A priori and a posteriori analysis of models for Large-Eddy simulation of particle-laden flow
AA priori and a posteriori analysis of models for Large-Eddysimulation of particle-laden flow
Christian Gobert and Michael Manhart
TU M¨unchen, Fachgebiet Hydromechanik, Arcisstr. 21, D-80333 M¨unchen, Germany
Abstract
In Large-Eddy simulation of particle-laden flow, the effect of the unresolved scales on theparticles needs to be modelled. In this work we analyse three very promising models, namelythe approximate deconvolution method (ADM) which was proposed for particle-laden flowindependently by Kuerten (Phys. Fluids 18, 2006) and Shotorban and Mashayek (Phys.Fluids 17, 2005) and two stochastic models, proposed by Shotorban and Mashayek (J. Turbul.7, 2006) and Simonin et al. (Appl. Sci. Res. 51, 1993). We present results from a priori anda posteriori analysis of these models in isotropic turbulence at Re λ = 52. This data allowsfor a direct quantitative comparison of the models. The analysis shows that ADM alwaysleads to improved statistics but that even for high Stokes numbers, the rate of dispersion isnot predicted correctly by ADM. Concerning the stochastic models, we found that with thecorrect choice of model parameters, the models perform well at small Stokes numbers. Onthe other hand, at high Stokes numbers the stochastic models show significant errors suchthat it may be recommendable to neglect the small scale effects instead of using one of thestochastic models. Keywords:
Large-Eddy Simulation, particle-laden flow, SGS effects, ApproximateDeconvolution, Langevin model
1. Introduction
Large-Eddy Simulation (LES) has become an important tool for the simulation of tur-bulent flow. State of the art methods provide reliable results and are capable to tackleapplication relevant challenges. One crucial component for LES is the correct choice of aturbulence model, i.e., a model for the effect of the unresolved subgrid scales (SGS) on theresolved scales. Such models are herein referred to as fluid-LES models .For Large-Eddy Simulation of particle laden flow, an additional model for the effect ofthe unresolved scales on the particles is needed, referred to as particle-LES models . Theworks of Yamamoto et al. (2001); Armenio et al. (1999); Kuerten and Vreman (2005); Fedeand Simonin (2006); Yang et al. (2008); Marchioli et al. (2008) show that neglection of smallscale effects is not an option.Most particle-LES models were developed in a Eulerian-Lagrangian framework, i.e., thecarrier fluid flow is computed by solving the Navier–Stokes equations and the particles are
Email address: [email protected] ( Christian Gobert and Michael Manhart)
November 12, 2018 a r X i v : . [ phy s i c s . f l u - dyn ] A p r omputed by tracing single particles through the domain. Then, modelling reduces to re-construction of small scale effect on a single particle.On this basis, a large number of models was proposed. Among these are for examplethe models of Simonin et al. (1993); Wang and Squires (1996); Shotorban and Mashayek(2006); Kuerten (2006b); Amiri et al. (2006); Gobert et al. (2007); Shotorban et al. (2007);Bini and Jones (2007, 2008); Pozorski and Minier (1998); Pozorski and Apte (2009), justto mention a few. Most of these models are stochastic models, often obtained by extendingmodels which were originally developed for inertia free particles in the context of ReynoldsAveraged Navier–Stokes (RANS) simulations, such as the generalised Langevin model byHaworth and Pope (1986).A deterministic alternative is the approximate deconvolution method (ADM) for particleladen flows (see Kuerten, 2006b; Shotorban et al., 2007; Shotorban and Mashayek, 2005).ADM is based on an approximate inversion of the LES filter and was originally developedin a Eulerian context.The present study focusses on three very promising particle-LES models, namely thestochastic models proposed by Shotorban and Mashayek (2006) and Simonin et