Local and global recurrences in dynamic gas-solid flows

Local and global recurrences in dynamic gas-solid flows T. Lichteneggera,b,∗ a

Department of Particulate Flow Modelling, Johannes Kepler University, Altenberger Str. 69, 4040 Linz, Austria b Linz Institute of Technology (LIT), Johannes Kepler University, Altenberger Str. 69, 4040 Linz, Austria

Abstract We investigate recurrent patterns in a lab-scale fluidized bed consisting of Np ≈ 57 000 glass particles by the means of computational fluid dynamics and the discrete element method (CFD-DEM). We generalize single-point measures for recurrence quantification analysis (RQA) to spatially extended particle density fields and compare the embedding dimensions necessary to unambiguously identify similar states. In accordance with many previous studies, we find locally very low-dimensional behavior (Dloc ≈ 5). Globally, the correlation dimension, i.e. the number of effective degrees of freedom, increases to Dglobal ≈ 18, which is still several orders of magnitude smaller than the 6Np microscopic translational degrees of freedom and explains the observed, characteristic structures. Distance plots reveal a combination of fast, recurrent motion interrupted by rare events like large eruptions. The temporal evolution of the average nearest-neighbor distance shows a rapid initial drop and follows a power law related to the correlation dimension for longer times. Given a sufficient tolerance, most states reappear after a few ∗

Corresponding author Email address: [email protected] (T. Lichtenegger)

Preprint submitted to International Journal of Multiphase Flow

May 18, 2018

seconds, which indicates why in recent, small-scale recurrence CFD (rCFD) simulations, the complicated long-term bed motion could be approximated with information from short-term studies within the scope of current computation techniques. From the perspective of numerical efficiency, the presented analysis allows to determine the shortest time series from which a significant amount of information on the underlying system may be obtained. Keywords: fluidized bed, recurrent patterns, distance plot, recurrence CFD 1. Introduction Gas-solid fluidized beds (Kunii and Levenspiel (1991)) in the bubbling regime are prime examples for multi-scale systems. On a microscopic, i.e. grain-scale level, particles collide with each other with subsequently diverging trajectories or temporary cluster formation. The surrounding fluid phase exerts stresses on each grain which in turn creates a complicated flow pattern in its wake that again interacts with the close-by solids. Taking a proverbial step back, this rapid, seemingly erratic motion with the huge number of 6Np translational and 3Np (assuming perfect sphericity) rotational degrees of freedom of the granular phase with Np particles, gains a certain amount of order on the mesoscale. The local number density exhibits variations between zero and the random close-packing limit over distances which can be orders of magnitude larger than a single grain – bubbles are formed, which are almost void inside, but push compacted granular matter in front of them and are surrounded by its downflow (Zenz (1978)). On even larger, macroscopic length scales encompassing many bubbles, their rise from the bed bottom towards its surface can be observed. During their ascent, they

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can create specific patterns (Coppens et al. (2002); Wu et al. (2017)), they split or they merge and grow (Kunii and Levenspiel (1968); Mori and Wen (1975)), leading to sometimes large eruptions (Levy et al. (1982, 1983)). For each of these spatial scales, one can identify characteristic times. Particle-particle collisions typically happen within microseconds (Hertz (1882); Johnson (1985)), while bubble motion is significantly slower. Depending on the operating conditions and grain size, it could take a void several hundredths of a second to pass a given point (a comparison of experimental findings and empirical correlations for bubble rise velocity can be found in the textbook by Kunii and Levenspiel (1991)) and correspondingly longer to make its way from the bottom to the surface. During this process, it will repeatedly encounter other bubbles and coalesce with them (Clift and Grace (1971)) leading to an increasing diameter with height (Horio and Nonaka (1987); Choi et al. (1988)). Splitting on the other hand is also possible but it is only relevant for rather small particles (Horio and Nonaka (1987); Choi et al. (1998)). If one studies the behavior even longer, recurrent meso- and macroscopic patterns become apparent. Intuitively, one could think that quite independent from the initial state, after a transient phase, the complicated bed motion starts to repeat itself in an irregular fashion. Putting it more formally, in dissipative, externally driven systems, a large portion of initial conditions is pulled towards a small, possibly fractal subset of phase space called attractor, where after a certain amount of time, each state is similar to previously visited ones without any obvious rules to exactly predict the long-term evolution.

3

This observation led to the question if fluidized beds can be regarded as chaotic systems (Stringer (1989)). A huge body of research emerged that aimed to calculate chaotic characteristics of such complex flows under various conditions (Daw et al. (1990); Schouten and Van den Bleek (1991); Van den Bleek and Schouten (1993); Hay et al. (1995); Bai et al. (1997, 1999); Ohara et al. (1999); Kim and Han (2006); Castilho et al. (2011); Llop et al. (2012); Salikov et al. (2015)), ranging from fundamental questions on collective particle behavior and self-organization (Daw et al. (1995)) and on similarities with turbulent flows (Ghasemi et al. (2011)) to more practical ones like issues of quality monitoring in industrial applications (Schouten and van den Bleek (1998); van Ommen et al. (2000)). Most of these investigations measured data – mostly pressure (c.f. van Ommen et al. (2011) for an extensive review) – only at a single or a few probing points and used delay coordinates (c.f. Sec. 2.2) to reconstruct the complicated local dynamics (Takens (1981)), leading to such low, non-integral attractor dimensions as in the range between 2 and 7. In this study, we want to take a global point of view along the lines of a recently suggested approach. Marwan et al. (2015) generalized the concept of recurrence plots (RP) introduced by Eckmann et al. (1987) to multi-component time series to examine spatially extended, high-dimensional, dynamic systems. While several authors (Chunhua et al. (2010); Babaei et al. (2012); Wang et al. (2012); Sedighikamal and Zarghami (2013); Llop et al. (2015); Norouzi et al. (2015); Tahmasebpoor et al. (2015); Savari et al. (2016, 2017); Tahmasebpoor et al. (2017); Zarghami et al. (2017); Ziaei-Halimejani et al. (2017)) have investigated reappearing signal structures from a small number of probes,

