Lattice Boltzmann method for the simulation of viscoelastic fluid flows

Transcription

1 Lattice Boltzmann method for the simulation of viscoelastic fluid flows O. Malaspinas a,b,, N. Fiétier a, M. Deville a a Ecole Polytechnique Fédérale de Lausanne, STI-IGM-LIN, 1015 Lausanne, Switzerland b Université de Genève, CUI - SPC, 1227 Carouge, Switzerland Abstract The simulation of viscoelastic fluids is a challenging task from the theoretical and numerical points of view. All the numerical models, uptonow have failed to solve the high Weissenberg number problem HWNP). There is therefore a need for novel techniques to tackle this class of problems. In this paper we propose a new approach based on the lattice Boltzmann method in order to simulate linear and non-linear viscoelastic fluids and in particular those described by the Oldroyd-B and FENE-P constitutive equations. We study the accuracy and stability of our model on three different benchmarks : the 3D Taylor Green vortex decay, the simplified 2D four-rolls mill, and the 2D Poiseuille flow. Keywords: lattice Boltzmann method, viscoelastic fluid flows 1. Introduction Nowadays the lattice Boltzmann method LBM) has established itself as a powerful tool for the simulation of a wide range of physical phenomena. One of its main applications is the field of computational fluid dynamics where it has proven successful to solve the weakly compressible Navier Stokes equations see Wolf-Gladrow [1], Succi [2], Chen and Doolen [3]), and models associated with more complex flows involving several phases or components see e.g. Shan and Chen [4], Reis and Phillips [5]). It has also been successfully applied to the simulation of flows of pseudoplastic e.g. Kehrwald [6], Malaspinas et al. [7]) and viscoplastic fluids e.g. Vikhansky [8]). This method does not solve directly the macroscopic conservation equations, but rather models the statistics of collision of particles and may offer more modeling freedom than the classical methods based on finite difference, finite volume or finite element to which it is a competitive alternative. Corresponding author addresses: orestis.malaspinas@unige.ch O. Malaspinas), nicolas.fietier@epfl.ch N. Fiétier), michel.deville@epfl.ch M. Deville) Preprint submitted to Journal of Non-Newtonian Fluid Mechanics October 25, 2013

2 In this paper, we apply the LBM to the simulation of viscoelastic fluid flows. This class of fluids is characterized by the fact that the deformation of an element of fluid induced by a stress does not only depend on the strain itself as it is the case for Newtonian fluids) but also on the history of the deformation memory effect). The polymer melts or solutions, which are present in many industrial and everyday life products, are good examples of such fluids. There exist numerous very different models to account for the behavior of viscoelastic fluids, ranging from molecular descriptions which consider all the microscopic interactions between particles, to macroscopic continuum mechanics models that ignore the small constitutive elements of the fluids and take only into account global effects see the books by Bird et al. [9, 10] and Larson [11] for example). In this work, we focus on two such models : the linear Oldroyd-B and non-linear FENE-P models see Oldroyd [12] and Peterlin [13]). These models have been investigated in many details with classical spatial discretization methods based on finite difference e.g. Tomé et al. [14], Vaithianathan et al. [15]), finite volumes e.g Oliveira [16]), finite element e.g. Purnode and Crochet [17]) or spectral element methods e.g. Chauvière and Owens [18], Fiétier [19]). Unfortunately, they all suffer from the same problem of representing correctly the fluids with strong viscoelastic interactions, which are characterized by a high Weissenberg Wi) or Deborah De) number Wi = 0 or De = 0 numbers correspond to Newtonian fluids). It is indeed reported in the literature as the HWNP high Weissenberg number problem) e.g. [20]) as all the numerical methods fail to converge as Wi increases. There is a need for novel modeling and simulation techniques. However, it is not yet clear which one of the physical model or of the numerical method, used for the resolution of the viscoelastic simulations, is responsible for the lack of accuracy and numerical stability in the HWNP e.g. Fattal et al. [21], Kupferman [22]). Here we will tackle the problem of the simulation of viscoelastic fluids, with a completely different approach based on the lattice Boltzmann method. Although we use here the standard Oldroyd-B and FENE-P models, we hope that in the future this approach may offer more freedom for the physical modeling of viscoelastic fluids. There have been already some attempts done by the lattice Boltzmann community in this direction, but all of them only partly take into account elastic effects in the flow and do not explicitly use a truly viscoelastic constitutive equation like the one corresponding to the Oldroyd-B or FENE-P models. In Qian and Deng [23] the elastic effect is obtained by modifying the equilibrium distribution, whereas in Ispolatov and Grant [24] the elastic effect is obtained by adding a Maxwell-like exponentially decaying) force to the system. In Giraud et al. [25, 26], and in Lallemand et al. [27] a scheme for solving the Jeffreys model is proposed, but it fails to exhibit some important elastic effects since the stress tensor is assumed traceless and is essentially linear. In recent works by Onishi et al. [28, 29] the Fokker Plank counterpart for the Oldroyd-B 2

