Rotating-surface-driven non-newtonian flow in a cylindrical enclosure

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1 Korea-Australia Rheology Journal Vol. 22, No. 4, December 2010 pp Rotating-surface-driven non-newtonian flow in a cylindrical enclosure Yalin Kaptan 1, *, Ali Ecder 2 and Kunt Atalik 2 1 Hansung University, Department of Mechanical Systems Engineering, Seoul, Korea 2 Bogazici University, Department of Mechanical Engineering, Istanbul, Turkey (Received April 30, 2010; final revision received July 26, 2010; accepted July 29, 2010) Abstract The numerical simulations of non-newtonian flows become difficult as the Weissenberg number increases. The main objective of this study is to generate a robust and efficient matrix-free Inexact Newton-Krylov solver (IN-GMRES) which can deal with the high nonlinearities arising from the increased Weissenberg numbers. In order to achieve this aim, non-newtonian flows of rotating surface driven cylindrical enclosure problem is numerically investigated by using three differential viscoelastic constitutive relations namely Upper Convected Maxwell (UCM), Oldroyd B and Giesekus. The results obtained by IN-GMRES solver are validated and compared with the POLYFLOW simulations. Additionally, the selection of the constitutive relation, effects of the Weissenberg number and effects of the Reynolds number are studied. The simulations indicate that the generated algorithm is capable of solving higher Weissenberg number problems (up to the Elasticity number limit of 130) when compared to the previous studies. Furthermore, it is shown that with the increasing Weissenberg number, the reversed flow can be observed in the flow domain and in some cases, depending on the Reynolds number, re-formation of the Newtonian like flow is possible at high Weissenberg numbers. Keywords : inexact Newton, GMRES, Giesekus, Oldroyd B, Upper Convected Maxwell 1. Introduction The investigation of the vortex flow in moving edge problems has been popular because of their physical applications such as lubrication, ocean circulation, turbo machinery, electronic cooling, rheology and coating. One of the most investigated types of such flows is rotating disc in a cylindrical enclosure problem, which continues to be an immensely-studied subject by many scientists. This problem is one of the main geometries of rheometry applications and it is modeled by Pao (1970, 1972) for Newtonian fluids. In the cited study, one of the plates (bottom or top) is held stationary while the other plate and the casing are rotated. The solutions for small Reynolds numbers are obtained for fixed aspect ratio (δ=1). Likewise, similar problems are considered by Lopez (1996) and Saci et al. (1998) and results for high Reynolds numbers and various aspect ratios for Newtonian fluids are achieved. The non-newtonian case of this problem is investigated by Moroi et al. (2001) both numerically and experimentally. Upper Convected Maxwell, Giesekus, Phan Thien-Tanner and power law models are used for the numerical part *Corresponding author: yalinkaptan@gmail.com 2010 by The Korean Society of Rheology where Particle Tracking Velocimetry (PTV) technique is adapted for the experimental part. The elasticity number limit for the numerical part of the cited study is Recently, Itoh et al. (2006) worked on the same problem by adopting a similar approach to that of Moroi et al. (2001). In the cited study, the solutions for variable aspect ratios and Weissenberg numbers are obtained by using Giesekus and Power Law constitutive models whereas the results are verified with the experimental data collected by laser Doppler anemometer. The dependence of flow structures on Reynolds number and aspect ratio is investigated up to the Elasticity number limit of 0.1 for the numerical part. Another numerical study is realized by Xue et al. (1999), in which UCM flows in a 3-D geometry for different aspect ratios are simulated. In the mentioned study, the Finite Volume Method is used for discretization while Semi-Implicit Method for Pressure Linked Equations (SIMPLE) method is utilized for the solution. The results point out that the flow patterns are altered from Newtonian like to reverse flow with the increasing Elasticity numbers where the limit of the elasticity number is for the numerical part of the study. The main objective of this study is to generate a robust and efficient matrix-free Inexact Newton-Krylov solver which can deal with the high Weissenberg number cases of the non-newtonian flow computations, specifically rotat- Korea-Australia Rheology Journal December 2010 Vol. 22, No

