Comparison between Lagrangian and Eulerian. mesoscopic modelling approaches for inertial. particles suspended in decaying isotropic.

Transcription

1 Comparison between Lagrangian and Eulerian mesoscopic modelling approaches for inertial particles suspended in decaying isotropic turbulence A. Kaufmann a,1 M. Moreau b O. Simonin b J. Helie b,2 a CERFACS, 42 Av. G. Coriolis, Toulouse, France, kaufmann@cerfacs.fr b IMFT,UMR 5502 CNRS/INPT/UPS, Toulouse, France, simonin@imft.fr Abstract The purpose of this paper is to evaluate an unsteady Eulerian approach from a direct comparison with a Lagrangian approach for the simulation of an ensemble of non-colliding particles interacting with a decaying homogeneous isotropic turbulence given by DNS. Comparison between both approaches is is carried out for two different particle inertia from global instantaneous quantities such as particle kinetic energy, local instantaneous Eulerian quantities such as particle number density and spectral quantities such as spectral particle kinetic energy. The proposed Eulerian approach yields instantaneous three dimensional velocity and number density fields which are in very good agreement, at large scales, with the discrete particle simulation results. However, appreciable discrepancies between the two approaches are measured at small scales for the particles with larger inertia. Indeed, for Stokes number value close to unity, segregation effects are very important and due to the stiff gradients in number density the Eulerian method is limited by spatial grid res- Preprint submitted to J. Comp. Physics 15 May 2006

2 olution. To overcome this numerical difficulty, a simulation approach using a bulk viscosity is proposed. Key words: particle laden flows, eulerian methods, preferential concentration, compressibility 1 Introduction A true direct numerical simulation (DNS) of dispersed two phase flows includes the simulation of the carrier phase around each single particle (or droplet), taking into account its deformation, to compute exactly the forces of the carrier phase acting upon it. Unfortunately such true DNS are today limited to few particles [1,2]. When studying collective physical phenomena such as particle segregation or turbulence modulation occurring in industrial applications or environmental flows, the full particle number cannot be accounted for and requires approximations to limit the numerical cost. For very small particle sizes compared to the typical energy containing scales, the momentum transfer with the carrier phase can be modeled by some semi-empirical interaction force expression that reduces to a drag law in the case of large density ratio. The characteristics of the particles can be described by several means. One possibility is the discrete particle simulation (DPS) approach in which every particle is treated in an Lagrangian framework. An alternative method is to consider only some collective characteristics of the particles in an Eulerian framework such as particle number density and mean particle velocity. 1 Present address : Siemens VDO, Regensburg Germany, Andre.Kaufmann@siemens.com 2 Present address : Siemens VDO, Toulouse France, Jerome.Helie@siemens.com 2

3 Equilibrium Eulerian approaches [3,4] consider a transport equation for the particle number density and compute the particle velocity using the carrier phase velocity and a Taylor expansion for the equation of particle motion. The expansion parameter is written in terms of the particle relaxation time to carrier phase characteristic time ratio and such approaches are well suited for small Stokes number values but lead to increasing error with increasing Stokes number. Computation of particle laden flow for particle relaxation time larger than the Kolmogorov time scale requires a transport equation for the particle velocity in order to account accurately for inertia effects. The purpose of the present paper is to compare predictions using an Eulerian approach, with a transport equation for the particle velocity, to the reference results obtained from discrete particle simulation (DPS) in a homogeneous isotropic decaying gaseous turbulence given by DNS. Following Février et al. [5], a probability density function (PDF) f p (1) (c p ; x p, t H f ) depending on the particle velocity c p, the particle location x p at time t may be defined for a given carrier flow realization H f. This function gives the local instantaneous probable number of particles with the given translation velocity u p = c p and obeys a Boltzmann-type kinetic equation, which may account for momentum exchange with the carrier fluid and inter-particle collisions. Alternatively, in analogy with the derivation of the Navier-Stokes equations in the frame of kinetic theory [6], transport equations of the first order moments (such as particle number density, mean velocity and fluctuant kinetic energy) may be derived directly by averaging from the PDF kinetic equation [7]. Such an approach has been evaluated in previous a priori test from DPS coupled with DNS or LES of forced homogeneous isotropic turbulence [8] and fully developed channel flow [9]. This study is the first attempt to solve the set of equations of the mesoscopic 3

4 Eulerian approach allowing a posteriori comparison of the model predictions with DPS results. In the present study, interaction forces are limited to Stokes drag and two-way coupling effects are neglected, that is the carrier phase is considered as uninfluenced by the presence of the dispersed phase. Particles are of identical size and the gravity force is not considered. However, the extension to evaporating droplets, gravity force, modelling non-linear effect in the drag force and other interaction forces is not in conflict with the derivation methodology of the Eulerian field equations. Also the introduction of interparticle collisions is without major difficulties [10]. The drastic assumptions are chosen in order not to obscure the effects related to particle turbulence dynamic interaction. Comparisons between Eulerian an Lagrangian simulation results are performed for three levels. First the integral properties of the dispersed phase such as volume averaged particle kinetic energy and particle-fluid velocity correlation are investigated. The second comparison concerns the local instantaneous field properties such as particle number density n p, correlated velocity ŭ p,i and random uncorrelated kinetic energy δ θ p. The third comparison concerns the spectral properties such as the particle kinetic energy spectrum E p (k) and the fluid-particle correlation spectrum E fp (k) computed from the correlated particle velocity field. To allow these comparisons a spatial filtering procedure is used to project Lagrangian quantities on a grid and thus to obtain continuous fields from the DPS results. To validate such a procedure, several projection methods are tested on a one-dimensional synthetic case. In the case of non-interacting particles, the number of particles per cell for a given carrier phase realization is not limited, a gaussian projector is then validated using DPS results with different simulated particle numbers at the kinetic energy spectrum level. The 4