3 and FENE-P models was introduced to carry out simulations with the help of the LBM. The numerical results presented were obtained for very simple shear flows and did not allow to assert that the viscoelastic effects were very strong. In Denniston et al. [30] and Marenduzzo et al. [31, 32] the LBM is used for the simulation of flows of liquid crystals with an approach that shares some similarities with the present one, although the constitutive equation is much more complex but is limited to very simple flow geometries. There is therefore still room for many improvements in this domain that we will discuss in this work. To our knowledge, this is the first attempt to solve the full set of equations including conservation and macroscopic constitutive equations of the extended Oldroyd-B type with non-linear effects for non-trivial flows using the lattice Boltzmann method. A decoupled approach is used to determine the velocity field by a classical lattice Boltzmann solver for the set of mass-momentum conservation equations where the divergence of the viscoelastic stress appears as an external forcing term whereas a modified advection-diffusion lattice Boltzmann scheme is used to obtain the viscoelastic stress tensor by solving the constitutive equation with a prescribed velocity field. Only isothermal and incompressible fluids are considered here. The outline of the paper is the following. In Section 2 we describe the mathematical models used for the simulation of viscoelastic fluids. In Section 3, the LBM model, used to solve numerically the equations of Section 2, is described. In Section 4, the model presented in Section 3 is validated by comparing results of simulations for a planar Poiseuille flow with the corresponding analytical solution for the Oldroyd-B model. We also use the semi-analytical results published by Thomases and Shelley [33] to test the proposed method on the 2D four-roll mill problem, and compare our results with data obtained with a high accuracy Fourier pseudospectral algorithm for the Taylor Green vortex benchmark. Finally, we draw conclusions of this work and evoke future prospects in Section The mathematical model In the present paper, we are interested in simulating the behavior of incompressible viscoelastic fluids, which are characterized by the Oldroyd-B and FENE-P models see Oldroyd [12] and Peterlin [13] respectively). Both constitutive equations can be used to model the behavior of polymer molecules surrounded by an incompressible Newtonian solvent. The polymer molecules are modeled by two beads connected by a spring. The viscoelastic model is depending on the interaction law of the spring linear for Oldroyd-B, non linear for FENE-P). The interaction between the polymer molecules and the solvent are taken into account by considering the drag force on the beads. The deformation of the polymer molecules is determined by two competing processes, namely the stretching by the velocity gradients and the relaxation due to elasticity of the molecule. Experiments have indicated that the relaxation process 3

4 is linear in a wide range of stretching of the molecules, which promotes the use of the Oldroyd-B equation Fouxon and Lebedev [34]). In strong flows, the finite extensibility of the polymer molecules must be accounted for. In these circumstances, an equation encompassing non-linear effects like FENE-P must be considered. The mathematical equations describing the solvent are given by u = 0, 1) t u+u )u = 1 ρ pi +2µ ss +Π), 2) where the operator represents the scalar product. The symbols ρ, µ s, p, u, I, S = 1/2 u+ u) T) are respectively the density, the solvent dynamic viscosity, the pressure, the velocity, the identity tensor and the strain rate tensor. The superscript T denotes the transpose operation. The viscoelastic stress tensor Π accounts for the effects of the polymers on the solvent. The constitutive equation, written in terms of the conformation tensor, A, which is a statistical indicator of the orientation of the polymer molecules, reads depending on the viscoelastic model) da dt = 1 λ A I)+A u+ u)t A Oldroyd-B), 3) da dt = 1 λ aa bi)+a u+ u)t A FENE-P), 4) where d/dt t +u ) is the material derivative. The inner product of the two tensors A and u is denoted by A u. The quantity λ is the relaxation time of the polymers. The two parameters a and b appearing in Eq. 4) are given by : 1 1 a = 1 tra/re 2, b = 1 3/re 2, 5) where r e is related to the maximum allowed length of the polymer molecules and tra is the trace of the tensor A. Depending on the model the relation between the viscoelastic stress and the conformation tensor is Π = µ p λ Π = µ p λ A I) Oldroyd-B), 6) aa bi) FENE-P), 7) whereµ p isthedynamicviscosityofthepolymerseee.g.birdetal.[9],purnode and Crochet [17]). 3. The numerical model In this section, the computational algorithm and the corresponding LBM schemes for the simulation of the flows of viscoelastic fluids are presented. A 4

5 decoupled approach has been selected to determine the velocity and viscoelastic stress fields as follows : 1. The mass-momentum set of conservation equations 1)-2) corresponding to the Navier-Stokes equations for incompressible fluids) is solved first with a classical LBM scheme described in Section 3.1 by considering the viscoelastic stress tensor Π as a given quantity determined by solving the constitutive equation, 2. The constitutive equation 3) or 4) is solved with a modified advectiondiffusion LBM scheme described in Section 3.2 by taking the velocity field computed with the Navier Stokes solver as a prescribed quantity. The two steps are solved simultaneously at each time iteration. Let us introduce some notations that will be used in the sequel. A difference is made between numerical and physical quantities these concepts will become clearer later) as the first ones are overlined, while the second ones are left bare. For example, the numerical which are the distributions used for the simulations) velocity distribution function is denoted f i and the physical one is denoted f i The incompressible Navier Stokes solver Figure 1: The D2Q9 lattice. The vectors represent the microscopic velocities ξ i. The rest velocity ξ 0 = 0,0)) is added to the 8 vectors of this lattice. In order to solve the Newtonian solvent equations, the forced BGKfor Bhatnagar, Gross and Krook, see [35]) lattice Boltzmann scheme is used see Guo et al. [36] and Appendix A). The lattice Boltzmann method solves the continuous BGK equation on a regular grid in two steps which are applied iteratively to the whole domain at each timestep x = t = 1 for simplicity in the notations) : 1. The collision step, which remains completely local : f out i x,t) = f i x,t) 1 ) f τ i x,t) f 0) i x, t) ) F i. 2τ 5

6 2. The propagation step, where the f i s are streamed to their neighbors f i x+ξ i,t+1) = f out i x,t). One can rewrite these two distinct steps in a more compact form as f i x+ξ i,t+1) f i x,t) = 1 ) f τ i x,t) f 0) i x, t) ) F i, 8) 2τ where the ξ i s are the discrete microscopic velocities in the i-th direction which connect a lattice grid point with its neighbors an example of the velocity set can be found in Figs. 1 and A.20), τ the relaxation time, F i a forcing term to be explicited later, and f 0) the discrete Maxwellian equilibrium distribution function given by f 0) i = w i ρ 1+ ξ i u c ) s 2c 4 Q i : uu, 9) s with Q i ξ i ξ i c 2 si whereas the colon : sign denotes full index contraction. The symbols w i and c s are respectively the weights and the sound speed of the lattice. The quantity aa represents the tensor product of the vector a by itself the α, β-th component of aa is [aa] αβ = a α a β where a α and a β are the α-th and β-th components of a). The forcing term F i is given by ξi u F i = w i ρ c 2 s ξ i u)ξ i c 4 s ) g, 10) where ρg represents the force density. As shown by Guo et al. [36] this scheme leads after a multi-scale Chapman Enskog expansion, see Chapman and Cowling [37]) to the weakly compressible limit of the forced Navier Stokes equations see Appendix A for a basic summary of the Chapman Enskog expansion for weakly compressible fluids) t ρ+ ρu) = 0, 11) t u+u )u = pi +2ν s S)+g, 12) with kinematic viscosity ν s see also Malaspinas [38]) 1 ν s = c 2 s τ 1 ). 13) 2 The lattices that are used in this paper are the D2Q9 see Fig.1) and the D3Q19 1 see Appendix B and Shan et al. [39] for a complete description of the lattices). For a lattice with q velocities the density ρ and the velocity field u are given by q 1 ρ = f i, u = 1 q 1 ξ i f ρ i + g 2. 14) i=0 i=0 1 The DdQq lattice stands for a d-dimensional lattice with q neighbors velocities) 6