2 Yalin Kaptan, Ali Ecder and Kunt Atalik Fig. 1. Geometry of the problem. ing disc in a cylindrical enclosure problem. What is more, the effects of the Weissenberg number, the Reynolds number and the selection of the constitutive relation for the flow of the rotating disc in cylindrical enclosure geometry will be studied in the following sections. The geometry of the problem can be seen in Fig. 1 which consists of a stationary cylindrical casing and two circular discs at the top and the bottom of the casing (stationary top disc and the rotating bottom disc). By utilizing the axial symmetry, the computation area is reduced to a rectangle which can be observed in the same figure with a hatched region. The aspect ratio for parallel plate problem is defined as δ=h/r and the bottom disc is rotating with an angular velocity Ω. 2. Governing Equations The vector form of the continuity, momentum and constitutive equations for the steady and incompressible flow can be given as follows: u = 0 Re( u )u = p + β 2 u + ( 1 β) τ τ+ We[ u τ ( u) T τ τ u + ατ τ] = u + ( u) T (3) Here τ is the viscoelastic stress tensor, u is the velocity vector, p is the pressure, β is the viscosity ratio (given that µ s is the solvent viscosity and µ p is the polymer viscosity, β = µ s /(µ s + µ p )) while α is the mobility factor. The realistic range of the mobility factor is between 0 and 0.5 (Bird et al. (1987)); however, the values between 0.05 and 0.15 are generally used to model the non-newtonian behavior in the literature (Bird et al. (1987), Itoh et al. (2006)). The definitions for the Reynolds number and the Weissenberg number are; Re = ΩR 2 /ν, We = λu/r where ν is the kinematic viscosity, λ is the relaxation time, U is the characteristic velocity (U = RΩ) and R is the radius. Increased Weissenberg number can be interpreted as the increasing (1) (2) elastic effects in the fluid. The elastic effects are compared by means of a dimensionless number, namely Elasticity number, which is the ratio of Weissenberg number to the Reynolds number (El = We/Re). In the equations above if β and α are zero, UCM constitutive relation can be obtained. If β is non-zero and α is zero Oldroyd B is reached. Lastly, if both β and α are nonzero the constitutive relation turns out to be Giesekus. With the aid of the axisymmetry, the problem is considered as 2.5 dimensional which means that the velocities in r, θ and z directions are solved for the flow domain shown in Fig. 1. The same approach is adopted by Pao (1970, 1972), Lopez (1996) and Saci et al. (1998) for the Newtonian problems. To ease up the solution stream function ( ψ ) vorticity ( ξ) circulation ( Γ) formulation is used. In addition, the viscoelastic stress tensor is splitted into Newtonian and non-newtonian parts as τ = S+2D. When this approach is used, the type of the vorticity equation is transformed to an elliptic one by which the convergence and the solution limit of the computations are enhanced (Crochet et al. (1991)). In the equation for splitting, S is the non-newtonian part of the stress tensor where D is the rate of deformation tensor. After some rearrangements, we need to solve nine equations for the rotating disc in a cylindrical enclosure flow (equations for ψ, ξ, Γ, S rr, S rθ, S rz, S θθ, S θz and S zz ). The full versions of the equations can be found in Appendix Boundary Conditions Zero stream function value is used for all of the edges. Vorticity boundary conditions are taken as; ξ = 2 ( ψ wall+1 ψ wall) / (dz 2 r) for the top and bottom discs, ξ= 2 ( ψ wall+1 ψ wall) / (dr 2 ) for the right wall and zero for the axis. In the vorticity boundary conditions, dr is the distance between the nodes in r-direction whereas dz is the distance in the z-direction in the similar way. The boundary condition for the circulation on the rotating disc is Γ = r 2 where all of the other edges have zero circulation value. Stress boundary conditions at the walls are applied in the way they are used in the study of Kawabata et al. (1990). It can be explained as equating the second derivatives of the non- Newtonian parts of the stresses to zero except the axis. At the symmetry axis, axially symmetric boundary condition is used for the stresses. 