MODELING INTERPHASE TURBULENT KINETIC … SPRAY COMPUTATIONS 807 ... for decaying turbulent flow...

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MODELING INTERPHASE TURBULENT KINETIC ENERGYTRANSFER IN LAGRANGIAN-EULERIAN SPRAY COMPUTATIONS

M. G. Pai and S. Subramaniam*M. G. Pai and S. Subramaniam*M. G. Pai and S. Subramaniam*M. G. Pai and S. Subramaniam*M. G. Pai and S. Subramaniam*Department of Mechanical Engineering, Iowa State University, Ames, IA 50011

Original Manuscript Submitted: 6/6/05; Final Draft Received: 1/9/06

Modeling turbulent multiphase flows, such as sprays, is a major challenge owing to droplet (or solid-particle) interactions with a widerange of turbulence length and time scales. In a broad class of Lagrangian-Eulerian models, the instantaneous Lagrangian dispersed-phase velocity evolves on a time scale that is proportional to the particle response time 2( ) /( 18 )p d p f fd . Numerical simulationsof a model from this class reveal a nonmonotonic and unphysical increase of the turbulent kinetic energy (TKE) in the dispersed phasekd that is not seen in direct numerical simulations (DNS) of decaying, homogeneous turbulence laden with solid particles. Analysisof this class of models shows that for a linear drag law corresponding to the Stokes regime, the entire class of models will predict ananomalous increase in kd for decaying turbulent flow laden with solid particles or droplets. Even though the particle response time isthe appropriate time scale to characterize momentum transfer between sub-Kolmogorov-size dispersed-phase particles and the smallestturbulent eddies (for droplet/particle Reynolds number of < 1), it is incapable of capturing the range of time- and length-scaleinteractions that are reflected in the evolution of kd. A new model that employs a time scale based on a multiscale analysis is proposed.This model succeeds in capturing the dispersed-phase TKE and fluid-phase TKE evolution observed in DNS. The model also correctlypredicts the trends of TKE evolution in both phases for different Stokes numbers.

INTRODUCTION

Turbulence in the ambient gas is important in determining the evolution of a spray. It affects the rate ofentrainment of ambient gas into the spray cone, which in turn strongly influences the spray angle and otherglobal characteristics, such as the spray penetration length. The turbulent two-phase flow at the edge ofa spray is a very complex physical phenomenon involving high shear rates, large fluctuations in instanta-neous liquid volume fraction, and interphase mass transfer (in the case of vaporizing sprays). It isrecognized that statistical models of sprays must represent the evolution of velocity fluctuations in the gas,as well as the droplets, in order to predict global spray properties. However, current models for thesequantities are still in need of improvement.

This study focuses on a considerably simpler turbulent two-phase flow problem of sub-Kolmogorov-size solid particles evolving in zero-gravity, constant-density, decaying homogeneous turbulence. The goalis to understand and assess current Lagrangian-Eulerian (LE) models and to propose model improve-ments. The choice of this simple problem is motivated by two reasons. One is that this problem isolatestwo important flow processes: (i) the interphase transfer of turbulent kinetic energy (TKE), and (ii) thedissipation rate of TKE in the carrier fluid, which enables a detailed evaluation of existing models. Thesecond reason is that direct numerical simulation (DNS) data are available from carefully controlledstudies of this flow in decaying turbulence [1]. Although turbulent flows laden with solid particles willbehave differently from droplet-laden turbulent flows, in general, all features of the models we considerare identical in the limit of sub-Kolmogorov-size nonvaporizing droplets evolving in zero-gravity, constant-density, homogeneous turbulence. Although in the more general case spray models must account forphenomena such as droplet vaporization and its effect on turbulence, we find that there is considerablescope for model improvement even in nonvaporizing cases, such as the simple flow considered here.

The LE approach is based on Williams’ spray equation [2], which is an evolution equation for thedroplet distribution function (DDF). The theoretical foundations of the LE approach are now rigorouslyestablished and understood [3, 4]. The evolution equation for the dispersed-phase velocity covariance in

*Corresponding author; e-mail: [email protected]. This work is partially supported by a U.S. Department of Energy, EarlyCareer Principal Investigator Program Grant No. DE-FG02-03ER25550.

Electronic Data Center, http://edata-center.com Downloaded 2007-1-17 from IP 129.186.22.168 by Shankar Subramaniam

M. G. PAI AND S. SUBRAMANIAMM. G. PAI AND S. SUBRAMANIAMM. G. PAI AND S. SUBRAMANIAMM. G. PAI AND S. SUBRAMANIAMM. G. PAI AND S. SUBRAMANIAM808808808808808

AAAAA acceleration of a drop in the exact sprayequation

AAAAA* modeled accelerationCD drag coefficientCε2 model constant in fluid-phase dissipation

equation [Eq. (14)]Cµ standard turbulence modeling constant

= 0.09Cps model constant equal to 3 4C

Cs model constant in fluid-phase dissipationequation [Eq. (14)]

dp diameter of solid particle/ droplet; used inthe definition of the particle response timescale τp

f droplet distribution functionf c

R conditional pdf of radiusf c

VVVVVR conditional joint pdf of velocity and radius~f c

VVVVVR volume-weighted conditional joint pdf ofvelocity and radius

g~ volume-weighted or r3-weighted dropletdistribution function

g~c volume-weighted or r3-weighted conditionaljoint pdf of fluctuating velocity and radius

ggggg acceleration due to gravitykd turbulent kinetic energy in the dispersed

phase (when superscripted with * indicatesa modeled quantity)

~kd volume-weighted turbulent kinetic energy

in the dispersed phasekf turbulent kinetic energy in the fluid phase

(when superscripted with * indicates amodeled quantity)

l characteristic length of eddies in the inertialsubrange

le characteristic length scale of an eddy3 21 2

fe fkl C

LE Eulerian integral length scale from DNSn drop number densityn(i)

s number of droplets represented by eachcomputational particle

⟨Ns⟩ mean number of spray droplets in theensemble

r radius phase spaceR random variable corresponding to radiusRp radius of the computational particle; refers

to the ith computational particle whensuperscriptedwith (i)

Rep particle Reynolds number⟨R3⟩ raw third moment of drop radiusStη Stokes number based on the Kolmogorov

timescale τηt time

NOMENCLATURE

tE lifetime of an eddytinf time where inflection in the evolution of

the dispersed-phase TKE is seentR traverse time of the particle through an

eddyTL Lagrangian integral time scale of turbulenceTref reference time scale from DNS Tref = u′/LE

u′ turbulence intensity in the fluid phase atinitial time

uuuuu ′f * modeled fluctuating fluid-phase velocity|uuuuu ′f *| absolute value of the modeled fluctuating

fluid-phase velocity|uuuuu ′f *|TTTTT transition value of the (modeled) absolute

fluctuating fluid-phase velocityUUUUU *

f instantaneous (modeled) fluid-phasevelocity

⟨UUUUU f ⟩* mean fluid-phase velocity obtained fromaveraged Navier-Stokes solution

v′ turbulence intensity in the dispersed phaseat intial time

vvvvv sample space variable corresponding to theinstantaneous velocity in the dispersedphase; used in the definition of the ddff (xxxxx, vvvvv, r, t)

vvvvv ″ random variable corresponding to thefluctuating velocity defined with respect tothe volume-weighted mean in thedispersed phase

vvvvv ′* modeled fluctuating velocity defined withrespect to the number-weighted mean(equal to that defined with respect to thevolume-weighted mean for monodisperseand nonevaporating droplets) in thedispersed phasevolume-weighted velocity covariance in thedispersed phase