al. (1993)and ADM as proposed by Kuerten (2006b) and Shotorban and Mashayek (2005). For allthree models, the respective authors present some results on the accuracy of their models.Their findings are summarised as follows.Kuerten (2006b) analysed ADM in particle-laden turbulent channel flow at a Reynoldsnumber based on friction velocity of Re τ = 150. He conducted an a posteriori analysis forparticles with Stokes numbers of St = 1, 5 and 25, based on the viscous time scale. Hisresults show that ADM significantly improves rms values of the wall normal component ofthe particle velocity. The improvement is greater for high Stokes number than for low Stokesnumber. In addition, Shotorban and Mashayek (2005) found that in a turbulent shear layer,ADM improves particle dispersion.Shotorban and Mashayek (2005) analysed their Langevin-based model in decaying isotropicturbulence and found that for small Stokes numbers ( St ≤ . St ≤
5, based on the Kolmogorov time scale. The presentstudy shows that at higher Stokes numbers the model does not perform very well.Concluding, all published results were obtained on different configurations and are there-fore not comparable. In particular, for the Langevin-based models only data at small Stokesnumbers is published. For all models, the available data density over the Stokes numberrange is not satisfactory. Data rather correspond to probes at specific Stokes numbers butfrom this data no Stokes number dependent behaviour of the models can be deduced.The present study aims at a clarification of that issue by providing data which allows adirect comparison of these three particle-LES models on a broad range of Stokes numbers.The data density on the Stokes number range is sufficiently high to allow the deduction ofa Stokes number dependence. The testcase is isotropic turbulence at Re λ = 52. All threemodels were originally developed such that they should perform well in that testcase but wewill show that even by tuning the model constants, the models do not always perform well.2ctually in some cases better results are obtained by neglecting SGS effects than using oneof the stochastic models.This paper is organised as follows. Sections 2 and 3 contain a description of the nu-merical methods used to compute flow and particle dynamics. Statistics of the single phasesimulations are also presented in section 2. In section 4, the three particle-LES models underconsideration are presented and section 5 contains results of an a priori and an a posteriorianalysis of the models.
2. Numerical Simulation of the carrier flow
In the present work we analyse particle dynamics in forced isotropic turbulence by DNSand LES. For the simulation of the carrier fluid, we use a second order Finite-Volume methodtogether with a third order Runge-Kutta scheme proposed by Williamson (1980) for advance-ment in time. The conservation of mass is satisfied by solving the Poisson equation for thepressure using an iterative solver proposed by Stone (1968). More details on the flow solvercan be found in Manhart (2004).The flow is driven using a slightly modified version of the deterministic forcing schemeproposed by Sullivan et al. (1994). Sullivan et al. propose a forcing scheme where the energyin the spectral modes below a certain wave number κ is held constant. We additionallyimposed a lower bound for the forced wavenumbers, i.e., only the modes in a given range[ κ , κ ] are forced. The Reynolds number in our simulations is always Re λ = 52, based onthe transverse Taylor microscale λ and the rms value of one (arbitrary) component of thefluctuations u rms .In all computations the flow was solved in a cube on a staggered Cartesian equidistantgrid. The size of the computational box and the cell width was chosen such that all scalesare resolved, based on the criteria stated by Pope (2000), cf. table 1. Table 1: Simulation parameters and Eulerian statistics from DNS of forced isotropic turbulence.
