Amir Esteghamatian

Florian Euzenat

Abdelkader Hammouti

Michel Lance

Anthony Wachs
email: wachs@math.ubc.ca

A Bernard

A stochastic formulation for the drag force based on multiscale numerical simulation of fluidized beds

Keywords: Multi-scale simulation, Fluidized Bed, Stochastic drag law, Discrete Element method, Fictitious Domain method, Euler/Lagrange method

Micro/meso simulation of a fluidized bed in a homogeneous bubbling regime, International Journal of Multiphase Flow, 92, 93-111, we performed a one-to-one comparison of micro-scale Particle-Resolved Simulation (PRS) results and meso-scale two-way coupled Euler-Lagrange (a. k. a. DEM-CFD for Discrete Element Method-Computational Fluid Dynamics) simulation results in a homogeneous bi-periodic liquid/solid fluidized bed. These data showed an acceptable agreement between micro-and meso-scale predictions for integral measures as, e.g., pressure drop and bed height. However, particles fluctuations are markedly underpredicted in DEM-CFD simulations, especially in the direction transverse to the main flow. The filtered DEM-CFD porosity appears to be a coarse descriptor of the local microstructure and

Introduction

Owing to their wide presence in environmental phenomena, pharmaceutical industry, energy sector, chemical processes, biological flows, etc., particle-laden flows have been subject of extensive analytical, numerical and experimental studies. Thanks to substantial advances in the computing resources, numerical approaches attracted much attention in the past decades. Accordingly, more care is given to the development of more advanced numerical models with higher levels of accuracy.

The nonlinear and multi-scale nature of fluid/particle interactions in such systems highlights the rich dynamics of the system and calls for well-adapted numerical models. A variety of numerical models depending on the required computational resources and desired level of accuracy have been proposed in the literature ( [START_REF] Van Der Hoef | Numerical Simulation of Dense Gas-Solid Fluidized Beds: A Multiscale Modeling Strategy[END_REF], [START_REF] Tenneti | Particle-resolved direct numerical simulation for gas-solid flow model development[END_REF]).

At the most refined end of the spectrum, micro-scale models based on PRS are able to resolve the flow at the scale of the particle boundary layer. Meso-scale DEM-CFD models filter the fluid field at scales at least one order of magnitude larger than the particle volume. Closure laws are hence required to estimate the momentum transfer between particle phase and fluid phase. Traditionally, this filtering operation is implicit as the volume of fluid grid cells is usually much larger than the particle volume. As contributions from a Lagrangian particle are simply distributed to the Eulerian fluid grid cells intersected by this particle, the filter size implicitly spans a maximum of two fluid grid cells in each direction. Over the past few years, a new and enhanced approach that decouples the filtering kernel from fluid grid cells has been developed [START_REF] Link | Flow regimes in a spout-fluid bed: A combined experimental and simulation study[END_REF], [START_REF] Pepiot | Numerical analysis of the dynamics of two-and three-dimensional fluidized bed reactors using an Euler-Lagrange approach[END_REF], [START_REF] Capecelatro | An Euler-Lagrange strategy for simulating particle-laden flows[END_REF]).

This methodology provides the model with the flexibility of using fluid grid cells with volumes comparable to the particle volume, or even smaller. The filter size is typically a few particle diameters and spans the required number of fluid grid cells as controlled by the particle diameter to fluid grid size ratio.

At the coarsest end of the spectrum, macro-scale models (either Two-Fluid models or moments based methods, see [START_REF] Fox | Large-eddy-simulation tools for multiphase flows[END_REF] for more details) consider both particle phase and fluid phase as continuous media described by Eulerian transport equations. In the particular framework of Two-Fluid models, the kinetic theory of granular media is employed to consider particle-particle and particle-wall interactions.

Meso-scale models have gained a large popularity in simulation of dense fluidparticle systems during the past two decades [START_REF] Tsuji | Discrete particle simulation of twodimensional fluidized bed[END_REF], [START_REF] Pepiot | Numerical analysis of the dynamics of two-and three-dimensional fluidized bed reactors using an Euler-Lagrange approach[END_REF]). In contrast to Two-Fluid models, a major source of inaccuracy is eliminated by directly taking into account particle-particle and particle-wall interactions. Also, employing closure laws for fluid-particle mo-mentum transfer leads to a drastic reduction of the computational load (by several orders of magnitude) as compared to micro-scale models. The reasonable compromise between level of accuracy and required computational effort favors this type of models in the simulation of meso-scale systems, i.e., systems that comprise 10 3 to 10 9 particles and the upper limit is increasing with advances in parallel algorithms, High Performance Computing and computational resources.