3. Solution Method The differential equations and the boundary conditions obtained at the previous section are discretized by using the finite difference method and as a result non-linear, sparse, coupled, large scale system (Equation 7) is obtained. Fx ( ) = 0 (7) 266 Korea-Australia Rheology Journal

3 Rotating-surface-driven non-newtonian flow in a cylindrical enclosure M 1 k Js k ( ) M 1 Fx ( k) k = x ( k+ 1) = x ( k+ 1) s ( k) ( ) (8) (9) Fx ( k) ( ) Js ( k) 2 η k Fx ( ( k) ) 2 (10) Here J is the Jacobian matrix, M k is the diagonal scaling matrix and s (k) is the linear step solution in the k th step and η is the inexact Newton parameter. For the solution of this system, Inexact Newton- GMRES(m) (Generalized Minimal Residual method with restart m by Saad et al. (1986)) algorithm is used (IN- GMRES(m)). In each step, Equation (7) is linearized by using Newton s method and the obtained system is scaled with respect to the diagonal term to enhance the convergence behavior. The result produced by this procedure is Equation (8). The system in Equation (8) is solved via GMRES(m) and the solution is updated as in Equation (9). This operation is carried out until the inequality in the Equation (10) is satisfied. Since the convergence of the Newton s method is strictly dependent on the initial guess, the continuation method is preferred. The continuation method can be described as using the previous solution as the initial guess. The reason for using GMRES(m) as the linear solver is its superiority over the other methods such as Gauss-Seidel and Successive Over Relaxation (SOR) in terms of its convergence rate, non-increasing residual property and easy application of the matrix-free versions. Since the computation of the Jacobian matrix is costly in terms of computation time and the matrix itself is sparse, the code is written as matrix-free. In the solution of the linear system, GMRES requires merely the products of Jacobian matrix and a vector instead of the Jacobian matrix itself. This operation is carried out by using the formula which is called Directional Differencing Discrete Newton. Fx ( + εv) Fx ( ) Jx ( )v = (11) ε In the Equation 11, ε is the perturbation parameter, the selection of which has an effect on the robustness of the nonlinear solver. This parameter can be selected either as a variable like in the studies of Brown et al. (1990) and Qin et al. (2000) or as a constant. In this study, it is chosen as a constant which is the square root of the machine epsilon. Upwinding is also used in the discretization processes. Additionally, in order to enhance the accuracy of the results, the fourth order difference stencils are used at the interior points. The convergence parameters for the linear and non-linear systems are taken as 1E Results Fig. 2. Streamline contours of the UCM flow with Re=0.32 and δ= Validation In this part of the study validation of the code is performed by comparing the results with the data available in the literature and the data obtained by using POLYFLOW which is commercial software designed for the flow of non-newtonian fluids. The study by Xue et al. (1999), in which UCM flows of rotating bottom disc in a cylindrical enclosure problem is selected as the first comparison case. In this problem, Reynolds number is fixed as 0.32 where aspect ratio is 1 and the constitutive relation is selected as UCM. Fig. 2 shows the streamline contours for different Weissenberg numbers. The streamline contours are in good agreement, therefore the contours of the study of Xue et al. (1999) are not given here. In Fig. 2, the positive values of the streamlines indicate the Newtonian like flow, in which the secondary flow in the r-z plane rotates radially outward from the disk (counter clockwise direction in the figure) whereas the negative values indicate the reversed flow pattern (clock- Korea-Australia Rheology Journal December 2010 Vol. 22, No