DNS Re λ N range of forced wavenumbers [ κ , κ ] [0 . , . /λ integral length scale L f . λ time scale of energy containing eddies k f /(cid:15) . λ/u rms Kolmogorov length scale η K . λ Kolmogorov time scale τ K . λ/u rms length of computational box L . L f cell width ∆ x . η K filter width ∆ 7∆ x kinetic energy of the filtered field ˆ k f . /k f The particle-LES models were assessed by a priori and a posteriori analysis. For the apriori analysis, we filtered the DNS field u by a box filter G with filter width ∆ = 7∆ x , ∆ x being the DNS cell width. The filtered DNS was sampled on a correspondingly coarse grid,3esulting in a field G u which is comparable to an LES field. The kinetic energy of the filteredfield ˆ k f = (cid:104)G u i (cid:105) / k f = (cid:104) u i (cid:105) /
2, cf. table 1. (cid:104)·(cid:105) denotes spatial and temporal averaging.In the a priori analysis, G u was used as input for the particle-LES models. Then, themodels were assessed with respect to the difference in statistics obtained from unfilteredDNS and filtered DNS with particle-LES model.For the a posteriori analysis, G u was computed by LES. As fluid-LES model we used theLagrangian dynamic Smagorinsky model proposed by Meneveau et al. (1996). The forcingparameters for LES were chosen in the same way as for DNS, i.e., the energy contained inthe range [ κ , κ ] is equal in LES and DNS. Beyond κ , the energy in LES is lower than inDNS due to the different grids and the fluid-LES model. With our choice of the grid, thekinetic energy resolved by LES ¯ k f is approximately equal to the kinetic energy of the filteredDNS field ˆ k f , cf. table 2. Instantaneous energy spectra E ( κ ) from DNS and LES are plottedin figure 1. In addition, a model spectrum proposed by Pope (2000) is shown. All data weremade dimension free by normalising with DNS quantities. Table 2: Parameters for LES of forced isotropic turbulence.
LES Re λ N cell width ∆ x . λ time scale of energy containing eddies ¯ k f / ¯ (cid:15) . λ/u rms resolved kinetic energy ¯ k f . k f κ η k E / u r m s / λ -2 -1 -6 -4 -2 DNSLESfiltered DNSmodel spectr.
Figure 1: Instantaneous energy spectrum functions together with lines proportional to κ − / and κ − .
3. Discrete particle simulation
In this study we consider dilute suspensions of small particles. Thus, effects of theparticles on the fluid and particle-particle interactions are neglected (one way coupling).The density of the particles was set to ρ p = 1800 ρ where ρ is the density of the fluid.In each simulation the particles were divided in 24 fractions with different diameter d . Themaximum diameter equals the Kolmogorov length scale. Consequently, the particles can betreated as point particles. 4he particle relaxation time τ p = ρ p ρ d ν (1)ranges from τ p = 0 . τ K to τ p = 100 τ K . Corresponding Stokes numbers St = τ p τ K based onthe Kolmogorov time scale τ K range from St = 0 . St = 100 .Based on the works of Armenio and Fiorotto (2001) and Kubik and Kleiser (2004), weassumed that in the given configurations the acceleration of a particle d v dt is given by Stokesdrag only, d v dt = − c D Re p τ p ( v − u f @ p ) . (2)Here, v ( t ) denotes the particle velocity and u f @ p the fluid velocity at the particle position.The particle Reynolds number Re p is based on particle diameter and particle slip velocity (cid:107) u f @ p − v (cid:107) which leads to a nonlinear term for the Stokes drag. The drag coefficient c D wascomputed in dependence of Re p according to the scheme proposed by Clift et al. (1978).The fluid velocity u f @ p must be evaluated at the particle position x p ( t ), i.e. u f @ p = u ( x p ( t ) , t ). Hence, these values must be interpolated. In the present work, a standard fourthorder interpolation scheme was implemented, following the recommendations of Yeung andPope (1988) and Balachandar and Maxey (1989).In the following, the notation ‘@ p ’ is adopted for arbitrary functions f ( x , t ), i.e., f @ p ( t ) = f ( x p ( t ) , t ) . (3)For example u f refers to the space- and time-dependent solution of the Navier–Stokes equa-tions whereas u f @ p refers to the time-dependent fluid velocity seen by the particle. Corre-spondingly, G u f refers to the space- and time-dependent solution of the filtered Navier–Stokesequations whereas ( G u f ) @ p refers to the time-dependent filtered fluid velocity seen by theparticle.Equation (2) is a stiff differential equation for small Stokes numbers. The numericalscheme for integrating equation (2) must be capable to handle this. Therefore, equation (2)was solved by a Rosenbrock-Wanner method (see Hairer and Wanner, 1990). This methodis a fourth order method with adaptive time stepping. The stiff term in equation (2) islinearised in each time step and discretised by an implicit Runge-Kutta scheme.The code was validated via probability density functions (PDFs) for the particle acceler-ation. To this end, a DNS of forced isotropic turbulence at Re λ = 265 on 1030 grid pointswas conducted. This data was then compared to data from a DNS conducted by Biferaleet al. (2004) and an experiment conducted by Ayyalasomayajula et al. (2006). Biferale et al.conducted a DNS at Re λ = 280 and traced inertia free particles (i.e. St = 0). Ayyalasomaya-jula et al.’s experiment was at Re λ = 250 with particle Stokes numbers St = 0 . ± . St = 0and another at St = 0 .