While meso-scale models have been widely applied to fluid-particle simulations since the 90s, only recently attention has been given to the accuracy of the model, particularly in terms of predictions of second-order particle statistics [START_REF] Subramaniam | Developing improved Lagrangian point particle models of gas-solid flow from particleresolved direct numerical simulation[END_REF], [START_REF] Kriebitzsch | Fully resolved simulation of a gas-fluidized bed: a critical test of dem models[END_REF], Esteghamatian et al. [2017a]). Since the fluid field remains unresolved at the level of particles, particle-induced fluctuations and in turn particle agitations are suppressed to some extent by meso-scale models. [START_REF] Subramaniam | Developing improved Lagrangian point particle models of gas-solid flow from particleresolved direct numerical simulation[END_REF] emphasized the underprediction of particles acceleration by point-particle models in decaying isotropic turbulent flow. [START_REF] Kriebitzsch | Fully resolved simulation of a gas-fluidized bed: a critical test of dem models[END_REF] also pointed out the underestimation of the drag force in DEM-CFD simulations as compared to PRS. In our previous work (Esteghamatian et al. [2017a]), we also observed the underestimation of the particles granular temperature particularly in the direction transverse to the mean flow.

In Esteghamatian et al. [2017b], we further analyzed the PRS of fluidization to have a better insight into the dominant mechanisms of particles motion.

In short, we have so far drawn the following conclusions: (i) the meso-scale model partially resolves fluid fluctuations and underpredicts particles fluctuations, and (ii) local fluid fluctuations around the particles depend on the filter size and smoothly increase with the averaging control volume particularly in a homogeneous system. (i) and (ii) emphasize that any information transfer from PRS to the meso-scale model requires to take into account the effect of the filter size. Strictly speaking, fluctuations captured by the meso-scale model are limited by the size of the filter. In that sense, the meso-scale model can be considered as the equivalent of the Large Eddy Simulation (LES) approach in an analogy with single-phase turbulence modeling. However, contrary to singlephase turbulence where instabilities are generated at the resolved scales, smallscale instabilities -also called pseudo-turbulence -are generated at the particle scale which is inherently not accessible with meso-scale model. The lack of a self-similar pattern and the generation of instabilities at unresolved scales prevent us from developing a universal formalism that relates momentum transfer to mean values at the filter scale. This has been previously addressed in the literature and accordingly, closures based on PRS or experiments have been proposed to correlate the mean momentum transfer to integral measures [START_REF] Hill | Moderate-reynolds-number flows in ordered and random arrays of spheres[END_REF], [START_REF] Ergun | Fluid flow through packed columns[END_REF], Di Felice [1994]). We hope that a similar procedure can be employed to recover momentum transfer fluctuations. Given that the PRS solution already accounts for all the relevant spatial scales of the system, it is technically possible to feed the meso-scale model with the missing information at the particle level in a multi-scale framework.