7 Finally the pressure p is computed with the help of the perfect gas law 3.2. The constitutive equation solver p = c 2 sρ. 15) The same type of approach has also been applied successfully by Denniston et al. [30] and Marenduzzo et al. [31, 32] for the simulation of flows of liquid crystals. The idea is to compute each component of the conformation tensor A αβ by its own distribution function set h iαβ by using a modified advection diffusion scheme see Guo et al. [40]) to which a source term is added. A detailed derivation can be found in Appendix C and in Malaspinas [38]. The lattice Boltzmann scheme reads h iαβ x+ξ i,t+1) h iαβ x,t) = 1 ) h iαβ x,t) h 0) iαβ ϕ A αβ,u) ) Gαβ h 0) iαβ 2ϕ A A αβ,u), 16) αβ whereϕistherelaxationtimeofthescheme,g αβ thesourceterm,whichdepends on the constitutive equation and will be explicited hereafter, and the equilibrium distribution function, h 0) iαβ, given by h 0) iαβ = w ia αβ 1+ ξ ) i u c 2, 17) l where c l is a scaling factor, and w i the weights of the lattice. It must be emphasized that in Eq. 16) no summation convention is used. The constitutive tensor A αβ is computed through the h iαβ for a lattice with q discrete velocities) A αβ = A αβ + G αβ 2, 18) where A αβ = q 1 i=0 h iαβ, and the velocity u is provided as a given field computed as explained in Section 3.1. It must be noted that the dependence of the equilibrium is linear in ξ i. Therefore the quadrature order see Appendix C) needed is of three instead of five for example for the classical incompressible limit of the Boltzmann equation). Therefore the lattices needed are the D2Q5 see Fig. 2) or D3Q7 depending on the physical dimension. The lattice description for these two lattices are given in Appendix B. Carrying out a Chapman Enskog expansion one can show that this scheme leads to the following equation see Appendix C and [38] for details on the Chapman Enskog expansion) da αβ dt = G αβ +κ 2 A αβ + κ c 2 l A αβ t u u A αβ u)), 19) 7

8 Figure 2: The D2Q5 lattice. The vectors represent the microscopic velocities ξ i. The rest velocity ξ 0 = 0,0)) is added to the 4 vectors of this lattice. where the diffusivity constant, κ, is κ = c 2 l 1 ϕ 1 ). 20) 2 The source term must be chosen according to the selected constitutive equation. For the Oldroyd-B or the FENE-P models, by identifying Eq. 19) with Eqs. 3) and 4), one obtains G = 1 λ A I)+A u)+ u)t A Oldroyd-B), 21) G = 1 λ aa bi)+a u)+ u)t A FENE-P). 22) Comparing Eqs. 3) or 4) with Eq. 19), we see that we have two extra terms in Eq. 19) which are proportional to κ κ 2 A αβ + κ c 2 l A αβ t u u A αβ u)). The existence of a diffusive term in the constitutive equation can be justified theoretically see e.g. El-Kareh and Leal [41]). The value of the diffusivity parameter κ can be estimated and should be very small in practice. Furthermore, it has been introduced in the constitutive equations by several authors to increase the stability of the simulations see e.g. Sureshkumar and Beris [42], Housiadas and Beris[43]). The second term in the previous equation has clearly no physical meaning and corresponds to an error induced by the LBM scheme. However, as the diffusivity parameter is small, this term remains actually also negligible and does not affect the results of the simulations. A value of the ratio κ/µ p equal to 10 6 has been selected for the simulations reported in this paper in order to find a compromise between stable simulations and no significant perturbation of the results that would be obtained without diffusive term see Subsection 4.3) Boundary conditions for the set of equations A well known difficulty of the lattice Boltzmann method is the treatment of boundary conditions. In the case of the incompressible fluid flows, classi- 8

9 cal Dirichlet boundary conditions consist in imposing the value of the velocity on walls e.g. no-slip condition) or at inlet. The problematic is the following. When one is located on a boundary node the collision-streaming steps see Subsection 3.1) cannot be applied anymore as is, since the boundary node is missing a certain amount of neighbors and therefore there are no f i incoming from these missing nodes see Fig.3). These missing f i have to be reconstructed Figure 3: A boundary node with the known f i s 0, 1, 2, 6, 7, 8) and the missing ones 3, 4, 5). The dashed part represents the wall. in some ad-hoc way. Solutions have been proposed by many authors e.g. Skordos [44], Inamuro et al. [45], Latt and Chopard [46] and see also the review by Latt et al. [47]) and can be readily applied to solve the mass-momentum set of equations 1)-2). In the test cases presented in Section 4, the regularized technique proposed in Latt and Chopard [46] is applied. In this case, all the distribution functions are replaced with their Chapman Enskog expression see Wolf-Gladrow [1], Latt and Chopard [46] among others) f i = w i ρ 1+ ξ i u c ) s 2c 4 Q i : uu 2c 2 sτs). 23) s On a Dirichlet wall the only quantity of this last equation that is imposed is the velocity u. Then from the known f i labeled 0,1,2,6,7,8 in Fig. 3) one can compute ρ and S. As reported in Zou and He [48] and Latt et al. [47] the density can be computed from only the known populations on a Dirichlet wall in the following way 1 ρ = 2ρ + +ρ 0 ), 24) 1+u where u is the velocity component perpendicular to the wall, and where for a wall with normal vector n pointing outwards of the computational domain ρ + = f i and ρ 0 = f i. 25) i {j ξ j n=1} i {j ξ j n=0} In the case of the boundary node of Fig. 3, ρ + and ρ 0 are given by ρ + = f 1 +f 7 +f 8, ρ 0 = f 0 +f 2 +f 6. 26) 9