VVVVVp(i ) instantaneous velocity of the ith

computational particle⟨VVVVV

~ ⟩ volume-weighted mean of the dispersed-phase velocity

wp(i ) statistical weight associated with the ith

computational particlewwwww sample space variable corresponding to the

dispersed-phase fluctuating velocityXXXXX position phase spaceXXXXXp

(i ) position of the ith computational particleαf fluid-phase volume fractionαd dispersed-phase volume fractionΓ vaporization rate scaled by radius; Γ = Θ/RΓ∼

volume-weighted vaporization rate scaledby radius

εf fluid-phase dissipationΘ rate of change of radius

'' '' i jv v



the LE approach has also been derived from the spray equation [5] and forms the theoretical basis of thisinvestigation. In this article, the focus is on testing and evaluating specific models in a simple flow todetermine whether the predicted evolution of the TKE in each phase is physically consistent with DNSresults. Based on these findings we propose an improved model.

It is important to note that all the LE models considered here are first-order models based on theaverage number density. This is a direct consequence of their being a solution approach to the sprayequation. A first-order model cannot represent certain physical phenomena, such as preferential concen-tration of droplets (or solid particles) in homogeneous turbulence. The proper description of suchphenomena will require the consideration of second-order statistics, such as the pair-correlation function.This is not to imply that second-order effects, such as preferential concentration, are not important, butrather that our current modeling capabilities are still in need of further development before they canrepresent these phenomena.

The rest of the article is organized as follows. A model problem involving particles (or nonevaporatingdroplets) evolving in homogeneous turbulence is described in the next section. The evolution equation forthe dispersed-phase TKE simplifies for the homogeneous problem and depends solely on the particleacceleration-velocity covariance, which needs to be modeled. Details of DNS results available from ahomogeneous particle-laden turbulent flow that are used to assess model predictions are given in thefollowing section. A drag model based on the particle response time that is widely used in LE implemen-tations is then presented. Evolution equations for the dispersed-phase TKE, as implied by this dragmodel, and the modeled evolution equation for the TKE in the fluid phase are derived. Model predictionsfor freely decaying particle-laden turbulence are reported. A theoretical analysis reveals that the particleresponse time is not an appropriate time scale for interphase TKE transfer. A multiscale interaction timescale is then proposed that improves model predictions for the decaying turbulence case. The implicationsof the study are discussed, and conclusions are drawn in the final section.

HOMOGENEOUS TWO-PHASE FLOW MODEL PROBLEM

A canonical problem that is useful in assessing the behavior of turbulent two-phase flow models consistsof sub-Kolmogorov-size particles evolving in zero-gravity, constant-density, decaying homogeneousturbulence. If gravity is neglected, then the mean pressure gradient must also be zero and the meanmomentum equation admits a trivial solution of zero mean velocity in each phase, which in turn impliesa zero mean slip velocity. The evolution of TKE in each phase can then be studied independent of the meanflow quantities.

Exact governing equations for the dispersed-phase velocity covariance are derived in Appendix Afor an inhomogeneous system of evaporating droplets with no coalescence, collisions, or breakup. The

µf absolute viscosity of the fluid phaseνf kinematic viscosity of the fluid phaseρd dispersed-phase thermodynamic densityρf fluid-phase thermodynamic densityτη Kolmogorov time scaleτint unaveraged interaction time scale⟨τint⟩ multiscale interaction time scaleτl characteristic time scale of the inertial-range

eddiesτp particle response time scaleω* modeled fluctuating particle response frequencyΩ*

p modeled particle response frequency⟨Ω*

p⟩ modeled mean particle response frequency

Subscriptsd dispersed phasef fluid phasep particle

Superscripts′ fluctuation defined with respect to the

number-weighted mean″ fluctuation defined with respect to the

volume-weighted mean* modeled quantitys spray

NOMENCLATURE (continued )



equation for the dispersed-phase TKE is then obtained by contracting the indices of the velocitycovariance equation. With the assumptions of zero interphase mass transfer, constant density, andstatistical homogeneity, Eq. (A.8) simplifies to

i j

i j j i

v vA v A v

t(1)

In the canonical homogeneous problem, the evolution of the covariance of particle velocity is solelydetermined by the model for the acceleration-fluctuating velocity covariance [the right-hand side ofEq. 1]. Taking half of the trace of Eq. (1) results in the evolution equation for the TKE in the dispersedphase (1/ 2)d i ik v v as

di i

kA v

t(2)

The tilde in Eqs. (1) and (2) represents mass weighting (or volume weighting for constant thermodynamicdensity in the dispersed phase).1 Mass-weighting of terms is necessary to consistently account for theinterphase TKE transfer terms that appear in the evolution equation for the TKE and dissipation in thefluid phase [cf. Eqs. (13), (14)]. Moreover, mass-weighted governing equations from the LE approachhave a direct correspondence with their counterparts in the Eulerian-Eulerian or two-fluid approach.Since the dispersed-phase thermodynamic density is constant, the distinction between volume weightingand mass weighting is not needed in the rest of the article. Furthermore, since this study focuses onmonodispersed particles with no evolution of their radii in time, number-weighted quantities are the sameas their volume-weighted counterparts. Hence, the tilde in Eqs. (1) and (2), and in the equations in therest of this work, can be dropped.

DNS RESULTS FOR THE HOMOGENEOUS MODEL PROBLEM

Several researchers [1, 6, 7] have performed DNS of particle-laden homogeneous turbulence. These DNSresults can be used to validate two-phase turbulence models. Sundaram and Collins [1] have performeda study on particle-laden freely decaying turbulence in the absence of gravity for several Stokes numbers.The Stokes number Stη is defined as the ratio of the particle response time scale τp to the Kolmogorovtime scale τη and characterizes the tendency of a particle to follow the turbulent fluctuations of the carrierphase. The particle response time scale is defined as 2( ) /( 18 )p d p f fd , and the Kolmogorov timescale is given by τη = (νf /εf )1/2. The system is volumetrically dilute, with particles in the sub-Kolmogorov-size range, and collisions among particles, if any, are assumed to be elastic. Particles are assumed to bepoint sources/sinks, and the simulation is two-way coupled, i.e., the effect of the particles on the fluid-phase momentum conservation is also accounted for. Parameters of the homogeneous model problem aregiven in Tables 1 and 2. In Table 2, u′ is the initial turbulence intensity in the fluid phase and v′ is theinitial turbulence intensity in the dispersed phase. These intensities are related to the respective TKE ineach phase at initial time through u ′2 = (2/3)kf (0) and v ′2 = (2/3)kd (0). The following section describesLE models that can be used to model this turbulent two-phase flow.