The scope of the present study is to answer the following questions: (a) to which extent the performance of meso-scale models are limited by the fluid field discretization?, (b) is there any methodology to quantify the sub-grid scale fluctuations from PRS results ? To answer (a), we perform a grid refinement study with our meso-scale model. A basic feature of many classical meso-scale models (including ours in [START_REF] Bernard | Controlling the quality of two-way Euler/Lagrange numerical modeling of bubbling and spouted fluidized beds dynamics[END_REF] Esteghamatian et al. [2017a]) is that the fluid grid cell is used as the fluid/particle averaging kernel. While pretty efficient when particles are much smaller than the fluid grid cell, it causes numerical stability issues when the size of particles are larger than (or even comparable to) the fluid grid cell. As suggested by [START_REF] Capecelatro | An Euler-Lagrange strategy for simulating particle-laden flows[END_REF], it is technically possible to decouple the fluid/particle averaging kernel from the fluid grid cell. We have adopted this approach to attain the flexibility of using fine grid cells and perform a full grid refinement study. In an effort to answer (b), we have characterized the drag coefficient experienced by particles in PRS as a Probability Density Function, in contrast to the classical deterministic drag laws. This provides us with a basis for our stochastic formulation of the drag force. Stochastic approaches have been widely applied in turbulence modeling particularly involving reactive flows [START_REF] Dreeben | Probability density function/Monte Carlo simulation of near-wall turbulent flows[END_REF] [START_REF] Pope | Lagrangian PDF methods for turbulent flows[END_REF] [START_REF] Muradoglu | A consistent hybrid finite-volume/particle method for the PDF equations of turbulent reactive flows[END_REF]). [START_REF] Sommerfeld | Recent advances in the numerical simulation of pneumatic conveying through pipe systems[END_REF] and [START_REF] Oesterle | Simulation of particle-to-particle interactions in gas solid flows[END_REF] introduced a probabilistic approach to model inter-particle collisions inspired by the kinetic theory of gases. In spray modeling, [START_REF] Subramaniam | Statistical representation of a spray as a point process[END_REF] proposed a statistical representation of a spray based on a stochastic point process model. Employing a stochastic approach to model the drag force is quite scarce in the literature, yet not nonexistent. In homogeneous isotropic turbulence and considering a oneway point-particle assumption, [START_REF] Fede | Stochastic modeling of the turbulent subgrid fluid velocity along inertial particle trajectories[END_REF] proposed a stochastic model to reconstruct the sub-filter fluid fluctuations based on a Langevin equation. DNS data were used to determine the constants of the model. The authors have shown that the correct level of particles kinetic energy can be attained by the LES of the fluid field and a discrete particle method incorporating a stochastic closure. A similar approach was adopted by [START_REF] Berrouk | Stochastic modelling of inertial particle dispersion by subgrid motion for LES of high Reynolds number pipe flow[END_REF] for the LES of inertial particles in a turbulent shear flow. [START_REF] Andrews | Coarse-grid simulation of gas-particle flows in vertical risers[END_REF] employed an ad hoc approach to formulate a stochastic drag coefficient in DEM-CFD coarse-grid simulations of vertical risers. The authors used DEM-CFD simulations with relatively refined grid sizes as a reference to determine the model's constants. In a more general approach, [START_REF] Tenneti | Stochastic Lagrangian model for hydrodynamic acceleration of inertial particles in gas-solid suspensions[END_REF] have recently proposed a stochastic formulation for the drag force based on PRS and the kinetic theory of granular flows.

The rest of the paper is organized as follows: In Section 2, we present the governing equations and numerical aspects of our numerical tool. In Section 3, we introduce the simulation parameters. Our stochastic formulation for the drag coefficient is detailed in Section 4. In Section 5, we present the main body of results in micro/meso simulations of low density ratio fluidization.

A single test case of high density ratio fluidization is subsequently studied.

Finally, the conclusions are summarized in Section 6.

Governing equations

Navier-Stokes and Newton-Euler equations are solved in a coupled fashion in both micro-and meso-scale models. The incompressible Newtonian fluid assumption is considered. The micro-scale model is based on the combined mass and momentum conservation equations extended in the solid phase [START_REF] Glowinski | A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow[END_REF]). The meso-scale model is based on locally averaged mass and momentum conservation equations proposed by [START_REF] Anderson | Fluid mechanical description of fluidized beds[END_REF].

In both models the solid phase is treated in a direct fashion by solving the Newton-Euler equations for each individual particle. All parameters in this paper are dimensionless except if distinguished by a "star" symbol. The following scales are employed to non-dimensionalize the set of equations:

L * c for length, V * c for velocity, L * c /V * c for time, ρ * f V * 2 c for pressure, V * 2 c /L * c for gravity acceleration, ρ * f V * 2 c /L * c
for distributed Lagrange multiplier (for the micro-scale model only) and ρ * f V * 2 c L * 2 c for forces.