4 Yalin Kaptan, Ali Ecder and Kunt Atalik Table 1. Maximum weissenberg numbers computed for different grids Grid Max. We stresses counterbalance the centrifugal force and generate a reversed flow field. In the Fig. 2(b), this reversed flow appears near the right bottom corner of the flow domain. With the further increase in Weissenberg number, elastic forces become dominant and the reversed flow field grows into the domain where it finally covers the whole domain as illustrated in Fig. 2(e). Furthermore, Fig. 3 reveals the comparison of the velocity plots with the POLYFLOW for the same problem. The reversing of the flow can also be observed in these figures. Moreover the solutions for UCM, Oldroyd B and Giesekus models are validated with POLYFLOW up to its limits and the results are turned out to be comparable. Grid dependence is also checked with these analysis and grid is found out to be useful for δ=0.5 cases where grid turns out to be useful for δ=1 cases. Table 2 reveals the grid dependence of the maximum Weissenberg numbers for the Giesekus flow with the parameters of; Re=0, δ=0.5, β = 0.5 and α = 0.1. The maximum Weissenberg number limits are decreasing with the refinement of the grid as expected. Fig. 3. Velocity plots of the UCM flow with Re=0.32 and δ=1. wise direction in the figure). The secondary rotation in the r-z plane for Newtonian fluids is formed because of the centrifugal force applied by the disk. Increasing Weissenberg numbers generate growing normal stresses in the opposite direction of the centrifugal force. After a certain point, approximately We = for this problem, these 4.2. Effects of the constitutive relation, weissenberg number and reynolds number The constitutive relation is the most prominent part of the modeling of the non-newtonian phenomenon. In this study, three differential type constitutive relations, namely UCM, Oldroyd B and Giesekus relations are used. Table 2 points out the comparison of maximum Weissenberg number limits of the matrix-free IN-GMRES code generated in this study and POLYFLOW for different constitutive relations when δ = 0.5. In this problem, the identical boundary conditions and continuation parameters are selected both for IN-GMRES(m) and POLYFLOW codes. This table demonstrates that the solver generated in this study is more efficient than the steady solver in POLYFLOW and they both exhibit similar behavior of maximum Weissenberg number limits in terms of increase or decrease depending on the Reynolds number. The maximum Elasticity number obtained by IN- GMRES code developed in this study is for a nonzero Reynolds number and it is possible to achieve higher Elasticity numbers for different aspect ratios and Reynolds numbers as well. This value is proven to be high when compared with the previous studies of Moroi et al. (2001) (Elasticity number limit is 0.11), Itoh et al. (2006) (Elasticity number limit is 0.1) and Xue et al. (1999) (Elasticity 268 Korea-Australia Rheology Journal

5 Rotating-surface-driven non-newtonian flow in a cylindrical enclosure Table 2. Maximum weissenberg numbers computed for different constitutive models UCM Oldroyd B β = 0.5 Giesekus β = 0.5, α = 0.1 This Study Polyflow This Study Polyflow This Study Polyflow Re = Re = Re = Re = number limit is 0.072). What is more, it exceeds the Elasticity number limits of the steady solver of POLYFLOW. Although the limits of IN-GMRES code are beyond the experimental limits, it is significant to generate a solver with this capability since the application of another problem is always possible. Although UCM is one of the earliest and simplest constitutive relations, its numerical solution is the most challenging one of all. Thus, the maximum Weissenberg number limit is very low compared to the others. For Oldroyd B model, β = 0 value coincides with the UCM relation where the β = 1 value coincides with the Newtonian flow. Numerical solutions of Oldroyd B model become easier when viscosity ratio approaches 1 because the elliptic property of the Oldroyd B model increases. The Giesekus constitutive relation is one of the most realistic