LAGRANGIAN MODELS FOR PARTICLE VELOCITY

LE models indirectly solve the DDF evolution equation using a particle method for reasons of compu-tational efficiency and ease of modeling. In this approach, an ensemble of N identically distributed

1See Appendix A for the definitions of volume weighting and number weighting.



computational particles is used to indirectly represent the modeled DDF. With each computationalparticle we associate a position vector ( )i

pX , velocity vector ( )ipV , radius ( )i

pR , and a statistical weight ( )ipw 2 The

evolution equation for the particle velocity implies a modeled evolution equation for the DDF of fluctuatingvelocity Eq. (A.6) and the velocity covariance Eq. (A.8). The particle velocity evolution equation

pp f p

d

dt⎛ ⎞⎜ ⎟⎝ ⎠

VA U V g (3)

defines a class of Lagrangian models that subsumes the vast majority of models [8–11] in the literature.In Eq. (3), AAAAA* is the modeled particle acceleration, UUUUU *

f and VVVVV *p are the modeled fluid-phase and dispersed-

phase instantaneous velocities, respectively, ggggg is the acceleration due to gravity, and Ω *p is a characteristic

particle response frequency.3 The particle response frequency depends on the drag coefficient CD, whichis a function of particle Reynolds number Rep. Models proposed in the literature for Ω *

p (see [8], forexample) can be cast in the following general form:

1(Re )p p

p

f (4)

where f (Rep) represents the functional dependence of the model for CD on Rep. This form [cf. Eq. (3)]of the particle acceleration model is based on the equation of motion of a sphere in a fluid under theinfluence of only drag and body forces [12]. The models in this class differ only in terms of the particleresponse frequency model and the model for the fluid-phase velocity.

The instantaneous fluid-phase velocity UUUUUf* is decomposed into a mean ⟨UUUUUf⟩*, and a fluctuation uuuuu′f *,

which are related through

ff f uU U (5)

Table 1Table 1Table 1Table 1Table 1 Parameters of the Particle-Laden Decaying TurbulenceTest Case

Dispersed-phase volume fraction αd 1.8 × 10–4

Fluid-phase thermodynamic density ρf (kg/m3) 1.1616Dispersed-phase thermodynamic density ρd (kg/m3) 1045.44Acceleration due to gravity g (m/s2) 0.0, 0.0, 0.0Initial mean slip (m/s) 0.0, 0.0, 0.0(kd /kf) ratio at initial time 1

Table 2Table 2Table 2Table 2Table 2 Particle-Laden Decaying Turbulence TestCase: Initial Values of the Turbulence Intensities u′and v′ in the Fluid Phase and Dispersed Phase,Respectively, and Dissipation Rate in the Fluid Phase,for Different Stokes Numbers

Stη = τp /τη u′(m/s) v′(m/s) εf (m2/s3)

1.6 0.80245 0.77250 0.362733.2 0.79371 0.73812 0.403096.4 0.79254 0.74360 0.43834

2The definition of the statistical weight w (i)p is not unique, but the sum of weights over all computational particles must sum

to unity: ( )

11

Ni

pi

w∑ . In KIVA [8], the statistical weight is defined as w (i)p = n (i)

s /⟨Ns⟩, where n(i)s is the number of droplets represented

by each computational particle and ⟨Ns⟩ is the mean total number of droplets represented by the ensemble.3The asterisk in Eq. (3) and in rest of this work is used to denote modeled quantities, which are only approximations to their

exact unclosed counterparts. For example, AAAAA***** in Eq. (3) is a model for AAAAA in Eq. (A.1).

In the Lagrangian-Eulerian approach, the solution tothe averaged Eulerian equations in the fluid phase yields amean fluid-phase velocity ⟨UUUUUf⟩*, whereas the fluctuation inthe fluid-phase velocity uuuuu′f * is modeled. Together the meanand fluctuating fluid-phase velocities form a model for theinstantaneous fluid-phase velocity UUUUUf

*.The particle-velocity evolution model implemented in

KIVA [8] also belongs to the general class of Lagrangian



models described by Eq. (3). The particle acceleration AAAAA***** in KIVA [8] is modeled as

38

ff pp fff p D

d p

dC

dt R

U VVU V

uu

+ggggg. (6)

The drag coefficient CD is given by

2 /3Re241 Re 1000

Re 6

0.424 Re 1000

pp

pD

p

C

⎧ ⎛ ⎞⎪ ⎜ ⎟⎪ ⎜ ⎟⎨ ⎝ ⎠⎪⎪⎩

(7)

where the particle Reynolds number Rep is

2Re .

ff f p pp

f

RU Vu(8)

The limit of Stokes flow results in the drag coefficient of CD = 24/Rep, corresponding to a particleReynolds number Rep << 1. Note that Stokes drag is a remarkably good approximation even for Rep ~1, sinceat this particle Reynolds number, the Stokes law predicts a drag that is only 10% in error (see [13], p. 61).

Models for Fluctuating Fluid-Phase Velocity

The fluctuating fluid-phase velocity uuuuu′f * is usually sampled from a joint-normal probability densitywith zero mean and covariance equal to (2kf /3)δij under the assumption that the turbulence is isotropic.This velocity is held constant over a time interval, called the turbulence correlation time, which is takento be the minimum of an eddy traverse time tR and an eddy-life time tE. At the end of the time interval,the renewal time is reached and a new value of fluctuating velocity uuuuu′f * is sampled. This is intended tocapture the effect of crossing trajectories as a particle shoots across successive eddies. Such models for thefluctuating fluid-phase velocity are commonly known as eddy lifetime models (ELT). Brown andHutchinson [10] and Gosman and Ioannides [11] used a linearized form of the equation of motion of adroplet to arrive at an eddy traverse time ln(1 0 ( ))R p e p f pt l U V , where the characteristic lengthscale of the eddy 3 21 2

fe fkl C . They also proposed a model for the eddy lifetime fE et l u .Ormancey [9] proposed that the time intervals over which uuuuu′f * remains constant be exponentially distrib-uted (Poisson model), with the mean time interval equal to the Lagrangian integral time scale ofturbulence TL. Amsden et al. [8] used a model similar to Hutchinson’s but with tE = kf /εf and

3 2 1( )f fR ps f f pkt C uU V , where Cps is a model constant equal to 0.16432 3 4( )C . This modelhas been incorporated into the popular KIVA family of codes [8].

