Cristòbal, Enric

Riera, Pau

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Viscoelastic Fluid Simulation with Lattice-Boltzmann MethodsEnric Cristòbal and Pau RieraProjectes d’Enginyeria F́ısica 2Universitat Politècnica de Catalunya(Dated: May 31, 2018)The Lattice Boltzmann (LB) method has established itself as a tool with growing acceptancefor the numerical simulation of fluid flows. It is a novel numerical method for fluid dynamics thatrelies on probability distribution functions for discrete particles moving on a lattice following simplecollision rules, recovering Navier-Stokes dynamics. Generally, fluids are grouped in Newtonian andnon-Newtonian, depending on the relation between viscosity and shear rate. The main goal in thisproject is to make a detailed study of non-Newtonian fluids and compare them with Newtonian fluidsresults. To this end, different simulations of fluids until the convergence to the stationary fluid in achannel will be done. As well, the velocity profiles of a Newtonian fluid and a non-Newtonian one,following the Carreau model, will be analyzed.I. INTRODUCTIONWhen talking about fluid motion, one rapidly comesacross the Navier-Stokes equations. These equations de-scribe how the mass, momentum, and energy of the fluidchange with position and time. Anyway, some advancesin the theory have made Lattice Boltzmann (LB) methodto be competitive when studying complicated fluid flows,as it is the case of complex fluids, the appearance of tur-bulence, etc. Thus, all this project will be developedwith the LB method in order to replicate the fluid flowof recent experiments on complex fluids [1].The LB method is a modern approach in Computa-tional Fluid Dynamics, concretely it is a discrete com-putational method based upon the Boltzmann equation,an analogue to the Navier-Stokes equation at a molecularlevel, where it describes the space-time dynamics of a sta-tistical quantity called probability distribution function.LB method finds its origin in a molecular description of afluid: it considers a typical volume element of fluid to becomposed of a collection of particles. The local flow ateach grid point is represented by a distribution functionfor each possible velocity value. The time is counted indiscrete time steps and the fluid particles can collide withone another as they move, possibly under applied forces.The rules governing the collisions are designed such thatthe time-average motion of the particles is consistent withthe Navier-Stokes equation [2].Considering the discretization of the fluid, it is crucialto explain the so called DnQm scheme: the moleculardescription of the fluid will define certain nodes, whereparticles can place, and different possible connections be-tween them. Then, based on the dimensions of the lat-tice involved and also on its corresponding nodes differ-ent values to n and m shall be given in order to definethe desired model. For instance, D2Q9 would be a 2-dimensional structure and each particle could either goto the vertices or the center of the sides, as well as stay-ing at the center of the square, having an overall of 9movement possibilities (see Fig. 1).Once the model that is going to be used along thisproject has been introduced, it is necessary to explainFIG. 1 Nodes and movement options of the lattice ofthe molecular description used in 2D and 3D simulationsof LB method for the D2Q9 and D3Q19 lattices.some theoretical differences between Newtonian and non-Newtonian fluids. On the one hand Newtonian fluids arethe ones that viscosity remains constant, no matter theamount of shear applied. However for non-Newtonianfluids viscosity can have a dependence on the shear rate[3]. As this varying behavior is not easy to fit many mod-els have appeared trying to relate viscosity and shear ratewith experimental results. One of them is the Carreaumodel, where viscosity and shear rate are related by thefollowing equation:ν = νinf + (ν0 − νinf )(1 + (λγ̇)2)N−12 (1)where ν0 and νinf are the fluid viscosity at zero and infi-nite shear rate, respectively. λ is the relaxation time, γ̇ isthe shear rate and N is the power. An example to provethat this equation fits perfectly with experimental resultsis shown in figure 2. Nevertheless, in the simulations νinfwas set to 0, as it was the only instance that had beenimplemented for the velocity inlet boundary conditions.As a consequence of these different behaviors, theirresponse to an applied force, or the introduction of aPoiseuille flow in one boundary of the volume shall bedifferent, too. So the main goal in this project was tocompare simulations carried out by a specific softwareto theoretical results. These simulations contained both2FIG. 2 Non-Newtonian viscosity of three polymersolutions as a function of the shear rate and a soapsolution fitted by the Carreau equation. Reproducedfrom [3].approaches: a Poiseuille flow in a Newtonian fluid andthe same flow for a fluid that obeys the Carreau model.II. METHODSAs stated before, a software was needed to reproducethe desired situation by coding. This way, two simi-lar softwares could have been used to develop the wholeproject: Palabos, which was finally the chosen one, andOpenLB. These two softwares are, as its own developersdefine them, open-source CFD (Computational Fluid Dy-namics) solvers based on the Lattice Boltzmann method.Even though the same final result should have been ob-tained with both of them, it was mutually agreed to workwith Palabos. The main reason may be the higher levelof detail provided by its user’s guide, that included moretheoretical references and was in general more intuitive,specially for people that are not used to working