5 (1) q 2 p q fp q 2 p q fp δq 2 p Eulerian DPS Simulation Volume Filtering Eulerian Field Variables (2) Mesoscopic Field Variables n p ū p,i δ θ p n p ŭ p,i δ θ p (3) ˆĒ p (k) ˆĒ fp (k) Ê p (k) Ê fp (k) Fig. 1. Methodology for comparison of DPS to Eulerian Simulation Approaches particle number densities n p and the correlated velocity ū p obtained by the volume filtering procedure can then be compared to it s counterpart from the mesoscopic Eulerian Simulation (fig. 1). The paper is organised as follow : the mesoscopic eulerian approach is presented in section 2, then the projection procedure to obtain Eulerian field from Lagrangian results is described and validated (section 3), the section 4 concerns the numerical test case, finally Eulerian simulation predictions are discused and compared with DPS results in section 5. 2 The mesoscopic Eulerian approach In turbulent particle laden flow the dispersed phase cannot be regarded as independent of the carrier phase turbulence [11]. Therefore several statistical 5

6 approaches rely on joint carrier-phase, dispersed-phase probability functions. In particular, the joint fluid-particle PDF equations may be used to establish transport equations for carrier phase and dispersed phase quantities such as number densities, momentum and energies as well as fluid-particle correlations [7,12]. Since the interaction of particles and turbulence is rather complicated, modelling of the unclosed terms is a difficult and challenging issue. Due to the statistical character of such approaches, three dimensional unsteady effects, as they can also be found in single phase turbulent flow, cannot be represented. One important example among others are segregation effects which need a two-point PDF approach [13]. To derive local instantaneous Eulerian equations in dilute flows (without turbulence modification by the particles), Février et al. [5,8,10] introduce an ensemble average over dispersed-phase realizations conditioned by one carrierphase realization. Such an averaging procedure leads to a conditional velocity PDF for the dispersed phase, f (1) p (c p, x, t; H f ) = W (1) p (c p, x, t) H f. (1) W (1) p is the fine grid PDF of the realizations of position and velocity in time of any given particle [14] and H f is the given carrier flow realization. Ignoring particle collisions, the conditional PDF equation takes the form : t f p + c p,j fp + x j c p,j [ Fp,j m p ] fp = 0 (2) This PDF equation can be used to derive transport equations for the moments (number density n p, mesoscopic velocity ŭ p,...) of the dispersed phase following kinetic theory methodology [6]. 6

7 2.1 Transport equations for particle properties Instead of resolving directly the Boltzman-type kinetic equation (2) one may solve the first order moments as done in kinetic theory of dilute gases when deriving the Navier Stokes equations [6]. Integration over the particle velocity phase space yields statistical mesoscopic properties of the dispersed phase. The mesoscopic particle number density is written n p = f (1) p (c p, x, t; H f ) dc p (3) Other averaged properties of the dispersed phase are obtained by integration of the product of the quantity with the conditional PDF. Φ = Φ H f p = 1 n p Φ (c p ) f (1) p (c p, x, t; H f ) dc p (4) With this definition one may define a local instantaneous particulate velocity field, which is here named mesoscopic or correlated Eulerian particle velocity field. ŭ p (x, t; H f ) = 1 n p c p f (1) p (c p, x, t; H f ) dc p. (5) For simplicity, the dependence of the above variables on the single carrier phase realization H f is not shown explicitly. The velocity of every individual particle V (k) i components : can be decomposed into its correlated ŭ p,i and uncorrelated δv (k) i V (k) i = ŭ p,i + δv (k) i (6) 7

8 This uncorrelated part of the particle velocity is here referred to as Random spatially Uncorrelated Velocity component (RUV) 3. By construction, δv (k) i is spatially uncorrelated. Indeed the two point correlation δr p,ij (x, x ) = [c p,i ŭ p,i (x, t)] [ c p,j ŭ p,j (x, t) ] ( ) (2) f p cp, c p, x, x, t; H f dcp dc p(7) is zero when x x > 0. This uncorrelated velocity distribution has similarity with the peculiar velocity distribution in the framework of kinetic theory of dilute gases which obeys the molecular chaos assumption. Nevertheless it has to be pointed out that the particle position distribution is not as random due to the segregation mechanism and the uncorrelated velocity distribution is not gaussian. Application of the conditional-averaging procedure to the kinetic equation (eq. (2)) governing the particle PDF leads directly to the transport equations for the first order moments consisting of number density and mesoscopic Eulerian velocity, t n p + n p ŭ p,i = 0 (8) x i n p + n p ŭ p,j ŭ p,i = n p [ŭ p,i u i ] + tŭp,i x j τ p x j n p δ σ p,ij (9) The first term on the right hand side of eq. (9) represents the drag force, where u i is the carrier phase velocity at the particle position. τ p is the particle 3 Random spatially Uncorrelated Velocity (RUV) has been referred to as Quasi Brownian Motion (QBM) in previous publications. We agree that the expression Quasi Brownian is misleading since the physical interpretation of the uncorrelated motion is not of Brownian nature. 8