1. Each fraction consists of 960000 particles. Figure 2 shows that theresults from the present simulations agree very well with the referenced data.5 igure 2: Probability density function P ( a ) of particle acceleration a for validation of the code. X-axis isnormalised with respect to the (Stokes number dependent) rms value of a . Triangles: reference DNS of St = 0 particles conducted by Biferale et al. (2004). Squares: reference experiment of St = 0 . ± . St = 0 and St = 0 .
1, respectively.
For model assessment, 24 fractions of particles were traced with 80000 particles perfraction. The particles were initialised at random positions (homogeneous distribution) insidethe computational box and traced until a statistical steady state was obtained. Then, 1000time records were taken within a time span of T = 250 λ/u rms for computing statistics. Thetemporal resolution of the statistics equals approximately the Kolmogorov time scale. Withthis temporal resolution, the Lagrangian correlation functions could be resolved for all Stokesnumbers. The time span was large enough to guarantee that averaging in time cancels outoscillations caused by the forcing scheme.In terms of particle time scales, T is large enough to guarantee reliable statistics. From (cid:15) = 15 νu rms /λ it follows that T /τ p = 250 √ /St ≈ . /St . In all simulations, St ≤ T /τ p ≥ .
68. Hence, statistics were sampled over at least 9.68 times the particlerelaxation time.
4. Analysed particle-LES models
In the present section, the three particle-LES models under consideration are presented.They are the approximate deconvolution method which was proposed for particle-laden flowindependently by Kuerten (2006b) and Shotorban and Mashayek (2005) and two stochasticmodels, proposed by Shotorban and Mashayek (2006) and Simonin et al. (1993). In thefollowing, the models are stated and the numerical implementation used in this work isexplained.
ADM is well established for incompressible single phase flows (see Stolzand Adams, 1999; Schlatter, 2004; Stolz, Adams and Kleiser, 2001). Kuerten (2006a,b),6hotorban and Mashayek (2005) and Shotorban et al. (2007) analysed the capabilities ofADM for particle-laden flow. With ADM, the fluid velocity seen by the particle u ADM f @ p iscomputed from u ADM f @ p = (cid:0) u ADM f (cid:1) @ p = N (cid:88) n =0 (( I − G ) n G u f ) @ p = (cid:0) H ADM G u f (cid:1) @ p . (4)Here, I stands for identity. N is the number of deconvolution steps. H ADM is calleddefiltering operator because it is supposed to approximate the inverse of G .Equation (4) is solved once per time step and the particle velocity is computed fromd u ADM p d t = c D Re p τ p (cid:0) u ADM f @ p − u ADM p (cid:1) . (5)The operator H = I − G can be interpreted as extractor of subgrid scales. With thisoperator, u ADM f can be written as u ADM f = N (cid:88) n =0 H n G u f = N (cid:88) n =0 H n ( I − H ) u f = (cid:0) I − H N +1 (cid:1) u f . (6)For N → ∞ the transfer function of H N +1 equals zero for the resolvable scales( (cid:107) k (cid:107) < κ c ) and one for the unresolvable scales ( (cid:107) k (cid:107) > κ c ). This shows that for large N , theeffect of ADM can be interpreted as improving the LES filter towards a sharp spectral filter. Implementation of the model in this work.