withthis kind of tools.Thus, to take a step forward it was first needed toget introduced into the programming world. First of all,many tutorials from the Palabos website had to be fol-lowed in order to understand as much as possible of thesoftware itself and how to make use of it. All this processwas vital to get into the simulation world and especially,into the Palabos world. Then, finding out what changesin the code had to be made in order to obtain the condi-tions for the desired experiment was the next step.After some research and a lot of acquired coding knowl-edge, it was decided to simulate how two different type offluids, Newtonian and non-Newtonian (Carreau model)behaved when a Poiseuille profile entered a 2D channel.Although many of the used parameters were dimension-less, the main idea was to simulate both experiments in achannel. Basically, this channel was build up by definingthe parameter n (typically 60) controlling the number ofcells. After that, there were another two dimensionlessparameters (lx and ly) that were used to define the sizeof the simulation. This way, any value given to thesetwo parameters, for instance lx, would mean that thesize of the channel on this direction would be lx · n inthe x-direction, equivalent to the number of cells. Fi-nally, it was decided to take lx = 3 and ly = 1, thusthe channel had a length of 180 cells and a height of60 cells, so it would look alike a rectangle. The realsystem size, and thus the physical width of each cell, isimplicitly determined by the Reynolds number and thespecific physical parameters of the problem. This casewas really interesting since an accurate theoretical solu-tion could be obtained in the laminar (i.e. small Re)regime and compared to the simulated results. This casewas really interesting since an accurate theoretical solu-tion could be obtained and compared to the simulatedresults. Moreover, this comparison between both fluidswould lead to the obtaining of two completely differentbehaviors with these conditions, confirming the theoreti-cal calculus. Many velocity profiles and center velocitieswere studied in detail to fully understand the physicalprinciple that was behind this project.It is noteworthy to know that in both scenarios thewalls were non-periodic, that is, they would not be con-nected to each other to keep regular flux. In addition tothat, the top and bottom walls had no-slip conditions,that is, the velocity on the highest and lowest points wasequal to zero. This boundary condition was implementedby means of the bounce-back algorithm [4].The main parameters of the simulation are as fol-lows: Reynolds number Re = 10, n = 60 and the max-imum velocity of the Poiseuille flow at the boundariesVmax = 0.05. For the non-Newtonian fluid: N = 0.5 andλ = 12000 (which corresponds to a Weissenberg numberWi = 10).After the harsh procedure of modifying the mentionedcodes in order to get the wished numerical approach, thedata was treated with MATLAB to acquire visible re-sults. It was also decided to have an insight into theconvergence of the codes depending on the size of thechannel and the divergence from the theoretical results.III. RESULTSA. Simulations convergenceFirst of all, a research on the computational efficiencyof the software was carried out changing the resolutionof the simulation. Apart from studying the change inthe center velocity of both fluids depending on the size ofthe lattice, it was also examined the minimum dimensionneeded to achieve an acceptable simulation error with re-spect to the analytical prediction of the velocity profile.3Surprisingly, both fluids showed a similar behavior if cen-ter velocity and simulation error were analyzed for grow-ing n. Nevertheless, the results obtained were not exactlyequal and some conclusions may be obtained from thiscomputational study.Considering the center velocity of the fluid behaviorone can observe the same tendency to the same veloc-ity as n gets higher. However, it reaches this final valuein a different way depending on the type of fluid. Onecan check this discrepancy in figure 3. From the graphFIG. 3 Center velocity of Newtonian andnon-Newtonian fluids as a function of n. In the case ofthe latter, the minimum n was 6 due to simulationissues.it might seen a pretty reliable tendency to the definitivevelocity, not requiring a high value of n. In contrast,not only the non-Newtonian fluid’s velocity for lower nis much distinct from the tendency value than the New-tonian one, but it also does not achieve a significant con-vergence until n = 20. On the other hand, Newtonianfluid reaches a difference of 0.05 from n = 10 to be seenin an smoother graph. Finally, it’s worth mentioning theconvergence for very low values of n: in spite of gettingresults with the Newtonian case, the Carreau adaptedmodel seemed to be divergent until n = 6, fact thatevinces the higher complexity of the model itself. Thefitting curves don’t suit any specific function type dueto the fact that their behavior isn’t a common structure.Nevertheless, they were plotted just to observe an overalltendency.Another aspect concerning center velocity could behow precisely it approaches a theoretical value. In orderto do that, the software included a command that allowedto compute an RMS (Root Mean Square) error that pro-vided a way to check the accuracy of the simulations. Tomake this clear, the error compares the simulated val-ues with those theoretically found and then calculates itsRMS.From figure 4, one could notice that both modelsare pretty accurate to their respective theoretical result,FIG. 4 Error for a Newtonian fluid as a function of n,chosen from 2 to 60 as well.since the error