CHAPTER NUMERICAL SIMULATION METHOD CHAPTER NUMERICAL SIMULATION METHOD 3.1 Introduction To better understand the flow phenomenon in the confined cylindrical container with one rotating end, a numerical simulation code was developed. The equations governing the flow are the axisymmetric Navier-Stokes equations, together with the continuity equation and appropriate boundary and initial conditions. Gelfgat et al. (2001) showed numerically that the onset of unsteadiness of the flow is via a supercritical axisymmetric Hopf bifurcation for H/R in the range of 1.6 to 2.8. Nonlinear computations (Blackburn and Lopez 2000, 2002, Blackburn 2002) have shown that this oscillatory state remains stable to three-dimensional perturbations for Re up to about 3400. That numerical finding is consistent with the experimental observations of Stevens et al. (1999). Thus, the problem under the conditions studied here (for most of the cases, the Reynolds number is less than 3000) can be solved by axisymmetric numerical simulations. The approach adopted follows the method which has been extensively used by Lopez (1990), Stevens et al. (1999), i.e. solving the axissymmetric Navier-Stokes and continuity equations in streamfunction / vorticity / circulation forms using a predictor-corrector finite difference method. A brief description of the numerical scheme is presented in this Chapter. It should be noted that with the exception of the numerical results in Chapters and 6, which were 33 CHAPTER NUMERICAL SIMULATION METHOD performed and provided by Prof J.M. Lopez using a different numerical scheme as part of the collaborative project, all the numerical results reported here are performed by the author using the axisymmetric scheme. The corresponding numerical method used by Prof J.M. Lopez will be introduced in the corresponding chapters. 3.2 Governing Equations and Boundary Conditions For the problem studied here, the system is axisymmetric, and the equations governing the flow are the axisymmetric Navier-Stokes equations, together with the continuity equation and appropriate boundary and initial conditions. A cylindrical container with radius R and height H is completely filled with an incompressible fluid of constant density ρ and kinematic viscosity ν (see Fig, 3.1). A cylindrical polar coordinate system (r, θ, z) is adopted, with the origin at the centre of the rotating end wall and the positive-z axis pointing towards the stationary endwall. R H z Ω θ r Fig. 3.1 Flow configuration in a confined cylinder with one rotating end. 34 CHAPTER NUMERICAL SIMULATION METHOD Two kinds of rotating motions were considered: a constant rotation only and a constant rotation with a superimposed sinusoidal modulation. In all cases, the bottom endwall is impulsively started from rest. For the sinusoidal modulation, the bottom disk rotates at a modulated rate of