A Lagrangian probability-density-function model for collisional turbulent fluid-particle flows. II. Application to homogeneous flows
Alessio Innocenti, Rodney O Fox, Maria Vittoria Salvetti, Sergio Chibbaro
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics A Lagrangian probability-density-functionmodel for collisional turbulent fluid–particleflows. II. Application to homogeneous flows
A. Innocenti , R. O. Fox , M. V. Salvetti and S. Chibbaro † Sorbonne Universit´e, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rondd’Alembert, F-75005 Paris, France Dipartimento di Ingegneria Civile e Industriale, Universit`a di Pisa, Via G. Caruso 8, 56122Pisa, Italia Department of Chemical and Biological Engineering, 618 Bissell Road, Iowa State University,Ames, IA 50011-1098, USA(Received xx; revised xx; accepted xx)
The Lagrangian probability-density-function model, proposed in Part I for dense particle-laden turbulent flows, is validated here against Eulerian-Lagrangian direct numericalsimulation (EL) data for different homogeneous flows, namely statistically steady anddecaying homogeneous isotropic turbulence, homogeneous-shear flow and cluster-inducedturbulence (CIT). We consider the general model developed in Part I adapted to thehomogeneous case together with a simplified version in which the decomposition ofthe phase-averaged (PA) particle-phase fluctuating energy into the spatially correlatedand uncorrelated components is not used, and only total exchange of kinetic energybetween phases is allowed. The simplified model employs the standard two-way couplingapproach. The comparison between EL simulations and the two stochastic models inhomogeneous and isotropic turbulence and in homogeneous-shear flow shows that inall cases both models are capable to reproduce rather well the flow behaviour, notablyfor dilute flows. The analysis of the CIT gives more insights on the physical natureof such systems and about the quality of the models. Results elucidate the fact thatsimple two-way coupling is sufficient to induce turbulence, even though the granularenergy is not considered. Furthermore, first-order moments including velocity of the fluidseen by particles can be fairly well represented with such a simplified stochastic model.However, the decomposition into spatially correlated and uncorrelated components isfound to be necessary to account for anisotropic energy exchanges. When these factorsare properly accounted for as in the complete model, the agreement with the EL statisticsis satisfactory up to second order.
Key words: particle-laden flow, multiphase turbulence, Lagrangian pdf model, turbu-lence modulation, homogeneous flows, cluster-induced turbulence
1. Introduction
Particle-laden flows represent an important class of natural and industrial flows (Crowe et al. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] M a r A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro,
Powell 2005; Forterre & Pouliquen 2008; Guazzelli & Morris 2011) and are often turbulent(Balachandar & Eaton 2010). Given the complexity of such phenomena, to put forwarda reduced model is mandatory for practical purposes. To guide this development it isvery useful to disentangle the different physical mechanisms at play, and in particularto understand how to cope with the effect of increasing the particle mass loading andthe consequent growing importance of collisions and two-way coupling. Unfortunately, itis hard to find clear-cut frontiers between the different regimes (Elghobashi & Truesdell1992), and thus some heuristic considerations are always needed.Generally speaking, two classes of modelling approaches can be chosen for turbulentflows, the Eulerian and the Lagrangian ones (Pope 2000). When the flow is dilute ormoderately dense, the Lagrangian approach is mature (Minier et al. et al. et al. et al. κ p = k p + (cid:104) Θ p (cid:105) where (cid:104) Θ p (cid:105) is the granular temperature. If collisionsare absent, the need to decompose the particle velocity into spatially correlated anduncorrelated components is less obvious. However, owing to the fact that particle–particlecollisions are driven by the spatially uncorrelated velocity component, this decompositionis thought to be crucial for collisional flows.In this work, we test two stochastic Lagrangian models describing the particle phase,coupled with Reynolds-average Eulerian equations for the fluid phase. The first model,derived in Part I, is based on velocity partitioning between correlated and uncorrelatedcomponents. The second one is a simplified version, where only the total particle velocity,derived as the sum of the two component, is resolved, leading to the lack of distinctionbetween the particle turbulent kinetic energy and the granular temperature. We focushere on statistically homogeneous turbulence. In particular, the goal of the paper is tounderstand if the stochastic models are able to deal with the momentum and energyexchange between phases, and the particle concentration fluctuations. These ingredientsare essential in all moderately dense particle-laden flows, and therefore it is importantto use the homogeneous cases in order to isolate their modelling from other complexfeatures present in non-homogeneous configurations (e.g. spatial fluxes). In particular,we are interested in cluster-induced turbulence (CIT), which occurs in fluid–particleflows when (i) the mean mass loading ϕ , defined by the ratio of the specific masses of theparticle and fluid phases, is of order one or larger; and (ii) the difference between the mean Lagrangian pdf model for collisional turbulent fluid–particle flows ρ p ρ f is very large in gas–particle flows,CIT is ubiquitous in practical engineering and environmental flows when body forces orinlet conditions generate a mean velocity difference, such as the gravity-driven flowsstudied herein. Some fundamental properties of such flows has been recently studied viaEulerian–Lagrangian numerical simulations (Capecelatro et al. et al. et al. et al. et al. § §
3, the models are applied to increasingly morecomplex particle-laden flows and the results compared to data from the literature. In §
2. Lagrangian pdf model for particle-laden flows
In Part I of this work, we have developed the general formalism for the Lagrangian pdfapproach to dense flows, and we have proposed a rather general stochastic model, whichshould be suitable for moderately dense flows where collisions play a role but are notcompletely dominant. We present in § A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, and does not distinguish between the correlated and uncorrelated parts of the particle-phase velocity field. 2.1.
Stochastic model for particle phase