becomes rapidly really small. Whereasthe error in the Newtonian case decreases faster, it takesa slightly higher number of n for a non-Newtonian fluidto be precise enough. Both fitting curves share a com-mon element: they are rational functions that tend to 0when n → ∞ but having different polynomial degrees.Despite this tendency, there’s a tiny marginal increase ofthe error by n = 40.B. Velocity profilesAs explained in section II, the main objective is thecomparisons of Newtonian and non-Newtonian fluids re-sponses to a Poiseuille profile. In principle, once the sim-ulation is done, the velocity profile of these two type offluids should be quite different. Basically, the Newto-nian fluid profile should be much peaked than the non-Newtonian one due to different dependence of viscosityon the local shear. The resulting profiles, obtained in thecenter of the box (x = 90) and all along the y-axis, areplotted in figure 5.Once both simulations had been run, the second goal ofthe work was to provide a clear vision of the response toa Poiseuille flow by a non-Newtonian fluid changing theλ parameter of the Carreau model. This parameter is amicroscopic relaxation time of the fluid, which affects thetendency of the fluid to reach an stationary state after anapplied disturbance. As a result, in figure 5 is seen thatfor a lower λ the fluid tends to resemble the behavior ofa Newtonian fluid.C. Transient of the flowAdditionally, another difference between these twotypes of fluids arises when speaking of convergence tothe final steady flow. Although in both cases there is a4FIG. 5 Comparison of the velocity profiles betweenNewtonian and non-Newtonian fluids simulations, andbetween non-Newtonian profiles changing λ.transient where velocity is not so stable, when some sta-bility is reached it appears a difference since Newtonianfluids tend to achieve a constant velocity after some itera-tions whilst non-Newtonian ones keep oscillating duringa much longer transient around a more or less definedvelocity value, as seen in figure 6.FIG. 6 Transient evolution of fluid’s center velocityfor Newtonian and non-Newtonian fluid.Also, some outcomes of the simulations for the Newto-nian fluid were recorded and can be observed in figure 7.From the images one could hypothesize about the com-pressibility of the fluid. Despite the fact that there is aflow of velocity, it is not constant, what would mean thatthe fluid itself is compressible. Instead, a non constantflow may indicate that it actually is uncompressible. Thiscould also be related to figure 6, since their initial oscilla-tions may be produced by stationary waves created at thebeginning. Finally, it’s worth mentioning that notwith-standing the quick convergence of the Newtonian fluid,the non-Newtonian one has a larger transient, probablydue to the factor λ introduced as a relaxation time.FIG. 7 Simulated Newtonian fluid response to aPoiseuille flow over time. As represented, the warmerthe color, the greater the modulus of velocity in thatpoint.IV. CONCLUSIONSAs a result of all the insight into the Lattice Boltz-mann method it is more than justified the reason whythis model is gaining an increasing interest for treatingfluid flows. As stated in section I, this method appearedin order to simplify some fluids problems. Moreover, ithas been shown that it is very stable and that by fol-lowing some simples procedures one can simulate manysituations as complex as desired.As well as a great knowledge of Lattice Boltzmannmethod, an incredible insight into fluids field has beenachieved. From all simulations the theoretical differ-ences between Newtonian and non-Newtonian fluids weredemonstrated throughout this work.Finally, in a more personal way, along this project alot of new knowledge has been acquired as it involvedmany new aspects never treated before. One of the mostenriching parts for our hypothetical professional futurewas when this project was first tackled, talking in termsof getting used to the software field, so necessary in sim-ulations environment. All this project was a discoveringprocess for both of us. As well as learning a lot about ofnew parts of simulation, much more that we could imag-ine, a completely new world in physics was opened forus, as it is Physics of fluids.ACKNOWLEDGMENTSThis project was born from the curiosity and ambitionthat Dr. Prof. Laureano Ramı́rez evokes at all time.We would like to thank him for all the support, evenwhen were lost he succeeded in finding a path to newdiscoveries.5[1] L. Casanellas. Oscillatory pipe flow of wormlike micellarsolutions. Universitat de Barcelona, 2013.[2] S. Succi. The Lattice Boltzmann equation : for fluid dy-namics and beyond. Oxford : Clarendon Press, 2001.[3] R. Byron Bird, Robert C.Armstrong, and Ole Hassager.Dynamics of polymeric liquids, Volume 1 Fluid mechanics,volume I. 1987.[4] Palabos. Lattice boltzmann method - the ker-nel of palabos. http://www.palabos.org/software/lattice-boltzmann-method, note = [Online; accessed 12-April-2018], 2012.[5] Palabos. User’s guide. http://www.palabos.org/documentation/userguide/, note = [Online; accessed 12-April-2018], 2011.[6] Dieter A.Wolf-Gladrow. Lattice-Gas Cellular Automataand Lattice Boltzmann Models: An introduction. Berlin,Heidelberg : Springer Berlin Heidelberg, 2000.[7] J. Meng, X.-J. Gu, and D.R. Emerson. Slip velocity oflattice boltzmann simulation using bounce-back boundaryscheme. 2015. [Online; accessed 28-May-2018].

Viscoelastic Fluid Simulation with Lattice-Boltzmann Methods

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Viscoelastic Fluid Simulation with Lattice-Boltzmann Methods

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