Ω(1+Asin(Ωft*)), where Ω (rad/s) is the mean constant rotation speed and Ωf (rad/s) is the angular modulation frequency, A is the relative amplitude of the modulation, t* is dimensional time in seconds. The angular modulation frequency can also be written as Ωf = 2πff = 2π/Tf, where ff and Tf are the frequency and period of modulation, respectively. In the present study, the system is non-dimensionalized using R as the length scale, and the dynamic time 1/Ω as the time scale. The flow thus can be specified by the following non-dimensional parameters: Reynolds number Re = ΩR2/ ν, aspect ratio Λ = H/R, forcing amplitude A, and forcing frequency ωf = Ωf /Ω. The velocity vector (u, v, w) in this coordinate system is: (u, v, w) = ⎛⎜ − 1ψ z , Γ, 1ψ r ⎞⎟ , ⎝ r r ⎠ r (1) where ψ is the Stokes streamfunction. Γ is defined as Γ = vr . Subscripts denote partial differentiation with respect to the subscripted variables. The corresponding vorticity field is: (ξ ,η ,ζ ) = ⎛⎜ − Γ z ,− ∇ ∗2ψ , Γ r ⎞⎟ , ⎝ r r r ⎠ (2) where ∇ ∗2 = ( )zz + ( )rr − ( )r r The axisymmetric Navier-Stokes equations, in terms of ψ, Γ, η, are 35 CHAPTER NUMERICAL SIMULATION METHOD DΓ = ∇∗ Γ Re (3) ( ⎛η ⎞ ⎧ ⎛η ⎞ ⎛η ⎞ ⎫ Γ D⎜ ⎟ = ⎨∇ ⎜ ⎟ + ⎜ ⎟ ⎬ + r4 ⎝ r ⎠ Re ⎩ ⎝ r ⎠ r ⎝ r ⎠ r ⎭ ) (4) z ∇ ∗2ψ = −rη where D=( (5) )t − 1ψ z ( )r + 1Ψ r ( )z , r r ∇2 = ( )zz + ( )rr + ( )r r The boundary and axis conditions corresponding to the flow are: (r = 0, ≤ z ≤ H / R ), ψ = Г = η = 0, η=− ∂ 2ψ r ∂r (r = 1, ≤ z ≤ H / R ), ψ = 0, Г = 0, η = − ∂ 2ψ r ∂z (z = H/R, ≤ r ≤ ), ψ = Г = 0, ∂ 2ψ ψ = 0, η = − , and Г = r ∂z (6) rv, (z = 0, ≤ r ≤ ), constant rotation speed r2[1+Asin(ωft)] (z = 0, ≤ r ≤ ), sinusoidal modulation The boundary condition at r = is due to the axial symmetry of the flow, the boundaries at z = H/R and r = are rigid and stationary, while at z = 0, the rigid endwall is in rotation for t > 0. 36 CHAPTER NUMERICAL SIMULATION METHOD 3.3 Method of Solution Equations 3-5 can be rewritten in explicit expressions as: ∂Γ ∂ψ ∂Γ ∂ψ ∂Γ ⎛ ∂ Γ ∂Γ ∂ Γ ⎜ − + = − + ∂t r ∂z ∂r r ∂r ∂z ∂z Re ⎜⎝ ∂r r ∂r ⎞ ⎟⎟ ⎠ ⎛ ∂ 2η ∂η η ∂ 2η ⎞ ∂η ∂ψ ∂η ∂ψ ∂η η ∂ψ ∂ ⎜ ⎟ − + − + − + + Γ2 = r ∂z r ∂z Re ⎜⎝ ∂r r ∂r r ∂z ⎟⎠ ∂t r ∂z ∂r r ∂r ∂z ( ) ⎛ ∂ 2ψ ∂ψ ∂ 2ψ − + ∂z r ⎝ ∂r r ∂r η = − ⎜⎜ ⎞ ⎟⎟ ⎠ (7) (8) (9) Combined with the boundary conditions, a second order central difference scheme can be used to solve the above well-posed equations. Mesh generation Uniform mesh was used for the present study. Assuming N+1 and M+1 are the number of grid points in the r and z direction, respectively, then Δr and Δz can be calculated as: Δr = so, HR and Δz = N +1 M +1 ri = i ∗ Δr , i = 0, 1, ., N + z i = j ∗ Δz , j = 0,1, , M + Solution of vorticity and angular momentum equations Equations 7-8 are discretized at all interior points ≤ i ≤ N, ≤ j ≤ M with the second order finite difference method: 37 CHAPTER dΓ i , j dt + = NUMERICAL SIMULATION METHOD ⎛⎜ ψ i , j +1 − ψ i , j −1 ⎞⎟⎛⎜ Γ i +1, j − Γ i −1, j ⎟⎜ ri ⎜⎝ Δz Δr ⎠⎝ ⎛⎜ Γ i +1, j − 2Γ i , j + Γ i −1, j Re ⎜⎝ Δr ⎞ ⎛ ψ i + , j − ψ i −1 , j ⎟− ⎜ ⎟ ri ⎜ Δr ⎠ ⎝ ⎞ ⎛ Γ i + , j − Γ i −1 , j ⎟− ⎜ ⎟ ri ⎜ Δr ⎠ ⎝ Setting RHS equals to G1 gives dΓ i , j dt ⎞⎛ Γ i +1, j − Γ i −1, j ⎟⎜ ⎟⎜ Δz ⎠⎝ ⎞ ⎛ Γ i , j + − Γ i , j + Γ i , j −1 ⎞ ⎟+⎜ ⎟ ⎟ ⎜ ⎟ Δz ⎠ ⎝ ⎠ ⎞ ⎟ ⎟ ⎠ (10.1) = G1 (10.2) Similarly, for vorticity η equation, we have: dη i , j dt = ⎛⎜ ψ i , j +1 − ψ i , j −1 ⎞⎟⎛⎜ η i +1, j − η i −1, j ⎟⎜ ri ⎜⎝ Δz Δr ⎠⎝ ⎞ ⎛ ψ i + , j − ψ i −1 , j ⎟− ⎜ ⎟ ri ⎜ Δr ⎠ ⎝ ⎞⎛ η i +1, j − η i −1, j ⎟⎜ ⎟⎜ Δz ⎠⎝ ⎞ ⎟ ⎟ ⎠ 2 η i , j ⎛ ψ i , j + − ψ i , j −1 ⎞ ⎛⎜ (Γ i , j +1 ) − (Γ i , j −1 ) ⎞⎟ ⎟⎟ + ⎜ − ⎟ Δz Δz (ri )2 ⎜⎝ ⎠ (ri ) ⎜ ⎝ + ⎛⎜ η i +1, j − 2η i , j + η i −1, j Re ⎜⎝ Δr ⎠ ⎞ ⎛ η i + , j − η i −1 , j ⎟+ ⎜ ⎟ ri ⎜ Δr ⎝ ⎠ Setting RHS equals to G2 gives dη i , j dt = G2 ⎞ η i , j ⎛ η i , j +1 − 2η i , j + η i , j −1 ⎞ ⎟ (11.1) ⎟− +⎜ ⎟ ⎟ (r )2 ⎜ Δ z i ⎝ ⎠ ⎠ (11.2) 38 CHAPTER NUMERICAL SIMULATION METHOD For equations 10 and 11, a second order predictor-corrector scheme was employed: ¾ Predictor step: η i∗, j = η ik, j + 0.5Δt ⋅ G1 k (12.1) Γ i ∗, j = Γ i ,kj + 0.5Δt ⋅ G2 k (12.2) ¾ Corrector step: η ik, +j = η ik, j + t ⋅ G1∗ (12.3) Γ i ,kj+1 = Γ i ,kj + t ⋅ G2 ∗ (12.4) In between predictor step and corrector step, the boundary conditions for vorticity η need to be updated by solving the streamfunction equation, which will be introduced in the following part. After the corrector step, the boundary conditions for vorticity also need to be updated for the next time step calculation. Solution of stream function equation Applying the central difference scheme to discretize the stream function equation gives: ψ i+1, j − 2ψ i, j + ψ i−1, j Δr − r i ψ i+1, j − ψ i−1, j 2Δr + ψ i, j+1 − 2ψ i, j Δz +ψ i, j−1 = −r η i i, j (13.1) This difference equation can be solved by normal Gauss-Seidel (SOR or SLOR) method, but we prefer a direct method here, since the matrix is not too large and the direct method will save much computational time. Since ri = i × Δr , the left two terms can be expressed as: 39 CHAPTER NUMERICAL SIMULATION METHOD ⎤ ⎡⎛ 1⎞ 1⎞ ⎛ ⎢⎜ + 2i ⎟ψ i − 1, j − 2ψ i, j + ⎜ − 2i ⎟ψ i + 1, j ⎥ , written as a matrix form, we have a ⎠ ⎝ ⎠ ⎦ Δr ⎣⎝ triangular matrix: ⎡ ⎢ −2 ⎢ ⎢ 1+ ⎢ ⎢ ⎢ Δr ⎢⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1− −2 0 0 1+ 2N ⎤ ⎥ ⎥ ⎥ ⎡ψ 1, j ⎤ ⎥ ⎢ψ , j ⎥ ⎥ ⎢ ⎥⎥ ⎢ ⎥ ⎥ j =1,…M ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥⎢ ⎥ 1− ψ N , j ⎥⎦ ⎢ ⎣ ⎥ 2N ⎥ −2 ⎥ ⎦ Similarly, for the third term in Equation 13.1, we have another triangular matrix: ⎡ ψ i ,1 ⎤ ⎡− ⎢ψ ⎥ ⎢ ⎢ i ,2 ⎥ ⎢ − ⎥⎢ ⎢ ⎥⎢ ⎢ Δr ⎢ ⎥⎢ ⎥⎢ ⎢ ⎥⎢ ⎢ ⎢⎣ψ i ,M ⎥⎦ ⎣⎢ 0 0 ⎤ ⎥⎥ ⎥ ⎥ i = 1,…N ⎥ 1⎥ ⎥ − ⎦⎥ Hence, the vorticity equation can be written as: ANNΨ NM +Ψ NM BMM = FNM (13.2) where FNM = − riη i , j Setting ANN Z NN = Z NN E NN , where E NN is diagonalized, and its entities are the eigenvalues of ANN . The columns of Z NN are the corresponding eigenvectors. 