum lift coefficient. This value of up limit of lift coefficient is close to the one argued by Prandtl, but the principles are totally different. KEY WORDS: rotating cylinder, CFD, centrifugal instability INTRODUCTION Because of the increasing attention to fuel consumption and anticipated legislation, the Flettner rotor, a spinning cylinder mounting on the deck, has attained renewed interest recently as an auxiliary ship propulsion device. In a design study to predict the thrust generated by the rotor, utilizing both numerical [1] and experimental [2] methods, we found some peculiarities in the numerical solution process. This motivated us to study also low Reynolds number flow simulations in order better understand the development of lift on the rotating cylinder. It is found that complex flow phenomenon develops in the spanwise direction, despite its simple geometry. Other applications such, as in flow control, has been presented by Tokumaru and Dimotakis [3][4]. Indeed, Prandtl [5] had studied the flow past a rotating cylinder in 1925 as the Flettner rotor suddenly aroused the popular interests at that time. Considering viscosity effect, he argued that an up limit exists, which was equal to the circulation at which upstream and downstream stagnation points join together. As a result he drew the conclusion that F the maximum lift coefficient, = L 0.5ρU 2 D, where F L is the lift and D is diameter of cylinder, is 4 π. He had found this up limit exists in his experiments as well. Also some other experimental studies [6] [7] [8] performed around the same time achieved similar conclusion. But later it has been reported that this limit can be exceeded [4]. With the purpose to study whether the up limit of lift exists, Glauert [9] proposed an asymptotic solution for the twodimensional Navier-Stokes equation for the flow past the rotating cylinder. By considering the irrotational rotary component of the flow inside the boundary layer, which is not included in Glauert s work, an improved analytical solution for the twodimensional Navier-Stokes equation were recovered by Wang and Joseph [10]. At the same time, Padrino and Joseph [11] used a Finite Volume Method (FVM) based commercial Computational Fluid Dynamic (CFD) software Fluent 6.1 to study the two-dimensional flow past the rotating cylinder. With the same purpose, Mittal and Kumar [12] had studied the flow numerically by utilizing a stabilized Finite Element Method (FEM). All of these theoretical and two-dimensional numerical analysis showed that the Prandtl s limit does not hold. But these are still not sufficient to give the answer to the query why lift coefficients in excess of 4 π had not been observed in the former experiments. In Mittal and Kumar s work, secondary instabilities had been discovered; these were later recovered by Akoury et al. [13] in their three-dimensional numerical simulation. In fact, also Mittal [14] later extended his numerical studies to three-dimensional conditions. Although Prandtl s limit does not hold in these three-dimensional simulations either, it was found that the centrifugal instabilities, instead of the former secondary instabilities, lead to loss of lift. Actually, Prandtl had mentioned the centrifugal effect in his report from 1925, but he didn`t carry out any further studies on the topic, even if Taylor [15] theoretically proved the existence of centrifugal instabilities in the flow between two concentric rotating cylinders. This leads us to hypothesize whether it is the centrifugal instabilities that prevent Prandtl and others to exceed an up limit in their measurement works and also our previous numerical work [1] for high Recondition.
S1 D y Inlet x Outlet L S2 Fig 1. Computational domain and the system of reference for numerical simulation (left). A panoramic (right top) and a close up view (right bottom) of the mesh of 2D fine mesh (it is of the cross section view of corresponding 3D mesh). Table 1. Details of meshes for grid refinement studies of three-dimensional simulation Mesh Nodes Nodes Nodes in radial in circumferential in span wise h 1 r D Elements Fine 126 128 241 5 10-4 3,809,280 Medium 88 89 171 7 10-4 1,301,520 Coarse 63 64 121 1 10-3 461,280 With this objective, we carried out numerical simulations for the flow past rotating cylinder for Re, defined as Re = VD, where V is the free ν stream speed, D is the diameter of the cylinder and ν is the coefficient of kinematic viscosity of the fluid, and spin ratio α, defined as α = 0.5Dω V, where ω is the angular velocity of the cylinder about its own axis, ranging from 0.01 to 800 and from 0 and 15 respectively. As all of these numerical simulations, especially the three-dimensional ones, only focused on some special Re and α, but not the wide range of these parameters, they don`t give a global view for the flow past a rotating cylinder. NUMERICAL CONFIGURATION To study the span wise flow, and especially its effect on the lift, three-dimensional steady incompressible