Implied Evolution of Dispersed-Phase TKE

The velocity covariance evolution implied by the class of particle velocity evolution models dis-cussed in the previous section (including the KIVA model) can be analyzed for the homogeneous modelproblem. With assumptions of statistical homogeneity4 and a monodisperse size distribution of solidparticles (or droplets), the evolution equations for the mean and covariance of velocity implied by suchdrag models are considerably simplified.

From Eq. (6), one can infer the instantaneous particle response frequency Ω*p to be

4The assumption of statistical homogeneity implies that the position property need not be retained.



38

ff pfp D

d p

CR

U Vu(9)

The evolution equation for the covariance of the dispersed-phase velocity, as implied by Eq. (6), is

j i

i jji f f

d v vvv U U

dt

p pji j iV Vvv

j ji ip pv vv v

j if fji p pu uvv (10)

where p p is the modeled fluctuating response frequency of the dispersed phase defined withrespect to the mean particle response frequency p , and p pj j j

V Vv is the modeled fluctuatingvelocity of the dispersed phase defined with respect to the number-weighted mean velocity p j

V in thedispersed phase. The modeled evolution equation for dk is then obtained by contracting indices in Eq. (10)

df pi i ipi i

dk U Vv v vdt

⎡ ⎤⎢ ⎥⎣ ⎦

ii p fv u (11)

For the case of zero mean slip, which is the case under consideration, Eq. (11) simplifies to

i

di i ip p f

dkv v v u

dt(12)

Comparing Eq. (12) with Eq. (2), one can infer that if Eq. (6) is used as a particle velocity evolutionequation, then the implied model for the acceleration-fluctuating velocity correlation i iA v is

ii i ip p fv v v u . One can expect that k *d in Eq. (12) could either decay or increase, depending

on the relative magnitudes of the terms on the right-hand side of Eq. (12). Since these terms involve triplecorrelations among fluctuating quantities, it is hard to enforce any physical constraint on them such thatk *

d evolves according to trends seen in DNS or experiments.

Evolution of Fluid-Phase TKE

In the LE approach, a modified k–ε model is used to evolve the TKE and dissipation in the fluidphase. The modeled equation for k*

f (for the statistically homogeneous, zero mean slip case) used here is[5, 8]

( )i i i

f f fif f f d d f p f p f

d kvu u u

dt⎡ ⎤⎣ ⎦

(13)

The term in square brackets on the right-hand side of Eq. (13) arises from a model that represents therate at which the turbulent eddies do work in dispersing the spray droplets. This term represents the TKEtransfer between the dispersed phase and the fluid phase. The modeled equation for the dissipation [5, 8]in the fluid phase εf is



2

2

( )i i i

f f f f fif f s d d f p f p f

f f

dvC C u u u

dt k k

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

(14)

The term in square brackets on the right-hand side of Eq. (14) represents the contribution to the evolutionof the modeled dissipation from the interphase TKE transfer.

It is important to note that most LE models (including KIVA) assume a volumetrically dilute sprayαd << 1, and because αf = 1 – αd, it follows that αf ≈ 1. On this basis, volume-displacement effects areneglected [8], and both Eqs. (13) and (14) are solved with αf = 1, but with αd ≠ 0.

MODEL PREDICTIONS FOR THE HOMOGENEOUS PROBLEM

Predictions of normalized TKE in the fluid phase k *f , as a function of scaled time t /Tref, are shown in Fig. 1

for increasing Stokes numbers Stη and a constant mass loading. These predictions are for the homoge-neous problem using the KIVA model. Here Tref = u′/LE is the large eddy turnover time scale from theDNS [1]. Also shown on the same plot is the evolution of k *

f from the DNS [1] for increasing Stokesnumbers. For a constant mass loading, it is expected that an increasing Stokes number quickens the decayof TKE in the fluid phase and, hence, the trend depicted by the DNS appears plausible. The predictedtrend of k *

f from KIVA for varying Stokes number does not match the trend depicted in the DNS.Model predictions of normalized TKE in the dispersed phase k *

d, as a function of scaled time t /Tref,are shown in Fig. 2 for increasing Stokes numbers Stη, alongside results from DNS. In KIVA, thedispersed-phase TKE at the end of every time step is computed as

12 i id v vk (15)

Again, for a constant mass loading, it is expected that the decay of TKE in the dispersed phase is morerapid for larger Stokes numbers. The model predictions for the evolution of k *

d for varying Stokes numbersdo not match with the trends seen in the DNS.

Fig. 1Fig. 1Fig. 1Fig. 1Fig. 1 Evolution of normalized k*f for the homogeneous model problem: (i) KIVA with Ω *

p and (ii) DNS of particle-laden freely decaying turbulence [1]. Not only is the decay rate fast compared to the DNS result, but also the trendof decay in k*

f is opposite to that seen in the DNS result. Arrows indicate direction of increasing Stokes number.



Fig. 2Fig. 2Fig. 2Fig. 2Fig. 2 Evolution of normalized k*d for the homogeneous model problem: (i) KIVA with Ω*

p (ii) DNS of particle-ladenfreely decaying turbulence [1]. The decay in k*

d as predicted by the KIVA model is significantly faster than the DNS result.An unphysical crossover in the predictions from KIVA is seen. Arrows indicate direction of increasing Stokes number.

One can make two important observations from the model predictions presented in Figs. 1 and 2.First, the time scale of decay of k *

f and k *d using the KIVA model is significantly lesser than that observed

in the DNS. Second, an anomalous increase at t/Tref = 0.1 (although slight, for the initial kd /kf ratio ofunity used in this study), after an initial steep decrease, is seen in the evolution of k*

d for Stη = 1.6. Later,it will be shown that this anomalous increase is accentuated at larger initial kd /kf ratios. This behavior isdeemed unphysical since the flow under consideration does not possess any mechanism to increase theTKE in the dispersed phase, and hence, k*

d should exhibit a monotonic decrease. On the other hand, theresults from DNS show a monotonic decay in the TKE in the fluid phase and dispersed phase, which isconsistent with the flow physics.

Lagrangian-Eulerian model predictions can exhibit statistical variability due to randomness ininitializing the particle properties (in this case, particle velocities). For different initial ensembles,model predictions of k *

f and k *d can be different. Multiple independent simulations are performed with

the model, and it is observed that the statistical variability in the model predictions due to randomnessin the initial conditions is < 0.2% of the mean. Statistical variability in the model predictions is foundto be insignificant compared to the differences observed due to the changing Stokes numbers Stη. It isfound that the 95% confidence intervals corresponding to each Stη do not overlap in the model predictionsshown in Figs. 1 and 2. Because these confidence intervals are extremely small, they have been omittedin these figures.

