The dispersion of a light solid particle in high-reynolds number homogeneous stationary turbulence: LES approach with stochastic sub-grid model

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1 Computational Methods in Multiphase Flow III 65 The dispersion of a light solid particle in high-reynolds number homogeneous stationary turbulence: ES approach with stochastic sub-grid model M. Gorokhoski & A. Chtab CORIA/UMR 6614 CNRS/Uniersity of Rouen, France Abstract In the framework of the ES approach, the simple stochastic model of turbulent cascade with intermittency is introduced in order to represent the interaction between a solid particle and turbulence at sub-grid scales. The computation of the agrangian statistics of a light particle in homogeneous stationary turbulence reproduced the results of measurements: (i) for the elocity statistics, the numerical results were in agreement with classical Kolmogoro 1941 phenomenology; (ii) the computation of the elocity increment at different time lags reealed, howeer, the strong intermittency. Keywords: dispersion, ES approach, turbulence, intermittency. 1 Introduction During the past years, it has been recognized that for computation of high- Reynolds turbulent gas flow, the large eddy simulation (ES) approach (integration of 3D filtered Naier-Stokes equations with effectie sub-grid eddy iscosity model) proides accurate local estimates of statistical quantities [1]. Recently [], this approach was rigorously formulated from the group-theoretical analyses. Applying the ES approach to computation of turbulent flow with dispersed phase, one can expect that at each time, the momentum transport from flow to particles can be predicted accurately. Howeer there is a problem: the interactions between gas and small particles are often characterized by length (or time) scales, which are not resoled by ES approach. Such interactions need to be modeled. 5 WIT Press

2 66 Computational Methods in Multiphase Flow III The objectie of this paper is to simulate the dispersion of light particle in the highly intermittent turbulent flow. This numerical study was motiated by recent experimental work done by Mordant et al. [3]. In this experiment, the on Karman swirling flow was generated by two counter-rotating discs. Remote from boundaries, such a flow was considered as a high Reynolds number homogeneous isotropic turbulence. The solid polystyrene sphere ( d = 5 µm; 3 τ St = 3.67ms ; ρ p = 1.6 g/ cm ) has been immersed in the flow in order to measure the agrangian elocity ariations across the inertial range of turbulence. It has been obsered that at Reynolds number Reλ = 57 1 ( λ is Taylor micro-scale), the elocity auto-correlation function and the time spectrum were in agreement with Kolmogoro, Howeer, the measured elocity increments reealed the strong intermittency: the periods of weak elocity increments alternated with moments, when the particle was subjected to intense accelerations (going up to 45m / s ). In this paper, the experiment of Mordant et al. [3] is simulated in the framework of ES approach. The particle/turbulence interaction at sub-grid scales is introduced assuming the scaling symmetry in turbulent cascade with intermittency [4]. Along with ES computation, can this model match the obserations from [3] - was the question raised before computation. Uniersalities of fragmentation under the scaling symmetry and turbulent cascade with intermittency Fragmentation plays an important role in a ariety of physical, chemical, and geological processes. Examples include atomization in sprays, crushing of rocks, polymer degradation, turbulence etc. Although each indiidual action of fragmentation is a complex process, the number of these elementary actions is large. In this situation, it is natural to abstract a simple scenario of fragmentation and to represent its essential features, independently of the details of elementary break-up actions. One of the models is the fragmentation under the scaling symmetry. Here each breakup action reduces the typical length of fragments, r α r, by an independent random multiplier α, which is goerned by the fragmentation intensity spectrum, say, q ( α), q ( α ) d 1 1 α =. The following question is raised: during the fragmentation under scaling symmetry, how does eole the distribution f ( r, t) to the ultimate steady-state solution f () r = δ() r? This question cannot be completely answered since such eolution requires the knowledge of the spectrum q ( α), which is principally unknown function. Howeer, as it has been shown in Gorokhoski and Saelie [5], due to scaling symmetry r α r f r, t goes, at least,, the eolution of ( ) 5 WIT Press

3 Computational Methods in Multiphase Flow III 67 through two intermediate asymptotic distributions. Ealuating these distributions does not require the knowledge of entire spectrum q ( α) - only its first two logarithmic moments (first uniersality), and further only the ratio of these moments in the long-time limit (second uniersality), determine the shape of f ( r, t). The first uniersality leads to the log-normal distribution of f ( r, t), and the second one erifies to be power (fractal) distribution (so far, the deltafunction distribution is neer achieed). In turbulence, due to intermittency, the energy of large unstable eddy is transferred to the smaller one at fluctuating rate. Here the fragmentation under scaling symmetry can be formulated in terms of Castaing et al. [6 1]: when the turbulent length scale r gets smaller, the elocity increment, δ r ( x) = ( x+ r) ( x), is changed by independent positie random multiplier, δ r = α δ l, with r l. et us introduce an eolution parameter τ = ln( ES / l) of penetration towards smaller scales ( ES is typical scale of finite-difference cell and l is a progressiely decreasing eddy scale). One can write an equation, which goerns the eolution from distribution f ( δ, τ l ) at scale l towards distribution f ( δ, τ r τ l ) at smaller scale r. Such an equation has been established in [5]: df ( δ; τ) dτ ( Iˆ + 1) f ( δ ; τ ) = (1) where ˆ ( ; dα δ I f δ τ) q( α) f ; + = τ α α is operator of fragmentation. The long-time limit solution of (1) is erified to be [5]: where f ( α δ, τ ) ( δ, τ) α ( α, τ) ( αδ, τ ) f = d B f () is the initial distribution and ( ) ( ) 1 lnα + ln α τ / ln α τ B ατ, = e. ln π α τ In the case of ES, the elocity increment, δ VES, is resoled exactly at ES and then the initial distribution of the elocity increment 5 WIT Press