9 relaxation time modeled by Stokes drag : τ p = ρ pd 2 18µ f (10) Here ρ p is the density of the particle, d is the particle diameter and µ f is the dynamic viscosity of the carrier phase. The stress term in eq. (9) arises from the averaging of the convective term in the PDF transport equation : n p δ σ p,ij = (c p,i ŭ p,i ) (c p,j ŭ p,j ) f (1) p (c p, x, t; H f ) dc p (11) = n p δu p,i δu p,j H f p. (12) This term account for the transport of mesoscopic momentum due to the uncorrelated part of the particle velocity. The time dependant fields of the particle number density and mesoscopic velocity can be predicted using eqs (8) and (9). Concerning the momentum conservation equation it is however necessary to specify an expression for the RUV stress tensor δ σ p,ij. One possibility is to use transport equations for the components of the velocity variance tensor. This shifts the difficulty however in finding an expression for the third order tensor appearing in the transport equation and increases the numerical cost because of six additional transport equations. Alternatively, one can model directly the velocity variances using the computed moments as done in the Navier-Stokes equations when deriving the viscosity assumption in the frame of kinetic theory. 2.2 The uncorrelated velocity variance tensor δ σ p,ij When the Euler or Navier-Stokes equations are derived from kinetic gas theory, the trace of δu p,i δu p,j p is interpreted as temperature (ignoring the Boltz- 9

10 mann constant and molecular mass) and defined as the uncorrelated part of the molecular kinetic energy. By analogy, the Random Uncorrelated particle kinetic Energy (RUE) is defined as half the trace of δ σ p,ij : δ θ p = 1 2 δu p,iδu p,i H f p. (13) In the frame of kinetic theory of dilute gas, pressure is linked to density and temperature by an equation of state. In the same manner, a Random Uncorrelated Pressure (RUP) may be defined by the product of uncorrelated kinetic energy and particle number density : P p = n p 2 3 δ θ p (14) With such a definition, the velocity stress tensor may be separated into an isotropic part corresponding to the RUP and a deviatoric trace-free term δ τ p,ij, n p δ σ p,ij = P p δ ij + δ τ p,ij (15) Then, the momentum transport eq. (9) is written, n p + n p ŭ p,j ŭ p,i = n p [ŭ p,i u i ] P p + δ τ p,ij (16) tŭp,i x j τ p x i x j and, in first approximation, the deviatoric stress tensor is written in terms of the mesoscopic rate-of-strain tensor and a dynamic viscosity, δ τ p,ij = µ p ( ŭp,i x j + ŭ p,j 2 ) ŭ p,k δ ij x i 3 x k (17) In such an approach the particle dynamic viscosity is written µ p = 1/3 n p τ p δ θ p [10] where τ p is the particle relaxation time. This expression can be obtained using the transport equation for the deviatoric part of the velocity variance 10

11 tensor δu p,i δu p,j and supposing weak shear [7]. This closure model for the dynamic viscosity corresponds to a characteristic mixing length proportional to λ p τ p 2/3δ θ p which is the stopping distance due to drag of a particle with relative instantaneous velocity δu p = 2/3δ θ p. The closure model (eq. (17)) and the dynamic viscosity µ p require the knowledge of the RUE δ θ p. Modelling approaches for this quantity are developed in the next section. 2.3 Models for Random Uncorrelated kinetic Energy (RUE) From the PDF equation (eq. 2) a general transport equation for RUE can be obtained in the same way than temperature equation in the frame of kinetic theory [6]. t n pδ θ p + n p ŭ p,j δ θ p = 2 n p ŭ p,i δ θ p + n p δ σ p,ij x j τ p x j x j n p 2 δu p,iδu p,i δu p,j H f p (18) The first rhs. term account for the RUE dissipation by drag force, the second is the production due to shear, the last term is the diffusion by RUV. In a first approach, transport of RUE and production of RUE by divergence of the mesoscopic velocity field are assumed to be dominant and the other contributions being negligible. In a second approach it is attempted to solve directly the transport eq.(18) with a closure model for the triple velocity correlations Isentropic Approach for δ θ p The first approach is assuming a quasi isentropic behavior of the dispersed phase leading to an algebraic expression for δ θ p depending on the particle 11

12 number density 4 : δ θ p = A n 2/3 p (19) In homogeneous turbulence A is only dependent on time and can be determined from the spatial average of δ θ p to insure the correct value for the mean RUE. The spatial average (over the computational domain) is defined by {φ} = 1/V V φdv. The particle weighted averages is defined by {φ} p = { n pφ} / { n p }. } This allows to define A = { n p } δqp/ 2. The mean uncorrelated particle { n 5/3 p kinetic energy is defined as δq 2 p = { δ θ p }p. DPS performed by P. Février [8] in stationary homogeneous isotropic turbulence suggest that the mean uncorrelated kinetic energy δq 2 p depends on the resolved dispersed phase kinetic energy q 2 p = 1/2 {ŭ p,k ŭ p,k } p, the fluid-particle correlation q fp = {u k ŭ p,k } p where u k is the carrier phase velocity, and the carrier phase kinetic energy weighted by the particle presence q 2 f@p = 1/2 {u k u k } p. This equilibrium expression is written, δq 2 p = q 2 p ( 4 q 2 p q 2 f@p q 2 fp 1 ) (20) Transport equation for δ θ p The second approach uses the full transport equation for random uncorrelated kinetic energy [10] and requires the closure of the triple correlation. The equation is written, t n pδ θ p + n p ŭ p,j δ θ p = 2 n p δ θ p (21) x j τ p 4 This quantity can be obtained by first order analysis [6,15] 12

13 [ ( ŭp,i + P p δ ij + µ p x j + [ ] κ p δ θ p x j x j + ŭ p,j 2 )] ŭ p,k ŭp,i δ ij x i 3 x k x j In this equation, the stress tensor δ σ p,ij from eq. (18) has been replaced by the approach already presented for the momentum equation (eq. (15)). The interpretation of the triple correlation as a diffusive flux motivates the gradient law approximation : n p 2 δu p,iδu p,i δu p,j H f p = κ p x j δ θ p (22) The diffusivity coefficient is modelled assuming κ p = 5/3 n p τ p δ θ p. This closure is equivalent to the Fick s law for the heat flux in the Navier-Stokes equations. 3 Computation of Eulerian mesoscopic fields from Discrete Particle Simulation results In DPS approach every single k-particle follows its individual trajectory X (k) i (t) and has its proper particle velocity V (k) i (t). Discrete particle position and velocity are given by the simple set of differential equations. d dt X(k) i d dt V (k) i = 1 τ p [ = V (k) i (t) (23) ] u i (X (k) (t), t) V (k) (t) (24) i Special care has to be taken when evaluating the carrier phase velocity u(x (k), t) at the particle location for the computation of Stokes drag law. In the present study high order interpolation methods (fourth order Lagrange polynomes) are used to ensure a minimal numerical error [16]. 13