Micro-scale model

The dimensionless and non-variational form of the combined continuity and momentum equations (written for simplicity for a system with a single particle) are as follows:

∂u ∂t + u • ∇u = -∇p + 1 Re ∇ 2 u -λ, over Ω (1) (ρ r -1)V p dU dt - P λ dx = j F cj + Fr(ρ r -1)V p g * |g * | , over P (2) (ρ r -1)(I p dω dt + ω × I p • ω) - P r × λ dx = j R j × F cj , over P (3)
u -(U + ω × r) = 0, over P (4)

∇ • u = 0, over Ω (5)
where u, U , ω stand for the fluid velocity vector, the particle translational velocity vector and the particle angular velocity vector, respectively. The particle domain and whole fluid/particle domain are denoted by P and Ω, respectively.

λ represents the distributed Lagrange multiplier vector, F c the contact force,

V p = M * p /(ρ * s L * d c )
Reynolds number Re = ρ * f V * c L * c µ * (6) Inverse Froude number Fr = g * L * c V * 2 c (7) Density ratio ρ r = ρ * s ρ * f (8) Stokes number Stk = ρ r Re p 18 (9)
Where µ * denotes the fluid viscosity. Please note that the Stokes number Stk is not an independent parameter and is given above for the sake of completeness only.

Meso-scale model

The Navier-stokes equations are locally averaged based on the set of equations proposed by [START_REF] Anderson | Fluid mechanical description of fluidized beds[END_REF]. Depending on the size of particles with respect to the fluid grid cells, the averaging process can be performed on either the fluid grid cells or an averaging kernel whose length scale is larger than the fluid grid cells. In both cases, the volume scale of the averaging operation is at least one order of magnitude larger than the particles diameter.

The dimensionless fluid conservation equations read:

∂ε ∂t + ∇ • εu = 0 (10) ∂εu ∂t + ∇ • (εuu) = -∇p -F f p + 1 Re ∇ • (2εD) (11)
where D and ε denote the strain rate tensor and the fluid volume fraction, respectively. The pressure gradient term contains the hydrodynamic pressure only and F f p represents the fluid/particle momentum transfer. Please note that in (11) we have neglected the additional unclosed Reynolds stress R u and viscous stress R µ terms that arise from volume filtering (see [START_REF] Capecelatro | An Euler-Lagrange strategy for simulating particle-laden flows[END_REF] for more details). Since the fluid field is unresolved at the particle scales, F f p needs to be closed by empirical correlations. Based on the equation proposed by [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF] and [START_REF] Gatignol | The faxén formulas for a rigid particle in an unsteady nonuniform stokes-flow[END_REF],

different contributions to the momentum transfer are:

F f p = F drag + F lif t + F am + F ba ( 12
)
Where the right hand side terms represent drag, lift, added mass and Basset forces, respectively [START_REF] Crowe | Multiphase flows with droplets and particles[END_REF]). The importance of each term depends on the flow configuration. For example, it can be shown that for high density ratios F am can be neglected [START_REF] Crowe | Multiphase flows with droplets and particles[END_REF]). In Esteghamatian et al. [2017a], we have shown based on PRS data that lift and added mass contributions are negligible as compared to the drag force in a liquid/solid configuration similar to the one investigated in this work. A description of the two closure laws used in this work is presented in Appx. A.

Newton-Euler equations and collision force

The Newton-Euler equations solved for each individual particle read:

ρ r V p dU dt = F b + F c + F f p (13) ρ r (I p dω dt + ω × I p ω) = T c + T f p ( 14
)
Where F b = (ρ r -1)FrV p g |g * | denotes the buoyancy force, F c inter-particle or particle-wall contact force and F f p the fluid/particle interaction force. Likewise, T c and T f p represent the contact-induced and the fluid-induced torque acting on the particle. A soft-sphere model is employed for the treatment of the inter-particle and particle-wall interactions. The model and its parallel implementation is introduced in detail in [START_REF] Wachs | Grains3D, a flexible DEM approach for particles of arbitrary convexshape -Part I: Numerical model and validations[END_REF], Rakotonirina and Wachs [2017], Rakotonirina et al. [2017]. In short, the soft-sphere model comprises a normal elastic restoring force, a normal dissipative force and a tangential friction force. The fluid/particle interaction force F f p and torque T f p are computed a posteriori using the Lagrange multipliers:

F f p = P λdx + V p dU dt (15)
T f p = P (r × λ)dx + I p dω dt + ω × I p ω (16)
As described in equation ( 12), the meso-scale model employs closure laws to estimate different contributions of fluid/particle interaction forces.