Reason for Anomalous Behavior in k*d

The unphysical increase in the k*d evolution can be explained by an exact analysis of the model

equations, which requires a few simplifying assumptions. The analysis assumes that: (i) the particleresponse frequency [cf. Eq.(9)] is constant and (ii) the fluctuating fluid-phase velocity [cf. Eq. (6)] isconstant (this is true if the decay in the TKE of the fluid phase k *

f is small over the time for which theanalytical predictions are valid). A constant particle response frequency corresponds to a linear drag model.It is observed in the current simulations that a significant number of particles have Rep < 1, a range whereinthe Stokes drag is accurate [13]. It must, however, be noted that in the KIVA model, Ω *

p does not remainconstant and changes with VVVVV *

p.



Let the particle velocity VVVVV *p be distributed joint normally with mean ⟨VVVVV *

p⟩ = 0 and covariance(2/3)kd (0)δij. For constant Ω *

p, the particle velocity evolution equation (6) can be solved to give anexpression for the particle velocity VVVVV *

p(t) at any time t as

( ) (0) (0) (0) ptp f f jj j j

V u u vt e⎡ ⎤⎣ ⎦ (16)

where u′f *j(0) and v′j *(0) are evaluated at initial time t = 0. The mean particle velocity remains at zero for

all time [cf. Eq. (B.1)]. From Eq. (16), one can derive the dispersed-phase kinetic energy k*d(t) at any time

t to be (see Appendix B for details)

2 2( ) (0) 1 2 (0)p p pt t td f dk t k e e k e (17)

It is worthwhile to note that Eq.(17) has an inflection point at tinfl given by

infl(0)1

ln 1(0)

d

p f

kt

k

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

(18)

The decay in normalized k*d as predicted by Eq. (17) is shown in Fig. 3 alongside the evolution of

predicted k*d from the KIVA model for Stη = 1.6 and for two different initial kd /kf ratios. As the initial kd /kf

ratio is decreased, the reversal in the evolution of k*d is more prominent. The analytical point of inflection

is close to the inflection point on the evolution curve of k*d from the KIVA model. The difference between

the analytical and numerical results until the point of inflection is because Ω *p and uuuuu′f * are not constant in

the numerical simulations. The analytical expression predicts the initial steep decay very accurately,thereby illustrating that the unphysical model behavior is not an artifact of the numerical simulation.

Fig. 3Fig. 3Fig. 3Fig. 3Fig. 3 Results from a simple analysis assuming constant Ω*p (dotted-dashed lines and subscript a in the legend) are

shown alongside predictions from KIVA (solid lines) for two initial kd /kf ratios and a Stokes number of 1.6. The insetshows the region where the reversal in the decay of k*

d (indicated by A and B) occurs. For a constant Ω*p, a decrease

in the initial kd /kf ratio tends to aggravate the unphysical behavior.



IMPROVED PARTICLE VELOCITY EVOLUTION EQUATION

We have shown that LE models for two-phase turbulence that are based on the particle response time[cf. Eq.(3)] result in anomalous evolution of averaged quantites like k*

d . It is interesting, therefore, tounderstand why such particle velocity evolution equations when used in DNS of sub-Kolmogorov-sizeparticle-laden turbulent flows [1] yield plausible results. The answer simply lies in the fact that in theDNS, the particles interact with a range of time and length scales, where UUUUU *

f appearing in Eq. (3) is nolonger modeled but is an adequately resolved solution to the Navier-Stokes equation, with additionalmomentum source terms due to the presence of the particles [1]. The particle response time scale is anappropriate time scale for the interphase momentum transfer terms that are added to the Navier-Stokesequations. Unfortunately, in LE models [8–11], the quantity uuuuu′f * represents a model for the fluctuatingfluid-phase velocity that does not represent all the velocity scales that are captured in the DNS velocityfield. Thus, the particle velocity evolution equation in LE computations needs modification to theinteraction time scale in order to achieve results comparable with DNS.

Multiscale Interaction Time Scale ⟨τ⟨τ⟨τ⟨τ⟨τintintintintint⟩⟩⟩⟩⟩

The fact that particles interact with a range of turbulence length and time scales—and that sucha complex interaction cannot be adequately characterized by the particle response time alone in LEcomputations—motivates the development of a mean multiscale interaction time ⟨τint⟩ in place of (1/Ω *

p)in Eq. (3). The angular brackets represent an interaction time scale averaged over all eddies, details ofwhich follow. The fluctuating particle velocity relaxes to the local modeled fluctuating fluid-phase velocityon the multiscale interaction time scale ⟨τint⟩ as

int

fddt

u vv(19)

The fluctuating fluid phase velocity is modeled as in the original KIVA proposal [8] by samplingfrom a Gaussian distribution with zero mean and variance (2/3)kf . In the homogeneous problem underconsideration, the mean velocity in either phase is zero for all time. Hence, there is no need to evolve themean velocities in this case. However, if the mean velocities are nonzero with nonzero mean slip, wehypothesize that the mean velocity in either phase would evolve over a time scale 1/⟨Ω *

p⟩ derived fromEq. (9) and CD would depend on a mean particle Reynolds number.

The multiscale interaction timescale ⟨τint⟩ was introduced by Pai and Subramaniam [14] and hasbeen successfully employed in the context of Eulerian-Eulerian two-phase turbulence modeling by Xu[15] and Xu and Subramaniam [16]. This time scale is derived from the fluid-phase velocity field by firstdefining a Stokes number valid in the inertial range as

St pl

l

(20)

where τl is computed as

2f

lf

u(21)

Let us assume that uuuuu′f * obeys a joint normal distribution with zero mean and covariance σ2f δij, where σ2

f =(2/3)kf . With this assumption, the pdf of |uuuuu′f *| is

2 2223

2 1( ) e fz

f

f z z (22)



where z is the sample space variable corresponding to |uuuuu′f *|. It is evident from Eqs. (20) and (21) that

2

1St l

fu∼ (23)

A mean time scale of interaction ⟨τint⟩ is derived from the pdf of |uuuuu′f *| as

int int0

( ) ( )T

f

Tf

pf z dz f z dz∫ ∫u

u(24)

where the time scale τint is hypothesized to be of the form

τint = Stl (τp – τ) + τ (25)

for |uuuuu′f *|T ≤ |uuuuu′f *| ≤ ∞. Here τ = k*f /ε*

f is the large eddy turnover time scale. The significance of the limit|uuuuu′f *|T and the rationale behind the choice of a weighted-average time scale ⟨τint⟩ in Eq. (24) is now discussed.

