20th European Symposium on Computer Aided Process Engineering – ESCAPE20 S. Pierucci and G. Buzzi Ferraris (Editors) © 2010 Elsevier B.V. All rights reserved.

Simulation of Free Surface Viscoelastic Fluid Flow Using the viscoelasticInterFoam Solver Jovani L. Favero,a Nilo S. M. Cardozo,a Argimiro R. Secchi,b Hrvoje Jasakc aChemical Engineering Depart. – EE/UFRGS - Universidade Federal do Rio Grande do Sul, R. Eng. Luiz Englert, s/n Centro – 90040-140 – Porto Alegre, RS – Brazil, favero@enq.ufrgs.br, nilo@enq.ufrgs.br bChemical Engineering Program – PEQ/COPPE – Universidade Federal do Rio de Janeiro, Av. Horácio Macedo, 2030 – CT - G116 – Ilha do Fundão – 21941-972 – Rio de Janeiro, RJ – Brazil, arge@peq.coppe.ufrj.br cFSB/Wikki – Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Croatia, and Wikki Ltd., London W8 7PU, United Kingdom, h.jasak@wikki.co.uk

Abstract The understanding of polymeric fluids flows is of essential importance for several industry sectors, including plastic and food processing industries. The rheological response of viscoelastic fluids is quite complex, including combination of viscous and elastic effects and highly non-linear viscous and elastic phenomena. In a previous work we presented a solver for internal viscoelastic fluid flows, called viscoelasticFluidFoam, which was developed with the OpenFOAM computational fluid dynamics package and validated with experimental and numerical data from the literature for the analysis of a planar 4:1 contraction flow. In the present work will be tested the extension of the viscoelasticFluidFoam solver to the viscoelasticInterFoam solver used for analysis of free surface viscoelastic fluid flows using the VOF methodology. A classical die swell flow phenomena used in the rheology literature to present the concept of viscoelastic effects was simulated. The results obtained using Giesekus model showed the great potential of the developed formulation, once all phenomena observed experimentally were reproduced in the simulations. Keywords: Viscoelastic Fluids, VOF, OpenFOAM, Die Swell, CFD.

1. Introduction The understanding of viscoelastic fluid flow characteristics is of great interest for polymer and food processing industries. Unlike the Newtonian flow that is well understood, the flow of viscoelastic fluids is still a subject of great discussion. Most of the works on viscoelastic fluids found in literature treats internal flows using the geometry known as abrupt planar contraction. This is due mainly because this geometry was adopted as standard for viscoelastic fluids flow studies in 1987 during the fifth international workshop on numeric methods for non-Newtonians fluids (Hassager, 1988). On the other hand, simulations using free surface viscoelastic flow are receiving a special attention mainly in the last two decades (Tomé et al., 2008). Flows considering free surface is particularly useful for industry that works with polymeric fluids, once processes as extrusion and injection molding of plastics, metallic leagues or fluids used in food industry depend always more on the knowledge of parameters that guarantee a high quality for their products (Khismatullin et al., 2006).

Favero et al.

Regarding the modeling, the study of free surface flow presents the same bases of internal flows plus an appropriate methodology for treatment of free surface motion, such as the MAC (Marker-and-Cell) (Harlow and Welch, 1965) and VOF (Volume-of-Fluid) (Hirt and Nicols, 1981). An overview of literature works about viscoelastic free surface simulation can be found in Harvie et al. (2008). There are in the literature several works treating viscoelastic fluid flow with free surface, and the most of them use finite element method to solve simplified cases. In this work we overcome that limitation by showing results of a generic and flexible tool developed for treatment of free surface viscoelastic fluid flow using finite volume method and VOF methodology. This tool can be used to simulate complex geometries encountered in real problems of polymer and food processing industries, using non-structured and moving meshes, and allowing to choose among several different interpolation schemes, solvers for linear algebra, possibility of data processing parallelization and several constitutive equations for treatment of viscoelasticity. This work is an extension of our previous development of a solver, called viscoelasticFluidFoam (Favero et al., 2009), for treatment of internal viscoelastic flow using the OpenFOAM software (Weller et al., 1998). Such extension consists of increasing the capability of the viscoelasticFluidFoam solver, already available to the user in OpenFOAM, for the treatment of free surfaces flow using the VOF methodology, creating a new solver called viscoelasticInterFoam. The discretization of the flow governing equations in OpenFOAM is based on the finite volume method formulated with collocated variable arrangement, with pressure and velocity solved by segregated methods. The software offers a wide range of interpolation schemes, solvers and preconditioners for the discretized algebraic equation system. The results, studying a classic experiment in viscoelastic fluids flow with free surface: the die swell in the exit of a channel, show the success of the applied methodology. The rod-climbing experiment was also reproduced in the simulations but is not shown here due to lack of space.