Navier-Stokes equations are solved numerically. For comparison, numerical solutions of two-dimensional steady incompressible Navier- Stokes equations are obtained as well. All the simulations are performed using the finite volume code OpenFOAM 1.7.0, [16], where the SIMPLE algorithm is used for pressure-velocity coupling and a blended scheme between upwind and central differenceing is used to discretize the convective term in the momentum equations. The computation domain (see figure 1) has a radial extension of L = 20D and the span wise extension of H =10D for three-dimensional simulation. Two symmetry boundary conditions ( S 1 and S 2 ) are applied on the outer edges far away from the cylinder, following Padrino and Joseph [11]. A uniform velocity inlet boundary condition is prescribed to the region on the left of the domain and an inletoutlet boundary condition, which switches velocity and pressure between fixed value type and zero gradient type depending on the direction of velocity, is chosen for the right side of the domain as outlet. The cylinder rotates in the counter-clockwise direction and a no-slip boundary condition with rotating speed is imposed at the cylinder surface. As flow in the span wise direction
but not the end wall effect is the major interest in selected for the both sides in span wise direction for present work, the cyclic boundary condition is Table 2. Error and uncertainties estimated from grid refinement studies for, and for cases of Re= 200, α =3, 4, 5 α =3 α =4 α =5 S 1 10.2970 0.4174 0.3487 17.1507 1.0121 0.4570 13.7633 1.5912 0.6988 δ * (%S 1 ) 1.83 11.34 -* 1.49 6.85-2.52 -** 0.00-5.10 U G (%S 1 ) 3.86 20.32 -* 7.61 13.27 5.36 3.81** 0.04 11.07 U I (%S 1 ) 0.04 0.13 0.12 0.44 2.00 0.26 5.18 17.96 1.95 * Grid refinement study gets diverged, so errors and uncertainties cannot be estimated. **Grid refinement study gets oscillatory convergence, and only uncertainty is estimated. Table 3. The comparison of the corrected solution (denoted as 3D) from grid study of three dimensional numerical simulations in this work for Re= 200, α =3.0, α =4.0, α =5.0 with two dimensional solution of this work (2D), J.Wang`s (A1) and Glauert`s (A2) analytical solution, Mittal`s (N1) and J.C.Padrino`s (N2) numerical solution α =3.0 α =4.0 α =5.0 3D 10.4851 0.3700 0.3487* 17.4054 0.9428 0.4453 13.7633** 1.5912 0.7345 2D 10.3014 0.2084 0.1778 17.4010 0.5042 0.2313 18.7508 0.6027 0.2854 A1 - - - 18.636 0.942 0.476 27.029 0.922 0.507 A2 12.4216 0.6092 0.2416 20.897 0.795 0.215 26.752 0.929 0.195 N1 10.3660 0.0350-17.5980-0.0550-27.0550 0.168 - N2 10.3400 0.0123-17.5820-0.1240 0.453 27.0287 0.0107 0.514 As grid refinement studies get divergence (*) and oscillatory convergence (**), no corrected solution can be given. So the solution of finest mesh is given instead. three-dimensional simulations. The initial condition for all the computations is assigned the value that corresponds to the potential flow past a stationary cylinder based on Mittal and Kumar [12]. COMPUTATIONAL GRID AND GRID REFINEMENT STUDIES Fully quadrilateral (for 2D) and hexahedral (for 3D) O-type grid, similar to Padrino and Joseph [11], are used for all simulations in the present work. The mesh elements are evenly distributed along the span wise direction for the three-dimensional simulations. Numerical errors δ * and uncertainties U are estimated firstly with cases Re= 200, α =3.0, α =4.0, α =5.0 by the procedure introduced by Stern [17]; while monotonic convergence is achieved, a newly developed factor of safety method by Xing and Stern [18] is used to estimate the numerical uncertainty U. To achieve this, a group of meshes consisting fine, medium and coarse grids with systematical refine rate r = 2 are created (See Table 1). A panoramic and close up view of the fine mesh are shown also in figure 1. All other cases discussed later are based on the fine mesh. The solutions of the fine mesh S1, iteration uncertainties U I and its numerical errors δ * and uncertainties U G from spatial discrete of grid for cases Re= 200, α =3, 4, 5 are shown in table 2. Only the momentum coefficient for case α =3 failed to converge when refining the grid, but the reason for that lies beyond the scope of this work; present work is mainly focused on the lift coefficient. Both uncertainties and error of lift coefficient will increase with the spin ratio α, and it even becomes difficult to get convergence while refining the mesh. The situation is totally different for the drag coefficient. This could be explained as the rotation of the cylinder will suppress the separation, which has been reported earlier, e.g. [10] [11] [12], which makes the prediction of drag easier, while the increasing effect of the instability in the span wise direction makes it difficult to predict the lift. This will be discussed in the next section. Furthermore, we compare the corrected solution from current mesh refinement study with twodimensional numerical solutions, analytical solutions, and also the two dimensional numerical solutions in the literatures in table 3. The discrepancies of the value of and among the methods are large, although for case α =4, those from present three-dimensional numerical simulation are agreeing well with Wang s analytical solution [10]. It was also concluded by Padrino and Joseph [11], starting from Mittal and Kumar s [12] stability analysis, that existence of multiple solutions is reasonable. They believed that the slightly disturbed pressure distribution would affect those results. On the other