4 68 Computational Methods in Multiphase Flow III ( l = τ = ) may be represented by the Dirac delta function, ES ( ) ( ) f αδ, τ = = δ αδ δv ES. In this case, solution () yields: f 1 1 lnα δ ; τ = exp τ t δves π ln α τ ln α ( ) δves ln exp δ δves ln α τ δ ln 1 ln α α (3) As it has been noted in [5], the second multiplier in log-normal asymptotic (3) tends to unity, as time progresses further, and only one uniersal parameter, ln α / ln α, determines the distribution at largest times (eolution from log-normal distribution towards power function with growing in time pick at small increments and stretched tails at larger increments). This parameter we assumed in the following form: ln α / ln α = ln ( λt / ES), where λ T is referred to as local Taylor micro-scale; the first logarithmic moment we introduced as ln α = A ln ( λt / ES). Here A is unknown constant. This constant has been adjusted comparing the eolution of the excess in (), ( ) ( ) ( ) 4 K τ = δ / δ 3, with alues measured in Mordant et al. [3]. To do this, the distribution, which was measured in [3] at integral turbulent time (Gaussian with ariance of 1 m/ s), has been introduced as initial distribution in (). The alue λ T =.76cm taken from [3] also was used. Different fitting cures with A =.7,.8,.9 are shown in fig.1; the alue A =.8 was taken in our further computations. 3 Computational model In this work, the filtered Naier-Stokes equations were integrated numerically in 3D box with periodic conditions in order to simulate the stationary homogeneous gaseous turbulence. The stationarity and statistical isotropy of turbulent flow was proided by the forcing scheme at large scales proposed by Oerholt and Pope [11]. In the computation, the Reynolds number, the kinetic energy, the microscales of Kolmogoro and Taylor were close to those mentioned in Mordant et al. [3] ( Reλ = 74 ; σ = 1 m/ s; η 14 k = µ m ; λ T =.76cm ; Tint = 3 ms; τ =.ms ). Examples of the flow configuration and the sub-grid iscosity K 5 WIT Press

5 Computational Methods in Multiphase Flow III 69 3 distribution computed by 3 grid points mesh at different times is gien in fig.. Fig. 3 shows the kinetic energy spectrum compared to the presumed spectrum of Oerholt and Pope [11]. It is seen that the presumed spectrum is well 1 represented up to the spatial scale k =.5cm. At smaller scales, the computed spectrum deiates from the presumed one; these scales will be simulated. K(τ) Mordant et al [3] A=.7 A=.8 A= τ/τ k Figure 1: Eolution of the excess in () compared to measured alues in Mordant et al. [3]. Along with ES computation of the flow, the rigid particle, same as in Mordant et al. [3], was tracked ( dp ηk and τ St τ K ). To simulate the drag force, one needs to estimate the gas elocity seen by the particle. We assume that this elocity is combined from the resoled one and its increment at spatial scales of order of the particle diameter, d p. The estimation of the last one can be obtained from distribution (3) by setting both the eolution parameter τ p = ln( ES / d p ) and the local resoled elocity increment δ VES. Hence, the equation for the dragging particle is: { } with δ rnd f ( δ δv ES, τ p ) 1 time S, where S = ( S ) 1/ ij S tensor. du p VES + δ U p = (4) dt τ = sampled from (3) after passage of local ij st is the norm of resoled elocity gradients 5 WIT Press

6 7 Computational Methods in Multiphase Flow III y, cm y, cm x, cm x, cm x, cm x, cm ν t,cm /s Figure : Examples of the flow configuration and the sub-grid iscosity distribution at two different times presumed spectrum computed spectrum E, cm 3 s k, cm Figure 3: Computed kinetic energy spectrum compared to the presumed shape in Oerholt and Pope [11]. 5 WIT Press