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we want to apply a similar methodology to the full flow fields of fluidized beds obtained from CFD-DEM simulations (c.f. Sec. 2.1) in order to better understand their recurrent nature as a whole. Besides the benefit of much larger amounts of information, we are convinced of the importance to capture global patterns because agitated granular systems can sustain structures with significant spatial extents (Coppens et al. (2002); Wu et al. (2017)). So far, only a few authors (Cizmas et al. (2003); Palacios et al. (2004)) have attempted to include global information in their analysis of the underlying dynamics in fluidized beds. At this point we want to stress that despite a possible, intuitive notion of recurrent behavior, i.e. the reappearance of a state similar to a given one after some time, a mathematically rigorous treatment always requires a norm (c.f. Sec. 2.2.1) to quantify the distance between states. When we talk about recurrence, it is implicitly connected to a specific norm which will influence the subsequently obtained results. Since there is no obvious “best” choice, we understand recurrent configurations in a very broad sense such that the point-wise, absolute difference of the fields connected to these states is small and their immediate short-term evolution is similar, too. The paper is organized as follows: After a brief introduction of the equations governing gas-solid flows (Sec. 2.1), we generalize the statistical tools for continuous field data (Sec. 2.2) which we obtained from simulations described in Sec. 3. Based on the results (Sec. 4), where we focus on the solid phase dynamics, we hope to answer two questions: a) Is there an advantage of global investigations in comparison to local ones, e.g. because of an increased complexity of the flow as a whole or because the distinction between truly

5

and seemingly similar states is easier? b) Given the high numerical costs of detailed simulations, how long does it take until recurring patterns have reappeared and no or little new information can be obtained? Especially the latter question has substantial impact on the feasibility of long-term studies of slow processes on highly dynamic backgrounds. In such diverse applications as (Lagrangian-based) species transport in bubble columns (Lichtenegger and Pirker (2016)), heat transfer in bubbling beds (Lichtenegger and Pirker (2017); Lichtenegger et al. (2017); Pirker and Lichtenegger (2018)) or particle-spray interaction in spouted beds (Kieckhefen et al. (2018)), the fast recurrent motion was successfully time-extrapolated using an iterated method of analogues (c.f. Cecconi et al. (2012)) under the assumption that the considered systems were composed of finite sets of reappearing states. Passive or weakly coupled processes could then be calculated with low numerical costs obtaining speed ups of two orders of magnitude and more. Here, we want to shed some light on the reasons why a methodology such as rCFD (c.f. Appendix A) can work and what limitations have to be expected. More generally, we aim to provide an approach that indicates when a computationally expensive calculation should be continued and when this would not give additional insights. 2. Theoretical background 2.1. The CFD-DEM method Particle-laden flows can be described on different levels of resolution. The most accurate methods resolve single particles as discrete elements (Cundall and Strack (1979)), often assumed to be perfect spheres, which interact with 6

each other and with the surrounding gas phase. The latter, in turn, feels the particles’ influence via the space they occupy and the exchanged momentum. Due to the increasing popularity of CFD-DEM simulations, their theoretical background has been reviewed extensively (e.g. Deen et al. (2007); Zhou et al. (2010)) and is explained in textbooks (Norouzi et al. (2016)) so that we present the governing equations in a very compact fashion. 2.1.1. Gas-phase equations of motion Anderson and Jackson (1967) showed that the compressible Navier Stokes equations for a fluid with density ρf and velocity field uf take the form ∂αf ρf + ∇ · αf ρf uf = 0 ∂t ∂αf ρf uf + ∇ · αf ρf uf uf = αf ∇ · σ f + fext + fdrag ∂t

(1) (2)

in the presence of a particulate phase with local volume fraction αp = 1 − αf . Besides internal stresses σ f = −Ipf + τ f

(3)

due to pressure pf and deviatoric stresses, e.g. 

†

τ f = µf ∇u + (∇u)



2 − µf I∇ · uf 3

(4)

for a Newtonian fluid with viscosity µf , external forces fext like gravity αf ρf g and the interaction with the solid phase fdrag can change the fluid’s local momentum density. In the present study, the latter source term was modeled with empirical correlations from highly resolved lattice-Boltzmann simulations (Van Der Hoef et al. (2005); Beetstra et al. (2007a)). To close Eqs. (1) and (2), we assume that pressure obeys the ideal gas law, which necessitates 7

the additional solution of an energy equation. More details can be found elsewhere (Lichtenegger and Pirker (2017)). 2.1.2. Particle equations of motion The trajectory and orientation of particle i with mass mi and moment of inertia Ji is governed by Newton’s second law (pp)

mi v˙ i = Fi

(pp)

Ji ω˙ i = τ i

(ext)

+ Fi

(pf)

+ Fi

(5) (6)

.

Particle-particle contacts are modeled with pairwise interactions in F(pp) , which are often approximated with spring-dashpot forces (Cundall and Strack (1979)). Besides external sources, most importantly gravity, the surrounding fluid transfers part of its momentum to the granular phase due to gradients of the local stresses and to drag, (pf)

Fi

(drag)

= Vi ∇ · σ f + Fi

(7)

.

The combined drag force F(drag) on all particles in a small volume Ω gives rise to the local drag force density fdrag in Eq. (2) by the means of fdrag = −

1 X (drag) F . Ω i in Ω i

(8)

While for monodisperse systems, Eq. (8) can be inverted uniquely to obtain (drag)

Fi

= −Vi fdrag /αp , additional assumptions are necessary for the polydis-

perse case (c.f. Van Der Hoef et al. (2005); Beetstra et al. (2007a,b)). Torques in Eq. (6) arise because of tangential components of particleparticle interactions F(pp) . Further contributions, e.g. due to surface roughness, are often termed “rolling friction“ (Ai et al. (2011)). 8

2.1.3. Coupling methodology The fluid and particle equations of motions Eqs. (1) – (2) and Eqs. (5) – (6) are coupled via the particle volume fraction field αp and the momentum exchange term Eq. (8). The former is calculated from the current particle positions {ri }, where information needs to be transferred from the discrete Lagrangian to the continuous Eulerian frame of reference and vice versa for the latter term. A graphical illustration is provided in Fig. 1. The importance of a proper coupling scheme has been discussed for example by Xiao and Sun (2011) and Peng et al. (2014). 2.2. Phase-space methods 2.2.1. Embedding dimension and delay time State vectors yi to describe a system’s dynamics at time ti are often obtained from finite-length sequences of a (scalar) signal x(ti ) by  