40 CHAPTER NUMERICAL SIMULATION METHOD Now, let Ψ NM = Z NN V NM (13.3) where V NN is to be determined. Equation 13.2 can be written as: ANN Z NN V NM + Z NN V NM BMM = FNM −1 Multiply the above equation by Z NN , we have −1 E NN V NM + V NM BMM = Z NN FNM Taking transpose of the above equation, we get T T V NM E NN + BMM V NM = H MN ( −1 T Z NN where H MN = FNM ) T . G K T , ei the eigenvalues of ANN , h the columns of H MN , Let v be the vectors of V NM then (BMM G K + ei I MM )vi = hi for i =1,…N (13.4) G Once initial vorticity values are determined, h can be calculated. Then, solving K Equation 13.4 gives v and Equation 13.3 gives the streamfunction. The LAPACK Routine was used in our code for solving Equations 13. Implementation of boundary conditions Since the boundary conditions for the stream function and the angular momentum are Dirichlet type, the boundary value can be used directly. However, the boundary conditions for the vorticity are Neumann type, and they need to be updated; this will be used for the computation of vorticity and stream function at interior points in the next time level. The top, bottom and wall boundary conditions for vorticity are second order 41 CHAPTER NUMERICAL SIMULATION METHOD of derivative of stream function, hence an one-sided second order finite difference scheme was used to approximate this derivative condition: ηi, j = − 8ψ N −1, j − ψ N − , j ri 2Δr ηi, j = − 8ψ i ,1 − ψ i ,2 ri Δz ηi, j = − 8ψ i ,M −1 − ψ i ,M − ri Δz (r = 1, ≤ z ≤ H / R ), (z = 0, ≤ r ≤ ), (14) (z = H/R, ≤ r ≤ ), 3.4 Method Verification The quality of the numerical simulation code was verified by the comparison with numerical simulation results obtained by independent calculations in other studies and with experimental results. Table presents the values and locations of three local maxima and minima of ψ and η at H/R = 2.5 and Re = 2494 with constant rotating speed. The finite differential results by Lopez and Shen (1999) are also included in the table for comparison, and it can be seen that our results show good agreement with theirs. Figure 3.2 shows the effects of grid density on the solution, in terms of a global value–the kinetic energy in the flow domain–Ek, which is defined as: 1 H Ek = ∫0∫0∫0 R (u + v + w2 )rdrdzdθ It can be seen from the figure that the solutions asymptotically approach a fixed value as the grid density is increased. Various tests have shown that the 160 x 400 uniform grid solutions is sufficiently accurate with acceptable computing time. Hence, this 42 CHAPTER NUMERICAL SIMULATION METHOD highly dense grid was used in this study, and the time-step δt = 0.01 was chosen to satisfy both the Courant–Friedrichs–Lewy condition and the diffusion requirement (Lopez 1990), Δt < 