14 Table 1 Projection procedure definitions in 3D Control volume Weight function Characteristic length Projection V c w(x (k) x) 2σ Box ( x) 3 1 x Large box (2 x) x Volumic ( x) 3 Π 3 i=1 Large volumic (2 x) Π 3 i=1 (2 2 X(k) i x i x/2 ) (1 X(k) i x i x ) x 2 2 x Gaussian (2 x) 3 (2 x) 3 erf( 6) 3 ( 6 π x 2 ) 3 2 exp( 6 X(k) x 2 x 2 ) (1 2 6e 6 πerf( 6) ) x x The Eulerian model presented above is based, in theory, on an ensemble average over all particle flow realizations for a given fluid flow realization. But, without inter-particle influences (directly by collisions, or through a modification of the fluid flow due to particles), an average over the Lagrangian quantities of a large number of particle is equivalent to the statistical conditional average on several particle realizations of a few number of particles. In addition, interpretation of Lagrangian results in term of Eulerian mesoscopic fields such as particle number density, correlated velocity or RUE require the use of a projection procedure based on volume filtering method. In this study, the DPS is performed with a large number of particles and field properties are obtained by projection on a regular grid with a given cell size x. Such projection procedure from Lagrangian to Eulerian quantities are widely used to handle the two-way coupling in DPS approach [11,17 19]. The salient feature of the configuration is the discontinuous distribution of the Lagrangian velocities due to the particle RUV. A control volume V c 14

15 around a computational node at the location x is defined. V c should be chosen small enough with respect to the characteristic length scale of variation of the mesoscopic variables but large enough to have sufficiently particles for the averaging. Mesoscopic fields such as particle number density n p, correlated velocity ū p and random uncorrelated energy δ θ p are measured in DPS as : n p (x, t) = 1 w(x (k) (t) x) (25) V c n p (x, t)ū p,i (x, t) = 1 V c k k k w(x (k) (t) x)v (k) i (t) (26) n p (x, t)δ θ p (x, t) = 1 1 w(x (k) (t) x)[v (k) i (t)] 2 2 V c 1 2 n p(x, t)ū 2 p,i(x, t) (27) where w(x (k) (t) x) is a weight function. Definitions of several different projections are presented in Table 1. The top-hat and volumic (based on Schoenberg M 2 spline) projectors are written for two different widths, a gaussian projector is also presented. We restrict our attention to second order space accurate projectors for two reasons. Firstly, higher order projector weight function should have negative loops leading for unphysical negative particle number when few particles are present. Secondly, the corresponding convergence rate in terms of the particle number is slower [19]. Integral of the different projector kernels over the control volume is unity to ensure the globaly (ie. for infinite particle number) quantity mean conservation by projection. It must be noticed that the box, large box and large volumic projector are also locally meanconservative for finite particle number densities [17]. The characteristic length scale of the projector is evaluated by twice the standard deviation of the weigth function σ : 15

16 Fig. 2. Weight function of the different projection in 1D σ 2 = 1 V c V c X x 2 w(x x)dx 1 dx 2 dx 3 (28) Leading to the characteristic length scale classification of the retained projectors : Large box > Large volumic > Box Gaussian > Volumic (29) 3.1 Validation on 1D synthetic case Tests of these projection procedures have been performed on one-dimensional synthetic case. The box, volumic and gaussian 1D projection kernels are presented on fig. 2. N p particles are randomly distributed on a one dimensional grid with an equidistant node spacing x of N nodes. The mean particle number per cell is N p /N = n p x in 1D. The particle distribution is projected on the grid (eq. 25) to obtain the mesoscopic particle number density. The conservation of mean quantities by gaussian and volumic projection is 16

17 Fig. 3. Evolution of the conservation of the projected particle number with the mean particle number per cell (error = ( n p n p ) 2 / n p 2 ) for the 1D synthetic case. evaluated for the mean projected number density n p (fig. 3). As expected the associated error is decreasing when the mean particle number per cell is increasing. It can easily be showed that this bias is avoided in the case of equally spaced particles and is due to non homogeneous particle repartition in every projection grid cell. We focuss now our attention on kinetic quantities. A turbulent-like velocity field (ŭ p ) and a gaussian uncorrelated white noise (δv (k) ) are given to the randomly placed particles : V (k) = ŭ p (X (k) ) + δv (k) (30) The velocity field follows a Passot-Pouquet type spectrum defined on 32 modes (presented in the study), but tests using a 5/3 decrease rate spectrum show similar results. The energy of the noise has been chosen to be 30% of the correlated energy, which correspond to the maximum of the particle random uncorrelated energy measured by Février et al. [5] in stationnary homoge- 17