constitutive relations in the area of the non-newtonian flow modeling and it can model the shear thinning behavior of non-newtonian fluids which is useful in the modeling of rotating disk in a cylindrical enclosure problem. Beneficially, the maximum Weissenberg number limits are higher for Giesekus constitutive relation compared to the UCM and Oldroyd B models. Since the Giesekus model is the most advantageous one compared with the other two, the simulations carried out in the following sections are performed by using the Giesekus constitutive relation. It is not advisable to compare the velocity fields of three constitutive relations mentioned above for a fixed Weissenberg number because the different viscosity ratios and mobility factors will result in dissimilar elastic effects and consequently the flow fields will not be similar. Fig. 4 reveals the streamline contours of non-newtonian flow computed by using Giesekus constitutive relation with the parameters β = 0.5, α = 0.1, δ = 0.5 and Re = 1. It can be observed from this figure that the elastic forces counterbalance the centrifugal force and the flow transformed from Newtonian like to the fully reversed flow with We = 0.5 case. After this point, the Weissenberg number is further increased and at nearly We = 25, a new flow spot which rotates in the direction of the Newtonian flow Fig. 4. Streamline contours of the Giesekus flow with Re = 1, δ = 0.5, α = 0.1 and δ = 0.5. (counterclockwise) appears. This re-formation means that the elastic forces now can not overcome the centrifugal force arising from the rotation of the disk. Continuous rise in Weissenberg number enlarges this new spot where it covers nearly 75% of the flow domain at We = 130. This phenomenon can be considered as a high Weissenberg number occurrence. The results are validated with POLY- FLOW up to We = 90 and they are in good agreement. Since the re-formation of the Newtonian like spot occurs nearly at We = 25 (El = 25) and the Elasticity limit in the Korea-Australia Rheology Journal December 2010 Vol. 22, No

6 Yalin Kaptan, Ali Ecder and Kunt Atalik Fig. 6. Streamline contours of the Giesekus flow with Re = 5, β = 0.5, α = 0.1 and δ = 0.5. Fig. 5. Velocity plots of the Giesekus flow with Re = 1, β = 0.5, α = 0.1 and δ = 0.5. literature is nearly 0.1, there is no result for the re-formation of the Newtonian flow spot in the literature for the authors knowledge. Although this limit is beyond the experimental studies, it is important to solve this type of problems by use of robust numerical methods such as matrix-free Inexact Newton-Krylov methods used in this study. Fig. 5 reveals the comparison of the velocity plots with the POLYFLOW for the same problem. The re-formation of the Newtonian like flow can also be observed from this figure. Fig. 6 demonstrates the streamline contours of non-newtonian flow computed by using Giesekus constitutive relation with the parameters β = 0.5, α = 0.1, δ = 0.5 and Re = 5. It can be observed from this figure that the reversed flow spot is formed at the right bottom corner of the flow domain while this time it can not cover the whole domain by growing into it. It gets bigger up to We = 2 and from this point on, it gets smaller and smaller again. The Newtonian like flow which is rotating in the counterclockwise direction covers nearly 95 % of the flow region at We = 20. The reversed flow can not cover the whole flow domain in this problem since the Reynolds number is increased to 5. Given the fact that the Reynolds number is the ratio of the inertial forces to the viscous forces, the increase in this number results in enhanced centrifugal forces which can not be overcome by the elastic forces. However the final shapes of the flow domains are similar for the problems in Fig. 4 and Fig. 6. The velocity plots of the flow of 270 Korea-Australia Rheology Journal