1/8 Re Δr2. Table Local maximum and minimum of ψ, η and their locations for Re = 2494, Λ = 2.5 at t = 3000 ψmax ψmin ηmax ηmin N, M (r, z) (r, z) (r, z) (r, z) 7.0776 × 10-5 -7.0723 × 10-3 0.52227 -0.50656 60, 150 (t = 0.025) (0.183, 1.95) (0.767, 0.800) (0.233, 2.033) (0.333, 2.283) 7.2530 × 10-5 -7.0842 × 10-3 0.53006 -0.51090 90, 225 (t = 0.025) (0.178, 1.956) (0.767, 0.800) (0.233, 2.033) (0.333, 2.278) 7.3326 × 10-5 -7.1017 × 10-3 0.53471 -0.51378 120, 300 (t = 0.01) (0.175, 1.958) (0.758, 0.808) (0.233, 2.033) (0.333, 2.275) 7.4323 × 10-5 -7.1161 × 10-3 0.53774 -0.51656 160, 400 (t = 0.01) (0.181, 1.956) (0.763, 0.800) (0.231, 2.038) (0.331, 2.281) 60, 150 (t = 0.05) (Lopez and Shen, 1999) 120,300 (t = 0.01) (Lopez and Shen, 1999) 7.1706 × 10-5 (0.183, 1.95) 7.3988 × 10-5 (0.183, 1.95) -7.0783 × 10-3 (0.767, 0.800) -7.1075 × 10-3 (0.758, 0.825) 0.52433 (0.233, 2.033) 0.53590 (0.233, 2.03) -0.50879 (0.333, 2.28) -0.51547 (0.333, 2.280) Fig. 3.2 Time history of kinetic energy Ek with various grid densities for Λ = 2.5, Re = 2494. 43 CHAPTER NUMERICAL SIMULATION METHOD Steady state Typical simulation results for steady state with the contours of ψ, Γ, and η are shown in Fig. 3.3 for Re = 1918, 1994, 2126, and 2494. These results agree well with the numerical results of Lopez (1990) and the well established experimental results of Escudier (1984). (a) Re = 1918 ψ [-0.0076, 4.3 x 10-6] Γ [0, ] η [-3.61, 18.27] Γ [0, ] η [-3.67, 18.68] (b) Re = 1994 ψ [-0.0076, 5.84 x 10-6] 44 CHAPTER NUMERICAL SIMULATION METHOD (c) Re = 2126 ψ [-0.0076, 2.53 x 10-5] Γ [0, ] η [-3.77, 19.37] Γ [0, ] η [-4.05, 21.18] (d) Re = 2494 ψ [-0.00712, 7.43 x 10-5] Fig. 3.3 Contours of ψ, Γ and η for the axisymmetric steady-state solution at H/R = 2.5 and Reynolds number as indicated; there are 20 positive and negative contour levels determined by c-level (i) = [min/max]x(i/20)3 respectively. Unsteady state As Reynolds number is increased beyond a critical value, the flow becomes a timeperiodic axisymmetric state, which is characterized by a large double vortex breakdown bubble undergoing large amplitude pulsations along the axis (Lopez, 1990). Figure 3.4 shows part of time history of kinetic energy Ek at Λ = 2.5, Re = 2765, from which the period can be determined to be about 36.2. The corresponding instantaneous streamlines over nearly one cycle of the periodic flow is presented in 45 CHAPTER NUMERICAL SIMULATION METHOD Fig. 3.5, with the time indicated in Fig. 3.4 with filled squares. These results agree well with those of Lopez (1990). From the above comparisons, it can be seen that the solutions calculated from this axisymmetric code are in good agreement with other independent numerical results and experimental results, allowing us to have confidence to explore the flow behavior, at least at the conditions of the Reynolds number below 3000, where the flow is still in an axisymmetric state. Note this numerical code is applied in Chapter only, while in other studies, the numerical calculations were performed by Lopez with a more advanced 3-Dimensional calculation. 