18 neous isotropic turbulence. Particle properties are then projected on a onedimensional 64 node grid of length 2π. Substituting eq. 30 in eq. 26 leads to : n p ū p = 1 w(x (k) x)ŭ p (X (k) ) + 1 w(x (k) x)δv (k) (31) V c V c k k n p ū p = n p ŭ p + n p δv (32) To identified the projected velocity ū p to the correlated velocity ŭ p, the projection procedure must be able to supprim the noise δv and must not affect the correlated velocity field. The difference between the projected velocity fields and the non projected correlated velocity distribution is evaluated by the quadratic error : Error(φ) = (φ(x) ŭ p(x)) 2 ŭ p (x) 2 (33) where φ is the projected velocity with added noise u p or without noise ŭ p. The same Lagrangian field is projected with the different projections. To obtain statistical convergence, average is performed on 5000 different realizations of the velocity field. This procedure is repeated for several values of the particle number per cell of the projection grid, from one to thousand. Fig. 4 shows the dependence of the velocity projection error (computed from eq. 33) on the particle number per cell for the different projection procedures. The error decreases when increasing the particle number, but a systematic error occurs even for a large particle number density. For more than 10 particles per cell, the related error arises in decreasing order from the large box projector, the large volumetric filter, the box filter and the volumetric filter. In the noisy case (fig. 5) more than 100 particles per cell are needed to reach the systematic projector error level, but the projector efficiency classification remains the same. 18

19 Fig. 4. Dependence of velocity quadratic error (given by eq. 33) due to the projection procedure on the particle number per cell for 1D synthetic case. Particle velocity distribution is given from the interpolation of a continuous turbulent field on the particle position. Upper figure : comparison between between box, large box, volumic, large volumic projections. Lower figure : comparison between box, volumic and gaussian projections at the bottom. 19

20 Fig. 5. Same caption as fig. 4. Particle velocity is the velocity of the continuous field at their location with an added gaussian white noise of 30% energy. In addition to the statistical error due to the finite number of particles in the averaging cell, Boivin et al. [17] showed that such projection procedures are equivalent to a filtering of the length scales smaller than the control volume size which limit high gradient values. This spatial error can be limited by using smaller control volume, but this increases however the statistical error due to 20

21 Fig. 6. Dependence of the projected correlated energy spectra using the gaussian projector on the mean number of particles per cell. 1D synthetic case. The line is the correlated energy spectra. particle number. A different alternative would be to use a projection kernel with a smaller characteristic length scale. It can be noticed that volumic-type projectors returns better results than box-type projectors. A compromise between small systematic error and accuracy for small particle number density is to use the gaussian projector on a large control volume (see Table 1) but having the box projection characteristic length scale. The Gaussian projector error is comparable to the box projection error for high particle numbers and to the large box for small particle number. To analyse more in details the projection error, the energy spectra of ŭ p, ŭ p, δv, and ū p are presented on fig. 6-8 with the gaussian projector for 2, 20 and 200 mean particle number per cell of the projection grid. For all the cases the projected correlated velocity energy (fig. 6) spectra follow the one of the correlated velocity at large scales. The projected correlated velocity (ŭ p ) energy spectra for the 200 mean particle number per cell case underestimates the 21

22 Fig. 7. Dependence of the projected noise energy spectra using the gaussian projector on the mean number of particles per cell. 1D synthetic case. The continuous line is the correlated energy spectra. The horizontal dashed lines correspond to the noise spectrum models (Eq. 35). energy at small scales comparing to the ŭ p energy spectra. This behaviour is the mark of the spatial error detailled before. In the case of 2 particles per cell case, the spectrum presents an unphysical overestimation of the small scale energy. This effect is avoided by increasing the mean particle number density, n p so that the spectra for 20 and 200 particles per cell are nearly identical for all waves numbers. For equally spaced particles, the error is removed, even for the case with the lowest particle number per cell (not presented here) proving that the overestimation of the velocity spectrum induced by the projection procedure is due to the random repartition of the particles. Fig. 7 presents the energy spectra of the projected noise (δv). These spectra are mainly flat and their level quasi-linearly decrease by increasing the particle number density. The noise spectra level are comparable to the correlated energy ones. More precisely, this level can be evaluated. The energy spectrum 22

23 Fig. 8. Dependence of the projected correlated energy spectra with added noise using the gaussian projector on the mean number of particles per cell. 1D synthetic case. The continuous line is the correlated energy spectra. The horizontal dashed lines correspond to the noise spectrum models (Eq. 35). of a discret white noise on N p equaly spaced particles is constant up to the Nyquist frequency (N p /2 = n p N x/2). Considering that the spectrum integral gives the Random Uncorrelated Energy, the energy spectrum of the discret noise is : E δv (k) = N x 2π (N p 2 1) 1 δqp 2 (34) E δv (k) 1 π n p δq2 p (35) For non homogeneous particle distribution, eq. 35 can be used as a noise spectrum level estimation. Comparing the projected noise spectra level with the non projected noise model level, the particle density effect is recovered and the mean level too (about 2% error). The projection procedure is just filtering the noise and cut off all the components with wave numbers bigger than N/2. 23

24 We now consider the projection of correlated velocity with added noise (fig. 8). At small scales the energy spectrum of ŭ p obtained with 2 or 20 mean particle number per cell follows the one of the noise. A statistical error occurs and the projection is not able to eliminate noise of the discret velocity field. By increasing the number of particle the energy of the projected noise becomes negligeable against the correlated energy at all length scales. An a-posteriory validity criteria in the computation of the correlated energy spectra with added noise is introduced : Eūp (k) > 1 π n p δq2 p for all k (36) To resume, when obtaining continuous eulerian fields from discret Lagrangian quantities by projection, errors are mainly due to a statistical error and to an intrisic filtering (or spatial error) at small scales. By using a well chosen projector (e.g. gaussian projection) with enough of particles (more than 10 particles per cell), it is possible to circumvent the problem in a satisfactory manner. 3.2 Validation on DPS results Finally to validate the gaussian projection, two DPS have been performed with 10 and 80 millions Lagrangian particles. Particles are randomly placed in the computational domain of length 2π with the fluid velocity at their position. Initial Lagrangian quantities are then projected with a gaussian filter on a 64 3 and node grids. For the coarser case, 10 millions of particles projected on a 64 3 grid cell (about 38 particles by cell) with a x equal to the computing mesh size, the projected spectrum is identical to the fluid one for k smaller than 24

