In this study we are interested on the statistics of finite sized particles. A sustained homogeneous isotropic turbulence with particles is numerically simulated in order to obtain those statistics

Brandle de Motta, Jorge

Climent, Éric

Estivalezes, Jean-Luc

Vincent, Stéphane

English

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   Open Archive TOULOUSE Archive Ouverte (OATAO)  OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.  This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 10473 To cite this version : Brandle de Motta, Jorge and Estivalezes, Jean-Luc and Climent, Eric and Vincent, Stéphane Simulation of fully resolved finite-size particles in a turbulent flow. (2013) In: Euromech Colloquium 555 : Small-scale numerical methods for multi-phase flows, 28 August 2013 - 30 August 2013 (Bordeaux, France). (Unpublished) Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr Simulation of fully resolved finite-size particles in a turbulent flowJ.C. Bra¨ndle de Motta, J.L. Estivalezes, E. Climent and S. Vincent1 IntroductionThe particle laden flows are presented in both natural and industrial frameworks. The point-wiseapproach to simulate particulate flows has been used during the last 30 years. Thanks to thisapproach many phenomena have been understood. Nevertheless, this approach can not be used forparticles which size is larger than the smaller length scale of the fluid flow. That is the reason thatit is necessary to analize those phenomena for finte-size particles.In this study we are interested on the statistics of finite sized particles. A sustained homogeneousisotropic turbulence with particles is numerically simulated in order to obtain those statistics.2 Numerical approachIn order to meet the physical needs a one-fluid approach for particulate flows is proposed. Thisapproach is based in a tensorial penalty method. The solid region is treated with a high viscosity(from 100 to 1000 times the fluid one). Then velocity field obtained by the Navier-Stokes solutionfulfill the solid constraint. After each Navier-Stokes Resolution the velocity of each particle iscomputed and the particle is tracked. With the new positions the density and viscosity field iscomputed in the same way that done on Volume of Fluid Methods [1].A collision algorithm is implemented in order to ensure the particle-particle rebound. Thisalgorithm includes a model that ensures the physical lubricated rebound [2].3 Description of the two-phase flowAs explained above our goal is to simulate statistical converged simulations. The way we maintainthe turbulent energy on the fluid is near the linear forcing proposed by [3]. In our implementationat each time step the velocity field is rescale in order to maintain the kinetic energy on the domain:u+ =√∫Ωu20∫Ωu2u.At the beginning of each simulation 512 particles are spread randomly in a sustained turbulencewhose spectrum is converged. The Taylor micro-scale Reynolds number of the initial turbulentflow computed from this spectrum is 73. The moderate Reynolds number gives a small spatialscale separation: the ratio of Taylor micro-scale over Kolgomorov scale is λ/η = 17. The particlesdiameter size is near the Taylor micro-scale: Dp/η = 20. The solid fraction is 3%.Three simulations are run with different density ratio: ρp/ρf = {1, 2, 4}. This ensures thatparticles respond to fluid fluctuations. Indeed, the Stokes number is small enough for particle torespond to fluctuations: based on the Kolmogorov time scale the simulations Stokes numbers areStk = {26, 52, 104} and based on eddy turnover time StE = {1.5, 3, 6}.The simulations are realized on a cube of 2563 grid points that ensures the solution of smallfluid scales (η/∆x = 0.56). For the simulation of one eddy turn-over time 10 real time hours areused over 512 processors. In order to obtain converged particles statistics around 12 eddy turn-oversimulation time are necessary.An study of the spatial spectrum and lagrangian temporal spectrum have shown that the in-clussion of particles have no major influence on the turbulence parameters.13.1 Particles statisticsThe particle distribution has been studied and converged statistics has been obtained. The table 1gives the agitation, q2p, the velocity autocorrelation time, TL and the diffusion coefficient obtainedwith the approximation D = 2q2pTL. As comparison, those values are given divided by the samemeasure obtained for fluid particles distributed on the single phase case.ρ q2pq2pq2f(q2pq2f)T.