10 It is then possible to compute the strain rate tensor. From the definition of f 0) i see Eq.9)) and from Eq. 23) one has f 1) i f i f 0) i = w iρτ c 2 Q i : S. 27) s From the properties of the sets of ξ i s and since the Q i is a symmetric tensor one has that f 1) i = f 1) oppi) oppi) represents the microscopic velocity ξ i pointing in the direction opposite to i) and S = 1 ρτc 2 2 s i {j ξ j n=1} ξ i ξ i f 1) i + i {j ξ j n=0} Finally all f i s are reconstructed with the help of Eq. 23). ξ i ξ i f 1) i ). 28) In the case of the constitutive equation, specifying boundary conditions is a more complicated task. An inherent feature of the lattice Boltzmann method is that the distribution functions must be determined everywhere, which is somehow in contrast with the common practice in classical methods of imposing no boundary condition for the viscoelastic stress or conformation tensors) on the walls of a domain as justified by the strong hyperbolic character of the constitutive equation. The presence of a diffusion term see Eq. 19)) does not allow this kind of treatment. Therefore, it is mandatory to impose a boundary conditions on walls with the LBM. Although the lattice used for the constitutive equation D2Q5 or D3Q7) is simpler than the one for the Navier Stokes equations D2Q9 or D3Q19) and therefore it is still needed to determine one unknown distribution function see Fig. 4). Unfortunately, there exists no equivalent to the velocity no-slip boundary condition for the viscoelastic stress or conformation tensor. In some rare cases where the problem geometry is very simple, analytical solutions are known for the conformation tensor on walls, A see Subsection 4.3), but to simulate flows in complex geometries, more generic implementations are needed. In order to explain the method, the upper flat 2D wall case is considered see Fig. 4). It can be generalized straightforwardly to any other configurations and to 3D cases. For the scheme associated with the constitutive equation, Figure 4: A boundary node with the known h i s 0, 1, 3, 4) and the missing one 2). the Chapman Enskog expansion see Eq. C.23)) implies that the numerical 10

11 conformation tensor is given by h 0αβ Furthermore, it can be inferred from Eq. C.22) that A αβ = h 0αβ w 0. 29) h iαβ = h oppi)αβ h 0) oppi)αβ A αβ)+h 0) i A αβ ), 30) where oppi) represents the velocity pointing in the opposite of the i-direction ξ oppi) = ξ i ) and thus, since A αβ is known, it is straightforward to compute h 2αβ h 2αβ = h 4αβ +h 0) 4αβ A αβ)+h 0) 2 A αβ). 31) This boundary condition is not the only one that can be implemented. We describe hereafter other methods that we have applied, but which turned out to be not better than the one presented above. The value of the conformation tensor can be also interpolated by using the term A αβ contained in h iαβ see Eq. C.22)). Then using a backward finite difference scheme, it is possible to compute the value of the conformation tensor. For the wall represented in Fig. 4, a backward second order finite difference scheme can be used to estimate the conformation tensor gradient x = 1) y A αβ x,y) = 1 2 Aαβ x,y 2) 4A αβ x,y 1) 3A αβ x,y) ). 32) From Eq. C.22), it can be seen that y A αβ x,y) is contained in h 4αβ and given by y A αβ = 1 h ) 4αβ +A αβ. 33) ϕ w 4 Replacing this relation in Eq. 32), one can solve for A αβ A αβ x,y) = 2ϕ 2 3ϕ h4αβ w 4 ϕ 1 Aαβ x,y 2) 4A αβ x,y 1) )) 34) 2 and then recompute the missing distribution function h 2αβ from Eq. 30). An alternative for steady state simulations and no-slip walls, is to simplify the Oldroyd-B constitutive equation as : 1 λ A αβ δ αβ )+A αγ γ u β + γ u α A γβ = 0, 35) which provides an algebraic system of equations to be solved, since the velocity gradient can be computed from the velocity field by using a finite difference scheme. One can compute the solution of this system numerically and get the values of the components of the conformation tensor A αβ on the boundaries. Then, from Eq. 30) one can recover h 2αβ. Unfortunately, it was found in practice that these last two methods did not improve the results obtained with Eq. 30). 11

12 3.4. Description of the algorithm In this section we will describe briefly the algorithm used for the simulation of viscoelastic fluids. We have shown how the lattice Boltzmann scheme for advection-diffusion with a source term can be used to simulate the constitutive equation. The algorithm described here is for an Oldroyd-B fluid but the procedure to extend it for the FENE-P model is straightforward. A time-step is composed of four distinct operations. The first and the second steps are the resolution, respectively, of the incompressible Navier Stokes and of the constitutive equations see Eqs. 1), 2) and 3)) u = 0, t u+u )u = 1 ρ pi +2µ ss +Π), t A+u )A = 1 λ A I)+A u+ u)t A. The third step is the coupling between the bulk of the two schemes : the advection-diffusion scheme with source term receives as input the velocity field and the source term see Eq. 21)) given by G = 1 λ A I)+A u)+ u)t A, from the Navier Stokes scheme, and the Navier Stokes solver receives an external forcing term see Eq. 6)) expressed by µp ) ρg = Π = λ A I), 36) computed from the advection-diffusion scheme with source term. Finally, in the fourth step, the boundaries have to be coupled. A more detailed explanation of the procedure is given hereafter the 2D case is presented for simplicity, but the generalization in 3D is straightforward) : 1. Collision and propagation of the advection diffusion with source scheme see Eq. 16)) h iαβ x+ξ i,t+1) = h iαβ x,t) 1 ϕ ϕ h iαβ x,t) h 0) ) Gαβ A αβ h 0) iαβ A,u). iαβ A,u) ) 2. Collision and propagation of the forced Navier Stokes schemesee Eq.8)) f i x+ξ i,t+1) f i x,t) = 1 ) f τ i x,t) f 0) i ρ, u) ) F i, 2τ where ρg = Π. 12