25 Fig. 9. Comparison of correlated energy spectra of DPS results at the initialisation with V (k) (t = 0) = u(x (k) (t = 0), (t = 0)). DPS with 10 7 or particles projected on a 64 3 or grid with the gaussian projector. 18 (fig. 9). For larger value of k, the energy spectrum of the projected velocity shows an unphysical increase but remains 10 4 smaller than the effective values measured in the energetic region. As expected, fig. 9 shows that this effect can be diminished by increasing the particle number and decreasing the projection cell size but leading to very expansive simulation costs. Both DPS are performed and the same analyse is realised on a case with RUE ( 16% of the correlated energy is measured) and particle segregation. The energy spectra of mesoscopic velocities obtained by projection on a 64 3 (fig. 10) are nearly identical for the two DPS results, no statistical bias is observed so that 10 million particles ( 38 particles per cell) is sufficient to analyse mesoscopic velocity fields in term of energy spectra. Indeed, these spectra follow the one obtained by projection on a finer grid (128 3 ) up to the higher wave number (fig. 10). The one-dimensional noise spectrum model (eq. 35) is extended for the three-dimensional spectrum by replacing the total number of 25

26 Fig. 10. Comparison of correlated energy spectra of DPS results. DPS with 10 7 or particles projected on a 64 3 or grid with the gaussian projector. Case St = 0.53 at time t = Horizontal dashed lines are the lower limit of spectrum validity given by Eq. 37 (upper line is for cases 10 7 particles projected on 64 3 and particles projected on grid; lower line is for case particles projected on 64 3 grid). particle by the number of particles in only one direction ( n p x 3 N) : Eūp (k) > 1 π n p x 2 δq2 p for all k (37) We notice on fig. 10 that the correlated energy spectrum validity criterion (eq. 37) is satisfied. This criterion is of course questionable because particle distribution is far from homogeneous and RUE is not uniform, but it could be considered as a spectrum validity indicator. The gaussian projector is really able to limit the intrinsic error of the projection procedure and is used to obtain mesoscopic fields from DPS results with 10 million particles for 64 3 projection cells (about 38 particles per cell) in the rest of the paper. 26

27 4 Homogeneous isotropic decaying turbulence test case Homogeneous isotropic turbulence is one of the classical cases where dynamics and dispersion of particle laden flows can be studied. This has been done extensively using the DPS formalism and encouraging results and insight are obtained by such methods. Comparison of DPS in decreasing homogeneous isotropic turbulence [20] with experimental measurements of particle dispersion in grid generated turbulence [21] shows that essential features of the particle dynamics can be captured. Preliminary computations with a simplified Eulerian formalism of this test case gave encouraging results [22]. In the case of tracer particles (small Stokes number limit) Eulerian methods are well suited to describe the dynamics [3,4]. With increasing inertia, the particle velocities become de-correlated from the gaseous carrier phase velocity and segregation occurs for particle relaxation times comparable or larger than the Kolmogorov time scale [23]. For the sake of simplicity, the case of decaying homogeneous isotropic turbulence is studied. The carrier phase is initially supposed to have uniform density, the velocity field to be divergence free and the kinetic energy to follow a Passot-Pouquet spectrum [24]. After roughly one turn over time of the energy containing eddies, the velocity field is supposed to represent a realistic turbulent flow and the dispersed phase is added. At the particle injection time, the Reynolds number is Re L = For DPS initial conditions, particles are randomly and homogeneously distributed in space and the initial particle velocities are given equal to the carrier phase velocity at the location of the particle. For the Eulerian computation, these conditions correspond to a uniform particle number density fieldand a mesocopic particle velocity 27

28 field identical to the carrier phase velocity field and zero uncorrelated velocity variances. So the initial uncorrelated kinetic energy (RUE) is zero at initialization and should develop during the simulation. The spatial resolution of the gaseous phase computation is 64 3 and a total of 10 million individual particles are traced in the computational domain. This corresponds to 38.1 particles per gaseous node. 4.1 Numerical Method The Eulerian simulation is performed using a different code (AVBP [25]) then the DPS reference solution (NTMIX [18]). AVBP offers several spatial and temporal schemes. In the present study a central second order spatial scheme with a second order temporal correction (Lax Wendroff) was used. Comparison to third order Runge Kutta time stepping showed no significant improvement of the numerical accuracy. NTMIX uses a high-order spectral like scheme [26] on cartesian grids and Runge Kutta time stepping. Both numerical tools use domain decomposition and MPI for parallel computation. For the test cases carrier phase solutions are identical and velocity spectra superpose. To ensure the accuracy of the DNS of the carrier phase a necessary condition is to satisfy the balance of the volume averaged kinetic energy (q 2 f) with the dissipation rate (ε) computed directly form the viscous stress tensor (τ ij ) on the computational grid. t q2 f = ε (38) This verification that dissipation of carrier phase kinetic energy results from laminar viscous effects and not from the numerical diffusion in AVBP code 28

29 Fig. 11. Time derivative of the carrier phase turbulent kinetic energy dqf 2 /dt computed from AVBP numerical simulation results compared to the dissipation rate computed directly from the viscous terms. Fig. 12. Comparison of the carrier phase turbulent kinetic energy q 2 f computed from AVBP simulations (line) using a second order difference scheme with the one computed from NTMIX simulations (triangles) using a spectral like scheme [26]. is shown by fig. 11. The carrier phase kinetic energy of the two codes are identical (fig. 12). Furthermore, the kinetic energy spectra at a adimentional time of 10.8 are very similar until the Kolmogorov wave number (1/η K ) which 29