 

yi ≡ x ti − (m − 1)τ , . . . x ti − τ , x ti

(9)

because for sufficiently large values of m, the evolution of y(t) in phase space is equivalent to that of the full system producing x(t) (Takens (1981)). The importance of proper choices for the embedding dimension m and the delay time τ was discussed by Kantz and Schreiber (2004). With too low or without embedding at all, coincidentally similar signal pieces that Kennel et al. (1992) called “false nearest neighbors” (see below) would be associated with each other. Particularly if τ is too short for the signal to vary significantly within mτ , the elements of yi are strongly correlated and the identification of false recurrences becomes increasingly difficult. Besides various more sophisticated suggestions (e.g. Fraser and Swinney (1986); Buzug and Pfister (1992); Kim 9

et al. (1999)), a popular choice for the delay is the correlation time τcorr of x(t), often obtained as first zero of the autocorrelation function gx (t) ≡

1 Z t0 +T 0 dt δx(t0 )δx(t0 + t), T t0

(10)

where δx(t) ≡ x(t)− x¯ is the fluctuation of the signal with respect to its timeaverage. Within τcorr , correlations drop from the signal’s variance gx (0) = σx2 to gx (τcorr ) = 0. Consequently, the mean-square difference between two values separated by τcorr is given by 2σx2 so that they will hardly be coincidentally similar. Alternatively, Schouten et al. (1994) suggested to resolve the cycle time (average time for a signal to cross its mean value twice; presumably in the same order of magnitude as τcorr ) with at least 50 to 200 points for noisy time series. Given a certain delay time, the embedding dimension can be determined by looking at the decrease of false nearest neighbors with increasing m. Near(m)

est neighbors yi (m)

(m)

∆i,j ≡

yi

(m)

and yj

are false if their distance

(m) − yj

(11) (m+1)

is small due to projection onto a sub-space but that of yi (m+1)

∆i,j

(m+1)

and yj

,

, is large as sketched in Fig. 2. Above the proper dimension, none

should be present. Different measures have been proposed to provide a straight-forward quantification based on a few characteristic numbers or curves (e.g. Kennel et al. (1992)), but as stressed by Hegger and Kantz (1999), one has to be very careful with the interpretation of these results. Especially for rather short time series which do not allow for a sufficiently dense coverage of the attractor and hence can show significant nearest-neighbor distances, 10

we believe that the inspection of the full point cloud is inevitable. Since in the present work, we use a scaled Euclidean norm

(m)

yi

−

(m) 2 yj

   2 X  1 m−1 ≡ 2 , x ti − kτ − x tj − kτ σx k=0

(12)

false neighbors reveal themselves with large single-component distances after the duration of the correlation time,    1  x t + τ − x t + τ (13) i corr j corr , σx √ on the order of 2 and larger. True nearest neighbors should remain signif√ icantly closer within τcorr than the mean variation 2σx of the signal within

(1)



∆i,j;corr ≡

this time. 2.2.2. Recurrence analysis Only with proper values of τ and m, an unambiguous investigation of actual recurrences is possible. In order to visualize the degree of a dynamical system’s self-similarity at two times, Eckmann et al. (1987) introduced the concept of RP. For a given sequence of N states yi separated by time steps ∆t, each one is compared with all the others based on some norm k.k via the Heaviside function, viz. 



Ri,j = Θ ε − kyi − yj k .

(14)

If the difference of states i and j is smaller than a threshold value ε, they are regarded as similar (Ri,j = 1) and the earlier of the two is said to have recurred at the time of the latter. Otherwise, yi and yj are dissimilar and Ri,j = 0. The N × N matrix R can be interpreted graphically by drawing dots at those coordinates (i, j) where Ri,j = 1. The resulting RP provides 11

information on the dynamics, particularly on the repeating appearance of states, slow drifts or abrupt regime changes in an intuitive way. It can be quantified with various measures known as recurrence quantification analysis (RQA) (Zbilut and Webber (1992); Webber and Zbilut (1994)). The recurrence rate RR(ε, N ) ≡

N 1 X Ri,j , N 2 i,j

(15)

for example, describes the density of recurrence points and is closely related to the correlation sum C(ε, N ) ≡

N X 1 Ri,j (N − nmin )(N − nmin − 1) |i−j|>nmin

(16)

which is the mean probability that states at two different times are similar. A sensible choice of nmin , e.g. nmin = τcorr /∆t ensures that only states which are not too close in time and hence automatically correlated are counted (Theiler (1986)). The intuitive notion that the average time until states reappear increases with decreasing their tolerable difference ε has been formalized by Grassberger and Procaccia (1983a,b) via lim lim C(ε, N ) ∝ εD .

(17)

ε→0 N →∞

Loosely speaking, the generally non-integral correlation dimension D is a measure for the number of effective degrees of freedom (c.f. for example the values of D found by Salikov et al. (2015) in the bubbling, spouting and instable regime in a particle bed: Bubbling allowed for the most variations in contrast to relatively “ordered” spouting and the instable state in-between.) 12

and therefore an indicator for the complexity of a system’s dynamics (Theiler (1990)). The larger it is, the faster the correlation sum drops with decreasing threshold and the lower the fraction of recurrent states one finds. The mean duration until a state i reappears can be obtained from the time-ordered set of recurrences, 







Ri (ε) ≡ i = j0 < j1 < j2 < . . . Ri,jk = 1 AND tjk+1 −tjk ≥ τcorr



. (18)

This definition excludes immediate recurrences due to temporal correlations (the second of the above two conditions) and gives rise to the recurrences times of second type {tj1 − tj0 , tj2 − tj1 , . . . } of state i (Gao (1999); Gao and Cai (2000); Marwan et al. (2007)). To characterize the behavior of the whole ensemble of observed states, one may alternatively consider the distribution of first return times Ti (ε) ≡ tj1 − tj0 = tj1 − ti

with j0,1 ∈ Ri (ε)