The set of stochastic equations for the particle phase, expressed for a homogeneousflow, is detailed in (2.1)–(2.5) below. dx p,i = V p,i dt = ( U p,i + δv p,i ) dt (2.1)where x p is the particle position and V p is the particle velocity. As explained in Part I,following F´evrier et al. (2005) and Capecelatro et al. (2015), the particle velocity isdecomposed in a spatially correlated part U p , and in a uncorrelated residual, δ v p . Theformer is governed by dU p,i = U s,i − U p,i τ p dt + g i dt − (cid:104) α p (cid:105) ρ p ∂ (cid:104) α p (cid:105) ρ p (cid:104) P ij (cid:105) ∂x j + δv p,j ∂ (cid:104) U p,i (cid:105) ∂x j dt − T Lp ( U p,i − (cid:104) U p,i (cid:105) ) dt + (cid:112) C p ε p dW p,i . (2.2)The first term of the RHS of (2.2) is the drag force related to the correlated part ofthe particle velocity, in which U s is the fluid velocity seen by the particle and τ p theparticle relaxation time (hereinafter taken as a constant). The second term is the effectof gravity, g , while the third is a pressure term, in which ρ p is the particle density, α p the particle-phase volume fraction and (cid:104) P ij (cid:105) = (cid:104) δv p,i δv p,j (cid:105) is the particle-phase pressuretensor. The brackets (cid:104)·(cid:105) denote phase-specific Reynolds average. The fourth and fifthterms are production and relaxation, respectively, in which T Lp is the particle Lagrangiantime scale (defined in the following). Finally, the last contribution is a diffusion term, inwhich C p is a model constant to be a priori assigned, ε p is the particle dissipation and dW p,i is a Wiener stochastic process.The uncorrelated residual velocity is modelled by d δv p,i = − δv p,i τ p dt + 1 (cid:104) α p (cid:105) ρ p ∂ (cid:104) α p (cid:105) ρ p (cid:104) P ij (cid:105) ∂x j − δv p,j ∂ (cid:104) U p,i (cid:105) ∂x j dt + B δ,ij dW δ,j − (1 + e )(3 − e )4 τ c δv p,i dt + (cid:114) τ c (1 + e ) (cid:104) Θ p (cid:105) dW c,i . (2.3)The first four terms in the RHS of (2.3) are analogous to the ones in (2.2). In particular, dW δ is a Wiener stochastic process and B δ is a diffusion matrix, whose expression isgiven in the following. The last two terms take into account collisions; e is a restitutioncoefficient, to be a priori specified, dW c is another Wiener process and (cid:104) Θ p (cid:105) is the granulartemperature, defined as (cid:104) Θ p (cid:105) = (cid:104) δ v p · δ v p (cid:105) . In particular, the following relation holdsfor the fluctuating energy partitioning: κ p = k p + (cid:104) Θ p (cid:105) , where κ p = (cid:104) v p · v p (cid:105) is thetotal particle-phase fluctuating energy and k p = (cid:104) u p · u p (cid:105) the turbulent particle-phasekinetic energy, v p and u p being the fluctuations arising from the Reynolds decompositionof V p and U p , respectively. Finally, τ c is a characteristic time for collisions, having thefollowing expression: τ c = √ πd p C c (cid:104) α p (cid:105)(cid:104) Θ p (cid:105) / , (2.4) d p being the particle diameter and C c a model parameter (Capecelatro et al. b ). Lagrangian pdf model for collisional turbulent fluid–particle flows dU s,i ( t ) = − ρ f ∂ (cid:104) p f (cid:105) ∂x i dt + G i,j ( U s,j − (cid:104) U f,j (cid:105) ) dt − ϕ (cid:18) U s,i − U p,i τ p (cid:19) dt + g i dt + (cid:104) ε f (cid:16) C f b i ˜ k f k f + (cid:16) b i ˜ k f k f − (cid:17)(cid:17) +2 ϕ (cid:104) U p,i − U s,i (cid:105) τ p ( (cid:104) U s,i (cid:105) − (cid:104) U f,i (cid:105) ) − (cid:104) α p (cid:105)(cid:104) α f (cid:105) ρ f ∂ (cid:104) p f (cid:105) ∂x i ( (cid:104) U s,i (cid:105) − (cid:104) U f,i (cid:105) ) (cid:105) / dW s,i . (2.5)The first term of the RHS is the pressure gradient term, where ρ f is the fluid densityand p f the fluid pressure; in general, the subscript f denotes a flow variable in the fluidphase. The second term is a relaxation term, where G ij = − T ∗ L,i δ ij + G aij . (2.6) T ∗ L,i is a modified fluid time-scale, which takes into account the anisotropy of the flowand particle inertia, defined by T ∗ L,i = T Lf (cid:113) ζ i β |(cid:104) U r (cid:105)| k f , T Lf = 2 (cid:0) C f (cid:1) k f ε f (2.7)where ζ = 1 in the mean drift direction and ζ , = 4 in the cross directions, β = T Lf /T Ef is the ratio of the Lagrangian and the Eulerian timescales and U r = U p − U s is therelative velocity. k f and ε f are the fluid turbulent kinetic energy and dissipation. G a isa traceless matrix to be added to generalize the model as shown in Part I: G aij = C f ∂ (cid:104) U f,i (cid:105) ∂x j . (2.8)It corresponds to the Isotropization-of-production contribution in the LRR-IP model,with C f being the IP constant. The value of the model constant C f is established bythe relation, see (Pope 1994): C f = 23 (cid:18) C Rf − C f P ε f (cid:19) . (2.9)where C Rf is the Rotta constant and P the mean shear production. The third term in(2.5) accounts for two-way coupling, ϕ being the mean mass loading, defined as ϕ = ρ p (cid:104) α p (cid:105) ρ f (cid:104) α f (cid:105) . Finally, the last term is a stochastic diffusion process extended to dense flows inwhich b i = T Lf /T ∗ L,i , ˜ k f = 32 (cid:80) i =1 b i (cid:104) ( U s,i − (cid:104) U f,i (cid:105) ) (cid:105) (cid:80) i =1 b i (2.10)and dW s is an additional Wiener process.When the correlation (cid:104) δv p,i δv p,j (cid:105) is evaluated, the diffusion matrix B δ must give theparticle-phase Reynolds-stress tensor multiplied by the proper coefficient together with A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, a diagonal isotropic part. Using a Choleski decomposition we obtain: B δ, = (cid:20) f s ε p k p (cid:104) u p, u p, (cid:105) + (1 − f s ) 23 ε p (cid:21) / ,B δ,i = 1 B δ, f s ε p k p (cid:104) u p,i u p, (cid:105) , < i (cid:54) B δ,ii = f s ε p k p (cid:104) u p,i u p,i (cid:105) + (1 − f s ) 23 ε p − i − (cid:88) j =1 B δ,ij / , < i (cid:54) B δ,ij = 1 B δ,jj (cid:32) f s ε p k p (cid:104) u p,i u p,j (cid:105) − j − (cid:88) k =1 B δ,ik B δ,jk (cid:33) , < j < i (cid:54) B δ,ij = 0 , i < j (cid:54) (cid:54) f s (cid:54) ε p = ε p (cid:20) f s (cid:104) u p ⊗ u p (cid:105) k p + (1 − f s ) 23 I (cid:21) (2.12)where ε p is one-half the trace of ε p .2.2. Statistically homogeneous particle-phase model
For statistically homogeneous flow, the Eulerian equations corresponding to thestochastic equation system (2.1)–(2.5) are the following (see Part I for their derivation): d (cid:104) U p (cid:105) dt = 1 τ p (cid:104) U s − U p (cid:105) + g , (2.13) (cid:104) δ v p (cid:105) = 0 , (2.14) d (cid:104) U s (cid:105) dt = − ρ f ∇(cid:104) p f (cid:105) + G · ( (cid:104) U s (cid:105) − (cid:104) U f (cid:105) ) + ϕτ p (cid:104) U p − U s (cid:105) + g , (2.15)The particle-phase pressure tensor, (cid:104) P (cid:105) = (cid:104) δ v p ⊗ δ v p (cid:105) , is found from d (cid:104) P (cid:105) dt = P P + ε p − τ p (cid:104) P (cid:105) + 12 τ c [(1 + e ) (cid:104) Θ p (cid:105) I − (1 + e )(3 − e ) (cid:104) P (cid:105) ] (2.16)where (cid:104) Θ p (cid:105) = trace ( (cid:104) P (cid:105) ) and the production term due to mean velocity gradients is P P = − ( (cid:104) P (cid:105) · ∇ (cid:104) U p (cid:105) ) † (2.17)where the symbol ( · ) † implies the summation of a second-order tensor with its transpose.For the particle-phase Reynolds-stress tensor, we obtain d (cid:104) u p ⊗ u p (cid:105) dt = P p + R p − ε p . (2.18)The redistribution term is expressed as R p = − C Rp ε p k p (cid:18) (cid:104) u p ⊗ u p (cid:105) − k p I (cid:19) (2.19)with k p = (cid:104) u p · u p (cid:105) and C Rp = 1 + 32 C