7 Rotating-surface-driven non-newtonian flow in a cylindrical enclosure depending on the Weissenberg and Reynolds numbers (up to El = 0.1) can be found in the study of Moroi et al. (2001). 5. Conclusions High Weissenberg number non-newtonian flows of rotating surface driven cylindrical container problem is investigated by generating a robust and efficient matrixfree Inexact Newton-Krylov solver (IN-GMRES). UCM, Oldroyd B and Giesekus constitutive relations are used to model the non-newtonian behavior and the results obtained are compared with the data obtained by POLY- FLOW. Additionally, the selection of the constitutive relation, the effects of the Weissenberg number and effects of the Reynolds number are studied. The simulations point out that the generated algorithm is capable of solving high Weissenberg number problems (up to the Elasticity number limit of 130) when compared to previous studies. The development of the reversed flow spot due to the counterbalancing of the centrifugal forces by elastic forces (with the increasing Weissenberg number) is observed for all of the simulated cases. Additionally, with the further increase in Weissenberg number re-formation of the Newtonian like flow is monitored. Furthermore, it is shown that the increase in the Reynolds number alters the centrifugal forces and changes the flow structure. Acknowledgements This work was funded by the Scientific and Technological Research Council of Turkey (Project: 107M390). Notation Fig. 7. Velocity plots of the Giesekus flow with Re = 5, β = 0.5, α = 0.1 and δ = 0.5. Giesekus constitutive relation with the parameters β = 0.5, α = 0.1, δ = 0.5 and Re = 5 can be seen in Fig. 7, in which the flow formations can also be examined. The comparison of Figs. 4 and 6 reveals that the change in Reynolds number effects the flow field by altering the centrifugal forces. The parametric study of the formation of the flow D Non-Newtonian part of the stress tensor El Elasticity number H Height J Jacobian matrix M Diagonal scaling matrix r Distance in the radial direction R Radius Re Reynolds number S Non-Newtonian part of the stress tensor s Linear step solution vector u u component of the velocity v v component of the velocity We Weissenberg number w w component of the velocity α Mobility factor β Viscosity ratio Γ Circulation δ Aspect ratio ε Matrix-free perturbation parameter ξ Vorticity Korea-Australia Rheology Journal December 2010 Vol. 22, No

8 Yalin Kaptan, Ali Ecder and Kunt Atalik η Inexact Newton parameter λ Relaxation time τ Stress Tensor τ Stream function ψ Angular velocity References Bird, R.B., R.C. Armstrong and O. Hassager, 1987, Dynamics of polymeric liquids, Vol. 1: Fluid mechanics, 2 nd ed., Wiley, New York. Brown, P.N. and Y. Saad, 1990, Hybrid Krylov methods for nonlinear-systems of equations, SIAM Journal on Scientific and Statistical Computing, 11, Crochet, M.J., A.R. Davies and K. Walters, 1991, Numerical simulation of non-newtonian flow, Elsevier, New York. Itoh, M., M. Suzuki and T. Moroi, 2006, Swirling flow of a viscoelastic fluid in a cylindrical casing, Trans. ASME, 128, Kawabata, N., T. Motoyoshi and A. Isao, 1990, A numerical simulation of viscoelastic fluid flow in a two-dimensional channel: application of Lax's Scheme to the constitutive equation [in Japanese], Transactions of the Japan Society of Mechanical Engineers B, 56, Lopez, J.M., 1996, Flow between a stationary and a rotating disk shrouded by a co-rotating cylinder, Phys. Fluids, 8(10), Moroi, T., M. Itoh, K. Fujita and H. Hamasaki, 2001, Viscoelastic flow due to a rotating disk enclosed in a cylindrical casing (Influence of Aspect Ratio), JSME International Series B, 44(3), Pao, H.P., 1970, A numerical computation of a confined rotating flow, Trans. ASME, J. Applied Mechanics, 37, Pao, H.P., 1972, Numerical solution of the navier-stokes equations for flows in the disk-cylinder system, Physics of Fluids, 15, 1, Qin, N., D.K. Ludlow and S.T. Shaw, 2000, A matrix-free preconditioned Newton/GMRES method for unsteady Navier Stokes equations, Int. J. Numer. Meth. Fl., 33, Saad, Y. and M.H. Schultz, 1986, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing, 7(3), Saci, R. and P.G. Bellamy-Knights, 1998, Diffusion driven rotating flow in a cylindrical container, Acta Mechanica, 126, Xue, S.C., N. Phan-Thien and R.I. Tanner, 1999, Fully threedimensional, time-dependent numerical simulations of Newtonian and viscoelastic swirling flows in a confined cylinder Part I. Method and steady flows, J. Non-Newtonian Fluid Mech., 87, Korea-Australia Rheology Journal