Statistical Analysis of the Effect of Small Fluctuations on Final Modes Found in Flows between Rotating Cylinders
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1 Statistical Analysis of the Effect of Small Fluctuations on Final Modes Found in Flows between Rotating Cylinders Toshiki Morita 1, Takashi Watanabe 2 and Yorinobu Toya 3 1. Graduate School of Information Science, Nagoya University, Nagoya, Japan 2. EcoTopia Science Institute, Nagoya University, Nagoya, Japan 3. Nagano National College of Technology, Nagano, Japan Abstract: We performed numerical computations of Taylor vortex flow at small aspect ratio. When the flow develops from its initial state at rest, non-unique flow patterns appear, which depend on the lengths of the cylinders and/or the final Reynolds number. In experiment, it is difficult to keep constant conditions because of the imperfection of the apparatus and the small disturbances of the flow field. Therefore, the experimental result may show its inherit variations. In this study, we focus our attention on the small fluctuations of flows. Two-dimensional numerical simulation is used to estimate the effect on the final modes of Taylor vortex flow. Statistical analysis of the results presented a favorable agreement with our prescribed experimental result. Keywords: Bifurcation, Mode Formation, Taylor Vortex Flow, Computational Fluid Dynamics 1. INTRODUCTION The study of the transition of the flow is important for understanding of the non-linear dynamics. Taylor vortex flow between a rotating inner cylinder and a fixed outer cylinder is one of the models which show non-uniqueness characteristic in the non-linear dynamics. As for Taylor vortex flow, it is experimentally clear that there are some stable states [1], [2]. When a flow develops from an initial state of rest, a non-uniqueness of patterns of various flows appears, which depends on the length of the cylinders and final Reynolds number. In the physical experiment, it is difficult to perform experiments with fixed conditions because of the incompleteness of the apparatus and minute fluctuations of the flow field. In fact, the experimental results showed that formation processes and final modes varied statistically [3], [4]. Numerical studies that focus on minute fluctuations of the flow have been carried out. Shin and Reed [5] used random initial values to investigate decaying properties of a nonlinear unsteady Burgers equation. In the case of Taylor vortex flow, Hirshfeld and Rapaport [6] presented that initial random velocities affected the growth of Taylor vortices. Rigopoulos et. al. [7] investigated new mode vortices caused by the Eckhaus instability. In their study, circular Couette flows with random perturbations were used as initial conditions. Bilson and Bremhorst [8] presented that final modes of turbulent Taylor-Couette flows were affected by the acceleration ratio and the velocity fluctuation field. However, these initial random values do not satisfy physical conditions. This study pays its attention to the minute fluctuations of the fluid. We use the initial velocity field that have minute fluctuations and meet the physical conditions, and investigate the influence on the final modes of Taylor vortex flow by two-dimensional numerical simulation. We compare final modes of the conventional experimental results with the calculation results. 2. COMPUTATIONAL METHOD 2.1. Dominant equations and formulation In this study, we use a model to show in Fig. 1. The dynamic parameters in the numerical analysis are Reynolds number Re and the acceleration of the inner cylinder, and the geometric parameters are the aspect ratio Γ and the radius ratio of the inner cylinder and the outer cylinder η. The Reynolds number is defined as Re=ωR 1 d/ν by the angular velocity ω of the inner cylinder, the gap d of the inner and outer cylinder, the radius R 1 of the inner cylinder and the kinematic viscosity ν of the working fluid. The aspect ratio Γ is given as Γ =L/d by the height of the working fluid L. The radius ratio η is set to be It is assumed that the inner cylinder turns and outer cylinder is fixed. The aspect ratio is 4.0, and the upper and lower ends of the cylinders are fixed. The Reynolds numbers are set from 1000 to 6000 with an interval of Table 1 shows the accelerations of the inner cylinder. The final modes are computed in thirty-six combinations of conditions in total. The basic equations are the Navier-Stokes equations in the cylindrical coordinate system (r,θ,z) and the equation of continuity : u + t 1 Re ( uu) = p + u, u = 0. We use the following function ψ for visualizing: 1 ψ 1 ψ u =, w =. r z r r This function is similar to the stream function of Stokes in the axisymmetry flow. We apply the no-slip condition at the upper and lower walls and the inner and outer walls. The 1101