p (2.20) Lagrangian pdf model for collisional turbulent fluid–particle flows C p is the model constant in (2.2). The production term in (2.18) is defined by P p = P Sp + P Dp where P Sp is the mean-shear-production term, given by P Sp = − ( (cid:104) u p ⊗ u p (cid:105) · ∇ (cid:104) U p (cid:105) ) † ; (2.21)and P Dp is the drag-production term, given by P Dp = 1 τ p ( (cid:104) u s ⊗ u p (cid:105) † − (cid:104) u p ⊗ u p (cid:105) ) . (2.22)The fluid-seen Reynolds-stress tensor is found from d (cid:104) u s ⊗ u s (cid:105) dt = P s + ( G · (cid:104) u s ⊗ u s (cid:105) ) † + (cid:104) B s B Ts (cid:105) (2.23)where B s is the diffusion matrix in (2.5) and k f @ p = (cid:104) ( U s − (cid:104) U f (cid:105) ) · ( U s − (cid:104) U f (cid:105) ) (cid:105) . Theproduction term in (2.23) is defined by P s = P Ss + P Ds where P Ss is the mean-shear-production term, given by P Ss = − ( (cid:104) u s ⊗ u s (cid:105) · ∇ (cid:104) U s (cid:105) ) † ; (2.24)and P Ds is the drag-production term, given by P Ds = ϕτ p ( (cid:104) u s ⊗ u p (cid:105) † − (cid:104) u s ⊗ u s (cid:105) ) . (2.25)The fluid–particle covariance Reynolds-stress tensor is found from d (cid:104) u s ⊗ u p (cid:105) dt = P sp + G · (cid:104) u s ⊗ u p (cid:105) T − T Lp (cid:104) u p ⊗ u s (cid:105) (2.26)where k fp = (cid:104) u s · u p (cid:105) . The production term in (2.26) is defined by P sp = P Ssp + P Dsp where P Ssp is the mean-shear-production term, given by P Ssp = −(cid:104) u s ⊗ u p (cid:105) · ∇ (cid:104) U p (cid:105) T − ( (cid:104) u p ⊗ u p (cid:105) + (cid:104) P (cid:105) ) · ∇ (cid:104) U s (cid:105) T ; (2.27)and P Dsp is the drag-production term, given by P Dsp = 1 τ p ( (cid:104) u s ⊗ u s (cid:105) − (cid:104) u p ⊗ u s (cid:105) ) + ϕτ p ( (cid:104) u p ⊗ u p (cid:105) − (cid:104) u s ⊗ u p (cid:105) ) . (2.28)The particle Lagrangian time scale, introduced in (2.2), is defined as T Lp = 2 (cid:0) C p + f s (cid:1) k p ε p . (2.29)The particle-phase dissipation is modelled through an Eulerian equation in analogy tosingle-phase flows (Fox 2014): dε p dt = ( C (cid:15) p P Sp − C (cid:15) p ε p ) ε p k p + C p τ p (cid:18) k fp k f @ p ε f − β p ε p (cid:19) (2.30)where C (cid:15) p , C (cid:15) p , C p and β p are model parameters. Finally, all the fluid-phase quantitiesare obtained through the RA equations presented in § Simplified model for particle phase
We propose here a simplified model for the particle phase, where collisions betweenparticles are neglected and only the total particle velocity is modelled, thus loosinginformation about its decomposition into the correlated and uncorrelated parts. Inparticular, this corresponds to assuming that the particle velocity coincides with the
A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, correlated part, i.e. V p = U p . With this hypothesis, we recover the model previouslyproposed by Minier et al. (2004) and Peirano et al. (2006) for the fluid velocity seen bythe particles, but with a modified diffusion term, as discussed in detail in Part I. Theresulting set of SDEs for the simplified model is dx p,i ( t ) = V p,i dt,dV p,i ( t ) = U s,i − V p,i τ p dt + g i dt,dU s,i ( t ) = − ρ f ∂ (cid:104) p f (cid:105) ∂x i dt − T ∗ L,i ( U s,i − (cid:104) U f,i (cid:105) ) dt − ϕ (cid:18) U s,i − V p,i τ p (cid:19) dt + g i dt + (cid:104) ε f (cid:16) C f b i ˜ k f k f + 23 (cid:16) b i ˜ k f k f − (cid:17)(cid:17) +2 ϕ (cid:104) V p,i − U s,i (cid:105) τ p ( (cid:104) U s,i (cid:105) − (cid:104) U f,i (cid:105) ) − (cid:104) α p (cid:105)(cid:104) α f (cid:105) ρ f ∂ (cid:104) p f (cid:105) ∂x i ( (cid:104) U s,i (cid:105) − (cid:104) U f,i (cid:105) ) (cid:105) / dW s,i (2.31)where all of the parameters were defined in the complete model above.The corresponding Eulerian RA equations for statistically homogeneous flow are d (cid:104) V p (cid:105) dt = 1 τ p (cid:104) U s − V p (cid:105) + g , (2.32) d (cid:104) U s (cid:105) dt = − ρ f ∇(cid:104) p f (cid:105) − T ∗ L ◦ ( (cid:104) U s (cid:105) − (cid:104) U f (cid:105) ) + ϕτ p (cid:104) V p − U s (cid:105) + g . (2.33)For the second-order moments, we obtain d (cid:104) v p ⊗ v p (cid:105) dt = P V p + 1 τ p ( (cid:104) u s ⊗ v p (cid:105) † − (cid:104) v p ⊗ v p (cid:105) ) , (2.34) d (cid:104) u s ⊗ v p (cid:105) dt = P V sp − T ∗ L ◦ (cid:104) u s ⊗ v p (cid:105) + 1 τ p ( (cid:104) u s ⊗ u s (cid:105) − (cid:104) v p ⊗ u s (cid:105) )+ ϕτ p ( (cid:104) v p ⊗ v p (cid:105) − (cid:104) u s ⊗ v p (cid:105) ) , (2.35) d (cid:104) u s ⊗ u s (cid:105) dt = P Ss − T ∗ L ◦ (cid:104) u s ⊗ u s (cid:105) + (cid:104) B s B Ts (cid:105) + ϕτ p ( (cid:104) u s ⊗ v p (cid:105) † − (cid:104) u s ⊗ u s (cid:105) ) (2.36)The mean-shear-production terms are P V p = − ( (cid:104) v p ⊗ v p (cid:105) · ∇ (cid:104) V p (cid:105) ) † (2.37)and P V sp = −(cid:104) u s ⊗ v p (cid:105) · ∇ (cid:104) V p (cid:105) T − (cid:104) v p ⊗ v p (cid:105) · ∇ (cid:104) U s (cid:105) T . (2.38)2.4. Fluid-phase model
The Eulerian RA equation describing the fluid phase mass balance for a statisticallyhomogeneous flow reduces to d (cid:104) α f (cid:105) dt = 0 (2.39)i.e., (cid:104) α f (cid:105) is constant. The fluid-phase velocity and Reynolds stresses are found from d (cid:104) U f (cid:105) dt = − ρ f (cid:104) α f (cid:105) ∇ (cid:104) p f (cid:105) + ϕτ p (cid:104) U p − U s (cid:105) + g , (2.40) Lagrangian pdf model for collisional turbulent fluid–particle flows d (cid:104) u f ⊗ u f (cid:105) dt = P f − C Rf ε f k f (cid:18) (cid:104) u f ⊗ u f (cid:105) − k f I (cid:19) − C f (cid:18) P Sf − P Sf I (cid:19) − ε f I (2.41)where k f = (cid:104) u f · u f (cid:105) . The production term is P f = P Sf + P Df where P Sf is themean-shear-production term, given by P Sf = − ( (cid:104) u f ⊗ u f (cid:105) · ∇ (cid:104) U f (cid:105) ) † ; (2.42)and P Df is the drag-production term, given by P Df = ϕτ p [ (cid:104) u s ⊗ ( u p − u s ) (cid:105) + (cid:104) U s − U f (cid:105) ⊗ (cid:104) U p − U s (cid:105) ] † . (2.43)In (2.41), C Rf is the Rotta constant for the redistribution (Pope 2000), and P Sf = trace ( P Sf ).The fluid-phase dissipation equation is dε f dt = ( C (cid:15) f P Sf − C (cid:15) f ε f ) ε f k f + C f ϕτ p (cid:18) k fp k f @ p ε p − β f ε f (cid:19) + C ε p k p P D (2.44)where C (cid:15) f , C (cid:15) f , C f , β f and C are model constants, and P D = ϕτ p (cid:104) U s − U f (cid:105) · (cid:104) U p − U f (cid:105) . (2.45)If the RA equations for the fluid phase are coupled with the simplified model describedin § U p must be replaced with V p . Moreover, the particle-phase Lagrangian time-scale k p /ε p is not specified, and it is thus replaced by a fluid time-scale through aproportionality constraint: ε p k p = α ε f k f @ p (2.46)Now, substituting (2.46) in (2.44) and incorporating α in the model constants, gives thefollowing equation for dissipation: dε f dt = ( C (cid:15) f P Sf − C (cid:15) f ε f ) ε f k f + C f ϕτ p (cid:18) k p k f @ p k fp − β f k f @ p (cid:19) ε f k f @ p + C ε f k f @ p P D (2.47)The values of C f and C in (2.47) may need to be adjusted as compared to (2.44) toaccount for the alternative time scale.