0.01470 0.01466 e d Ek 0.01462 a 0.01458 b f g h i j c k l 0.01454 0.01450 6200 6220 6240 6260 6280 t Fig. 3.4 Time history of kinetic energy Ek at Λ = 2.5, Re = 2765, showing the timeperiodic flow state. The filled squares and the alphabets correspond to the images in Fig. 3.5. 46 CHAPTER NUMERICAL SIMULATION METHOD (a) t = 6229.78 (b) t = 6232.92 (c) t = 6236.06 (d) t = 6239.20 (e) t = 6242.34 (f) t = 6245.49 (g) t = 6248.63 (h) t = 6251.77 (i) t = 6254.91 47 CHAPTER (j) t = 6258.05 NUMERICAL SIMULATION METHOD (k) t = 6261.19 (l) t = 6264.34 Fig. 3.5 Instantaneous streamline contours of ψ, for the axisymmetric time-periodical solution at Λ = 2.5, Re = 2765; there are 20 positive and negative contours determined by c-level (i) = [min/max] x (i/20)3, with ψ ∈[-0.007, 0.0002]. 48 [...]... -7.07 23 × 10 -3 0.52227 -0.50656 60, 150 (t = 0.025) (0.1 83, 1.95) (0.767, 0.800) (0. 233 , 2. 033 ) (0 .33 3, 2.2 83) 7.2 530 × 10-5 -7.0842 × 10 -3 0. 530 06 -0.51090 90, 225 (t = 0.025) (0.178, 1.956) (0.767, 0.800) (0. 233 , 2. 033 ) (0 .33 3, 2.278) 7 .33 26 × 10-5 -7.1017 × 10 -3 0. 534 71 -0.5 137 8 120, 30 0 (t = 0.01) (0.175, 1.958) (0.758, 0.808) (0. 233 , 2. 033 ) (0 .33 3, 2.275) 7. 432 3 × 10-5 -7.1161 × 10 -3 0. 537 74 -0.51656... 1.956) (0.7 63, 0.800) (0. 231 , 2. 038 ) (0 .33 1, 2.281) 60, 150 (t = 0.05) (Lopez and Shen, 1999) 120 ,30 0 (t = 0.01) (Lopez and Shen, 1999) 7.1706 × 10-5 (0.1 83, 1.95) 7 .39 88 × 10-5 (0.1 83, 1.95) -7.07 83 × 10 -3 (0.767, 0.800) -7.1075 × 10 -3 (0.758, 0.825) 0.52 433 (0. 233 , 2. 033 ) 0. 535 90 (0. 233 , 2. 03) -0.50879 (0 .33 3, 2.28) -0.51547 (0 .33 3, 2.280) Fig 3. 2 Time history of kinetic energy Ek with various grid... respectively Unsteady state As Reynolds number is increased beyond a critical value, the flow becomes a timeperiodic axisymmetric state, which is characterized by a large double vortex breakdown bubble undergoing large amplitude pulsations along the axis (Lopez, 1990) Figure 3. 4 shows part of time history of kinetic energy Ek at Λ = 2.5, Re = 2765, from which the period can be determined to be about 36 .2 The... corresponding instantaneous streamlines over nearly one cycle of the periodic flow is presented in 45 CHAPTER 3 NUMERICAL SIMULATION METHOD Fig 3. 5, with the time indicated in Fig 3. 4 with filled squares These results agree well with those of Lopez (1990) From the above comparisons, it can be seen that the solutions calculated from this axisymmetric code are in good agreement with other independent numerical... 43 CHAPTER 3 NUMERICAL SIMULATION METHOD Steady state Typical simulation results for steady state with the contours of ψ, Γ, and η are shown in Fig 3. 