Equation (21) is based on an inertial subrange scaling for eddies with a characteristic length scale l.The Stokes number Stl defined in Eq. (20) using the characteristic time scale τl determines how thedroplets respond to these eddies. For a value of Stl > 1, it is hypothesized that the droplet responds slowlyto the eddies and the time scale of energy transfer is influenced more by the particle response time τp. Onthe other hand, if Stl < 1, it is hypothesized that the droplet responds immediately to the flow and thetime scale of energy transfer is influenced more by the eddy turnover time scale τ. Thus, the pdf of |uuuuu′f *|(see Fig. 4) can be divided into two regions: one that represents Stl > 1 and the other that represents Stl < 1with |uuuuu′f *|T representing the transition between the two regions at Stl = 1. Therefore, |uuuuu′f *|T is uniquelydetermined by the relation (|uuuuu′f *|T )2 = τp ε*

f .It is interesting to note that Eq. (24) has the correct behavior under limiting conditions of Stl and

|uuuuu′f *|T . In the limit |uuuuu′f *|T → 0, there are no eddies in the system with Stl > 1. The droplets are simplyconvected by the flow, and the correct time scale for interphase TKE transfer in this limit is τ. In the limit|uuuuu′f *|T → ∞, practically all the eddies in the system satisfy Stl > 1, which implies that there are no eddiesenergetic enough to convect the droplets. The correct time scale for interphase TKE transfer in this limitis the particle response time scale τp.

Implementation of Multiscale Interaction Time Scale in LE Computations

The following algorithm outlines the procedure that can be used to implement the multiscaleinteraction time scale ⟨τint⟩ in LE computations of particle-laden flow:

1. An ensemble of N computational particles with velocity and radius ( ) ( ), , 1, ,i ip pR i NV … is

sampled from a specified initial joint pdf of velocity and radius. The TKE kf and dissipation rateεf in the fluid phase are initialized.

2. The particle response time scale τp for each particle is computed using 2( ) /( 18 )p d p f fd . Fora monodispersed ensemble, all particles will have an identical particle response time scale.

3. The transition value Tf p fu is computed for each particle. All particles will have an

identical value of |uuuuu′f *|T for a monodispersed ensemble.4. The multiscale interaction time scale ⟨τint⟩ is computed by numerically integrating Eq. (24).5. Each particle’s velocity is evolved in time using Eq. (19).6. Quantities k*

d, k*f, and ε*

f are calculated at the new time step, and steps 2–5 are repeated.

Note that for a polydispersed ensemble of droplets, and for a spray with drop radii changing in time (asin the case of an evaporating spray), τp changes in time. In either case, |uuuuu′f *|T will be different for eachparticle. Note also that if ε*

f evolves in time, |uuuuu′f *|T will also change in time.



Fig. 4Fig. 4Fig. 4Fig. 4Fig. 4 Sketch of the probability density function of |uuuuu′f *| used in the derivation of the multiscale interaction time scale⟨τint⟩. Here, z is the random variable corresponding to |uuuuu′f *|. The transition value |uuuuu′f *|T is also indicated.

Model Results with Multiscale Interaction Time Scale

Predicted evolution of normalized k*f and k*

d is shown in Figs. 5 and 6, respectively, for the KIVAmodel with ⟨τint⟩, alongside results from DNS. It can be inferred that the time scale of decay has improvedsignificantly compared to the results using the KIVA model with τp (cf. Figs. 1 and 2). The decay trendsof k*

f and k*d with increasing Stokes number are also in the same direction as those depicted in the DNS,

and the anomalous reversal in the evolution of k*d is also absent. Statistical variability due to randomness

in the initial conditions is again found to be insignificant compared to the differences observed due to the

Fig. 5Fig. 5Fig. 5Fig. 5Fig. 5 Evolution of normalized k*f for the homogeneous model problem: (i) KIVA with ⟨τint⟩ and (ii) DNS of particle-

laden freely decaying turbulence [1]. Not only has the time scale of decay in the evolution of k*f improved, but also the

trend of decay with increasing Stokes number matches the DNS result. Arrow indicates direction of increasing Stokesnumber.



Fig. 6Fig. 6Fig. 6Fig. 6Fig. 6 Evolution of normalized k*d for the homogeneous model problem: (i) KIVA with ⟨τint⟩ and (ii) DNS of particle-

laden freely decaying turbulence [1]. The time scale of decay with increasing Stokes numbers has improved signifi-cantly and is now closer to DNS results, and the trend of decay matches the DNS result. Arrow indicates directionof increasing Stokes number.

changing Stokes numbers Stη. Again, it is observed that the 95% confidence intervals for the modelpredictions corresponding to each Stη are extremely small and, thus, are omitted from Figs. 5 and 6.

A simple modification to the existing particle velocity evolution equation (6) that replaces theparticle response time scale with a multiscale interaction time scale has improved the decay characteristicsof the TKE in the fluid phase and dispersed phase significantly.

DISCUSSION

The new multiscale interaction time scale correctly reproduces the trends in the decay of TKE in thefluid phase and dispersed phase, as observed in DNS of a homogeneous particle-laden turbulent flow.Implicit in the above statement is the assumption that the DNS data are themselves accurate represen-tations of the physical system. The point-particle assumption for the particle drag in such DNS isjustified in a limited flow regime where particle Reynolds numbers Rep << 1, dispersed-phase to fluid-phase density ratios ρd /ρf are O(1000), and particles are sub-Kolmogorov size with negligible wakeeffects. Volume-displacement effects are neglected in such DNS and the fluid-phase velocity field isassumed to be solenoidal.

The homogeneous problem that forms the basis of the investigation in this work, and for whichDNS datasets exist, corresponds to a flow regime where the assumptions mentioned earlier are valid.However, it is important to note that a good approximation to the particle drag in the DNS does notnecessarily guarantee accurate calculation of the fluctuating velocity-acceleration correlation [cf. Eq.(2)]or the fluid-phase dissipation in the presence of particles. In the point-particle approximation, particle-particle interaction effects are not accounted for and the effect of the point-particle approximation on thetrue pressure field is also not quantified. The only way to test these approximations is to perform true DNSwhere the flow around each particle is fully resolved and exact boundary conditions are imposed on eachparticle surface. Solenoidality of the fluid phase (which, in turn, affects the fluid-phase pressure field) andneglect of particle-particle interaction effects can be tested in a true DNS. A recent study by Moses andEdwards [17] seeks to assess the consequences of the point-particle approximation. However, their studyis in two dimensions for considerably large cylinders (Reynolds number based on the diameter of cylinder



= 26), with an emphasis on evaluating the effects of filtering the velocity field. Their study is relevant toexamining the validity of LES based on the point-particle assumption. Similar studies are needed forDNS, although such simulations are still limited by computational expense. Therefore, the DNS datasetsperformed with the point-particle approximations that are used in this study are the best data availablefor model testing and validation. It appears very likely that the existing DNS datasets do capture the majortrends of the TKE variation with important nondimensional parameters, such as Stokes number and massloading. It is possible that true DNS might lead to revision in the exact quantitative predictions.Nevertheless, since our principal conclusions concern qualitative trends rather than an exact quantitativematch between model predictions and DNS, it is reasonable to assert that the incorporation of the newmultiscale interaction time scale leads to a better representation of the problem physics.

It is worthwhile to examine whether any experimental data can be used for model validation.Experimental investigations of nearly isotropic particle-laden turbulence include work by Friedman andKatz [18] and Fallon and Rogers [19]. Although the former [18] reports only the rise rate of droplets inthe presence of two levels of turbulence intensity in the carrier phase, and the latter [19] reportspreferential concentration of particles for varying Stokes numbers, both do not report the TKE in eitherphase, which is required for model validation. Although the data they report are useful for models thatinvolve buoyancy effects and those that predict preferential concentration, information on the covariancesof fluid-phase velocity and dispersed-phase velocity is not reported.