3. Numerical Results
If the fluid–particle flow is spatially homogeneous as in the cases that we are going totest below, the equations can be simplified, since hydrodynamic variables are invariantin space (see § § et al. et al. P Sf ) without0 A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, a mean velocity difference, and the third at validating the model for production due toa mean velocity difference (i.e., (2.45)).3.1.
Homogeneous isotropic turbulence
In order to illustrate the effectiveness of the decomposition of the particle velocity,we apply the models developed for the particle phase to the homogeneous isotropicturbulence simulations of F´evrier et al. (2005) for non-collisional particles. For thisexample, the mean velocities U p , U f , U s are null, and ϕ = 0. At a first glance it mayappear odd to compare the results of a model developed for collisional flows to DNS datafor non-collisional particles. However, the crucial point for the modelling is the correlationbetween the fluid and particle velocities as captured by k f and k p , respectively. Theapplicability of the proposed models to non-collisional flows depends on the model usedfor turbulent dissipation, since the relative balance between k p and Θ p is determined by ε p , for both dilute and dense flows. The scope of this section is thus to verify if in thedilute case, where collisions do not play any role, energy budgets are well predicted.The mesoscale DNS simulations of F´evrier et al. (2005) use one-way coupling withstationary fluid turbulence, and a particle-Reynolds-number-dependent drag coefficient f D (instead of a constant τ p ). Therefore, the drag time scale is Stokes-number-dependent,and only qualitative comparisons can be made.When a cloud of particles is put into a box filled with a homogeneous and isotropicturbulent flow and is being agitated by the fluid turbulence, then, after a transientperiod, the statistics of particle velocities reach equilibrium values. These limit valuesare of course functions of the (constant) statistics of the fluid (its mean kinetic energy,the Lagrangian timescale, among others). The relations giving the equilibrium values interms of the fluid statistics are called the Tchen’s relations. They were first obtainedby Tchen (1947) and later reformulated by Hinze (1975). In Tchen or Hinze’s works,the determination of the equilibrium values was obtained through spectral analysisand manipulation of the fluid and particle energy spectra, where the fluid spectrumis assumed to have an exponential form. This derivation can be cumbersome and thephysical meaning of the exponential form is not obvious. On the other hand, the samerelations are derived from the Lagrangian pdf model in a straightforward way.In forced, homogeneous, isotropic turbulence without body forces, all mean veloci-ties are zero and the Reynolds-stress and particle-phase pressure tensors are isotropic.Moreover, k f @ p = k f . With one-way coupling and fixed k f and ε f , the relevant momentequations from the complete model for the particle phase reduce to dk p dt = 2 τ p ( k fp − k p ) − ε p , (3.1 a )32 d (cid:104) Θ p (cid:105) dt = − (cid:104) Θ p (cid:105) τ p + ε p , (3.1 b ) dk fp dt = − (cid:18) T Lf + 1 T Lp (cid:19) k fp + 1 τ p ( k f − k fp ) , (3.1 c ) dε p dt = − C (cid:15) p ε p k p + C p τ p (cid:18) k fp k f ε f − β p ε p (cid:19) (3.1 d )where T Lf is given by (2.7) and T Lp by (2.29). After a transient period, all the statistics Lagrangian pdf model for collisional turbulent fluid–particle flows k fp − k p τ p − ε p = 0 , (3.2 a ) − (cid:104) Θ p (cid:105) τ p + ε p = 0 , (3.2 b ) k f τ p − (cid:18) τ p + 1 T Lf + 1 T Lp (cid:19) k fp = 0 , (3.2 c ) St p − C p C (cid:15) p (cid:18) k fp k p St f − β p St p (cid:19) = 0 (3.2 d )where St p = τ p ε p /k p and St f = τ p ε f /k f . Summing (3.2 a ) and (3.2 b ) then yields k fp = κ p , (3.3)which can be used together with (3.2 c ) to obtain a Tchen-like relation: κ p = 11 + τ p /T (cid:48) L k f (3.4)with C = C p = C f and1 T (cid:48) L = 1 T Lf + 1 T Lp = (cid:18)
12 + 34 C (cid:19) (cid:18) ε f k f + ε p k p (cid:19) . (3.5)Here, τ p /T (cid:48) L is an effective integral-scale Stokes number for the particles. Furthermore, St p is constant, and can be related to St f using (3.2 a ) and (3.2 d ). With β p = 1, thisrelation depends only on the parameter ratio C p C (cid:15) p , and thus St p = St f when C p = 2 C (cid:15) p .Note that the value of St p controls the ratio k p /κ p = 2 / (2 + St p ) and, as expected, allof the particle-phase kinetic energy is spatially correlated when St p = 0.F´evrier et al. (2005) presented time-dependent DNS results of particle-laden homo-geneous and isotropic turbulence for St f = 0 .
81 and ϕ = 0, for three sets of initialconditions: (i) κ p = k fp = 1, k p = 1; (ii) κ p = k fp = 0, k p = 0; and (iii) κ p = 0 . k p = k fp = 0. We reproduced the same cases by solving the dimensionless forms of system(3.1) with the following values of the model constants: C = 1, C (cid:15) p = 1 . C p = 3 . β p = 1. For consistency with k p , ε p is initially set to zero when k p = 0 and forcase (i) the initial value of dissipation is ε p = 2. Figure 1(a) shows the time evolution of κ p , k p and Θ p obtained with the present model for the three different sets of consideredinitial conditions, while the evolution of κ p obtained with the simplified model for thesame cases is reported in figure 1(b). Moreover, 1(c) shows the same quantities as infigure 1(a) obtained with the model proposed in Fox (2014) and 1(d) the results of theDNS of F´evrier et al. (2005). In all cases, after a transient a steady state is reached, asexpected. It can be seen how in DNS the total particle kinetic energy is distributed inthe correlated part and in the uncorrelated granular temperature. This energy partitionis satisfactorily captured by our complete model as well as by the model proposed in Fox(2014). Clearly, the simplified model can only give the total energy κ p , which is however ingood agreement with that of DNS and of more complete models. The transient behaviouris also in very good qualitative agreement with that obtained in DNS.3.2. Decaying and homogeneous-shear flow
In this section we focus on the particle–turbulence interactions in homogeneous flows,and, in particular, on the cases simulated by DNS in Sundaram & Collins (1999) for2
A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, (a) (b) q p , q p , δ q p t / τ Ffp ~ (c) (d) Figure 1.
Time evolution of the dimensionless particle-phase energy components for St f = 0 . κ p is plotted in blue, k p in green and Θ p in red. (a) Simulationswith the complete particle model (3.1). (b) Simulations carried out with the simplified particlemodel (2.31) with C = 2 .