7 4 Numerical results and discussion In the framework of Kolmogoro 1941 scaling in the inertial range, the secondorder structure function, D ( τ) = ( t+ τ) ( t), is expected to be ( ) ( ) D τ ετ σ τ T, where σ is ariance of the Eulerian elocity field and T proides the measure of the agrangian elocity memory. For τ << T, this is consistent with Taylor 191 theory, according to which 1 / T D τ = t+ τ t = σ e τ. At ery large times, the ( ) ( ) ( ) ( ) ( ) D τ σ =, second-order structure function tends to its constant magnitude, ( ) while at ery small times, this function is expected to behae as 3/ 1/ D ( τ ) ε ν τ [1]. Along with Mordant et al. [3], our computation of the second-order structure is in agreement with these classical results. In fig. 4, we show the second-order structure function for tracking particle at different time lags. Here, the time lags are scaled by Kolmogoro s time τ K and υ K is the Kolmogoro s elocity scale ( a and C are constants, for illustration). The quadratic dependency at small scales, and the linear dependency at larger scales, with saturation towards the prescribed in experiment ariance σ / υ K can be seen in this figure. Note that the range, where D ( τ ) τ, is not wide. This has been also mentioned by Mordant et al. [3]. Another confirmation of Kolmogoro 1941 phenomenology is demonstrated in fig. 5. This figure shows that the computed elocity spectrum, presented as Fourier transform of autocorrelation function, proofs the orentz form, which is proportional to ω, at small ω τ 1 4, and to ω at larger frequencies. At the same time, the computed statistics of the elocity increment displayed a strong non-gaussianity. In fig. 6, we show the computed and measured distributions of particle elocity increment at different time lags. It is seen that at integral times, the elocity increment is normally distributed. This confirms the result of Oboukho s agrangian model of turbulence [1, 13]. Howeer at smaller time lags, similar to experimental obseration, the distributions exhibit a growing central peak with stretched tails: small amplitude eents alternate with eents of large acceleration. Such a manifestation of intermittency at small scales is seen also from the computed distribution of the particle acceleration (fig. 7): the accelerations of particle may attain ery large alues. In the same figure, we present the measured distribution and the distribution, which was computed without sub-grid δ p Computational Methods in Multiphase Flow III 71 modeling (zero ). One can see that in the last case, the computed acceleration is of order of Kolmogoro s scaling 5 WIT Press

8 7 Computational Methods in Multiphase Flow III a K one. 3 4 ( ε / ) 1/ 3m / s, which is substantially less than the measured 1 σ /υ υ k C τ/τ k 1 1 a (τ/τ k ) D (τ)/υ k ES with sub-grid model τ/τ k Figure 4: Second order structure function (solid line) at different time lags scaled by Kolmogoro s time τ K ; υ K is Kolmogoro s elocity scale) /ω 1-1 E (ω) 1-3 1/ω ω Figure 5: Computed agrangian elocity spectrum. 5 WIT Press

9 Computational Methods in Multiphase Flow III 73-1 log 1 (PDF) τ τ /<( τ ) > 1/ Figure 6: Velocity increment distribution at different time lags (computation - on the left; measurements of Mordant et al. [3] at the same lags - on the right) log 1 (PDF) ES with sub-grid model ES without sub-grid model -5-5 a(m/s 5 5 ) Figure 7: PDF of acceleration (computation - on the left; measurements of Mordant et al. [3] - on the right). References [1] Moin, P. & Kim, J., Numerical inestigation of turbulent channel flow. J. Fluid Mech., 118, pp , 198. [] Saelie, V.. & Gorokhoski, M.A., Group-theoretical model of deeloped turbulence and renormalization of Naier-Stokes equation. Physical Reiew E 7, 163, 5. [3] Mordant, N., eeque, E. & Pinton, J.-F., Experimental and numerical study of the agrangian dynamics of high Reynolds turbulence. New Journal of Physics, 6, pp. 116, 4. [4] Gorokhoski, M., Scaling Symmetry Uniersality and Stochastic Formulation of Turbulent Cascade With Intermittency. Annual Research 5 WIT Press

10 74 Computational Methods in Multiphase Flow III Briefs -3, Center of Turbulence Research, Stanford Uniersity, NASA, pp , 3. [5] Gorokhoski, M & Saelie, V., Further analyses of Kolmogoro s model of breakup and its application in air blast atomization. J. Physics of Fluids, 15(1), pp , 3. [6] Castaing, B., Gagne, Y. & Hopfinger, E.J., Velocity proprobability density functions of high Reynolds number turbulence. Physica D 46, pp 177-, 199. [7] Castaing, B., Gagne, Y. & Marchand, M., og-similarity for turbulent flows? Physica D 68, pp 387-4, [8] Castaing, B., Temperature of turbulent flows. Journal De Physique II, 6(1), pp , [9] Kahalerras, H., Malécot, Y., Gagne, Y. & Castaing, B., Intemittency and Reynolds number. Phys.Fluids, 1(4), pp 91-91, [1] Naert, A., Castaing, B., Chabaud, B., Hebral, B. & Peinke, J., Conditional statistics of elocity fluctuations in turbulence. Physica D: Nonlinear Phenomena, 113(1), pp 73-78, [11] Oerholt, M.R & Pope, S.B., A deterministic forcing scheme for direct numerical simulations of turbulence. Computers & Fluids, 7(1), pp. 11-8, [1] Monin, A.S & Yaglom, A.M., Statistical fluid mechanics: mechanics of turbulence, The MIT press, 9 pp., [13] Obukho, A.M., Atmospheric diffusion and air pollution. Ad. In Geophys., 6, pp , WIT Press