3 for Re = 1918, 1994, 2126, and 2494 These results agree well with the numerical results of Lopez (1990) and the well established experimental results of Escudier (1984) (a) Re = 1918 ψ [-0.0076, 4 .3 x 10-6] Γ [0, 1 ] η [ -3. 61, 18.27] Γ [0, 1 ] η [ -3. 67,... numerical results and experimental results, allowing us to have confidence to explore the flow behavior, at least at the conditions of the Reynolds number below 30 00, where the flow is still in an axisymmetric state Note this numerical code is applied in Chapter 6 only, while in other studies, the numerical calculations were performed by Lopez with a more advanced 3- Dimensional calculation 0.01470 0.01466... 10-6] 44 CHAPTER 3 NUMERICAL SIMULATION METHOD (c) Re = 2126 ψ [-0.0076, 2. 53 x 10-5] Γ [0, 1 ] η [ -3. 77, 19 .37 ] Γ [0, 1 ] η [-4.05, 21.18] (d) Re = 2494 ψ [-0.00712, 7. 43 x 10-5] Fig 3. 3 Contours of ψ, Γ and η for the axisymmetric steady-state solution at H/R = 2.5 and Reynolds number as indicated; there are 20 positive and negative contour levels determined by c-level (i) = [min/max]x(i/20 )3 respectively...CHAPTER 3 NUMERICAL SIMULATION METHOD highly dense grid was used in this study, and the time-step δt = 0.01 was chosen to satisfy both the Courant–Friedrichs–Lewy condition and the diffusion requirement (Lopez 1990), Δt < 1/8 Re Δr2 Table 1 Local maximum and minimum of ψ, η and their locations for Re = 2494, Λ = 2.5 at t = 30 00 ψmax ψmin ηmax ηmin N, M (r, z) (r, z) (r, z)... 0.01462 a 0.01458 b f g h i j c k l 0.01454 0.01450 6200 6220 6240 6260 6280 t Fig 3. 4 Time history of kinetic energy Ek at Λ = 2.5, Re = 2765, showing the timeperiodic flow state The filled squares and the alphabets correspond to the images in Fig 3. 5 46 CHAPTER 3 NUMERICAL SIMULATION METHOD (a) t = 6229.78 (b) t = 6 232 .92 (c) t = 6 236 .06 (d) t = 6 239 .20 (e) t = 6242 .34 (f) t = 6245.49 (g) t = 6248. 63 (h)... 6245.49 (g) t = 6248. 63 (h) t = 6251.77 (i) t = 6254.91 47 CHAPTER 3 (j) t = 6258.05 NUMERICAL SIMULATION METHOD (k) t = 6261.19 (l) t = 6264 .34 Fig 3. 5 Instantaneous streamline contours of ψ, for the axisymmetric time-periodical solution at Λ = 2.5, Re = 2765; there are 20 positive and negative contours determined by c-level (i) = [min/max] x (i/20 )3, with ψ ∈[-0.007, 0.0002] 48 . studies and with experimental results. Table 1 presents the values and locations of three local maxima and minima of ψ and η at H/R = 2.5 and Re = 2494 with constant rotating speed. The finite. cylindrical container with radius R and height H is completely filled with an incompressible fluid of constant density  and kinematic viscosity  (see Fig, 3. 1). A cylindrical polar coordinate. CHAPTER 3 NUMERICAL SIMULATION METHOD 33 CHAPTER 3 NUMERICAL SIMULATION METHOD 3. 1 Introduction To better understand the flow phenomenon in the confined cylindrical container