In the present work, the ADM defiltering operator H ADM was computed in three different ways. First, it was computed as proposed by Kuerten(2006b). Second, it was computed making use of the DNS spectrum and third, a modelspectrum was used.If a dynamic Smagorinsky model is used as fluid-LES model, then Kuerten (2006b)proposes to compute H ADM as approximate inverse of the corresponding test filer. In hiswork and in the present work, this is a box filter. Kuerten (2006b) approximates its inverseby a second-order Taylor expansion in the filter width. The transfer function of this filter isshown in figure 3. In the following this approach is referred to as ADM
Kuerten .However, it is not clear whether an inverted box filter gives highest accuracy. ThereforeADM was tested by two more approaches. In both approaches, the ADM filter is constructedsuch that the product of filter transfer function and LES spectrum is as close as possibleto a target spectrum under the constraint that the filter stencil covers up to 5 LES cells.The target spectrum is either the DNS spectrum or the model spectrum proposed by Pope(2000). The results from the corresponding defiltering operators are referred to as ADM
DNS and ADM mod , respectively. The corresponding transfer functions are also shown in figure 3.Evidently ADM mod leads to a very much stronger amplification around κ c than ADM DNS .This was to be expected because around κ c the model spectrum is higher than the DNSspectrum, cf. figure 1. 7 / κ c t r a n s f e r f un c t i on ADM mod
ADM
DNS
ADM
Kuerten
Figure 3: Transfer functions of the defiltering operators for the three implemented ADM approaches.
ADM does not take explicitly into account that the model itself affects the particle path.More precisely, the model inherently assumes that the resolved spectrum seen by the particleis not modified by the model itself. In order to differentiate between this model assumptionand other approximation errors of the model, the a priori analysis was conducted such thatthe model does not affect the particle path.More precisely, in the a priori analysis for ADM, for each particle two different values forthe particle velocity were computed simultaneously. One value, referred to as DNS particlevelocity, is the velocity obtained from the DNS flow field. The second value, referred to asmodelled particle velocity, is the velocity obtained from a filtered DNS field and ADM. Theparticles were tracked with the DNS particle velocity and statistical samples were taken fromthe modelled velocity. This approach basically tests whether ADM is capable to do what itis supposed to do, neglecting the effect of ADM on the particle path.
ADM cannot reconstruct scales smaller than the LES grid. In order to circumvent this,Shotorban and Mashayek (2006) and Simonin et al. (1993) propose stochastic models basedon a Langevin equation for the fluid velocity seen by a particle. Such models were originallydeveloped for inertia free particles by Pope (1983), Heinz (2003) and Gicquel et al. (2002),referred to as generalised Langevin models.
Statement of the model proposed by Shotorban and Mashayek (2006).
Shotorban and Mashayekadopted generalised Langevin models for inert particles. They propose to compute the fluidvelocity seen by the particles u Sho f @ p from the stochastic differential equation (Langevin equa-tion) d u Sho f @ p,i = (cid:18) G (cid:18) ∂u f,i ∂t + u f,j ∂u f,i ∂x j (cid:19)(cid:19) @ p d t − u Sho f @ p,i − ( G u f,i ) @ p T L d t + (cid:112) C (cid:15) d W i (7)8nd the particle velocity fromd u Sho p = c D Re p τ p (cid:0) u Sho f @ p − u Sho p (cid:1) d t. (8)The reader is reminded that ‘@ p ’ denotes ‘at the particle position’, cf. equation (3). Thefirst term on the right hand side of equation (7) is the filtered material derivative of the fluidvelocity and can be computed from the right hand side of the filtered Navier–Stokes equation.The second term is a drift term for the random variable u Sho f @ p , leading to a relaxation of u Sho f @ p against ( G u f ) @ p . The last term is a diffusion term for u Sho f @ p . W denotes a Wiener processand (cid:15) is the (modelled) dispersion of subgrid scale kinetic energy. The model parameters T L and C are specified below. Statement of the model proposed by Simonin et al. (1993).