30 Fig. 13. Kinetic energy spectra of the carrier phase from simulations with the two numerical codes used in this study at time t = is 7.7 in non-dimensional units. In the Eulerian simulation, the dispersed phase is computed using the same numerical method as the carrier phase, imposing an additional limit on the time step due to particle relaxation time (τ p ). The Eulerian results are compared to DPS results at different levels : average quantities over the compational domain, kinetic energy spectra and snapshot of instantaneous local fields. 4.2 Integral Properties The integral properties of particle kinetic energy qp 2 and fluid-particle correlation q fp should be identical in the two simulations and therefore constitute a first step of comparison. In the present test case of decreasing homogeneous isotropic turbulence those quantities vary in time. The properties are averaged over the computational volume and defined for the Eulerian simulation by 30

31 q 2 f = 1 2 {u iu i } (39) q fp = {u i ŭ p,i } p (40) q 2 p = 1 2 {ŭ p,iŭ p,i } p (41) and for the DPS by q fp = 1 N q 2 p = 1 N k k V (k) i u i (X (k) ) (42) 1 2 V (k) i V (k) i (43) The volume averaged uncorrelated kinetic energy (RUE) from DPS is given by δq 2 p = { δ θ p } p (44) with the local RUE δ θ p measured by eq Spectral treatment Using the Fourier transformed velocities of the carrier phase û i (k) = F(u i (x)) and dispersed phase û p,i (k) = F(ŭ p,i (x)) one can construct three dimensional fluid and particle energy spectra and a fluid-particle correlation spectrum. E f (k) = 1 2ûi(k)û i (k) (45) E p (k) = 1 2ûp,i(k)û p,i (k) (46) E fp (k) = û i (k)û p,i (k) (47) For established turbulent flow the undisturbed carrier phase kinetic energy follows the standard Kolmogorov spectrum. Here the interest lies on the behavior of the spectrum of the correlated particle kinetic energy and the fluid-particle 31

32 correlation. Whereas the carrier phase is considered incompressible, segregation effects measured in DPS show that there must be a compressible part in the correlated dispersed phase velocity spectrum. With the definition of Kraichnan [27] operators in the spectral velocity can be divided into a compressible and an incompressible (solenoidal) component. û c p,i = κ iκ j κ 2 ûp,j (48) ( û s p,i = 1 κ ) iκ j û κ 2 p,j (49) This orthogonal decomposition allows to construct a compressible and a solenoidal spectral energy such that the sum equals to the total spectral energy. E c p(k) = 1 2ûc p,i(k)û c p,i(k) (50) E s p(k) = 1 2ûs p,i(k)û s p,i(k) (51) 5 Simulations results and discussion Eulerian methods are well suited to describe the particle dynamics [3] for particle relaxation times small compared to the Kolmogorov time scale. It is therefore interesting to study how the Eulerian description behaves outside this range. Preliminary tests with the decaying homogeneous turbulence described above, showed that the numerical simulation with any of the two models proposed for the RUE is possible up to a Stokes number (St = τ p /τ + ) of St = based on the macroscopic dissipative time scale τ + (= k/ε) at the particle injection time. This corresponds to a Stokes number of St K = 0.17 based on the Kolmogorov time scale. In contrast, mesoscopic Eulerian Simulations with larger Stokes numbers failed. Detailed analysis revealed that, due 32

33 Fig. 14. Comparison of correlated particle kinetic energy and fluid particle correlation from DPS and Eulerian simulation for the test case of St = to segregation effects, the particle number density field has very stiff local gradients that caused dispersion errors in the numerical scheme. Increasing grid resolution for the Eulerian simulation up to (128 3, 192 3, ) only allowed a small increase in attainable Stokes number. Therefore in a first step the results of DPS and the Eulerian model are compared for the limiting Stokes number of St = In a second step an heuristic extension of the proposed model overcoming the numerical difficulties encountered due to massive segregation is presented and results are compared for a Stokes number based on the macroscopic dissipative time scale of St = 0.53 according to a Stokes number value, based on the Kolmogorov time scale, St K = 2.2. Eulerian results presented here are obtained with the RUE transport equation (eq. 21). 33

34 Fig. 15. Comparison of volume filtered DPS and Eulerian number density PDF for the test case with St = at t= Comparison at St = At Stokes numbers as small as St = based on the dissipative time scale τ + (= k/ε) the particles are expected to follow closely the carrier phase velocity and the uncorrelated velocity contribution to the particle dynamics is very small. In fig. 14 the integral quantities of correlated particle kinetic energy q p, 2 fluid-particle correlation q fp of the Eulerian simulation are compared to the DPS results. Temporal evolution of particle kinetic energy and fluid-particle correlation are well predicted by the Eulerian Simulation. RUE measured in DPS ( 10 2 qp) 2 is of the same order of numerical error. Then the predicted RUE is of two orders of magnitude lower than the correlated kinetic energy, the contribution of RUE can be neglected in this case. Fig. 15 shows the probability to find a computational cell with a given value for the number density in the Eulerian prediction and the DPS reference result at t=10.8. The distribution of the Eulerian number density is less wide than the DPS distribution. If the correlated velocity is well predicted in the 34

35 Eulerian simulation, a possible cause of the non-matching number density distributions may be the numerical scheme dispersion leading to a more homogeneous droplet number density. 35

36 Fig. 16. Comparison of the normalized droplet number n/ n from the Lagrangian simulation (upper graph, resolution 64 3 ) with the one from the Eulerian simulation (lower graph, resolution 64 3 ) at the non-dimensional time t=10.8, case St = The drawing plane is parallel to the x-y reference plane and crosses the computational domain center. 36