2 calculation conforms to MAC method. The calculation grid is staggerd grid. The Poisson equation of the pressure is solved in SOR. The time development of the velocity use the fractional step method Definition of the modes The modes of the Taylor vortex flow are decided by the direction of the flow near the end walls, and they are a normal mode and an anomalous mode. The normal mode has a structure where flows through the outer cylinder from the inner cylinder on both end walls in which fluid flows from the inner cylinder to the outer cylinder. The anomalous mode has a structure in which fluid flows from the inner cylinder to the outer cylinder on both end walls or one end wall. The symbol Nn represents normal n cell mode and An is anomalous n cell mode, where integer n stands for the number of cells Generation of initial velocity field First, we rotate the inner cylinder and generate the normal 4-cell mode or anomalous 4-cell mode. Then we completely stop the inner cylinder and take out the flow at the time when the deviations of disturbances relative to the original peripheral velocity of the inner cylinder are 10-5 ~ 10-8 during deceleration. Figure 2 shows contours of the stream function at the initial flow field. The left sides of the contours are the inner cylinders and the right sides are outer cylinders. The starting points of the allows are the contours before deceleration and the ending points of the allows are the contours after deceleration. Fig. 2 shows that original and do not seen. We use each initial flow field and compute well-developed flow and get final modes. Table 1. Definition of the symbol of the angular acceleration Re Symbol Re = 1000 ~ 6000 Angular acceleration of Acceleration AC1 15 rad/s 2 AC2 30 rad/s 2 AC3 45 rad/s 2 AC4 60 rad/s 2 AC5 75 rad/s 2 AC6 90 rad/s 2 Fig.2 Initial flow field at and. Fig.3 Bifurcation with decrease of the Reynolds number (Γ=4.0). Al3 Au3 Fig.1 Cylindrical coordinate system Al5 Au5 Fig.4 Final modes of Taylor vortex flow at Γ=
3 (a) Numerical result (decelerated n4). (a) Numerical result (decelerated n4). Fig.6 The probability of each modes at Γ=4.0, Re=1000. Fig.8 The probability of each modes at Γ=4.0, Re=3000. (a) Numerical result (decelerated ). (a) Numerical result (decelerated ). Fig.7 The probability of each modes at Γ=4.0, Re=2000. Fig.9 The probability of each modes at Γ=4.0, Re=
4 (a) Numerical result (Initial n4). Fig.10 The probability of each modes at Γ=4.0, Re=5000. (a) Numerical result (Initial n4). 3. RESULTS 3.1. Numerical results of flow mode Figure 3 shows the experimental results of the bifurcation of the Taylor vortex flow of experiment at Γ=4.0. The arrow shows the bifurcated direction when the Reynolds number decreases slowly. Six kinds of vortex structures appeared at Γ=4.0:,,,, and. Figure 4 shows the numerical results about contours of the stream function at Γ=4.0. In this figure, the arrows denote the directions of the flow. Figure 4 includes whole modes found in Fig. 3. The minute disturbances of initial velocity field reproduce experimental modes The probability of each modes Figures from Fig. 6 (a) to Fig. 11 (a) show the probability of each mode obtained from numerical computations at Γ = 4.0, Re= Vertical axis shows the probability of each mode in percentage and horizontal axis shows the acceleration of the inner cylinder. The initial flow fields are made by deceleration the flow of. In case the initial flow fields are generated from the flow of, no significant differences is found. Figures from Fig.6 (b) to Fig.11 (b) show the results that are same conditions as those of experiments. The mod of and are classified into to patterns according whether anomalous cell is on upper or lower end walls. However, the figures from Fig.6 (a) to Fig.11 (a) don t classify the modes to adapt the experiment. Figure 6 shows only is found in the results of experiment at Re=2000, but, and are found in those of numerical computation. At Re=2000 in Fig.7, the results of numerical computation generate more probability of than those of experiment. At Re=3000 in Fig. 8, both of the results generate high probability of. Figure 9 shows that the probability of increases. When the Reynolds number is 5000 in Fig. 10,, and are often generated and other modes are hardly gotten. Figure 11 shows, and are often generated in the results of calculation and experiment. In all Reynolds numbers, the final modes appear in the experiment are gotten by the present calculation. 4. CONCLUSIONS We give some pseudo-disturbances to the initial velocity field and find influences on the final modes of the Taylor vortex flow by the two-dimensional numerical computation. We get all final modes shown in bifurcation diagram by the minute difference of the initial flow field in the numerical computation. We reproduce the prior experiment by the present calculation. The calculation results accord with the experiment results qualitatively because the modes appeared in the experiment are made by the numerical computation. Fig.11 The probability of each modes at Γ=4.0, Re=
5 REFERNCE 1. I. Nakamura et al., Japan Society of Mechanical Engineers, , B(1990), p Y. Toya et al., Japan Society of Mechanical Engineers, , B(1990), p I. Nakamura et al., Japan Society of Mechanical Engineers, , B(1994), p J. E. Burkhalter and E. L. Koschmieder, Phys. Fluids, vol. 17 (1974), p Y-C Shin and X B Reed, Jr., Phys. Fluids, vol. 28 (1985), p D. Hirshfeld and D. C. Rapaport, Phys. Review E, vol. 61 (1999), p J. Rigopoulos, J. Sheridan and M. C. Thompson, J. Fluid Mech, vol. 489 (2003), p M. Bilson and K. Bremhorst, J. Fluid Mech., vol. 579 (2006), p
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