Simonin et al. (1993) also proposeto model the fluid velocity seen by the particles by a stochastic process. Fede et al. (2006)presented in detail how to deduct Simonin et al.’s model for particle-laden flow starting fromthe Navier–Stokes equations. This results in a different Langevin equation than the equationproposed by Shotorban and Mashayek.In contrast to Shotorban and Mashayek, Simonin et al. propose to transport the resolvedscales by particle velocity (and not by fluid velocity). The model can be formulated via aLangevin equation for the unresolved scalesd u Sim (cid:48) f @ p,i = (cid:32) − u Sim (cid:48) f @ p,j (cid:18) ∂ G u f,i ∂x j (cid:19) @ p + (cid:18) ∂τ i,j ∂x j (cid:19) @ p + Γ ij u Sim (cid:48) f @ p,j (cid:33) d t + (cid:112) C (cid:15) d W i . (9) τ ij = G ( u i u j ) − G u i G u j is the SGS stress tensor. The model constant C is equivalent to C of Shotorban and Mashayek’s model. For isotropic turbulence, Fede et al. (2006) recommendΓ ij = − (cid:0) + C (cid:1) (cid:15)k sgs δ ij = − T L δ ij , (10) δ ij denoting the Kronecker delta function. This form of Γ was adopted in the present work.The fluid velocity seen by the particles is then computed from u Sim f @ p = ( G u f ) @ p + u Sim (cid:48) f @ p and the particle velocity is computed fromd u Sim p = c D Re p τ p (cid:0) u Sim f @ p − u Sim p (cid:1) d t. (11) Implementation of the model in this work.
For model closure, two parameters need to bespecified, namely the time scale T L and the Kolmogorov constant C . Based on the recom-mendation of the model’s authors, in the present work T L was set to T L = k sgs (cid:0) + C (cid:1) (cid:15) , (cid:15) = C (cid:15) k / ∆ . (12) k sgs denotes the subgrid kinetic energy and was computed from the DNS data. (cid:15) denotes theSGS rate of dispersion. The model constants C (cid:15) and C were set to C (cid:15) = 1 and C = 2 . τ was computed in accor-dance with the fluid-LES model, i.e., using an eddy viscosity hypothesis.As mentioned above, for the a priori analysis of ADM, the particles were traced alongthe path computed from DNS. For the stochastic models, this would be in contradictionto the model assumptions because the model takes explicitly into account that the particlepath depends on the modelled small scale fluctuations (see Shotorban and Mashayek, 2005;Fede et al., 2006). Therefore here the particle paths were computed from the modelled fluidvelocity.The stochastic differential equations (7) and (9) were solved by an Euler-Maruyamascheme (see e.g. Kloeden and Platen, 2010). The stiff terms − u Sho f @ p,i /T L and Γ ij u Sim (cid:48) f @ p,j werediscretised implicitly. Shotorban and Mashayek (2005) and Fede et al. (2006) used an explicitEuler-Maruyama scheme. These authors focussed on small Stokes numbers. In the presentsimulations no significant differences between explicit and implicit discretisation was foundat small Stokes numbers. At high Stokes numbers, the explicit scheme was found to producesignificantly worse results. In particular, the kinetic energy seen by the particles explodes athigh Stokes numbers when using an explicit scheme. It should be noted that the terms underconsideration are linear and therefore implicit schemes do not produce any computationaloverhead. In the following, ‘Sho’ denotes results from the model proposed by Shotorban andMashayek (2005) and ‘Sim’ denotes results for the model proposed by Simonin et al. (1993).
5. A priori and a posteriori analysis of particle-LES models
The present section contains results from a priori and a posteriori analysis for assessmentof the three models presented in section 4. The analysis comprises the kinetic energy seenby the particles, particle kinetic energy and rate of dispersion.