1. Only total kinetic energy κ p is computed. (c) Results from Fox(2014) with the same initial conditions as in (a). (d) Results of point-particle DNS from figure8 of F´evrier et al. (2005), where the results are in dimensional form. The notation in F´evrier et al. (2005) is q p = κ p , (cid:101) q p = k p , δq p = Θ p and τ Ffp ∝ τ D . decaying turbulence and in Ahmed & Elghobashi (2000) for homogeneous-shear flows.For these examples, the mean velocities U p , U f , U s are null. In both cases the particle-phase volume fraction is such that two-way interactions need to be considered: flowmodification by non-collisional point particles reveal a non-trivial dependence on theparticle Stokes number and mass loading ϕ . It is thus interesting to verify if our modelis able to reproduce such physics and if the same dependencies on the particle Stokesnumber are found.The Eulerian RA equations describing the fluid and particle phases for the consideredcases are summarized below for the sake of completeness. In these equations, the flow isstatistically homogeneous with a constant shear S f = ∂ (cid:104) U f, (cid:105) /∂x ( S f = 0 in decayingturbulence) and with gravity and collisions neglected. The non-zero components of thesecond-order moments are ( i, j ) = (1 , , (1 , , (2 , , (2 , , (3 , Lagrangian pdf model for collisional turbulent fluid–particle flows d (cid:104) u f,i u f,j (cid:105) dt = P f,ij − C Rf ε f k f (cid:18) (cid:104) u f,i u f,j (cid:105) − k f δ ij (cid:19) − C f (cid:18) P Sf,ij − P Sf δ ij (cid:19) − ε f δ ij , (3.6 a ) dε f dt = ( C (cid:15) f P Sf − C (cid:15) f ε f ) ε f k f + C f ϕτ p (cid:18) k fp k f @ p ε p − β f ε f (cid:19) (3.6 b )with production terms due to mean shear and drag: P f,ij = P Sf,ij + P Df,ij , (3.7 a ) P Sf,ij = −(cid:104) u f,i u f, (cid:105)S f δ j − (cid:104) u f,j u f, (cid:105)S f δ i , (3.7 b ) P Df,ij = ϕτ p ( (cid:104) u s,i u p,j (cid:105) + (cid:104) u s,j u p,i (cid:105) − (cid:104) u s,i u s,j (cid:105) ) . (3.7 c )For the particle phase, the pressure tensor and Reynolds stresses are found from d (cid:104) P ij (cid:105) dt = P P,ij + ε p (cid:20) f s (cid:104) u p,i u p,j (cid:105) k p + (1 − f s ) 23 δ ij (cid:21) − τ p (cid:104) P ij (cid:105) , (3.8 a ) d (cid:104) u p,i u p,j (cid:105) dt = P p,ij − C Rp ε p k p (cid:18) (cid:104) u p,i u p,j (cid:105) − k p δ ij (cid:19) − ε p (cid:20) f s (cid:104) u p,i u p,j (cid:105) k p + (1 − f s ) 23 δ ij (cid:21) , (3.8 b ) dε p dt = ( C (cid:15) p P Sp − C (cid:15) p ε p ) ε p k p + C p τ p (cid:18) k fp k f @ p ε f − β p ε p (cid:19) (3.8 c )with production terms: P P,ij = −(cid:104) P i (cid:105)S p δ j − (cid:104) P j (cid:105)S p δ i , (3.9 a ) P p,ij = P Sp,ij + P Dp,ij , (3.9 b ) P Sp,ij = −(cid:104) u p,i u p, (cid:105)S p δ j − (cid:104) u p,j u p, (cid:105)S p δ i , (3.9 c ) P Dp,ij = 1 τ p ( (cid:104) u s,i u p,j (cid:105) + (cid:104) u s,j u p,i (cid:105) − (cid:104) u p,i u p,j (cid:105) ) . (3.9 d )Note that when S f is null (i.e., decaying turbulence), all second-order tensors will beisotropic so that only their traces are needed.The mean gradients for the particle phase and fluid seen obey d S p dt = 1 τ p ( S s − S p ) , (3.10 a ) d S s dt = 1 τ p ( S f − S s ) . (3.10 b )In the following, the particle-phase velocity is initially the same as the fluid-phase velocitysuch that S p ( t ) = S s ( t ) = S f and thus system (3.10) is not needed.4 A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro,
The Reynolds stresses involving the fluid seen by the particles are found from d (cid:104) u s,i u p,j (cid:105) dt = P sp,ij + (cid:18) G ij − T Lp (cid:19) (cid:104) u s,i u p,j (cid:105) , (3.11 a ) d (cid:104) u s,i u s,j (cid:105) dt = P s,ij + 2 G ij (cid:104) u s,i u s,j (cid:105) + ε f (cid:20) C f k f @ p k f + 23 (cid:18) k f @ p k f − (cid:19)(cid:21) δ ij (3.11 b )with production terms due to mean shear and drag: P s,ij = P Ss,ij + P Ds,ij , (3.12 a ) P Ss,ij = −(cid:104) u s,i u s, (cid:105)S s δ j − (cid:104) u s,j u s, (cid:105)S s δ i , (3.12 b ) P Ds,ij = ϕτ p ( (cid:104) u s,i u p,j (cid:105) + (cid:104) u s,j u p,i (cid:105) − (cid:104) u s,i u s,j (cid:105) ) , (3.12 c )and P sp,ij = P Ssp,ij + P Dsp,ij , (3.13 a ) P Ssp,ij = − ( (cid:104) u s,i u p, (cid:105)S p + (cid:104) P i (cid:105)S s + (cid:104) u p,i u p, (cid:105)S s ) δ j − ( (cid:104) u s,j u p, (cid:105)S p + (cid:104) P j (cid:105)S s + (cid:104) u p,j u p, (cid:105)S s ) δ i , (3.13 b ) P Dsp,ij = 1 τ p [ (cid:104) u s,i u s,j (cid:105) − (cid:104) u s,j u p,i (cid:105) + ϕ ( (cid:104) u p,i u p,j (cid:105) − (cid:104) u s,i u p,j (cid:105) )] . (3.13 c )In (3.11), T Lf is given by (2.7) and T Lp by (2.29).It is worth noting that, even in these simple flow conditions, the model still retainssome of its features, as, for instance, the distinction between the fluid kinetic energy andthe fluid–particle velocity correlation, which can be computed from Lagrangian quantitiesby averaging, i.e. k fp = (cid:104) u p,k u s,k (cid:105) , while in Eulerian models that do not account for thefluid seen by the particles (see Fox (2014)), it is modeled as k fp = ( k f k p ) / . Moreover,having derived our model from the one proposed for dilute flows by Peirano & Minier(2002), it should be remarked that only a part of the crossing trajectory effect is takeninto account, that is when there is a mean drift, and thus, a mean relative velocitybetween fluid and particles. This means that in the case that we are testing, the modifiedLagrangian timescale equals the fluid Lagrangian timescale, T ∗ L = T L , for all Stokesnumbers. Conversely, particle inertia should affect the Lagrangian timescale of the fluidvelocity seen by the particles. In particular, if we consider the limit cases, we have twosituations: particles with very low inertia, i.e. τ p /T L (cid:28)
1, follow almost exactly the fluid,yielding T ∗ L = T L for the fluid velocity seen. Particles with high inertia, i.e. τ p /T L (cid:29) T ∗ L = T E . This inconsistency hasalready been pointed out by Pozorski & Minier (1998), and, even if it can be neglectedin flows where a mean drift drives the particles, becoming secondary, here it is of crucialimportance, especially if we are interested in finding the trends of the decay-rate withrespect to particle inertia. For this reason we propose to add a Stokes dependence in C (cid:15) of the kind C (cid:15) = C (1 − ϕSt ) with C = 1 .
92, in order to retrieve the good trend with St .Note that this simple model, which was also used in Fox (2014) for the same test case, isjust qualitative and valid for the range of conditions considered herein. A more refinedanalysis may be necessary for general situations.3.2.1. Decaying turbulence
Concerning the values of other model constants, they are the same as in the stationarycase, i.e. C (cid:15) = C (cid:15) p = C (cid:15) f , β = β f = β p = 1, C = C f = C p = 3 . C = C f = Lagrangian pdf model for collisional turbulent fluid–particle flows C p = 1. As S f = 0, the isotropic model equations for k f , k f @ p , k p and k fp are solveddirectly: dk f dt = 2 ϕτ p ( k fp − k f @ p ) − ε f , (3.14 a ) dk f @ p dt = 2 ϕτ p ( k fp − k f @ p ) − ε f , (3.14 b ) dk p dt = 2 τ p ( k fp − k p ) − ε p , (3.14 c ) dk fp dt = 1 τ p [ k f @ p + ϕk p − (1 + ϕ ) k fp ] − (cid:18)
12 + 34 C (cid:19) (cid:18) ε f k f + ε p k p (cid:19) k fp , (3.14 d ) dε f dt = − C (cid:15) ε f k f + C ϕτ p (cid:18) k fp k f @ p ε p − ε f (cid:19) , (3.14 e ) dε p dt = − C (cid:15) ε p k p + C τ p (cid:18) k fp k f @ p ε f − ε p (cid:19) . (3.14 f )In the decaying turbulence test, initial conditions for the simulation are k f (0) = k f @ p (0) = k p (0) = k fp (0) = 1 . ε f (0) = ε p (0) = 1 . ϕ = 0 . k f (0) = k f @ p (0), the first two equations in system (3.14)will yield k f ( t ) = k f @ p ( t ) so that only k f is required to model decaying turbulence forthis case.Figure 2(a) shows the time evolution of the fluid turbulent kinetic energy obtainedwith the particle models, for particle sets characterized by four different Stokes numbers,namely St = τ p /T e = 0 (fluid tracers), St = 0 . St = 0 .