hand, lift coefficient are agree well with solution of others, as the difference are within the theoretical Cl 80 70 60 50 40 Re0.01 Re0.1 Re1 Re10 Re20 Re40 Re100 Re200 Re400 Re500 Re700 Re800 Mittal Re200 Glauert analytical Cl 80 70 60 50 40 order of accuracy of numerical strategy, except for Re0.01 Re0.1 Re1 Re10 Re20 Re40 Re100 Re200 Re400 Re500 Re700 Re800 Mittal Re200 Glauert analytical 30 30 20 20 10 10 0 0 2 4 6 8 10 12 14 16 α 0 0 2 4 6 8 10 12 14 16 Fig 2. Flow past a rotating cylinder: variation of lift coefficient, results from finest mesh, for different Re and rotation rate α. The left is from present two-dimensional simulation, and the right is from present threedimensional simulation. α α =3 α =4 α =5
Fig 3. Re= 200, α =3,4,5 flow past a rotating cylinder: stream line. The first row is from J.C.Padrino [11], the second is from present two-dimensional computation and the third is from present three-dimensional computation at h = H 2. Fig 4. Re= 200, α =5 flow past a rotating cylinder: iso-surface of the span wise component of vorticity (=0.4). The left is from Mittal [14] and the right is of the present three-dimensional computation. Fig 5. Re= 200, α =3.0, α =4.0, α =5.0 flow past a rotating cylinder: distribution of pressure p on the cylinder. Fig 6. Re=1, 40, 200, α =8.0 flow past a rotating cylinder: distribution of normalized pressure %p = cylinder. p max(p) on the
the case α = 5, and especially for the 3D solution. If we consider the flow structures in the span wise direction discussed in the next section, this difference could also be reasonable from a theoretical point of view. It is however still difficult to draw a conclusion on which approach is most accurate. RESULTS AND DISCUSSIONS The comparison is made for the solution of lift coefficient of all Re and spin ratio α studied in the present work between the two-dimensional and three-dimensional numerical simulation, all data plotted in figure 2. For the two-dimensional numerical simulations, these curves can be divided in to three groups based on the shape, that is α <1, α =1 and α >1 respectively. For α <1, will linearly increase with α and while the case α =1 exhibits close to quadratic increase. Most of the curves in the group α >1 have three distinctive stationary points. This could be explained as the secondary instabilities mentioned by Akoury et al [13]. However, Akoury et al. used unsteady simulation, but we here seem to be captured the effect in steady simulations. We also note that before the first stagnation point is reached, the rate of increase of lift with spin ratio tends towards the analytical solution of Glauert as Re increases. For the three-dimensional simulations, these curves can also be divided in to three groups in the same way as the two-dimensional simulation. For case α <1, the increasing rate of two spin ratios in the current study are almost the same, and it is about half of the Glauert analytical solution and that of potential theory. The most important discovery here is the secondary and third stationary points no longer exist when α >1. Instead, the curve of lift coefficient is flattened at a certain α for different Re. With the increase of Re, the flattened values decrease, finally a maximum value of lift coefficient is achieved for cases of Re >500. This maximum value of lift coefficient is around 15, which of same order of magnitude to that in the Prandtl s argument. Although it is slightly larger than the value of the Prandtl, the phenomenon that lift coefficient is limited by a maximum value is recovered here in the numerical simulation As discussed before in the grid refinement studies, the case Re= 200, α =5 behaves totally different for the three-dimensional and the two-dimensional simulation. From figure 2, case α =5 is a stationary point of curve Re= 200 in the two-dimensional simulation, and it is flattened in the threedimensional simulation. So we carefully study the flow structure here with this case. First of all, we compared the streamline for cases Re= 200, α =3,4, 5 as shown in figure 3. For the lower two spin ratios, the three-dimensional and the two-dimensional flow are quite similar, also in comparison with results of Padrino [11]. However, in the three-dimensional simulation at α =5 only, a complex structure appears on one side of the cylinder, where the rotating speed has an opposite direction component of inlet velocity. Secondly, from