13 3. Coupling of the bulk of the two schemes. On the one hand, with the Navier Stokes scheme, we compute the velocity and its gradients which will be used for the computation of the equilibrium and the source of the advection diffusion scheme. The velocity gradients are determined with a centered finite difference scheme x ux,y) = ux+1,y) ux 1,y), 2 ux,y +1) ux,y 1) y ux,y) =. 2 On the other hand the advection diffusion scheme enables one to estimate the conformation tensor, which is transformed into the viscoelastic stress tensor,π,usingeq.6). ThedivergenceofΠiscomputedusingacentered finite difference scheme x Πx,y) = Πx+1,y) Πx 1,y), 2 Πx,y +1) Πx,y 1) y Πx,y) =. 2 The quantity ρg = Π is then transferred to the Navier Stokes solver. The evaluation of the derivatives could be made more precise by using more advanced schemes such as the ones found in Tomé et al. [49] and Fattal and Kupferman [50]. 4. Coupling of the boundaries of the two schemes. On the boundaries, the regularized technique of Latt and Chopard [46] is applied for the Navier Stokes solver and the scheme discussed in the preceding subsection see Eq. 29) and 30)) is applied for the constitutive equation. This boundary condition was chosen because of its improved stability compared to more classical boundary conditions see Latt et al. [47]). The divergence of the viscoelastic stress and the velocity gradients cannot be computed using a centered finite difference for the direction perpendicular to the wall. Therefore a backward finite difference scheme is used. For a wall, as depicted in Fig. 4, this leads to: 3ux,y)+4ux,y 1) ux,y 2) y ux,y) =, 2 3Πx,y)+4Πx,y 1) Πx,y 2) y Πx,y) =. 2 At this point, all the needed pieces of information for the schemes are provided to proceed to the next time-step. Nevertheless, we point out that the macroscopic quantities that are used here are taken at time t since the LBM scheme is explicit in time. 4. Validation of the model In this section the numerical model presented in the previous section is validated on two- and three-dimensional benchmarks. We first compare the energy 13

14 decay of the 3D Taylor Green vortex with a high accuracy pseudo spectral Fourier code devised by Boeckle [51], then with the results obtained for a simplified 2D four-roll mill with analytical approximations. Finally we test the order of accuracy of our method in the 2D steady Poiseuille case, where there exists an analytical solution for the case of an Oldroyd-B fluid. In order to characterize the viscoelastic interaction, two non-dimensional numbers are defined, the Weissenberg Wi number and the kinematic viscosity ratio, R ν Wi = λ γ, 37) R ν = ν s ν p +ν s, 38) where λ is the relaxation time of the polymer and γ the characteristic shear rate of the flow, while ν p and ν s are respectively the polymer and solvent kinematic viscosities. The Reynolds number, Re, is defined as Re = UL ν s +ν p, 39) with U and L respectively the characteristic velocity and length of the flow. In the lattice Boltzmann method these non-dimensional numbers are only used to recover the macroscopic quantities in physical units, since the equations are solved in lattice units see Appendix D for details about the units conversion) Taylor Green vortex In order to validate the numerical scheme discussed in Section 3, we will use the Taylor Green vortex benchmark see Brachet et al. [52]). The simulation setup is the following. The physical domain is a periodic box [0,2π] [0,2π] [0,2π], where we have a prescribed velocity profile at time t = 0 u x x,t = 0) = 1 3 sinx)cosy)cosz), 40) u y x,t = 0) = 1 3 cosx)siny)cosz), 41) u z x,t = 0) = 2 3 cosx)cosy)sinz). 42) Initially, the viscoelastic stress tensor is taken equal to zero everywhere. At various time steps the computed energies of the solvent and of the dumbbells are compared to the ones determined with a high accuracy Fourier pseudospectral algorithm designed by Boeckle [51]. The average kinetic and the spring energies are computed as E kin = 1 M E spring = 1 M k k u 2 x k ), 43) 2 trπx k )), 44) 2 14

15 where k ranges over all the nodes of the simulation, and M is their total number. As in Boeckle [51], the benchmarks will be run until t = 5 dimensionless units for Re = 1, Wi = 1,10, and R ν = 0.1. In the reference solution of Boeckle, the number of Fourier modes is on 64 3 for Wi = 1 and 96 3 for Wi = 10. The characteristic length of the flow is taken to be the length of the periodic box L p = 2π, L lb = N), the characteristic velocity U p = 1, U lb = 0.01 and the characteristic shear rate γ = U/L. Before proceeding to the simulation, special care about the initial condition must be taken. As discussed in Mei et al. [53] or Latt [54] for a proper initialization of the problem the distribution functions, f i must be given by their Chapman Enskog expansion. For the incompressible Navier Stokes model, they are given by f i = f 0) i +εf 1) i = w i ρ 1+ ξ i u c 2 + Q ) i : uu s 2c 2 s w iτρ c 2 Q : S. 45) s Thevelocityfieldandthereforethestrainratetensorareknown. Weareonlyleft with the computation of the initial density field ρ. Itis determined by taking the divergence of the momentum conservation equation. Using the incompressibility condition, u = 0, we find that the pressure field must be solution of a Poisson equation 2 p = u : u. 46) In our case this equation has an analytical solution given by px,t = 0) = cos2y)+ 1 4 cos 2y +2x)+ 1 4 cos2x+2y)+cos2z) cos 2z +2y)+ 1 1 cos2z +2y)+ cos2z +2x) cos 2z +2x)+ 1 ) 4 cos2x) 1. 47) Then, from the perfect gas law, the initial density is given by : ρ = p c 2. s Since the viscoelastic stress tensor Π is zero initially, the conformation tensor A is given by A = I, 48) which states that the dumbbells are at equilibrium. As the initial velocity field must be incompressible, the initial h iαβ are simply given by their equilibrium distribution h iαβ x,t = 0) = w i A αβ 1+ ξ ) i u c 2. 49) l As can be seen on Fig. 5 the LBM energy curves for all Wi numbers tested coincide with the Fourier results for a number of lattice grid points in each direction 15