(19)

for all i in some analogy to the “global approach” discussed by Anishchenko et al. (2015). For visualization purposes and with an eye on ε-independent measures, it can be convenient to generalize binary RPs to continuous degrees of selfsimilarity, e.g. by the means of distance plots based on Eq. (11) (e.g. McGuire et al. (1997); Iwanski and Bradley (1998); Sipers et al. (2011)). It is then possible to define the average distances to the most similar and most distinct states, ∆sim ≡

N 1 X minj ∆i,j N i

(20)

∆dis ≡

N 1 X maxj ∆i,j . N i

(21)

13

In order to deal with extended systems, all of the above definitions can be generalized for spatially resolved data ϕ(r, t) by the means of

ϕi

2

− ϕj

2 1 Z 3  ≡ d r ϕ(r, t ) − ϕ(r, t ) , i j V0 σϕ2

(22)

where V0 is the volume occupied by the field ϕ(r, t) with variance σϕ2

1 Z 3 ≡ d rσϕ (r)2 V0 2 1 Z 3 1ZT  , ≡ dr dt ϕ(r, t) − ϕ(r) ¯ V0 T 0

(23) (24)

and ϕ(r) ¯ is the temporal average of ϕ(r, t). Besides spatial embedding over the system’s whole domain, which is automatically contained in Eq. (22), ϕ(r, ti ) can be embedded temporally in the sense of Eq. (9) if needed. In this case, vectors of fields  









yi ≡ ϕ r, ti − (m − 1)τ , . . . ϕ r, ti − τ , ϕ r, ti



(25)

are compared with each other. Of course, other exponents in Eq. (22) are equally valid and should not lead to qualitatively too different results. However, we think that for spatially extended fields, it is important that multiple points in a domain contribute to the measure of difference. The maximum norm, on the other hand, would assign the same, large distance to pairs of states differing in large and in small subdomains, making it a not very suitable choice for our purposes. 3. Simulation details The theoretical framework presented in Sec. 2 was applied to study a lab-scale fluidized bed very similar to that used by Patil et al. (2015) and Lichtenegger et al. (2017), where more details can be found. 14

Approximately 57 000 spherical particles with diameter dp = 1 mm, density ρp = 2500 kg/m3 and contact-mechanical parameters of glass were charged into a cuboid with dimensions lx = 8 cm, ly = 1.5 cm and lz = 25 cm pictured in Fig. 3. Gas with properties of air entered through the bottom face in the xy-plane with a velocity of uin = 1.33ez m/s, fluidized the grains and left through the top face at ambient pressure. Inflow was restricted by a small obstacle in the center of the inlet plane to split the flux and create smaller bubbles. Side walls were assumed to be impermeable. The granular phase, the inflowing gas and the walls had a temperature of T0 = 20 ◦ C. The fluid and particle equations of motion outlined in Sec. 2.1 were solved on a regular, hexahedral grid consisting of 35 × 6 × 110 equal cells. A combination of the PISO and SIMPLE algorithms (Patankar and Spalding (1972); Issa (1986); Peric and Ferziger (2002)) and velocity-Verlet integration (Allen and Tildesley (1987)), respectively, employing the CFD library OpenFOAM coupled to the DEM code LIGGGHTS (Kloss et al. (2012)) via CFDEMcoupling (Goniva et al. (2012)) was used. Sufficiently small time steps of ∆tDEM = 2.5 · 10−6 s and ∆tCFD = 2.5 · 10−5 s ensured numerical stability. Data were gathered for tmax = 60 s of simulated process time. Probe values from a single point were recorded every 2.5 · 10−4 s in the center of the bed 3 cm above the inlet (c.f. Fig. 3), the full flow fields every 5.0 · 10−3 s. Patil et al. (2015) have demonstrated that the CFD-DEM method captures the experimentally found behavior of the fluidized bed under consideration very well, a validation of our own implementation has been provided by Lichtenegger et al. (2017).

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4. Results and discussion While the complete specification of a gas-solid flow’s state comprises information on the distribution and motion of the dispersed and on the pressure, the velocity and possibly the temperature fields of the continuous phase, we focused our investigations on the particle volume fraction. Since its evolution is intimately connected to the solid phase momentum and it causes resistance to which the gas flow has to adapt, it implicitly contains a lot of meso- and macroscopic information on the whole system. Nevertheless, the following analysis may be applied to any characteristic field of interest in case a more detailed description is necessary. A first, intuitive impression of the fluid dynamics is provided by the snapshots shown in Fig. 4. Most of the time, several bubbles were present, which rose through the particle bed. If they merged, they could cause infrequent, strong eruptions at the bed’s surface. While a few such images allow for a rough regime identification and a visceral distinction between similar and dissimilar states, they clearly do not contain the same information as highly time-resolved data. Single-point probe values in Fig. 5a reveal, for example, an initialization phase that lasted approximately 0.5 s during which the particles had first to be lifted from their compact resting positions before entering the fluidization cycle. To avoid any tainting of our calculations due to this equilibration process, we discarded any values from the first tequil = 1.0 s to obtain the subsequent results. After the initialization period, fast variations that corresponded to the passing of a bubble dominated the local dynamics at the probing point and led to minimum values of zero and maxima near the random-close-packing 16

limit. It can be seen in Fig. 5b that these oscillations caused a quick decay of the volume fraction autocorrelation function gα (t). It went down from σα2 = 0.039 to 0 within τcorr = 0.055 s which is approximately the time it took a bubble to pass the probing point. Close inspection of the time series in Fig. 5a shows even more rapid changes with less pronounced minima and maxima, which were due to particles entering a bubble and medium dense, solid aggregates, respectively. In order to resolve this fast local dynamics, we chose τlocal = 2.5 · 10−4 s ≈ τcorr /200 as delay time for embedding point-data along the lines of Schouten et al. (1994). Regarding the global bed behavior, on the other hand, we discarded processes below the typical bubble time scale so that we used τglobal = τcorr for field data. In the next step, we constructed the (false-)nearest-neighbor point clouds for both the single-point signal and the global volume fraction fields displayed in Figs. 6 and 7, respectively. For each state yi found at time ti , we compared the distance to its nearest neighbor yj with their subsequent divergence tendency. As expected, the former case showed a large number of false nearest neighbors for embedding dimensions too low to satisfy mτlocal & τcorr . For m = 100, even smallest distances could lead to strong separations after τcorr . √ In almost half of all cases, it was larger than the average deviation 2 after this duration, which underlines that too small embedding values do not allow to identify truly similar states. However, increasing it to m = 200 caused a significant drop of this fraction and hence eliminated practically any false nearest neighbors. A completely different picture was found for the comparison of spatially resolved data. Even without temporal embedding, i.e. m = 1, small nearest-