2. Methodology In this section, the mathematical formulation, the strategy used to solve free surface viscoelastic fluid flow, and a description of the test problems are presented.

2.1. Mathematical Formulation The governing equations of laminar, incompressible, and isothermal flow of viscoelastic fluids are the mass conservation (continuity equation):

0 u (1)

and momentum conservation:

p gS Pt

uuu τ τ

(2)

together with a mechanical constitutive equation that describes the relation between the stress and the deformation rate for a given fluid. In the above equations ρ is the density, u the velocity vector, p the pressure, τS the solvent Newtonian contribution, and τP the polymeric contribution.

The solvent contribution is given by:

2S Sτ D (3)

Simulation of Free Surface Viscoelastic Fluid Flow Using the Software OpenFOAM

where ηS is the solvent viscosity and D is the deformation rate tensor, given by:

1

2

T D u u (4)

The extra elastic contribution, corresponding to the polymeric part, is obtained from the solution of an appropriate constitutive differential equation. Various constitutive equations are found in literature, as for example, Maxwell, Oldroyd-B, Phan-Thien-Tanner (PTT), Finitely Extensible Nonlinear Elastic and so on. A well known and widely used is the Giesekus constitutive model described as (Giesekus, 1982):

( ) 2PP P P pp

τ τ τ τ D

(5)

where λ and ηP are the relaxation time and the polymer viscosity at zero shear rate,

respectively, α is the dimensionless ‘mobility factor’, and P

τ is the upper convected

derivative of the elastic stress tensor defined as (Bird et al., 1987):

( )P

TP P P Pt

τ τ u τ u τ τ u (6)

Another important subject is the treatment of the free surface. In the developed solver it was used the VOF methodology (Hirt and Nicols, 1981). In this method, the volume of fluid in a cell is computed as Fvol = γVcell where Vcell is the computational cell volume and γ is the liquid fraction in this cell. The value of γ in a cell varies between 0 and 1. A cell totally filled with one fluid presents γ = 1 and another totally filled with one different fluid have γ = 0. In the interface the value of γ is an intermediate value between 0 and 1, satisfying the following equation:

(1 ) 0rt

u u

(7)

where ur is the velocity field subject to interface compression (Ubbink and Issa, 1999; Rusche, 2003). The last artificial term is only activated in the region of the interface due to the properties of the term γ(1-γ). The VOF methodology allows to solve the flow of two different fluids (two-phase flow) solving as for one fluid only, in other words, the same equations are solved for both fluids, the only difference is regarding to the physical properties. To make this possible, the physical properties in any control volume of the domain are given by the corresponding fraction of the fluid in that volume. It is mathematically obtained, considering a generic property θ, as:

1 2(1 )fluid fluid (8)

2.2. Test Geometry To test the implemented methodology, the reproduction of a classic example used to demonstrate viscoelasticity effects are carried out. The test consists in reproducing the die swell phenomena. Figure 1 shows the effect of die swell due to the fluid elasticity (Bird et al., 1987). An effect due to viscoelasticity is the presence of normal stress differences in shear flow. In addition to shear stress, viscoelastic fluids present an extra stress due to stretching and

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alignment of polymeric chains. In this experiment, it is considered a fluid flowing out of a capillary with diameter d. For Newtonian fluids the swell can vary 13% above or below of the diameter d, whereas for polymeric fluids, an increase in more than 300% relative to d can be found, depending on the fluid elasticity. To simulate this experiment a shear rate of 60 s-1 was used in the flowing channel.