37 Fig. 17. Comparison of total and compressible kinetic energy spectra from DPS and Eulerian Simulation for the test case with St = at t=10.8. As an instantaneous local quantity the normalized droplet number density is shown in fig. 16 for the x-y plane crossing the computational domain center at the simulation time t=10.8. This corresponds to roughly 18 particle relaxation times (τ p ) and two dissipative times scales (τ + ) of the carrier phase. It shows that the Eulerian number density prediction is quantitatively close to the DPS result. Regions with high and low particle number densities are well correlated in the two approaches. Fig. 17 shows the total particle kinetic energy spectra and the compressible kinetic energy spectra computed from Lagrangian and Eulerian simulation results. The correlated particle kinetic energy spectrum follows closely the spectrum of the carrier phase at large scales. Up to the kink in the DPS energy spectra, the spectra computed from the Eulerian Simulation match well those from the DPS. As shown in section 3, this kink is probably unphysical and is due to a numerical error induced by the non-homogeneous particle distribution. 37

38 In case of small Stokes number, the mesoscopic simulation is able to predict integral quantities such as particle energy and also local quantities such as segragation. Results with Eulerian mesoscopic approach are very similar to the ones obtained using the Eulerian equilibrium approach propose by Rani and Balachandar [4,28]. 5.2 Comparison at St = 0.53 Preliminary tests with Stokes numbers larger than St = failed since large segregation effects imply shock like in the number density distribution that could not be handled by the numeric scheme [15]. Supposing that the numerical resolution of the model is insufficient, several possibilities exist to circumvent this difficulty : a different numerical scheme using up-winding or flux limiters are clearly able to capture those strong gradients but imply some type of numerical diffusion. For the computation of the number density in the equilibrium Eulerian approach with a limited Stokes number range Rani and Balachandar [4,28] use a spectral viscosity to overcome this difficulty. Increasing spatial resolution increases strongly numerical cost. The origin of the preferential concentration is due to the compressibility of the particle number density field. A pure diffusivity added to the evolution of the number density field would bias the conservative transport of the particle velocity. The origin of the compressibility is the compressible component of the mesoscopic particle velocity field. At this point it was found preferable to act on the compressible component of the correlated velocity field to circumvent the measured difficulties associated with steep gradients. This leads to compute for a modified correlated velocity 38

39 ŭp,i transport equation and consequently a modified droplet number density field n p. The compressible component of the correlated velocity is modified by introducing a subgrid bulk viscosity, ξ sgs. t n p ŭp,i + np ŭp,i ŭp,j = PQB + x j x i + ξ sgs x i x j τ ij n p τ p ( ŭp,i ũ f,i ) x k ŭp,k (52) The subgrid model has the form of a bulk viscous term ξ sgs ŭ k / x k δ ij which is added to the shear viscosity term δ τ p,ij in eq. 17. The subgrid bulk viscosity is mesh-size dependent (ξ sgs n p ( x) 2 ŭ k / x k ). The proportionality coefficient for the bulk viscosity used for all simulations with St > is 50. The RUE transport equation (eq. 21) is not modified. In homogeneous turbulence, the spatial average of this bulk viscous term is zero, still it acts locally and leads to a more homogeneous number density field. Simulations have been carried out for several particle relaxation times. The following computation with a Stokes number of St = 0.53 is performed with this heuristically introduced bulk viscosity and compared to the DPS reference results Integral quantities Figure 18 shows the temporal evolution of carrier phase kinetic energy, particle kinetic energy and fluid-particle correlation. The carrier phase kinetic energy decreases due to viscous dissipation. Particle kinetic energy follows the carrier phase kinetic energy with a delay of the order of the particle relaxation time. Due to particle inertia, the particle velocities become partially uncorrelated in space and the RUE begins to increase. The behavior of the integral 39

40 Fig. 18. Temporal evolution of the fluid, particle and fluid-particle velocity correlations for DPS (symbols) and Eulerian simulations (lines). For Eulerian approach the total particle kinetic energy qp 2 is computed as the sum of the kinetic energy due to correlated motion q p 2 and the random uncorrelated energy δqp. 2 quantities of correlated and uncorrelated particle kinetic energy as well as the fluid-particle correlation are well predicted by the Eulerian simulation using the transport equation for RUE. Simulations using the isentropic approximation show results of similar quality. Comparison of particle number density PDF is provided on fig. 19. As expected, the Eulerian approach underestimates particle segregation Instantaneous local fields Fig 20 shows a snapshot of number density in DPS and the Eulerian simulation in the upper and the lower graph. The number density field shows the same kind of structures for both the Lagrangian and Eulerian simulations. But the instantaneous distribution is less heterogeneouss for the Eulerian simulation due to the heuristic bulk viscosity which reduces compressibility effects. 40

41 Fig. 19. Comparison of DPS and Eulerian number density PDF for the test case St =0.53 at t=10.8. The heuristically introduced bulk viscous term tends to render the spatial particle number density more uniform. Without this bulk viscous term, Eulerian simulations can currently not be carried out : the physical particle segregation is too large as it could be resolved by the numerical scheme. Since the spatial average of the volume viscosity is however zero, it does not effect the temporal evolution of the averaged kinetic energy of Random Uncorrelated Motion of the particles δqp. 2 In contrast, local instantaneous values of δ θ p may differ notably from the values obtained in the DPS (fig. 21). 41

42 Fig. 20. Comparison of the normalized droplet number n p / n p from the Lagrangian simulation (upper graph, resolution 64 3 ) with the one from the Eulerian simulation (lower graph, resolution ) (St = 0.53) at time (t=10.8). Same cut-plane as fig

43 Fig. 21. Comparison of RUE (δ θ p ) in the DPS (upper graph,resolution 64 3 ) and the Eulerian computation (lower graph,resolution ) after one particle relaxation (St = 0.53) time (t=10.8). Same cut-plane as fig