-H.TLTLT fLD DDf a2rmsτ thcτkτmcτk- (m2 s−2) - - (s) - (m2 s−1) - (m2 s−4) - -1 1.27 0.85 1 1.22 1.31 3.10 1.13 4.00 49.3 103.62 1.17 0.79 0.5 1.35 1.45 3.16 1.15 2.12 51.4 89.14 1.10 0.74 0.2 1.53 1.65 3.37 1.23 1.05 52.9 79.0Table 1: Particle and collision statistics.As observed for point-wise simulations the agitation decreases with the density ratio where thevelocity autocorrelation time increases. Although the behaviour is the same, the values obtainedcan not be compared to those obtained for point-wise simulations. That is because the particles arenot influenced by all fluctuations as for the point-wise simulations. The agitation expected by theTchen-Hinze theory [4, 5] are also given on table 1.In order to analyze the phenomena of preferential concentration he particle radial distributionfunction has been stablished. After 12 eddy turnover time the distribution stays normal, there, nopreferential concentration are highlighted. This result can be a physical result or could be explainedby different reasons: The simulation domain is not large enough to enable concentration areas, thesimulation time is not long enough to allow particles to migrate to the concentration zones.As show by figure 1 the probability density function can be approximated by an exponential dis-tribution. In our case this p.d.f. is obtained without the accelerations of particles during collisions.This distribution is common for inertial particles. In our case as for the experiments presented by[6] the particles’ acceleration r.m.s. (a2rms on table 1) can be related to the density ratio ρ. Thecorrelation that takes into account the turbulent parameters : a2rms−2/3ν1/2 = αρ−β.10−510−410−310−210−1−20 −15 −10 −5  0  5  10  15  20p .d .f .a2 / a2rmsExponentialρ = 1ρ = 2ρ = 4Figure 1: Probability density function of the acceleration of particles.23.2 Collision regimeIn the first simulations we considered the results of collisions without lubrication model. The thecollision regime is compared to the kinetic theory. On the case of the kinetic theory the intercollisiontime (i.e. the mean time between two collisions) can be explaide dependig on the particles densitynp, their diameter and the agitation by the formula:1τ thc= nppiD2p√16pi23q2p. On the table 1 thisapproximation is given and compared to the measured values. We can see that when density ratioincrease the behaviour reach the kinetic theory. That is because the particles are more inertial andthen the regime are more similar.For the kinetic theory the p.d.f. of the intercollision time can be approximated by an exponentialform: P (∆tcoll) =1τcexp(−∆tcollτc). On figure 2 we compare the obtained values for each simulationto the theory. The same conclusion is given: more the particles are heavy more the regime is similarto the kinetic theory.10−310−210−1100 0  2  4  6  8  10  12p .d .f .∆tcoll/τcKinetic theoryρ = 1 ρ = 2 ρ = 4 Figure 2: Probability density function of collision times.The results given below are not satisfactory because they doesn’t explain why the regime aredifferent. To understand the different the p.d.f. of the angle at the collision is analyzed andcompared to the kinetic theory on figure 3. It can be observed that there are less frontal collisionsthat for the kinetic theory. That is because the particle velocity is correlated to the surroundingfluid and then the velocity of pairs that are in collision are correlated too. Again, this effect is lessimportant when the density increases.One effect of this correlation at collision is that the energy on the collision is less important.Then the lubrication effect can not be neglected. An additional simulations have been done with thelubrication model activated for the neutrally buoyant particles case. The statistics of the particlesstay similar. The major difference concern the intercollision time that increase drastically (from103.6 τk to 11.6 τk). This can be understand by analyzing the p.d.f. of the intercollision time,figure 4. As we can see the probability of short times between two collisions increases. This increasehas been explained by [7]: secondary collisions appears. For the case with lubrication model thep.d.f. of the time between two encounters is given too. For this purpose, we consider an encountera pair that activate the lubrication model. We can see that it is similar to the solid collisions onthe cases without lubrication model. The typical sequence is that two particles that are near eachother activate the lubrication model, and then, before the definitive separation many solid collisionsoccurs. The same study has been realized for the case ρp/ρf = 4 and similar behaviour has beenobserved. Nevertheless, this effect is more important for buoyant particles because the energy on3 0 0.2 0.4 0.6 0.8 1 1.2 0  20  40  60  80  100  120  140  160  180  200p .d .f .