(a) (b)

Figure 1: (a) Representation of the die swell effect for a Newtonian fluid (left) and a polymeric fluid (right) (Source: Bird et al., 1987). (b) 2D geometry used to simulate the experiment.

In Table 1, the physical properties of the fluids used in the simulations are listed. In the analyzed case, a Newtonian and two polymeric fluids are considered flowing to an atmosphere containing air. The gravitational acceleration used was -9.8 m/s2 in Y direction. To the Newtonian fluid was used the viscosity value equal to the sum of the polymer and solvent viscosities (ηP + ηS). The difference between the two polymeric fluids analyzed is only on the value of λ parameter. The value of λ is responsible to the deviation of the Newtonian like fluids and then an increase of this value results in an increase of the elasticity phenomena for the same flow conditions. To Newtonian’s like fluids the value of λ is assumed as being null.

Table 1: Physical properties of fluids in simulations.

Fluid Parameters Die Swell

ρ[kg/m3] 803.87 ηS [Pa.s] 0.002

Polymer ηP [Pa.s] 1.20 λ [s] 0.03 and 3.0 α [--] 0.15

ρ[kg/m3] 1.0 ηS [Pa.s] 0.0 Air ηP [Pa.s] 1.48x10-5 λ [s] 0.0 α [--] 0.0

3. Results and Discussion Analyzing Figure 2, we can observe three main differences comparing Newtonian and viscoelastic behavior: 1) The occurrence of jet ruptures for the viscoelastic case with high λ value. This could be attributed to the fluid accumulation at the nozzle due to elastic behavior. When there is enough accumulation of polymer a sudden acceleration going down is observed, causing a local increase of elongation rate and producing, in this way, the rupture. When jet break, recoil of fluid is also observed due to elasticity.

Simulation of Free Surface Viscoelastic Fluid Flow Using the Software OpenFOAM

Newtonian Polymer (λ = 0.03 s) Polymer (λ = 3.0 s)

time = 0.5 s

time = 1.0 s

time = 1.5 s

time = 2.0 s

time = 2.5 s

time = 3.5 s

Figure 2: Results for die swell simulation. 2) The viscoelastic fluid presents a jet that oscillates in the X direction, and due to this effect air bubbles are occluded in the mass of fluid. 3) The die swell only happens for the polymeric fluid and it is also noticed that the die swell happens mainly next to the exit of the channel, where exists the relaxation of the polymeric chains.

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In Figure 3, a zoom on the exit of the channel region is shown for a better visualization of the die swell, together with the respective stress field for the time instant 3.5 s. The larger values of stress happen inside the channel and soon at the channel exit, where the die swell exists. Newtonian Polymer (λ = 0.03 s) Stress field

Figure 3: Die swell and stress field at time = 3.5 s.

For all the presented simulations the Crank-Nicholson method was used for time derivatives, with zero initial condition being specified for all variables unless the γ field where the value of 1.0 was specified on the channel. The resulting linear discretized systems were solved by conjugated gradient method (CG), using CG with AMG preconditioning for pressure and BiCGstab with an Incomplete Lower-Upper (ILU) preconditioning for γ, velocity and stress. The absolute tolerance for pressure was 1.0×10−7 and for γ, velocity, and stress was 1.0×10−6.

4. Conclusion Simulation results of viscoelastic fluid flow with free surface were presented. The analyzed cases show results with qualitative agreement with experimentally observed. Die swell was observed in the polymer jet at the channel exit, and jet rupture and air bubbles occlusion in the fluid were verified. The obtained results were satisfactory, justifying the usage of the methodology for free-surface viscoelastic fluid flow using the software OpenFOAM.

5. Acknowledgements We want to thanks CAPES for the financial support.

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