SUMMARY AND CONCLUSIONS

Particle-turbulence interaction occurs over a range of length and time scales. A simple turbulent two-phase flow is one that consists of monodispersed sub-Kolmogorov-size solid particles (or nonevaporatingdroplets) evolving in zero-gravity homogeneous decaying turbulence. Direct numerical simulations per-formed for this flow report that, for a constant mass loading, the rate of decay of TKE in both the fluidphase and the dispersed phase increases with increasing Stokes number. This simple two-phase flow isused to assess a class of LE turbulence models that use the particle response time scale as the time scalefor interphase TKE transfer. Such LE models fail to reproduce the correct trend of decay in TKE, asobserved in DNS, for both the fluid phase and the dispersed phase. When the particle response time scaleis replaced with a multiscale interaction time scale derived from an assumed fluid-phase turbulence field,the trends of decay of fluid-phase and dispersed-phase TKE match those seen in the DNS. This lendsto support the hypothesis that the particle response time scale is inadequate to represent the multiscaleeffects inherent in a two-phase flow system.

The principal conclusions of this study are as follows:

1. LE models based on the particle response time scale do not capture the correct trends of decayin TKE with varying Stokes number in freely decaying particle-laden turbulence. The KIVAmodel with the particle response time scale also predicts an unphysical increase of dispersed-phase TKE in freely decaying turbulence. A simplified analysis assuming a constant particleresponse time reproduces the unphysical behavior, thereby illustrating that the nonmonotonicbehavior is not an artifact of the numerical simulation.

2. LE models with an improved multiscale interaction time scale correctly predict the trends ofdecay in TKE with varying Stokes number for freely decaying particle-laden turbulence.

Predictions from the LE model with the multiscale interaction time scale can be assessed in othercanonical flows, such as droplet-laden homogeneous shear and mixing layers.

REFERENCES

1. S. Sundaram and L. R. Collins, A Numerical Study of the Modulation of Isotropic Turbulence by SuspendedParticles, J. Fluid Mech., vol. 379, no. 105–143, 1999.

2. F. A. Williams, Spray Combustion and Atomization, Phys. Fluids, vol. 1, no. 6, pp. 541–545, 1958.



3. S. Subramaniam, Statistical Representation of a Spray as a Point Process, Phys. Fluids, vol. 12, no. 10,pp. 2413–2431, 2000.

4. S. Subramaniam, Statistical Modeling of Sprays using the Droplet Distribution Function, Phys. Fluids,13(3):624–642, 2001.

5. S. Subramaniam, Modeling Turbulent Two-Phase Flows, In Proc. of 16th Annual Conference on LiquidAtomization and Spray Systems, Int. Liquid Atomization and Spray Systems Soc., Monterey, CA, 2003.

6. F. Mashayek, F. A. Jaberi, R. S. Miller, and P. Givi. Dispersion and Polydispersity of Droplets in StationaryIsotropic Turbulence, Int. J. Multiphase Flow, vol. 23, pp. 337–355, 1997.

7. M. Boivin, O. Simonin, and K. Squires, Direct Numerical Simulation of Turbulence Modulation by Particlesin Isotropic Turbulence, J. Fluid Mech., vol. 375, pp. 235–263, 1998.

8. A. A. Amsden, P. J. O’Rourke, and T. D. Butler, KIVA–II: A Computer Program for Chemically ReactiveFlows with Sprays, Technical Report No. LA–11560–MS, Los Alamos National Laboratory, May, 1989.

9. A. Ormancey and J. Martinon, Prediction of Particle Dispersion in Turbulent Flows, PhysicoChem. Hydrodyn.,vol. 3/4, no. 5, pp. 229–244, 1984.

10. D. J. Brown and P. Hutchinson, The Interaction of Solid or Liquid Particles and Turbulent Fluid FlowFields—A Numerical Simulation, J. Fluids Eng., vol. 101, pp. 265–269, 1979.

11. A. D. Gosman and E. Ioannides, Aspects of Computer Simulation of Liquid Fueled Combustors, J. Energy,vol. 6, no. 7, pp. 482–490, 1983.

12. M. R. Maxey and J. J. Riley, Equation of Motion for a Small Rigid Sphere in a Nonuniform Flow, Phys. Fluids,vol. 26, pp. 883–889, 1983.

13. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley, New York, 2002.14. G. M. Pai and S. Subramaniam, Analysis of Turbulence Models in Lagrangian-Eulerian Spray Computations,

In Proc. of 17th Annual Conference on Liquid Atomization and Spray Systems, Int. Liquid Atomization and SpraySystems Soc., Arlington, VA, 2004.

15. Y. Xu, An Improved Multiscale Model for Dilute Turbulent Gas Particle Flows Based on the Equilibrationof Energy Concept, Master’s thesis, Iowa State University, Aug., 2004.

16. Y. Xu and S. Subramaniam, A Multiscale Model for Dilute Turbulent Gas-Particle Flows Based on theEquilibration of Energy Concept, Phys. Fluids, vol. 18, no. 3, Art. No. 033301, 2006.

17. B. Moses and C. Edwards, LES-Style Filtering and Partly Resolved Particles, In Proc. of 18th Annual Conferenceon Liquid Atomization and Spray Systems, Intl. Liquid Atomization and Spray Systems Soc., Irvine, CA, 2005.

18. P. D. Friedman and J. Katz, Mean Rise Rate of Droplets in Isotropic Turbulence, Phys. Fluids, vol. 14, no. 9,pp. 3059–3073, 2002.

19. T. Fallon and C. B. Rogers, Turbulence-Induced Preferential Concentration of Solid Particles in MicrogravityConditions, Exp. Fluids, vol. 33, pp. 233–241, 2002.