Numerical aspects

Temporal and spatial discretization of Navier-Stokes equations in micro-and meso-scale models are described in [START_REF] Wachs | Accuracy of finite volume/staggered grid distributed lagrange multiplier/fictitious domain simulations of particulate flows[END_REF], Esteghamatian et al. [2017a]. In general, for both micro-and meso-scale models the set of equations are discretized with a second-order Finite Volume/Staggered Grid scheme.

Both models employ a first order Marchuk-Yanenko operator-splitting algorithm for temporal discretization. Depending on the model, sub-problems are resolved in the following order: (II) As a predictor step, a purely granular problem is solved and an intermediate particle velocity is obtained.

(III) A fictitious domain problem is solved. By the end of this subproblem, rigid-body motion is imposed in the particle domain.

(IV) As a corrector step, a purely granular problem is solved and the final particle velocity and position are computed.

(b) Meso-scale model:

(I) A modified L2-projection scheme is employed to solve Navier-Stokes equations for a known particle reaction force and a known porosity field.

(II) A granular problem with a known hydrodynamic force is solved and the updated porosity field is computed.

(a) ∆x 2 ∆x 1 x 0,0 x 1,0 x 0,1 x 1,1 (b)
Figure 1. Bounding Cube averaging method in meso-scale model: (a) Lagrange→Euler and (b) Euler→Lagrange information transfer.

In the meso-scale model, the averaging kernel plays an important role in both the stability and the accuracy of the computed solution. In our meso-scale model we employ two different methods to compute locally averaged quantities such as porosity and slip velocity. The choice of the method depends on the ratio between the grid cell size ∆x and particles diameter d:

(a) Bounding Cube (BC) method: If ∆x ≥ 1.7d, we use the fluid grid cell for Euler Lagrange information transfer. A bounding cube method, as described in [START_REF] Bernard | Controlling the quality of two-way Euler/Lagrange numerical modeling of bubbling and spouted fluidized beds dynamics[END_REF] and schematically illustrated in Fig. 1 is used to compute the porosity field and distribute the particle reaction forces at the fluid grid cells. Additionally, a simple linear interpolation is used to transfer the information from the fluid grid cells to the particles centers.

(b) Gaussian Kernel (GK) method: If ∆x < 1.7d, a truncated Gaussian kernel is used to transfer information from the fluid grid cells to the particles centers, back and forth. The main motivation behind employing this method is to decouple the fluid discretization from the averaging kernel. Furthermore, it smooths out the presence of particles on the background grid in a monotone fashion. This method is totally equivalent to the one proposed in [START_REF] Capecelatro | An Euler-Lagrange strategy for simulating particle-laden flows[END_REF]. A generic mathematical description of an Euler-Lagrange information transfer with a spatial filtering is detailed in appendix B. In short, for the reconstruction of any Eulerian quantity φ at an -5 -4 -3 -2 -1 0 1 2 3 4 5

ÈË Ö Ö ÔÐ Ñ ÒØ× x W ∂W/∂x δ 0.5
Figure 3. A Gaussian filtering kernel and its derivative. δ 0.5 denotes the width of the kernel where it reaches 50% of its maximum value.

arbitrary Lagrangian position r i we have:

φ(r i ) = Ω φ(x)W(x -r i )dx Ω W(x -r i )dx (17)
Where W denotes a symmetric kernel function centered at the Lagrangian position r i . For the reverse operation, viz. projecting particles information on the Eulerian grid reads as follows:

φ(x j ) = Np i=1 φ(r i )W(r i -x j )V p,i Ω W(x -x j )dx (18)
While V p,i denotes the volume of particle i. A 3D Gaussian kernel centered at the origin reads as follows:

W(x) = exp(-x 2 /2σ 2 k ) (σ k √ 2π) 3 , (19)
σ k = δ 0.5 /2 √ 2 ln 2 (20)
As shown in Fig. 3, δ 0.5 denotes the width where the kernel reaches 50% of its maximum value. Although the shape of the kernel may impact the smoothing properties and the numerical performance of the projection, we rely on a Gaussian function and do not further analyze the influence of the kernel shape on projection properties. For the Euler→Lagrange information transfer, the method implicitly handles the truncation of the domain by a simple normalization (c.f. equation 17, for instance). Hence, the interpolated values near the boundaries are less accurate, yet conservative. For the reverse operation Lagrange→Euler, we employ the mirror method suggested by [START_REF] Capecelatro | An Euler-Lagrange strategy for simulating particle-laden flows[END_REF]. For the particles close to the wall, the truncated part of the projection kernel is redistributed over the interior domain in a conservative fashion as shown in Fig. 2(b). We consider an image particle in symmetry with the respective wall for all near-wall particles. The kernel for a near-wall particle W nw is hence defined as the superposition of the original kernel and the kernel corresponding to the respective image particle:

W nw (x -r i ) = W(x -r i ) + W(x -r img i ) (21)
Where r img i is the position vector of the image particle. This simple method not only guarantees mass-conservation of the Euler→Lagrange projection step, but also enforces a Neumann boundary condition for the particle variables (c.f. Additionally, a heterogeneous fluidization in a high density ratio system is considered. Due to the larger length scales of the system, a larger domain is chosen: 2000 particles in a bi-periodic domain. Limited by the computational cost of this configuration, we have only chosen one set of parameters. The purpose of this simulation is mostly to compare for a critical test case, the performance of meso-scale models with a classical drag law as compared to PRS in a high density ratio regime.

Assuming uniform axial inlet velocity and an a priori estimate of ε mf , the minimum fluidization velocity for a homogeneous pack of spherical particles is derived from [START_REF] Gidaspow | Multiphase flow and fluidization: continuum and kinetic theory descriptions[END_REF]:

(1 -ε mf )(ρ * f -ρ * s )g * = βu * mf ( 22
)
Where ε mf and u * mf denote the fluid volume fraction of the system and the inlet fluid velocity at the onset of fluidization, respectively. β is the famous Ergun friction coefficient for a dense pack of spherical particles [START_REF] Ergun | Fluid flow through packed columns[END_REF]).

In all cases, we made sure that the inlet velocity is much larger than the minimum fluidization velocity. F f p denotes the total hydrodynamic force, F drag and F drif t are the components in parallel and perpendicular directions with respect to the slip velocity vector.

In this section, we first shortly present the post-processing and analysis of drag force acting on particles based on the PRS results. Next, we focus on the time-behavior of the drag force exerted on particles. These two steps serve as a basis for the description of our stochastic drag force, which will be subsequently explained. Statistical mean and standard deviation are denoted as < . > and σ(.), respectively, and defined for a quantity φ as follows:

< φ > = 1 N data N data i=1 φ i (23) σ φ = 1 N data N data i=1 (φ i -< φ >) 2 (24)
where N data is the number of data φ i in the sample.

Drag coefficient mean values and fluctuations

Here, we average the PRS fluid field on the equivalent DEM-CFD grid cells with a simple box-hat filter g(x) = 1, as shown in Fig. 5(a). For an Eulerian PRS quantity φ, its average value denoted φ in a DEM-CFD grid cell of volume V DEM -CF D is thereby computed as follows:

φ = V DEM -CF D χ(x)g(x)φ(x)dx V DEM -CF D χ(x)g(x)dx (25)
where χ(x) is a binary indicator function that is 1 in the fluid and 0 in any particle. Fluid Eulerian averaged variables are then interpolated at the particles position. In order to compute the drag coefficient of a particle, we estimate local porosity and fluid velocity at the particle position using the 2-step procedure (averaging followed by interpolating) described above. Such a filtering procedure provides us with an estimation of local fluid variables from PRS results which is most similar to its counterpart in a DEM-CFD simulation.