Figures 4 to 6 show the kinetic energy seen by the particles, particle kinetic energy andthe rate of dispersion for the three ADM implementations under consideration. In addition,results from filtered DNS and LES without particle-LES model are shown. In order to obtaincomparable results, the presented data from filtered DNS without model corresponds to thedata which ADM receives, i.e., in particular in the filtered DNS the particles were tracedalong unfiltered paths.The kinetic energy seen by the particles (figure 4) shows a Stokes number dependencedue to particle clustering. The Stokes number dependence can be observed in all simulationsalthough the results from filtered DNS and LES are shifted towards higher Stokes numbersin comparison to the unfiltered DNS. This shift is not corrected by ADM.As expected, the kinetic energy seen by the particles is lower in LES or filtered DNSthan in unfiltered DNS. ADM leads to a clear improvement of the kinetic energy seen by theparticles although a clear gap between results from ADM and DNS persists. This gap wasto be expected because ADM does not reconstruct the smallest scales.Among the three ADM approaches, Kuerten’s model shows least improvement. This isnot surprising because Kuerten’s approach corresponds to a single defiltering step, N = 1.The other two approaches must perform better here because the wider stencils allow for10arger values of N , leading to higher kinetic energy. On the other hand it must be mentionedthat ADM Kuerten is computationally less expensive than ADM
DNS or ADM mod and somewhatmore generalistic in the sense that ADM
Kuerten only assumes a specific filter whereas ADM
DNS and ADM mod are based on the spectra of this specific flow.On first sight, the comparison of ADM
DNS against ADM mod is surprising. ADM mod shows closer resemblance to the unfiltered result than ADM
DNS although ADM
DNS is basedon the unfiltered field. Actually this is an effect of two errors cancelling out each other.Around the cutoff wavenumber the model spectrum is higher than the DNS spectrum butinterpolation of the fluid velocity on the particle position leads to strong damping around thecutoff wavenumber. In other words, for ADM mod the damping properties of the interpolationscheme brings the spectrum seen by the particles closer to the spectrum of the DNS flowfield.Figure 5 shows k u , the kinetic energy of the particles themselves. As expected from theobservations on k u @ p , k u is underestimated by ADM. With St → ∞ the error vanishes.Most interesting is the rate of dispersion, shown in figure 6. It was computed fromthe product of particle kinetic energy and integral time scale. The results from a prioriand a posteriori analysis differ qualitatively . In the a priori analysis, ADM leads to anunderprediction of the rate of dispersion for all Stokes numbers. The reader is remindedthat in the a priori analysis of ADM, the particles were traced along DNS particle paths.The a posteriori analysis shows that actually in LES the rate of dispersion is overesti-mated by ADM as a result of too high integral time scales. In the a posteriori analysis ofADM, the particles were traced with the modelled particle velocity. Thus, the qualitativedifference between a priori analysis and a posteriori analysis could be twofold; either due tothe difference in particle paths or due to a defect in the fluid-LES model. Additional sim-ulations showed that the principal reason is that the fluid-LES model leads to too high lifetimes for the large eddies and thus to an overprediction of particle dispersion. Consequently, St k u @ p / k f -1 a priori analysis St k u @ p / k f -1 a posteriori analysis (unfiltered) DNS,filtered / LES,ADM Kuerten ,ADM
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Figure 4: A priori (left) and a posteriori (right) analysis of ADM, kinetic energy seen by particles.