17

neighbor distances induced only minor separations after τcorr in most cases. Only a negligible fraction had larger deviations. Since an almost identical picture was obtained for m = 2, we conclude that the high degree of spatial embedding renders any temporal embedding unnecessary, or to put it differently, the current particle distribution suffices to uniquely describe the granular phase state. While when looking only at a single point, a few consecutive values of the particle volume fraction cannot determine its evolution in the immediate future, a single snapshot of the whole system provides enough information to do so. This observation is also confirmed by the dependence of the correlation sum Eq. (16) and its low-ε limit Eq. (17) on the embedding dimension. Since Taken’s theorem (Takens (1981)) ensures that an embedding dimension of 2D +1 is sufficient to unfold an attractor of dimension D, one expects saturation of the low-ε exponent with increasing m. Indeed, the double-logarithmic plots in Fig. 8 exhibit scaling regions for both correlation sums based on the local and the global dynamics with embedding-independent slopes. The former seems to be located on an approximately Dlocal = 5-dimensional subspace, which is in the range of attractor dimensions of bubbling beds found in other studies (e.g. Salikov et al. (2015)). As was to be expected, the latter is more complex and takes place in about Dglobal = 18 dimensions. Only if spatial correlations were such long-ranged that each sub-part of the system was tightly connected with all the others, one could expect similar dimensions for the local and the global behavior. We stress that the ε → 0 limit can never be reached with a finite data set. Although Figs. 8a and 8b exhibit scaling regions above the lowest admissible

18

thresholds, one cannot rule out that for even lower values, a different scaling exponent would be present (e.g. Kapitaniak (1993); Collins and De Luca (1994); Cecconi et al. (2012); Sreekumar et al. (2014)). As a matter of fact, a hint on a corresponding second region might be seen in the lowest threshold values in Fig. 8a. Therefore, our estimates for the correlation dimensions are tightly connected to the smallest ε values for which we found a sufficiently large number of pairs of similar states in our relatively short time series. Nevertheless, it compares well with mode decomposition results of Cizmas et al. (2003); Palacios et al. (2004) who found in a not too different gas-solid flow that most energy is contained within a similar number of modes and those of higher order played a minor role. In this regard, we point out that it was argued (Smith (1988); Ruelle (1990); Eckmann and Ruelle (1992)) that a series with 105 elements does not allow to accurately resolve higher dimensions than 5. While there have been serious discussions on this “pessimistic view” (Essex and Nerenberg (1991)), also more optimistic estimates (Nerenberg and Essex (1990)) agree that the required amount of data to reliably cover high-dimensional systems quickly grows beyond any limits of experimental or computational feasibility (c.f. Tab. 2 of Llop et al. (2015) for an overview on the number of sample points in recent investigations). Based on the findings of Nerenberg and Essex (1990), we estimate that our limited data induced an uncertainty of ∆Dglobal ≈ 2 on the global correlation dimension. Within these error bounds, it is still valid to conclude that compared to the huge number of 6Np microscopic translational degrees of freedom, our system shows a remarkably organized, low-dimensional motion on a global scale, which in turn is more complicated

19

than the local behavior. Clearly, local similarity does not imply large-scale similarity: States at times ti , tj , which were globally closer than e.g. 10% of the distance between the closest and the most distant configurations, (global)

∆i,j

(global)

≤ ∆10%

(global)

≡ ∆sim



(global)

+ 0.1 ∆dis

(global)

− ∆sim



,

(26)

were five times as likely to be also locally closer than (local)

∆10%

(local)

≡ ∆sim



(local)

+ 0.1 ∆dis

(local)

− ∆sim



(27)

than vica versa. Locally similarity carried over to the global perspective less often. Based on the above findings, in the following, we focus on global over local field information to detect recurring patterns. Besides the much larger amount of available information, which is particularly critical for short time series within the scope of detailed simulations, we believe that it is simply more natural. Due to the high degree of spatial correlations, no embedding is necessary – each snapshot determines the current state of the system and allows for the comparison with other states. Additional insight into the bed dynamics may be obtained from the distance matrices ∆i,j shown in Fig. 9 for the first 4 s and 20 s of process time, respectively. A pattern of short local minima parallel to the main diagonal which is always present (identical states have no difference) can be seen. They correspond to similar particle distributions and since they will also evolve in a similar way for a certain duration, they are oriented parallel. The pattern is crossed by pronounced vertical and horizontal maxima reappearing quite irregularly with lower frequencies. Their origin are rare states, i.e. configurations with infrequent recurrences, for example eruptions of very 20

large bubbles like in Fig. 4c which depicts that state in our simulation with the largest nearest-neighbor distance. Of course, close inspection of Fig. 9a reveals that these events also reappear: Some crossings of horizontal and vertical lines are broken by fine, but pronounced local minima with 45◦ orientation. Neither the close look Fig. 9a nor the long-term analysis Fig. 9b give a hint that the general degree of recurrence fades with time. Minima close to the diagonal do not look qualitatively different than those farther away, which means that there is no more complicated transient behavior superposed to the rather rapid recurrent one. On the other hand, within the investigated time scale, there is no obvious advantage to look for states with the highest degree of similarity after very long times in comparison to those found earlier. This is underlined by the plots of ∆sim and ∆dis over observation time in Fig. 10. Initially, the average distance to the most similar state drops and that to the most distinct one increases, but both curves seem to saturate quickly. After about 1 s − 2 s, which corresponds to several bubble passages through the bed, the former decreases only very slowly anymore. We interpret this behavior such that most states have recurred at least approximately after this time and only much longer observation would produce significantly more similar reappearances to satisfy ∆sim → 0 in the limit of infinite time. The behavior of the curve after a few recurrences can be approximated very well with a simple power law. Given an attractor with dimension D, one could naively expect that doubling the number of recurrences of a certain state, i.e. going from t to 2t, leads to a decrease of the mean nearest-neighbor distance