35 and St = 0 .
69 (where T e = 1 . St = 0 was obtained from the particle equations as the limit casefor τ p /T L (cid:28)
1, as described in Appendix A. Figures 2(b)–(d) show the same quantitiesas in figure 2(a), obtained by the simplified version of the present model, the Eulerianmodel by Fox (2014) and the DNS by Sundaram & Collins (1999) respectively. The samecomparisons for the particle-phase turbulent kinetic energy are reported in figure 3. It canbe seen that the effect of the Stokes number on the decay of the turbulent kinetic energyof both the fluid and the particle phases is qualitatively well captured by the presentstochastic model, in its complete version as well as in the simplified one, although theinitial stages of the time evolution are quite different from the DNS results.3.2.2.
Homogeneous-shear flow
We consider now the case of a homogeneous shear flow with S f = 0 .
6, and solvethe anisotropic model equations given in § ϕ = 0 .
162 and theinitial conditions ε f (0) = ε p (0) = 0 .
25 (see Fox (2014)). As in the previous decayingcase, simulations have been carried out for the following four Stokes numbers: St =0 , . , . , .
69. The values of the constants in our model are the same as in the previouscase of homogeneous decaying turbulence with C (cid:15) f = C (cid:15) p = 1 .
44, which are standardvalues for single-phase turbulence models (Pope 2000). For the simplified model C (cid:15) f =1 .
2. Figure 4 shows the time evolution of the fluid turbulent kinetic energy obtained withthe particle model, both in its complete and simplified versions, with the same quantityobtained from the Eulerian model in Fox (2014). Comparison should also be made withthe DNS data in figure 45 of Ahmed & Elghobashi (2000). For all the models, the timebehaviour is qualitatively similar to that observed in DNS, with an initial decrease of6
A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, (a) (b) ( a ) St = 0 (F2) St = 1.6 (A) St = 3.2 (B) St = 6.4 (C) t / T e T f ( t ) / T f ( ) t / T e (c) (d) Figure 2.
Fluid turbulent kinetic energy as a function of the non-dimensional time, t/T e , indecaying fluid–particle turbulence: (a) complete particle model, (b) simplified particle model, (c)Fox (2014) Eulerian model and (d) DNS of Sundaram & Collins (1999). The curves correspondto four different Stokes number: St = 0, solid line; St = 0 .
17, dashed lines; St = 0 .
35, dottedlines; St = 0 .
69, dash-dotted lines. In panel (c) the light dotted line is relative to an additionalvalue of the Stokes number, not reported in the other panels; in panel (d) T f ∝ k f . the fluid turbulent kinetic energy followed by an increase. The value of the minima of k f given by the complete particle model are closer to those obtained in DNS. Moreover,the effect of particle inertia on the time evolution of k f is also correctly captured, i.e.,the rate of increase of the fluid turbulent kinetic energy after the minimum is reduced asthe inertia of the particles increases. The same effect is found also for the particle-phasefluctuating energy, κ p , in agreement with Fox (2014), as it can be seen in Figure 5. NoDNS data are available for this quantity.3.3. Cluster-induced turbulence
To isolate the effect of turbulence generated by particles through two-way coupling,we consider a flow initially at rest laden with a random distribution of finite-sizeparticles of diameter d p subject to gravity oriented in the downward x direction. Thephysical parameters are chosen to correspond to the Euler–Lagrange (EL) point-particlesimulation of Capecelatro et al. (2015) as summarized in table 1. The dimensionlesstwo-phase parameters that characterize the flow include the particle-to-fluid densityratio ρ p /ρ f = 1000, the average particle-phase volume fraction (cid:104) α p (cid:105) = 0 .
01 and theparticle Reynolds numbers Re p = τ p gd p /ν f = 1 where τ p = ρ p d p / (18 ρ f ν f ) is the particle Lagrangian pdf model for collisional turbulent fluid–particle flows (a) (b) ( b ) t / T e T p ( t ) / T p ( ) St = 1.6 (A) St = 3.2 (B) St = 6.4 (C) t / T e (c) (d) Figure 3.
Particle-phase fluctuating energy as a function of the non-dimensional time, t/T e ,in decaying fluid–particle turbulence: (a) complete particle model, (b) simplified particle model,(c) Eulerian model of Fox (2014) and (d) DNS of Sundaram & Collins (1999). The curvescorrespond to the following Stokes number: St = 0 .
17, dashed lines; St = 0 .
35, dotted lines; St = 0 .
69, dash-dotted lines. In panel (c) the light dotted line is relative to an additional valueof the Stokes number, not reported in the other panels; in panel (d) T f ∝ k f . relaxation time, ν f is the fluid-phase kinematic viscosity and g is the magnitude of thegravity vector. Combination of these non-dimensional numbers yields the mass loading ϕ = ρ p (cid:104) α p (cid:105) / ( ρ f (cid:104) α f (cid:105) ) = 10 .
1, where (cid:104) α f (cid:105) = 1 − (cid:104) α p (cid:105) is the average fluid-phase volumefraction. Finally, V = gτ p is the settling velocity for a single particle.The CIT case is statistically homogeneous in all directions with periodic boundaryconditions; therefore, in the context of the present formalism, it reduces to a 0-Ddescription, with only the time dependency. Moreover, as in the previous consideredcases, since the RA equations obtained from the stochastic ones are in closed form fora homogeneous configuration, we can limit ourselves to solving a system of coupledODEs, instead of carrying out a Lagrangian Monte-Carlo simulation. The simulation isperformed starting from an initial condition where both the particle and fluid phases areat rest and it is evolved in time up to the steady state. The fluid-phase pressure gradientis dynamically adjusted in order to keep the mean fluid velocity (cid:104) U f (cid:105) equal to zero.The model constants have been set in order to obtain a good prediction of the steady-state values for first-order moments. The values, so obtained, are reported in tables 2–3.A comment is in order concerning the values of C f and C p . These values are takendifferent from those used in the isotropic cases previously analysed. The results obtained8 A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, (a) (b) (c) t / T e Figure 4.
Homogeneous shear flow. Fluid turbulent kinetic energy as a function of thenon-dimensional time, t/T e with (a) the complete stochastic model; (b) the simplified stochasticmodel and (c) the Eulerian model by Fox (2014). The curves correspond to four different Stokesnumbers: St = 0, solid line; St = 0 .
17, dashed lines; St = 0 .
35, dotted lines; St = 0 . Physical parameters d p Particle diameter 0.09 mm ρ p Particle density 1000 kg m − ρ f Fluid density 1 kg m − ν f Fluid kinematic viscosity 1 . · − m s − g Gravity magnitude 8 m s − Non-dimensional parameters e Restitution coefficient 0.90 (cid:104) α p (cid:105) Mean particle volume fraction 0 . ϕ Mean mass loading 10 . Re p Particle Reynolds number 1
Dimensional parameters τ p Drag time 0.025 s V Settling velocity 0.20 m s − Table 1.
Fluid–particle parameters used in CIT simulations (Capecelatro et al. in the CIT test-case with the previous values are in reasonable agreement with the fullnumerical simulation, but show some discrepancy which has been eliminated using thevalues proposed in table 2. In fact, C p has an insignificant effect on the asymptoticresults, but the present higher value smooths the transient dynamics. In contrast, thevalue of C f turns out to be key to get the correct level of turbulent kinetic energy. Lagrangian pdf model for collisional turbulent fluid–particle flows (a) (b) (c) Figure 5.
Homogeneous shear flow. Particle-phase fluctuating energy as a function of thenon-dimensional time, t/T e with (a) complete particle model; (b) simplified particle model and(c) Eulerian model in Fox (2014). The curves correspond to the Stokes numbers: St = 0 . St = 0 .