θcoll (o)Kinetic theoryρ = 1ρ = 2ρ = 4Figure 3: Probability density function of angles at contact.the encounters are less important.10−210−1100101 0  0.5  1  1.5  2  2.5  3  3.5  4p .d .f .∆tcoll/τcS.L.Kinetic theorySolid collisions (without lubrication) − ρ = 1Solid collisions (with lubrication) − ρ = 1Encounters (with lubrication) − ρ = 1Figure 4: Probability density function of collision times with and without lubrication.3.3 Fluid surrounding particlesAs it is said before the particles are correlated to the surrounding fluid. In order to verify this pointthe average fluid around particles has been done taking into account the velocity of each particle.The figure 5 shows the dimensionless velocity and the streamlines around particles. The flow issimilar to a Stokes flow but the velocity decreases slowly and is not symmetric.As shown by [8] the dissipation increases around the particle. Nevertheless, in their studythe structure of the dissipation is not given. In our case, the average local dissipation allows toconclude that there are an average dissipation film around particles, figure 6. This dissipation is4ρ = 1 ρ = 4Figure 5: Streamlines obtained from the averaged velocity field. Color field by velocity divided by< |Vp| >. (for space constraint ρ = 2 is not presented)less important on the back of the particle. That can be explained by the weak created by particles.4 ConclusionThe presented simulations had permit to better understand the behaviour of finite-size particleson a turbulent flow. Many results can be retained. The particle statistics have the same trendthat the statistics obtained for point-wise simulations. The particle velocity is correlated to thesurrounding fluid velocity. This have an effect on the velocity angle at collision that is smallerthat for the kinetic theory. The lubrication film between particles can not be neglected for thissimulations. The introduction of a lubrication model bring secondary collisions out. As a finalresult, the structure of the dissipation around particles is analyzed.References[1] S. Vincent, J. C. Bra¨ndle de Motta, A. Sarthou, J.-C. Estivalezes, O. Simonin, and E. Cli-ment, “A Lagrangian vof tensorial penalty method for the dns of resolved particle,” Journal ofComputational Physics (Submitted), 2013.[2] J. C. Bra¨ndle de Motta, B. Breugem, W.-P.and Gazanion, J.-L. Estivalezes, S. Vincent, andE. Climent, “Numerical modelling of finite-size particle collisions in a viscous fluid,” Physics ofFluids (in press), 2013.[3] C. Rosales and C. Meneveau, “Linear forcing in numerical simulations of isotropic turbu-lence: Physical space implementations and convergence properties,” Physics of Fluids, vol. 17,p. 095106, 2005.[4] C. Tchen, Mean value and correlation problems connected with the motion of small particlessuspended in a turbulent fluid. PhD thesis, University of Delft, The Hague, 1947.[5] J. O. Hinze, Turbulence. McGraw-Hill, 1975.5ρ = 1 ρ = 4Figure 6: Averaged dissipation around particles divided by the dissipation without particles.[6] N. M. Qureshi, U. Arrieta, C. Baudet, A. Cartellier, Y. Gagne, and M. Bourgoin, “Accelerationstatistics of inertial particles in turbulent flow,” The European Physical Journal B, vol. 66,pp. 531–536, Dec. 2008.[7] A. T. Cate, J. J. Derksen, L. M. Portela, and H. E. A. Van Den Akker, “Fully resolved simulationsof colliding monodisperse spheres in forced isotropic turbulence,” Journal of Fluid Mechanics,vol. 519, pp. 233–271, Nov. 2004.[8] F. Lucci, A. Ferrante, and S. Elghobashi, “Modulation of isotropic turbulence by particles oftaylor length-scale size,” Journal of Fluid Mechanics, vol. 650, p. 5–55, 2010.6

A Lagrangian vof tensorial penalty method for the dns of resolved particle,&quot;

Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties,&quot;

Mean value and correlation problems connected with the motion of small particles suspended in a turbulent uid.

Numerical modelling of  particle collisions in a viscous uid,&quot; Physics of Fluids (in press),

Simulation of fully resolved finite-size particles in a turbulent flow

http://oatao.univ-toulouse.fr/10473/1/brandle_10473.pdf

Simulation of fully resolved finite-size particles in a turbulent flow

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