21

from ∆sim (t) to 2−1/D ∆sim (t), or more generally ∆sim (t) ∝ t−1/D

(28)

with D obtained from Fig. 8b. Besides a satisfying consistency check for the correlation dimension, Eq. (28) underlines that after the initial rapid drop, any further substantial decrease of ∆sim takes very long. This behavior gets clearly more pronounced with increasing correlation dimension. The average largest distance between states saturates even sooner within 0.5 s and remains almost constant after a few wiggles. As one would expect, strongly dissimilar configurations are found before reappearances begin. On closer inspection, a very weak increase of ∆dis can be seen over longer times, which means that throughout our simulation, new configurations kept appearing. A complementary point of view on the time recurrences took is provided by the distribution of first return times T displayed in Fig. 11a for thresholds ε = 0.7, ε = 0.8 and ε = 0.9. At least the former two values were located the scaling region in Fig. 8b as recommended by Webber Jr and Zbilut (2005). An intuitive impression of this degree of similarity may be obtained from Fig. 12 that shows two snapshots with a corresponding distance. (Note that the assumption stated at the beginning of Sec. 4 – recurrence of the volume fraction field implies also similar solid and similar fluid velocities – is confirmed by their comparison in Fig. 12.) To avoid any bias towards shorter return times, we focused on the recurrence of all states observed within the first 30 s. This way, each state had at least another 30 s to reappear. While hardly any structure can be seen for ε = 0.7, local maxima are present for larger thresholds. For ε = 0.9 corresponding to ∆sim (t ≈ 1.6 s), it comes as no 22

surprise that the relative majority of states reappeared within 1 s. However, also for ε = 0.8 which is on the order of ∆sim (t ≈ 16 s), a qualitatively similar behavior was found. Even though it took about 16 s until on average every state had recurred with this tolerance, a relative majority had already done so between 1 s and 2 s. At the same time, the cumulative return time distributions in Fig. 11b show that for all thresholds, some configurations remained unique within the observation time. While it was to be expected for ε = 0.7 given a time of about 150 s it took to reach the corresponding degree of similarity (estimated via Eq. (28)), this finding underlines how long-ranged the distributions are even for larger thresholds (Gao (1999); Anishchenko et al. (2015)). In addition to frequently visited regions of the attractor, a significant set of states with continuously distributed, longer return times exists. 5. Conclusion In this study, we investigated the recurrent behavior of a lab-scale, bubbling, fluidized bed from the perspectives of a single probing point and of spatially extended particle number-density fields. Because of their connection with the solid momentum field via the continuity equation, similar evolutions of the former imply comparability of the latter. The same holds for the fluid velocity which adapts quickly to the varying resistance of the bed. Both assertions are substantiated by Fig. 12. In order to distinguish between truly and seemingly similar configurations, i.e. those with strongly differing short-term evolutions, we employed temporal embedding. While at least 150 – 200 consecutive local probe val23

ues were necessary for the proper identification of recurrent states, instantaneous global field snapshots turned out to be equally sufficient. However, the dependence of the correlation sum on the threshold value underlined that the global bed motion is more complex than the local one. In both cases, we found clearly visible scaling regions C(ε) ∝ εD with Dlocal ≈ 5 and Dglobal ≈ 18. In principle, these exponents may be identified with the correlation dimension of the system, but we stress that our limited amount of data induces a significant uncertainty, especially on Dglobal . Therefore, we refrain from attempting to provide an accurate prediction of the underlying attractor’s correlation dimension. Nevertheless, our results show a behavior of very low dimension on a local scale, which is clearly higher on a global scale but still extremely low if compared to the number of microscopic degrees of freedom. An interpretation of Dlocal and Dglobal is not straightforward and needs to be taken with care. Regarding the local perspective, we note that the behavior of the particulate phase will largely be determined by its volume fraction αp , the solid velocity up and to some extent also the by granular temperature Tp , i.e. the velocity fluctuations, which altogether add up to five dimensions. Globally, since temporal embedding turned out to be unnecessary, the velocity field follows from the current particle distribution which already contains the full information to describe the granular phase state. Assuming that most of the time, two to four bubbles were present as indicated by Figs. 4 and 12, and that each of these voids in the pseudo-2D bed may be approximated with an ellipse in terms of five variables (two coordinates of its center, two axis lengths, one orientation angle), one ends up with a number in the same range

24

as Dglobal to specify the current bubble configuration. It seems reasonable that this “reduced-order” view can describe the large-scale behavior of the bed at least with a certain accuracy. The rough estimation we obtained for Dglobal is consistent with the temporal evolution of the average nearest-neighbor distance ∆sim . After an initial rapid drop which corresponds to the intuitive notion of a few cycles on the attractor, ∆sim decreases very slowly like t−1/Dglobal . The higher the system’s complexity, the longer it takes to observe recurrences with a much higher degree of similarity than those after the initial phase. Conversely, if for a specific application, ∆sim is at this point already sufficiently small, does the so far obtained sequence of fields provide an equally good approximation of the attractor as one of much longer duration? The weak, but undeniably existing growth of ∆dis over time and the long tail of the first return time distribution prove that states which had not already been encountered were constantly appearing. Clearly, it is highly unlikely that a short-term simulation of a complex system contains every possible configuration and reaches the whole attractor with respect to a sensible threshold. However, the fast initial and the much slower further increase of the cumulative first return time distribution lead us to conclude that even quite short simulations can provide an approximation of those regions of the attractor where most time is spent, which will also be the most important parts for many applications. This explains why relatively short time series could be extrapolated to much longer times in previous studies on rCFD mentioned in Sec. 1. More generally, we are convinced that often the relevant information on the fast dynamics of flows is already contained within data from brief