35, dotted lines; St = 0 .
69, dash-dotted lines. In panel (c) the light dottedline is relative to an additional value of the Stokes number, not reported in the other panels. C f C p C (cid:15) C f C p C f s β f . .
18 1 .
92 3 . . .
81 0 . Table 2.
Values of the model constants used in CIT simulations for the complete model. C f C (cid:15) C f C β f . .
02 0 . . Table 3.
Values of the model constants used in CIT simulations for the simplified model.
The first-order moments are found by solving d (cid:104) U f, (cid:105) dt = − ρ f (cid:104) α f (cid:105) d (cid:104) p f (cid:105) dx + ϕτ p (cid:104) U p, − U s, (cid:105) − g = 0 , (3.15 a ) d (cid:104) U p, (cid:105) dt = 1 τ p (cid:104) U s, − U p, (cid:105) − g, (3.15 b ) d (cid:104) U s, (cid:105) dt = − ρ f d (cid:104) p f (cid:105) dx − T ∗ L, (cid:104) U s, (cid:105) + ϕτ p (cid:104) U p, − U s, (cid:105) − g. (3.15 c )Here, (3.15 a ) fixes the fluid pressure gradient in (3.15 c ). At steady state, (3.15 b ) yields (cid:104) U s, − U p, (cid:105) = V , which agrees with the EL simulations of Capecelatro et al. (2015), and0 A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, (3.15 c ) yields (cid:104) U s, (cid:105) = −(cid:104) α p (cid:105) (1 + ϕ ) T ∗ L, τ p V (3.16)where, using the definition in § T ∗ L, τ p = (cid:34)(cid:18)
12 + 34 C f (cid:19) (cid:16) β V k f (cid:17)(cid:35) − / St f (3.17)and St f = τ p ε f k f . β = T Lf /T Ef is set equal to 0 .
8. In fully developed CIT, the turbulence isgenerated by the clusters and the resulting Stokes number is nearly constant (Capecelatro et al. b ). The complete model therefore predicts that the steady-state value of (cid:104) U s, (cid:105) / V depends on the particle volume fraction, the mass loading, and the dimensionlessfluid-phase turbulent kinetic energy 2 k f / V . The prediction of (cid:104) U s (cid:105) is perhaps the mostimportant contribution of the Lagrangian pdf model for CIT because information onthe fluid seen by the particles is not available in most multiphase turbulence models forfluid–particle flows (see, e.g., Fox 2014, for details).The second-order moments have two independent, non-zero components, i.e., thevertical (1 ,
1) and horizontal (2 , d (cid:104) u f, (cid:105) dt = P f, − C Rf ε f k f (cid:18) (cid:104) u f, (cid:105) − k f (cid:19) − ε f , (3.18 a ) d (cid:104) u f, (cid:105) dt = P f, − C Rf ε f k f (cid:18) (cid:104) u f, (cid:105) − k f (cid:19) − ε f (3.18 b )where k f = ( (cid:104) u f, (cid:105) + 2 (cid:104) u f, (cid:105) ), P f = ( P f, + 2 P f, ), P f, = 2 ϕτ p [ (cid:104) U s, (cid:105)(cid:104) U p, (cid:105)−(cid:104) U s, (cid:105)(cid:104) U s, (cid:105) + (cid:104) u s, u p, (cid:105) − (cid:104) u s, (cid:105) ] , (3.18 c ) P f, = 2 ϕτ p ( (cid:104) u s, u p, (cid:105) − (cid:104) u s, (cid:105) ) . (3.18 d )In CIT, the fluid-phase Reynolds stresses are anisotropic because of the mean velocitiesappearing in P f, . In general, redistribution is weak so that (cid:104) u f, (cid:105) (cid:28) (cid:104) u f, (cid:105) .For the particle-phase pressure tensor, the complete model yields d (cid:104) P (cid:105) dt = ε p, − τ p (cid:104) P (cid:105) + 12 τ c [(1 + e ) (cid:104) Θ p (cid:105) − (1 + e )(3 − e ) (cid:104) P (cid:105) ] , (3.19 a ) d (cid:104) P (cid:105) dt = ε p, − τ p (cid:104) P (cid:105) + 12 τ c [(1 + e ) (cid:104) Θ p (cid:105) − (1 + e )(3 − e ) (cid:104) P (cid:105) ] (3.19 b )where (cid:104) Θ p (cid:105) = ( (cid:104) P (cid:105) + 2 (cid:104) P (cid:105) ), ε p, = ε p (cid:34) f s (cid:104) u p, (cid:105) k p + (1 − f s ) 23 (cid:35) , (3.19 c ) ε p, = ε p (cid:34) f s (cid:104) u p, (cid:105) k p + (1 − f s ) 23 (cid:35) . (3.19 d )In CIT, the anisotropy of the particle-phase pressure tensor arises due to the source terms ε p, , ε p, , whose anisotropy is controlled by f s . For example, if f s = 1 and collisions Lagrangian pdf model for collisional turbulent fluid–particle flows d (cid:104) u p, (cid:105) dt = P p, − C Rp ε p k p (cid:18) (cid:104) u p, (cid:105) − k p (cid:19) − ε p, , (3.20 a ) d (cid:104) u p, (cid:105) dt = P p, − C Rp ε p k p (cid:18) (cid:104) u p, (cid:105) − k p (cid:19) − ε p, (3.20 b )where k p = ( (cid:104) u p, (cid:105) + 2 (cid:104) u p, (cid:105) ), P p = ( P p, + 2 P p, ), P p, = 2 τ p ( (cid:104) u s, u p, (cid:105) − (cid:104) u p, (cid:105) ) , (3.20 c ) P p, = 2 τ p ( (cid:104) u s, u p, (cid:105) − (cid:104) u p, (cid:105) ) . (3.20 d )In CIT, the anisotropy of the particle-phase Reynolds stresses arises due to the productionterms, i.e., due to the anisotropy of (cid:104) u s,i u p,j (cid:105) . The latter are found by solving d (cid:104) u s, u p, (cid:105) dt = P sp, − (cid:32) T ∗ L, + 1 T Lp, (cid:33) (cid:104) u s, u p, (cid:105) , (3.21 a ) d (cid:104) u s, u p, (cid:105) dt = P sp, − (cid:32) T ∗ L, + 1 T Lp, (cid:33) (cid:104) u s, u p, (cid:105) (3.21 b )where k fp = ( (cid:104) u s, u p, (cid:105) + 2 (cid:104) u s, u p, (cid:105) ), P sp = ( P sp, + 2 P sp, ), P sp, = 1 τ p (cid:2) (cid:104) u s, (cid:105) − (cid:104) u s, u p, (cid:105) + ϕ ( (cid:104) u p, (cid:105) − (cid:104) u s, u p, (cid:105) ) (cid:3) , (3.21 c ) P sp, = 1 τ p (cid:2) (cid:104) u s, (cid:105) − (cid:104) u s, u p, (cid:105) + ϕ ( (cid:104) u p, (cid:105) − (cid:104) u s, u p, (cid:105) ) (cid:3) . (3.21 d )Likewise, the Reynolds stresses for the fluid seen by the particles are found from d (cid:104) u s, (cid:105) dt = P f, + 2 T ∗ L, (cid:18) (cid:104) U s, (cid:105) + 23 ˜ k f − (cid:104) u s, (cid:105) (cid:19) − ε f , (3.22 a ) d (cid:104) u s, (cid:105) dt = P f, + 2 T ∗ L, (cid:18)
23 ˜ k f − (cid:104) u s, (cid:105) (cid:19) − ε f (3.22 b )where ˜ k f and T ∗ L,i are defined in § T ∗ L,i . The dissipation rates ε p and ε f are found by solving (2.30) and (2.44) with themean-shear-production terms set to zero.Figures 6–8 show the time evolution of some mean velocities and second-order momentsof both the fluid and the particle phase, obtained with the complete and simplified models.It can be seen that all the quantities, after a transient of about 80–100 τ p due to the non-trivial coupling between particles and fluid, tend to a steady value. The dashed horizontalline in the figures is the steady-state value obtained in the EL simulation by Capecelatro et al. (2015). It can be seen that the mean velocities (figure 6) are well captured by boththe complete and simplified models. In particular, an important feature of the Lagrangian2 A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, k f / V (cid:104) u f, (cid:105) / (2 k f ) (cid:104) u f, (cid:105) / (2 k f )EL simulation 8 .