the vortex point of view, three-dimensional centrifugal instabilities in the span wise direction, which is mentioned by Mittal 14], is present as shown in figure 4 and according to Mittal [14], this will lead to loss of lift. Lastly,a strong pressure wave is detected along the span of the cylinder surface in this case, shown in figure 5. The pressure distribution around the surface of the cylinder generates the lift, thus the forming of this pressure wave could be used to explain the loss of lift. The forming of the pressure wave is due to the centrifugal instability. This flow structure clearly plays a key role in the flow past the rotating cylinder, especially with respect to the maximum lift coefficient. So although we get a maximum value of lift coefficient, which is close to that in the Prandtls argument, the principles are totally different. Indeed this could be also experienced when one fixes α and varies Re as shown in figure 6. Considering the centrifugal instability effect, the magnitude of lift coefficient of present threedimensional simulation is more reasonable. CONCLUSION Flow past a rotating cylinder is studied via numerical simulation for Reynolds number Re and spin ratio α ranging from 0.01 to 800 and from 0 and 15 respectively. An open source finite volume method based software OpenFOAM is used to solve steady-state incompressible Navier-Stokes equations. It is found that compared with two-dimensional simulation the secondary and third stationary points in the lift curve no longer exists in the three dimensional simulation. The centrifugal instabilities in the span wise direction along the cylinder play an important role in the generation of the lift. This effect contributes to the loss of the lift. As a result, the lift will stop increasing with the spin ratio α at certain values for different Re, and the phenomenon that lift coefficient is limited by a maximum value is recovered here in the numerical simulation. Although this maximum value of lift coefficient is
close to that in the Prandtl s argument, the principles are totally different. As a conclusion the centrifugal instabilities could be a potential explanation for the query why lift coefficients in excess of a maximum value have not been observed in the former experiments and also the author`s previous numerical simulation. ACKNOWLEDGEMENTS The authors want to thank for the discussion with L.Zou, Dr. V.Chernoray and Dr. M.Golubev from of Chalmers University of Technology for discussion. This work is funded by Chalmers strategic funding in Area of Advance Transport. The computations were performed on resources provided by Swedish National Infrastructure for Computing (SNIC) at C3SE. REFERENCES [1] Zhang, W and Bensow, R. 14 th Numerical Towing Tank Symposium[C], Poole, UK, 2011. [2] Chernoray, V., Golubev, M., Bensow, R. and Zhang, W.. The 9 th Eurpean Fluid Mechanics Conference [C], Rome, Italy, 2012. [3] Tokumaru, P.T. and Dimotakis, P.E.. Rotary oscillation control of cylinder wake [J], J. Fluid Mech., 1991, 224, 77-90. [4] Tokumaru, P.T. and Dimotakis, P.E.. The lift of a cylinder executing rotary motion in a uniform flow [J], J. Fluid Mech., 1991, 255, 1-10. [5] Prandtl, L.. Application of the Magnus effect to the wind propulsion of ships [J], Die Naturwissenschaft, 1925, 13, 93-108. [6] Bets, A.. Der Magus-Effekt, Die Grundlage der Flettner-walze [J], Zeitschrift des vereines deutscher Ingeneure, 1925, 69, 11. [7] Redi, E.G.. Tests of rotating cylinder[j], Tech. Notes Nat. Adv. Comm. Aero., Wash., 1924, 209, 17-20. [8] Thom, A.. Effects of discs on the air forces on a rotating cylinder [J], Aero.Res. Counc. R&M, 1934, 1623. [9] Glauert, M.B.. The flow past a rapidly rotating circular cylinder [J], Proc. R. Soc. Lond. A., 1957, 242, 1228, 108-115. [10] Wang, J. and Joseph, D.D.. Boundarylayer analysis for effects of viscosity on the irrotational flow induced by a rapidly rotating cylinder in a uniform stream [J], J. Fluid Mech., 2006, 557, 145-165. [11] Padrino, J..C and Joseph, D.D.. Numerical study of the steady state uniform flow past a rotating cylinder [J], J. Fluid Mech., 2006, 557, 191-233. [12] Mittal, S. and Kumar, B.. Flow past a rotating cylinder [J], J. Fluid Mech., 2003, 476, 303-334. [13] Akoury, R.. The three-dimensional transition in the flow around a rotating cylinder [J], J. Fluid Mech., 2008, 607, 1-11. [14] Mittal, S.. Three-dimensional instabilities in flow past a rotating cylinder [J], J. Appl. Mech., 2004, 71, 89-334. [15] Taylor, G.I.. Stability of viscous liquid contained between two rotating cylinder [J], Proc. R. Soc. Lond. A., 1923, 223, 289-343. [16] www.openfoam.com [17] Stern, F., Wilson, R. V., Coleman, H.W., and Paterson,.E.G., Comprehensive approach to verification and validation of CFD simulations Part 1: Methodolody and Procedures [J], ASME J. Fluids Eng., 2001, 123,4, 793-802. [18] Xing, T. and Stern, F., Factors of Safety for Rickardson Extrapolation [J], ASME J. Fluids Eng., 2010, 132,6, 061403.