25

simulations, where the here introduced approach may be used to assess a sufficient duration. In these and also in the present work, the systems under consideration were quite small. Above a certain size related to the chaos correlation length (Cross and Hohenberg (1993)), the number of effective degrees of freedom grows proportionally to the volume (Egolf and Greenside (1994); Egolf et al. (2000)). Since the average return time is expected to be inversely proportional to the probability of finding a pair of similar states, it increases massively like T ∼ C(ε)−1 ∝ ε−D ,

(29)

which is an alternative way to read Eq. (28) (explaining its very good agreement with ∆sim obtained directly from distance matrices) and is closely related to Kac’s lemma (Kac (1947)) for ergodic, phase-space volume preserving systems. This necessitates a much larger distance matrix leading to a drastic increase in computational costs. It remains to be explored in future investigations to which extent simple measures such as filtering out small-scale structures for a more generous classification of similar states can counterbalance this trend and which more fundamental simplifications are needed. Nomenclature Greek letters α

volume fraction

∆

distance of two states

26

∆t

time step

ε

threshold

µ

viscosity

ρ

density

σ

variance

σ

stress tensor

τ

duration

τ

torque; viscous stresses

ω

angular velocity

Latin of symbols C

correlation sum

D

correlation dimension; diffusion coefficient

f

force density

F

force

g

autocorrelation function

I

unit tensor

m

embedding dimension; particle mass

Np

particle number 27

p

pressure

R

recurrence matrix

R

set of recurrence times

S

source term

T

first return time; temperature

u

velocity field

v

particle velocity

V

volume

Subscripts/Superscripts coll

collision

corr

correlation

dis

dissimilar

ext

external

f

fluid

global

global

loc

local

p

particle

pf

particle-fluid 28

pp

particle-particle

rec

recurrent

sim

similar

List of abbreviations CFD

computational fluid dynamics

DEM

discrete element method

rCFD

recurrence CFD

RQA

recurrence quantification analysis

Acknowledgement This work was partly funded by the Linz Institute of Technology (LIT), Johannes Kepler University (project LIT-2016-1-YOU-007). Furthermore, support from K1-MET GmbH metallurgical competence center is acknowledged. Appendix A. Recurrence-based CFD The simulation of a slow, passive or weakly coupled process on a dynamic, recurrent flow can be significantly sped up with the following procedure.

29

Recurrence statistics A detailed simulation (e.g. CFD-DEM) is carried out for a short but sufficiently long duration τrec such that typical flow features have started reappearing. Field data are stored with a sampling step ∆trec (creation of database) and a distance matrix based on a characteristic variable, e.g. the local solid fraction αp (for the sake of simplicity here without embedding), ∆i,j ∝

Z

2



d3 r αp (r, i∆trec ) − αp (r, j∆trec )

(A.1)

is calculated. For each time tm = m∆trec , that past instant tn = n∆trec when the system was most similar to its current state can be found by minimizing ∆m,n . Recurrence process In the next step, the evolution of the flow is time-extrapolated based on ∆i,j . Starting with an interval of consecutive time steps t1 , t2 , . . . ta , the following one is obtained by identifying the state tb most similar to ta and adding tb+1 , tb+2 , . . . tc . Clearly, this operation can be repeated arbitrarily often to create a time series which is orders of magnitudes longer than τrec . Besides other methods, interval sizes can be drawn from a random number distribution with reasonable bounds related to the size of the database. In the above case, the approximate time evolution e.g. of the solid velocity field up is given by up (r, t) ≈ u(rec) (r, t) ≡ p 



up (r, t1 ), up (r, t2 ), . . . up (r, ta ), up (r, tb+1 ), up (r, tb+2 ), . . . .

30

(A.2)

Recurrence CFD With an approximation of the fast degrees of freedom (volume fraction, velocity etc.) available, one can simulate passive or weakly coupled processes according to models A, B (Lichtenegger and Pirker (2016) and C (Pirker and Lichtenegger (2018)): Eulerian perspective (“Model A”) In the Eulerian frame of reference, a passive transport equation ∂ (rec) α (r, t)c(r, t) + ∇ · u(rec) (r, t)c(r, t) ∂t − ∇ · α(rec) (r, t)D(r)∇c(r, t) = S(r, t)

(A.3)

for a quantity c(r, t), e.g. some species concentration, subject to convection, diffusion and source terms, is solved. Lagrangian perspective (“Model B”) If a discrete-particle representation is desired, non-interacting tracers following the solid velocity field by the means of dri = u(rec) (ri , t)dt + drcoll

(A.4)

are used. The second term mimics collisions and prevents overpacking without costly contact detection. The tracers can carry additional information like chemical composition or temperature and interact with the surrounding fluid, which amounts in the solution of one or more transport equation(s) of type Eq. (A.3).

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44

CFD

DEM
αp {ri } ,
fdrag {ri , vi } {ri (t), vi (t)}

uf (t), pf (t) (pf)

Fi

uf

t + ∆t




t + ∆t

Figure 1: CFD-DEM coupling scheme. In each time step, particle positions {ri } and velocities {vi } are communicated from the DEM to the CFD side to calculate the particle volume fraction αp and particle-fluid interaction field fdrag needed to propagate the fluid (pf) equations of motion for uf and pf . In turn, the force Fi acting on each particle due to fluid flow is transferred back to the DEM code. After each solver has performed one single or multiple (sub-)steps, the coupling procedure is repeated.

45

z B A

x

A’ B’

y

Figure 2: Sketch of the importance of embedding. Two well-separated points A and B on a curve (blue) might appear to be neighbors A’ and B’ if projected onto a subspace with too small a dimension (gray).

46

z

x

y

Figure 3: Simulation geometry. Gas flowed in through the xy-plane around a small obstacle (gray). Local probe values were taken at a point (blue) in the center of the bed which had an initial filling height sketched with the red line.