04 0 .
82 0 . .
74 0 .
93 0 . .
60 0 .
98 0 . Table 4.
Steady-state values of fluid-phase velocity statistics. EL simulation data are takenfrom Capecelatro et al. (2015). (cid:104) U p, (cid:105) / V EL simulation − . − . − . κ p / V (cid:104) v p, (cid:105) / (2 κ p ) (cid:104) v p, (cid:105) / (2 κ p )EL simulation 5 .
41 0 .
78 0 . .
13 0 .
81 0 . .
71 0 .
96 0 . k p /κ p (cid:104) u p, (cid:105) / (2 k p ) (cid:104) u p, (cid:105) / (2 k p )EL simulation 0 .
89 0 .
81 0 . .
99 0 .
81 0 . (cid:104) Θ p (cid:105) / (2 κ p ) (cid:104) P (cid:105) / (3 (cid:104) Θ p (cid:105) ) (cid:104) P (cid:105) / (3 (cid:104) Θ p (cid:105) )EL simulation 0 .
11 0 .
51 0 . .
01 0 .
49 0 . Table 5.
CIT - Steady-state values of particle-phase velocity statistics. EL simulation dataare taken from Capecelatro et al. (2015). models is to provide a prediction of the fluid velocity seen by the particles, as compared toEulerian models in which it must be a-priori specified. As for the second-order moments(figures 7 and 8), the complete model still gives a good agreement with EL simulations forboth the fluid and particle phases, while the simplified version significantly overestimatesthe steady-state values.Tables 4–6, in which the steady-state values of particle and fluid statistics are reported,confirm the previous observations. Table 5 also shows the repartition of the particleturbulent kinetic energy, κ p , into the coherent part, k p , and granular temperature, Θ p .For the simplified model, by definition, Θ p = 0 and κ p = k p . The complete model alsounderestimates the granular temperature, most likely due to underestimating the valueof ε p through the choice of C p . Nonetheless, the decomposition of the particle turbulentkinetic energy appears to be essential to well predict second-order statistics, as doneby the complete model in contrast to the simplified one. The complete model also wellreproduces the fact that for all the quantities, except for the uncorrelated part of theparticle velocity, turbulence fluctuations in the vertical direction are much higher thanthose in the horizontal directions. The complete model is able to correctly reproduce theanisotropy of the second-order tensors while the simplified model only gives a qualitativeagreement.Finally in figure 9 it is shown the time evolution of the correlation coefficient ρ fp = k fp (cid:112) k f @ p k p (3.23) Lagrangian pdf model for collisional turbulent fluid–particle flows (cid:104) U s, (cid:105) / V EL simulation − . − . − . k f @ p / V (cid:104) u s, (cid:105) / (2 k f @ p ) (cid:104) u s, (cid:105) / (2 k f @ p )EL simulation 8 .
32 0 .
85 0 . .
06 0 .
88 0 . .
36 0 .
96 0 . k fp / V (cid:104) u s, u p, (cid:105) / (2 k fp ) (cid:104) u s, u p, (cid:105) / (2 k fp )EL simulation 5 .
45 0 .
82 0 . .
13 0 .
82 0 . .
71 0 .
96 0 . Table 6.
CIT - Steady-state values of fluid-phase turbulence statistics seen by particles. ELsimulation data are taken from Capecelatro et al. (2015). (a) (b)
Figure 6.
Time evolution of the vertical mean fluid velocity seen by the particles (a) and of thevertical mean particle velocity (b) from the complete (red line) and from the simplified (bluedot-dashed line) stochastic model. (a) (b)
Figure 7.
Time evolution of second-order moments of the vertical mean fluid velocity seen bythe particles (a) and of the vertical mean particle velocity (b) from the complete (red line) andfrom the simplified (blue dot-dashed line) stochastic model. which proofs the importance of having a stochastic model that predicts k fp , leading to acorrelation coefficient ρ fp that can vary in time, instead of setting it to a constant value.4 A. Innocenti, R. O. Fox, M. V. Salvetti and S. Chibbaro, (a) (b)
Figure 8.
Time evolution of (cid:104) u p, u s, (cid:105) (a) and of k f (b) from the complete (red line) and fromthe simplified (blue dot-dashed line) stochastic model. Figure 9.
Time evolution of ρ fp = k fp / ( k f @ p k p ) / from the complete stochastic model.
4. Conclusions and discussion
In this work, the stochastic models developed in Part I have been applied to statisticallyhomogeneous particle-laden flows of increasing complexity. Compared to previous modelsfor strongly coupled flows where such statistics are approximated (Fox 2014; Capecelatro et al. b ), the models used here explicitly account for the fluid statistics seen by theparticles. While this approach introduces more variables, namely U s , it eliminates theneed to close the coupling terms between the particle and fluid phases. For homogeneousflows with one-way coupling, the differences between the stochastic models and previousmodels is small. However, for the cases with two-way coupling, and especially with non-zero mean-slip velocity as in CIT, the correct prediction of U s is crucial for successfuloverall predictions. Interestingly, the steady-state model for (cid:104) U s (cid:105) given in (3.16) isrelatively simple (compare, for example, the correlation used in (Capecelatro et al. b )), with the Lagrangian time scale T ∗ L, playing a prominent role.In future work, it would be interesting to test (3.16) for CIT over a wide range of (cid:104) α p (cid:105) and ϕ values to determine whether the parameters in the model for T ∗ L, should dependon these quantities. More generally, the complete model developed in Part I should betested for inhomogeneous particle-laden flows wherein the spatial transport terms playan important role. For example, the particle-laden channel flow of Capecelatro et al. (2016 a ) would be a challenging test case. In particular, for channel flows the correlatedand uncorrelated particle velocity components generate separate spatial fluxes for allstatistics. From the model developed in Capecelatro et al. (2016 b ), it is known that,depending on the Stokes number, one or the other of these fluxes may be dominant. As aresult, the wall-normal distribution of (cid:104) α p (cid:105) , as well as other statistics, is very sensitive tohow the spatial fluxes are modelled. In any case, as shown in this work, it can be expected Lagrangian pdf model for collisional turbulent fluid–particle flows U s the resulting models for the spatial fluxeswill provide more robust closures for inhomogeneous turbulent particle-laden flows. Acknowledgements
R.O.F. gratefully acknowledges support from the U.S. National Science Foundationunder Grants CBET-1437865 and ACI-1440443.
Appendix A. Fluid–particle limit
The limit behaviour of the equations is only shown in homogeneous isotropic conditionsfor the sake of simplicity. In the limit case of tracer particles, i.e., τ p →
0, we know fromthe equations for the stochastic model that U p → U s and U s → U f , but we do not knowif the model equation for U s is exactly the same as U p . At the same time we have that theparticle-phase uncorrelated velocity goes to zero, which is consistent. When the particleinertia becomes very small where β p = β f → k fp = k f @ p → k f , the particle-phasedissipation tends to ε p → ε f , as we can see from (A 1) and (A 2): dε f dt = ( C (cid:15) P − C (cid:15) ε f ) ε f k f + C ϕτ p (cid:16) k fp k f @ p ε p − β f ε f (cid:17) , (A 1) dε p dt = ( C (cid:15) P − C (cid:15) ε p ) ε p k p + C τ p (cid:16) k fp k f @ p ε f − β p ε p (cid:17) . (A 2)Now we can check what happens to the stochastic equation for U s . From the spatiallyhomogeneous Lagrangian model, we can obtain d U s + ϕd U p = − T L U s dt − ϕT Lp U p dt + (cid:112) C ε f dW s + ϕ (cid:112) C p ε p dW p . (A 3)Now, when τ p →
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