16 N = 100. Let us discuss briefly the simulated physical phenomenon. At the beginning of the simulation the dumbbells are at equilibrium. Then due to the friction between the solvent and the beads, the dumbbells start to stretch. When all the kinetic energy is transformed in spring potential energy, the velocity gradients are no longer strong enough to continue stretching the beads. As a consequence, the dumbbells start to give back their energy to the solvent. Therefore a peak of kinetic energy is reached which in turn coincides with a minimum of the spring energy. This process continues until all the energy of the dumbbells and of the solvent has been dissipated through viscous processes. As can also be noticed from the plots, the higher the Wi number is the longer the period between two peaks can be observed. This effects can be easily understood since a high Wi number means a high relaxation time of the dumbbells. A simulation Figure 5: Kinetic left column) and spring right column) energy evolution with time in nondimensional units for R ν = 0.1 and Wi = 1 top row) and Wi = 10 bottom row) for the Oldroyd-B constitutive model. The solid line is the reference solution while the circles and the dots are solutions of the LBM simulation for two numbers of lattice grid points in each directions N = 25 and N = 100. with the non-linear FENE-P model was performed for Wi = 1 and R ν = 0.1 with the maximum dumbbells length r 2 e = 10,25. As can be seen in Fig. 6 the energy curves have a shape similar to those obtained with the Oldroyd-B model 16

17 corresponding to r 2 e =. Furthermore, as expected, the FENE-P energy curves are approaching those of the Oldroyd-B model for r e. After testing the Figure 6: Kinetic left) and spring right) energy evolution with time in non-dimensional units for R ν = 0.1 and Wi = 1. The solid line is the solution obtained with the Oldroyd-B model while dotted and dashed lines are the FENE-P model with r 2 e = 10 and r2 e = 25. accuracy of the scheme by comparing it to the pseudo-spectral code, a stability test has also been performed. Accuracy is not the main issue, but rather the numerical stability at high Wi numbers. A fixed value R ν = 0.1 has been used for all tests. The simulations are stopped when the total energy falls below 0.1% of its initial value, which means E = E kin + E spring All the tests were carried out with a resolution of N = 25 and up to Wi = The scheme was found numerically stable for all values of Wi in that range Simplified four-roll mill This test case consists in simulating the effect of four cylinders that rotate in such a way that they create an elongational flow in the vicinity of a central stagnation point between the rollers. A sketch of the configuration is given in Fig. 7. Here the rollers are replaced by a body force to drive the flow, which has Figure 7: The 2D projection of the four roll mill. the effect of producing four vortices at the location of the rollers as considered in Thomases and Shelley [33]. This enables one to consider periodic conditions over the domain boundary and suppresses the need for boundary conditions on 17

18 the cylinder walls. At the central hyperbolic point, the state of the fluid is described by a simple elongational flow, which enables us to compute analytically local solutions at steady state that will be compared with our simulations. The geometry is a 2D periodic box of size [0,2π] [0,2π]. The flow is supposed to be at very small Re number and is imposed by a body force g = 2ν s sinx)cosy), cosx)siny)), 50) computed in order to impose a velocity field given by u = sinx)cosy), cosx)siny)). 51) The initial viscoelastic stress tensor is set to zero. In the central region, near the stagnation point x = π,y = π), the velocity field is given by u = εx, εy) = x u x π,π), y u y π,π)), 52) where ε is the local elongational rate. An effective Weissenberg number Wi eff can be defined as Wi eff = εwi, 53) with Wi = λu max /L with U max the maximum velocity of the flow and L = 2π). It is called effective because it scales the actual Weissenberg number by the local rate of strain at the hyperbolic point and appears simply in the equations. When the simulation reaches steady state t ), as shown by Thomases and Shelley [33] the components of the conformation tensor can be expressed as a good approximation by : 1 A xx π,y) = 1 2Wi 1 A yy π,y) = eff ) +C y 1 2Wi eff + y 2+1/Wi eff 1+2Wi Wi eff, 54) eff), 55) A xy π,y) = 0. 56) where C is a constant. These expressions can be used for comparison with the results of our simulations. As an illustration, in Fig. 8 the vorticity and tra at time t = 6 are depicted. As can be seen there is a major difference between the Wi = 0.6andWi = 5. WeseethatathighWinumbersasecondaryflowappears between the rolls since the vorticity field changes sign. In our simulations, the velocity field is initialized with Eqs. 51) and the density is taken as ρ = 1. The body force of Eq. 50) is applied on the domain. Before solving the full system of equations for the viscoelastic fluid, the simulation is started by considering the Newtonian solvent only i.e. by solving only the Navier-Stokes problem without retaining the contribution of the viscoelastic stress tensor. Once this step is completed, the real simulation is started by coupling the solver of the massmomentum conservation equations with the one for the Oldroyd-B constitutive 18

19 Figure 8: The vorticity left column) and tra right column) for Wi = 0.6 and Wi = 5 top and bottom row respectively) at time t = 6. equation. The values of the conformation tensor and of the elongational rate ε are then taken when the simulation has reached steady state, which happens between time t = 10 and t = 50 depending on the Wi number. The simulations were run for Wi = From Fig. 9 it can be seen that ε is decreasing with increasing Wi. This result is qualitatively very close to the results found in Thomases and Shelley [33]. Figs. 10 and 11 show a very good agreement between our simulations and the analytical results of Eqs. 54)-56) for the conformation tensor values. Nevertheless, one can notice that the component A xy of Fig. 11, at values of Wi eff close to one starts to deviate from zero. This discrepancy should be compared with the characteristic value of A xy at other locations in the flow for the given Wi eff, which is several orders of magnitude higher Steady 2D Poiseuille In order to test the numerical scheme on domains with boundaries, the planar Poiseuille benchmark consisting of the steady shear flow between infinite parallel plates has been selected see Fig. 12). We also selected this test case in order to verify the influence of the polymer diffusivity κ. The length and height of the channel are respectively L x and L y. The flow is usually driven by a constant pressure gradient, which is replaced here by a constant body force 19