47

0.15

0.15

0.15

0.6

0.6

0.2

0.00 0.0 -0.04 0.00 0.04

0.05

0.2

0.00 0.0 -0.04 0.00 0.04

0.4 αp

αp

αp

z [m]

0.05

0.10 0.4

z [m]

0.10 0.4

z [m]

0.10

0.6

0.05

0.2

0.00 0.0 -0.04 0.00 0.04

x [m]

x [m]

x [m]

(a)

(b)

(c)

Figure 4: Snapshots of the particle volume fraction αp . At t = 5 s (a) and t = 10 s (b), several medium-sized bubbles are present while at t = 39.29 s (c), a very large one has just erupted and some small ones are created. The particle configuration (c) is those with the largest nearest-neighbor distance found in our simulation.

48

0.8

αp

0.6

0.4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

t [s]

(a) 0.04

0.03

gα

0.02

0.01

0.00

−0.01 0.0

0.2

0.4

0.6

0.8

t [s]

(b) Figure 5: (a) Particle volume fraction αp and (b) the corresponding autocorrelation function gα . After a short initialization phase of about 0.5 s, αp changes rapidly between 0 due to a bubble at the probing point and the random-close-packing limit of maximum compacted particles. These fast variations cause a decay of gα within a duration on the same order of magnitude as that of oscillations of αp .

49

m=100 m=200

2

(1)

∆i, j;corr

3

1

0

0

1

2

3

4

(m)

∆i, j

(a)

fraction of distances larger than ∆

1.0 m=100 m=200

0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

∆

(b) Figure 6: (False) nearest neighbors for local probe values of αp . (a) shows the nearestneighbor point clouds for two embeddings. Points above the green line have a larger √ (1) single-component distance after τcorr , ∆i,j;corr , than 2 and are thus regarded as false nearest neighbors. (b) The fraction of nearest neighbors with a single-component separation larger than a given value. Too small embedding dimensions cause a slow decay whereas sufficiently high embedding leads to a fast drop.

50

2.0

(1)

∆i, j;corr

1.5

1.0

0.5 m=1 m=2

0.0 0.0

0.4

0.8

1.2

1.6

(m)

∆i, j

(a)

fraction of distances larger than ∆

1.0 0.8 0.6 0.4 0.2 0.0 0.0

m=1 m=2

0.5

1.0

1.5

2.0

2.5

∆

(b) Figure 7: Same as Fig. 6 for global fields. Additional temporal embedding (m = 2) does not give any advantage over only spatially embedded states (m = 1).

51

100 10−1

C

10−2 10−3 10−4

m = 150 m = 200 m = 250 ∝ ε 4.88

10−5 10−6

100

101

ε

(a) 100 10−1

C

10−2 10−3 10−4

m=1 m=2 ∝ ε 17.67

10−5 10−6

100

ε

(b) Figure 8: Correlation sums for (a) local and (b) global field information. In both cases, the investigated embedding dimensions led to consistent low-ε slopes each. The gray lines are fits to the scaling range in the log-log plots. With decreasing threshold values, decreasing numbers of state pairs were available for the computation of C(ε) so that the ε → 0 limit needs to be taken with care.

52

2.2

distance

t [s]

4

2

0

0.8 0

2 t [s]

4

(a)

2.2

distance

t [s]

20

10

0

0.8 0

10 t [s]

20

(b) Figure 9: Distance plots for (a) 4 s and (b) 20 s of process time. A pattern of rather short local minima which are parallel to the main diagonal is visible in (a). Their 45◦ orientation resembles a temporarily similar evolution of similar states. Horizontal and vertical lines are caused by rare events, i.e. configurations like strong eruptions which do not reappear as frequently as other states. It can be seen in (b) that the relatively fast, recurrent motion is repeatedly interrupted by such rare events.

53

2.2

∆sim ∆dis

distance

1.8

∝ t −1/D 1.4

1.8 1.4 0.0

1.0

0.6

0

10

0.5

1.0

20

30

40

50

60

t [s]

Figure 10: Nearest- and farthest-state distances. The average distance to the nearest neighbor and the most distant configuration vary rapidly within a short duration of about 1 s − 2 s after which both change very slowly anymore. ∆sim follows a power law (green line) related to the correlation dimension of the attractor obtained from Eq. (17).

54

ε = 0.7 ε = 0.8 ε = 0.9

pdf(T )

0.2

0.1

0.0

0

5

10

15

20

25

30

T [s]

(a) 1.0

ε = 0.7 ε = 0.8

cdf(T )

0.8

ε = 0.9

0.6

0.4

0.2

0.0

0

10

20

30

T [s]

(b) Figure 11: Probability and cumulative distribution functions of first return times. (a) For not too small tolerable distances ε, the relative majority of states returns approximately after 1 s − 2 s (c.f. Fig. 10). However, the distributions of return times exhibit a long tail and become increasingly flat with decreasing threshold. (b) The cumulative distributions show that depending on ε, a significant fraction of states does not recur at all within the observed time.

55

0.05

z [m]

αp

z [m]

0.4

0.0 0.05

0.2

0.00 0.0 -0.04 0.00 0.04

−0.2

0.00 -0.04 0.00 0.04

0.10

2

0.05

1

z [m]

0.2

0.10

3

−0.4

0.00 -0.04 0.00 0.04

x [m]

x [m]

x [m]

(a)

(b)

(c)

0.15 0.6

αp

z [m]

0.4

0.05

0.2

0.00 0.0 -0.04 0.00 0.04

0.0 0.05

−0.2

0.00 -0.04 0.00 0.04

−0.4

0

3

0.10

2

0.05

1

z [m]

0.2

0.10 z [m]

0.10

0.15

0.4

up,z [m/s]

0.15

uf,z [m/s]

0.6 0.10

0.15

0.4

0.00 -0.04 0.00 0.04

x [m]

x [m]

x [m]

(d)

(e)

(f)

uf,z [m/s]

0.15

up,z [m/s]

0.15

0

Figure 12: Particle volume fraction αp (a), (d), z-component of solid velocity up,z (b), (e) and z-component of fluid velocity uf,z (c), (f) at t = 3.0 s (upper row) and t = 11.035 s (lower row). Their distance ∆a,b ≈ 0.8 is representative for the average most similar states observed during the simulation. Even though recurrence was established solely via the volume fractions, also the particle and fluid velocity fields show clear indications of similarity each. The existing deviations are mainly found in regions of little importance with hardly any grains present.

56