11t depends on the fluid-LES model whether the rate of dispersion is under- or overestimatedby LES.However, a priori and a posteriori analysis show that even for high Stokes numbers wheresmall scale effects should be negligible (see Yamamoto et al., 2001), the rate of dispersion isnot predicted correctly by ADM. St k p / k f -1 unfilteredfilteredADM Kuerten
ADM
DNS
ADM ms a priori analysis St k p / k f -1 DNSLESADM
Kuerten
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Figure 5: A priori (left) and a posteriori (right) analysis of ADM, particle kinetic energy. St D / u r m s / λ -1 a priori analysis St D / u r m s / λ -1 a posteriori analysis (unfiltered) DNS,filtered / LES,ADM Kuerten ,ADM
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DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine
Figure 6: A priori (left) and a posteriori (right) analysis of ADM, rate of dispersion. Result from LESwithout particle-LES model (not shown for reasons of clarity) is almost identical to the results from LESwith ADM. .2. Assessment of the Langevin-based models The testcase for the Langevin-based models is again isotropic turbulence at Re λ = 52.Figure 7 shows the kinetic energy of the fluid seen by the particles. In particular the resultsfrom the a priori analysis are very disappointing. The model of Shotorban and Mashayek(2006) shows too little k u @ p for St = 0 . k sgs was computed from the DNS data.We found that this is an effect of the convective term in the model, (cid:16) G (cid:16) ∂u f,i ∂t + u f,j ∂u f,i ∂x j (cid:17)(cid:17) @ p .If this term is neglected, then k u @ p is correctly predicted by the model.Concerning the a priori analysis, for low Stokes numbers the model of Simonin et al.(1993) leads to a more accurate prediction of k u @ p than the model of Shotorban and Mashayek(2006) but for high Stokes numbers one can observe the inverse. In the a posteriori analysis,the model of Simonin et al. (1993) gives a satisfactory prediction for k u @ p whereas the modelof Shotorban and Mashayek (2006) leads to an overestimation.The error in the kinetic energy of the particles is simply a consequence of the error inthe kinetic energy seen by the particles, cf. figure 8. In the a priori analysis of the modelof Simonin et al. (1993), the excess in k u @ p for high St is not apparent in k u . On the otherhand, in the a posteriori analysis of the model of Shotorban and Mashayek (2005), the excessin k u @ p for high St is also visible in k u . This is probably due to the different time stepping.In LES, the time step size is higher than in DNS. For the particles St = 100, the ratiobetween particle relaxation time and time step size is about 1000 for DNS and about 200for LES. Thus, the St = 100-particles can rather follow the modelled fluctuations in LESthan in DNS. Therefore, the excess in k u @ p at high Stokes number is rather reflected in thea posteriori analysis than in the a priori analysis.Concerning the rate of dispersion, figure 9, the result from the a priori analysis is verydiscouraging, the a posteriori results are somewhat better concerning accuracy of the models.Nevertheless, these tests show that both models do not necessarily improve the result of theLES in comparison to an LES without particle-LES model or LES with ADM. In particularat high Stokes numbers it might be recommendable to use no model instead of one of the St k u @ p / k f -1 a priori analysis St k u @ p / k f -1 a posteriori analysis (unfiltered) DNS,filtered / LES,ADM Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine
Figure 7: A priori (left) and a posteriori (right) analysis of the Langevin-based models, kinetic energy seenby particles. St D / u r m s / λ -1 a priori analysis St D / u r m s / λ -1 a posteriori analysis (unfiltered) DNS,filtered / LES,ADM Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine (unfiltered) DNS,filtered / LES,ADM
Kuerten ,ADM
DNS ,ADM mod
Sho,Simfiltered DNS,filtered DNS coarse,filtered DNS fine
Figure 9: A priori (left) and a posteriori (right) analysis of the Langevin-based models, rate of dispersion.
6. Conclusions
We have presented data from DNS, filtered DNS and LES of particle-laden isotropicturbulence with three different models for the effect of the unresolved scales on the parti-cles (particle-LES models). The models under consideration are approximate deconvolution(ADM) as proposed by Kuerten (2006b) and two stochastic models, based on the works of St k p / k f -1 unfilteredfilteredShoSim a priori analysis St k p / k f -1 DNSLESShoSima posteriori analysis
Figure 8: A priori (left) and a posteriori (right) analysis of the Langevin-based models, particle kineticenergy.