20 Figure 9: The elongational rate ε with respect to the Wi number left) and the effective Weissenberg number Wi eff with respect to the Wi number. Figure 10: The A xx components as a function of the y position for x = π at Wi = 0.3 left) and Wi = 1.0 right). The dots are the results of the LBM simulation, while the plain line is the analytical result of Eq. 54), where we fitted the constant coefficient C. g x to obtain the same effect. In order to avoid corners in the geometry, the domain is made periodic in the x-direction. The top and bottom walls have no-slip boundary conditions. There exists an exact solution for the Oldroyd-B constitutive equation which is given by : u x = g x 2 y 2 L y y ν s +ν p ), u y = 0, 57) A xx = 1+ g2 xλ 2 ν p 2y +L y ) 2 2ν s +ν p ) 2, 58) A xy = g x 2 λ2y L y ) ν s +ν p, 59) A yy = 1. 60) Defining the reference velocity as the maximum velocity of the flow U lb, which by symmetry is located at y = L y /2 as it can be seen from the previous equation 20

21 Figure 11: The A xy left) and A yy right) components as a function of the effective Weissenberg number Wi eff at the stagnation point. The dots are the LBM results while the full line is the analytical solution of Eqs. 56) and 55). Figure 12: The 2D planar Poiseuille, where U lb is the maximum velocity of the flow. giving the expression for u x, the force is given by : g x = 4 L 2 ν s +ν p )U lb. 61) y For our simulations we chose L x = L y = N. Before showing the accuracy results obtained for this flow, let us first discuss thechoiceofthediffusivityconstantκ. WemadetwosetsofsimulationsforN = 200 and Wi = 0.1,1 and measured the variation of the error of our simulations for κ/µ p [10 7,10 3 ] with respect to the analytical solution for A xx see Eq. 58)). The results are depicted in Fig. 13. One can see that for κ/µ p = 10 6 theerrorisassmallaspossibleforwi = 0.1andWi = 1. Whenκ/µ p < 10 6 the simulations for Wi = 1 became numerically unstable. This compromise between accuracy and stability lead us to chose κ/µ p = q0 6 for all our benchmarks. In Figs , the velocity field and the components of the conformation tensor obtained from the numerical simulations for Wi = 0.1,1 and from the analytical solution are plotted. The error between the lattice Boltzmann solution and the reference solution for this flow has also been computed. In addition, both orders of accuracy for the velocity field and the conformation tensor have 21

22 Figure 13: The error of A xx with respect to κ/µ p Figure 14: The velocity field with respect to the y position in lattice units for Wi = 1, R ν = 0.1, and Re = 1 for the LBM simulation and the analytical solution. been measured. The errors are defined as E u = 1 M 1 2 u lb x k ) M u analyt. x k ) U lb, 62) k=0 E Aαβ = 1 M 1 2 A lb x k ) M αβ A analyt. x k ) αβ, 63) k=0 and again the sum is performed on the M nodes of the lattice, which are located on x k. The simulations were run at Re = 1, R ν = 0.1,0.7 and Wi = 0.1,1. The numerical stability of the code has also been tested. In this case R ν = 0.1 was kept fixed, the Wi number was varied in the interval [0.1,10] and we tested the numerical stability of the code by varying the resolution from N = 10 up to N = 200 lattice sites. The accuracy of the solution was not taken into account in this analysis. 22

23 Figure 15: The A xx component of the conformation tensor with respect to the y position in lattice units and the analytical solution for R ν = 0.1, Wi = 0.1 left) and Wi = 1 right). Figure 16: The A xy left) and A yy right) components of the conformation tensor with respect to the y position in lattice units and the analytical solutions for R ν = 0.1 and Wi = 0.1,1. As can be seen in Figs. 17 and 18, the error of the velocity field, in the Wi = 0.1 and Wi = 1 cases, is decreasing with order two. The errors are of the same order magnitude for R ν = 0.1 or R ν = 0.7. For the conformation tensor, it can be observed that the errors on the components A xx and A xy are decreasing more slowly, in fact the order is of 1.5 and 1.4 for respectively Wi = 0.1 and Wi = 1. For the A yy component the accuracy seems to have reached its minimum value for all the lattice resolutions since it does not really decrease or increase) when the resolution is increased. As can be seen in Fig. 19, the maximum stable Wi number depends on the resolution used, ranging from Wi = 1 for N = 25 to Wi = 10 for N = 200. The numerical stability increases with the increase of resolution. All these tests were performed for κ/µ p The possible reasons for the failure of the simulations at higher values of Wi are discussed in the next section. 23

24 Figure 17: The L 2 -error of the conformation tensor and of the velocity with respect to the resolution in lattice units for Re = 1, R ν = 0.1 and Wi = 0.1 left) and Wi = 1 right). Figure 18: The L 2 -error of the conformation tensor and of the velocity with respect to the resolution in lattice units for Re = 1, R ν = 0.7 and Wi = 0.1 left) and Wi = 1 right) Summary and further comments To summarize, the three test cases described in the previous section have enabled us to validate the numerical schemes for Oldroyd-B and FENE-P class of viscoelastic fluids. The results found were in good agreement with analytical and numerical results. The scheme could therefore be an alternative to classical schemes. Furthermore, the order of accuracy of the method was between 1.5 and two depending on the value of the Wi number. We have been able to compare the scheme with a high accuracy Fourier pseudospectral algorithm for the Taylor Green vortex case and found identical results for the energy decay of the solvent and of the polymers, using the Oldroyd-B constitutive equation. The difference between the results found with the pseudospectral code was of less than 0.1%. Furthermore the code was shown to be unconditionally stable, even at a very high unphysical) value 10 6 ) of the Wi number. We have also implemented the FENE-P model but our approach 24