APPENDIX A. EVOLUTION EQUATION FOR THE DISPERSED-PHASE VELOCITYCOVARIANCE IN THE LAGRANGIAN-EULERIAN APPROACH

The droplet distribution function f (xxxxx, vvvvv, r, t) [2] gives the probable number of droplets with positions inthe range xxxxx + dxxxxx, instantaneous velocities in the range vvvvv + dvvvvv, and radii in the range r + dr, and its evolutionforms the basis of the LE modeling approach. The phase space may also include other quantities, suchas droplet temperature, droplet distortion from sphericity, and rate of change of distortion from sphericity(see [8], for example). For an inhomogeneous system of evaporating droplets, neglecting collisions,breakup, and coalescence, one can derive the spray equation as [4]

[ ] [ ] [ ] 0k kk k

fv f A r t f r t f

t x v rx v x v (A.1)

where ⟨Ak|xxxxx, vvvvv, r ;t⟩ represents the expected acceleration (rate of change of velocity) conditional on location[xxxxx, vvvvv, r] in phase space, ⟨Θ|xxxxx, vvvvv, r ; t⟩ represents the expected rate of change of radius conditional on [xxxxx,vvvvv, r]. Subramaniam [3, 4] has also shown that the DDF can be decomposed into a conditional joint pdfof velocity and radius f c

VVVVVR(vvvvv,r|xxxxx ;t), and drop number density n(xxxxx;t) as

f (xxxxx, vvvvv,r, t) = f cVVVVVR(vvvvv,r|xxxxx ;t)n(xxxxx ; t) (A.2)

One can define an instantaneous fluctuating velocity w with respect to the volume-weighted (r3-weighted)mean velocity in the dispersed phase ⟨VVVVV

~~~~~|xxxxx ; t⟩ as

wwwww = vvvvv – ⟨VVVVV~~~~~|xxxxx ; t⟩ (A.3)

Volume-weighted averages of any smooth function Q(vvvvv, r) are defined as

33

[ ]

3 3

[ ]

( )

( )

cR

r

cR

r

r Qf r t d drR QQ

R r f r t d dr

∫∫

Vv

Vv

v x v

v x v(A.4)

We can similarly define a volume-weighted DDF of fluctuating velocity g~(xxxxx, wwwww, r, t) as

( )g r t f t r tx w x V x w

= r3 f (xxxxx, vvvvv, r, t)

3( ) ( ) c

RR t n t t r tf V

x x V x w x

3( ) ( ) ( )cgR t n t r tx x w x (A.5)

where g~c(wwwww,r|xxxxx,t) is the r3-weighted joint pdf of fluctuating velocity and radius conditioned on position.The evolution equation for g~ can be derived from Eq. (A.1) using Eq. (A.5) as

lk kk

k l k

g g g Vw wVt x w x

⎛ ⎞⎜ ⎟⎝ ⎠

) l ll k

l k

V VA r t g g g Vw t x

⎡ ⎤⎢ ⎥⎣ ⎦

x v

3r t g r t gr

x v x v (A.6)



where

t tR

x x (A.7)

From the evolution equation for g~, one can derive the evolution of the dispersed-phase velocitycovariance as

3 3i j k i j i j k

k k

n R v v v v n R v v vVt x x⎧ ⎫ ⎡ ⎤⎨ ⎬ ⎢ ⎥⎣ ⎦⎩ ⎭

3j k i i k j

k k

n R v v v vV Vx x⎧ ⎫⎨ ⎬⎩ ⎭

3i j j in R A v A v

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

3 3 i jn R v v tx

3 0 ( 0 )ci j Rn R v v r t f r t⎡ ⎤

⎣ ⎦x x

3 0 ( 0 )ci j Rn R v v r t f r tx x (A.8)

where

[ ]( )c

i j i jr

gv v w w r t d dr∫ vw x w (A.9)

The description of various terms appearing in Eq. (A.8) is as follows:

Material derivative of the dispersed-phase velocity covariance and triple velocity correlation

3 3i j k i j i j k

k k

n R v v v v n R v v vVt x x⎧ ⎫ ⎡ ⎤⎨ ⎬ ⎢ ⎥⎣ ⎦⎩ ⎭

(A.10)

Production due to mean velocity gradients

3j k i i k j

k k

n R v v v vV Vx x⎧ ⎫⎨ ⎬⎩ ⎭

(A.11)

Acceleration-fluctuating velocity correlation

3i j j in R A v A v

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

(A.12)

Net velocity covariance change due to interphase mass transfer

3 '' '' 3 i jn R v v tx

3 0 ( 0 )ci j Rn R v v r t f r t⎡ ⎤

⎣ ⎦x x

3 0 ( 0 )ci j Rn R v v r t f r tx x (A.13)



In particle method solutions to the DDF evolution equation (see, for example, [8]), the triple velocitycorrelation is in closed form. If there is no interphase mass transfer, then the terms representing the changein the velocity covariance due to interphase mass transfer are zero, and the only remaining term to bemodeled in the LE approach is the correlation of acceleration with fluctuating velocity.

Since this study focuses on monodisperse particles with no evolution of their radii in time, number-weighted quantities are the same as their volume-weighted counterparts. The number-weighted averageof any smooth function Q(vvvvv, r) is defined as

[ ]( )c

Rr

Q Qf r t d dr∫ Vv

v x v (A.14)

and for the special case of monodisperse particles ⟨Q~ ⟩ = ⟨Q⟩.



APPENDIX B. ANALYSIS OF THE SIMPLIFIEDPARTICLE VELOCITY EVOLUTION EQUATION

Assuming a constant particle response frequency Ω*p and a constant fluid-phase fluctuating velocity uuuuu′f , the

solution to the particle velocity evolution equation Eq. (6) can be derived [cf. Eq.(16)]. The mean particlevelocity ⟨Vp i

* ⟩ at any time t can be computed from Eq. (16) as

*( ) (0) (0) (0)

0

ptp f f ii i i

V u ut v e⎡ ⎤⎣ ⎦

(B.1)

showing thereby that the mean particle velocity remains zero for all time. Using Eq. (16), one can computethe dispersed-phase TKE as

* **

* *

22* *

121

( ) ( ) ( ) ( )21

(0) (0) (0) (0) (0) (0)2

1(0) (0) 1 (0) (0)

2

p p

p p

d i i

p p p pi i i i

t tf f f fi ii i i i

t tf f i ii i

k v v

V V V Vt t t t

u u u uv e v e

u u e v v e

⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

*2 (0) 1 (0)p pt tf ii

u e v e

= kf (0)(1 – 2e–Ω*pt + e–2Ω*pt) + kd(0)e–2Ω*pt (B.2)

The last equality follows from the following relations:

* *

*

(0) (0) 2 (0)

(0) (0) 2 (0)

(0) (0) 0

f f fi i

i i d

f ii

u u k

v v k

u v

We know that the samples u′f *i and v′i * are independent at initial time, so the last relation follows from the

fact that their covariance is equal to zero. We now have an analytical expression for the evolution of k*d

in Eq. (B.2) that is applicable until the first renewal of uuuuu′f * .Using Eq. (B-2), it is straightforward to compute the slope of the k*

d evolution curve at time t = 0,which could explain the reason for the unusually steep initial descent in the predicted evolution of kd fromthe KIVA model. Differentiating Eq. (B.2) with respect to time results in

2 2(0) 2 2 2 (0)p p pt t tdf p p d p

dkk e e k e

dt(B.3)

At t = 0,

2 (0)dp d

dkk

dt(B.4)

Thus, k*d decays initially over a time scale that scales like the inverse of the particle response frequency Ω*

p–1

and is the reason for the steep decay not seen in the DNS results [1].


MODELING INTERPHASE TURBULENT KINETIC … SPRAY COMPUTATIONS 807 ... for decaying turbulent flow...

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