25 Figure 19: The maximum Weissenberg number Wi max with respect to the resolution N. could be easily extended to other models. In the simplified four roll mill problem, we were able to reproduce accurately the analytical results for an elongational flow that were obtained by Thomases and Shelley [33] in the steady state limit. In the Poiseuille flow case, it has been shown that the numerical result was in close agreement with the analytical solution and therefore the boundary condition treatment for flat walls suggested in Section 3.3 see Eq. 30)) seems to be correct. For stability reasons, we were not able to reach a value of the Wi number higher than 10 for R ν = 0.1. Although results were better with higher R ν, one of the major issues here is the lack of understanding of the behavior of the viscoelastic stress tensor at the walls. The tests performed by prescribing the analytical solution for the conformation tensor on the boundaries allowed to simulate flows with Wi = O100). It must also be noted that increasing the polymer diffusion κ also increased the stability of the model. This is simply because the relaxation time of the advection diffusion scheme departs from 1/2 as κ is increased 2. In addition, we were unable to determine suitable boundary conditions for more complicated geometries, including corners, the simulation of the 4:1 contraction were always unstable even at low Wi number). In order to improve this defect, we are now investigating more deeply the modeling of the boundaries for such flows. A possible solution for the boundary conditions problem would be to couple the LBM solver with a molecular dynamics solversee Hernández-Ortiz et al.[55] and Izmitli et al. [56]). Having a physically relevant value for the conformation tensor on the boundaries would allow one to impose a physically meaningful 2 The limit ϕ 1/2 is well known stability issue for the lattice Boltzmann. 25

26 boundary condition for the viscoelastic constitutive scheme. Thus, there would be no need to resort to interpolated or extrapolated values that are mainly responsible for the lack of stability of our code. 5. Conclusion In this paper, a novel lattice Boltzmann scheme for the simulation of viscoelastic fluid flows which exhibit memory effects, has been presented. Our approach was found to be in good agreement with analytical results and other numerical techniques for very different kind of flows. It would therefore be an alternative solution for the simulation of viscoelastic fluids of the Oldroyd-B class. Although only the Oldroyd-B and FENE-P constitutive equations have been considered, other models can be easily implemented as the advection diffusion with source model that was used for performing all these simulations can account for any constitutive equation which has the same form. The methodology proved successful in solving the Taylor Green decay vortex case, the steady simplified 2D four-roll mill and the 2D channel flow. In the Taylor Green vortex benchmark, the results obtained with a high accuracy spectral method for values of the Weissenberg number Wi = 1, 10 have been reproduced accurately with the proposed lattice Boltzmann method. At higher Wi numbers, the simulation was shown to be also stable and the overall shape of the energy curves remains the same, but the simulation time becomes much longer since the bounces observed in Fig. 5 take longer time to develop as Wi increases. The analytical results available for the 2D four-roll mill case were reproduced accurately. A method for taking into account flat wall boundaries in our simulations has also been described, which represents a first step towards the study of more complex geometries. Results of simulations have been compared to the analytical solution given for a 2D Poiseuille flow. It could be observed that at moderate values of the Weissenberg number Wi 0.1 the accuracy was second order in space, while it was decreased to order 1.5 for Wi = 1. In agreement with results of other methods, we also noticed that the stability of the computational model strongly depends on the viscosity ratio. Our computational tool could now be applied to more challenging cases like the tracking of elastic instabilities for shear flows such as the Kolmogorov flow see Berti et al. [57]) or simplified elongational flows without walls. Other more sophisticated constitutive equations, which would contain more physical ingredients and therefore reproduce better the behavior of the viscoelastic fluids in specific situations, could also be easily implemented. The next issue to be investigated is to obtain a better understanding of the boundaries in order to be able to use the ability of the lattice Boltzmann to easily simulate complex geometries. To this effect, it could be interesting to try to couple molecular dynamics models with the lattice Boltzmann method, since in this case it is easier to introduce more refined mesoscopic modelings see Hernández-Ortiz et al. [55] and Izmitli et al. [56] for example). 26

27 6. Acknowledgements We would like to thank Drs. J. Latt, E. Grandjean and A. Malaspinas as well as R. Vonlanthen for the enlightening discussions shared and their useful comments. O. Malaspinas also thankfully acknowledges the support of the Swiss National Science Foundation SNF under grant number FN ). Appendix A. Lattice Boltzmann scheme for weakly compressible Newtonian fluids It is well known see Huang [58] for example) that the continuous BGK approximation of the Boltzmann equation df dt = tf +ξ f +F = 1 τ f f0) ), A.1) where τ is the relaxation time, F is the forcing term, taken in the small Knudsen number limit leads to the compressible Navier Stokes equations with the dynamic viscosity given by µ s = c 2 sτρ, A.2) where c s is the speed of sound and ρ the density. This equivalence is shown through a Chapman Enskog expansion where the distribution function is assumed to be made of an equilibrium part plus a small perturbation proportional to the Knudsen number f = f 0) +f 1), f 1) f 0). A.3) We will not go into the details of the derivation but rather mention that the macroscopic equations recovered from the Boltzmann equation do not depend on the actual form of f and f 0) but only on their moments. This implies that when discretizing the microscopic velocity space, one only wants that the integrals used to compute the moments are exactly recovered by sums. In our case, we want to simulate only the weakly compressible limit of the Navier Stokes equations, thus it is sufficient to impose mass and momentum conservation and require that the following moments of the equilibrium distribution are given by dξ f 0) = f 0) i = ρ, A.4) i dξ ξf 0) = i dξ ξξ I)f 0) = i ξ i f 0) i = ρu, A.5) ξ i ξ i c 2 si)f 0) i = ρuu, A.6) 27