Once the local fluid velocity around each particle is estimated, we can identify the direction of slip velocity and hence of drag force. As shown in Fig. 5(b), the drag force magnitude is estimated by projecting the PRS-based fluid particle force, F f p , on the slip velocity vector. The PRS drag coefficient for each particle is hence estimated as:

C d (Re p , ε) = 8F * drag πρ * f d * 2 ε 2 (u * -U * )|u * -U * | (26) Re p = ρ * f |u * -U * |εd * µ * (27)
The drag coefficient is considered to be a function of Re p and ε following the classical formulation of the drag law [START_REF] Ergun | Fluid flow through packed columns[END_REF]. Consequently, the drag coefficient experienced by each particle is instantaneously estimated. One might legitimately argue that the drag coefficient estimated by the instantaneous locally-averaged fluid field is different from the classical drag laws in two ways: a classical drag law (i) is derived in a stationary state e.g. flow past a fixed assembly of particles, and (ii) is based on far-field/integral properties e.g. pressure drop through the bed, inlet velocity, global fluid volume fraction.

About (i), our interpretation of drag is in coherence with the DEM-CFD basic assumption, i.e., the mean drag considers local stationarity [START_REF] Fan | Principles of gas-solid flows[END_REF]). In that sense, a meso-scale model assumes that the particles can be considered to be in a quasi-steady state at each instant of time where there is no acceleration of the relative velocity1 . Concerning (ii), the presence of meso- distribution with a rather large dispersion of the data. To shed more light on the origins of these variations we investigate their dependence on the local values of Re p and ε. Before further detailing the characteristics of these PDFs, it is noteworthy to mention that as a general trend PDFs in the liquid/solid regime tend to a quasi-normal distribution, while they are mildly skewed toward larger values in the gas/solid regimes. The skewness is even more evident when considering the non-normalized absolute drag force (c.f. Fig. 11). In general, the convective gradients and heterogeneous meso-scale structures are responsible for such asymmetries.

We analyze the statistical dependence of the drag coefficient PDF to the number of realizations N real (=N data in ( 23)-( 24)). As shown in Figs. [START_REF] Beetstra | Drag force of intermediate reynolds number flow past mono-and bidisperse arrays of spheres[END_REF] and Di Felice [1994] for (a) liquid/solid regimes and (b) gas/solid regimes, respectively) and drag coefficients C PRS d predicted by PRS are represented by lines and symbols, respectively.

we acknowledge that the graphical linearity of the trend in gas/solid regime is an artifact of the short Re p range. However, recalling that the preceding fits are to be employed in DEM-CFD simulations in which particles presumably experience an equivalent range of Re p as compared to PRS, such simplification is deemed to be acceptable. Indeed, a full description of the correlation be- (spatial) and over a particle trajectory (temporal). First, differences between distributions in time and space are quite limited, emphasizing the validity of the ergodicity assumption. Second, distribution forms are quite different in the gas/solid and liquid/solid regimes. The former is close to normal, while the latter is clearly skewed toward larger values. Such asymmetries in drag force PDFs highlight the significance of large-scale motions in the gas/solid regime.

In order to have a complete picture of the drag force fluctuations, we study its characteristic time. Although particles motion is indeed far from Brownian, we assume that it consists of some random modes representing the pseudothermal agitation of particles. This hypothesis can be further examined by studying the drag force auto-correlation function defined as:

R F F (τ ) = < F f p,i (t + τ )F f p,i (t) > < F f p,i (t)F f p,i (t) > , i = x, y, z (28)
F f p,i = F f p,i -< F f p,i >, i = x, y, z (29)
Fig. 12 shows the auto-correlation function for the drag force magnitude in liquid/solid and gas/solid regimes. As expected, the co-existence of random motion at early separation times and large-scale convective motion at larger separation times are present in both regimes. Nevertheless, the convective motion in the form of a harmonic wave is obviously dominant in the gas/solid regime. The integral time of the auto-correlation function is classically defined as:

T = τ∞ 0 R F F (τ )dτ (30)
Where τ ∞ denotes an infinitely large separation time. For a purely random motion, the autocorrelation function is expected to decay exponentially. In that hypothetical condition, the integral in equation 30 exponentially tends to the characteristic time of fluctuations. This is not the case in the presence of convective gradients. The integral in equation 30 as a function of the upperbound, τ ∞ , oscillates around a steady level. We have simply chosen this steady level as the fluctuations characteristic time. Hence, in our particular analysis we re-define the equation 30 as:

T =< τ∞ 0 R F F (τ )dτ > τ∞ (31)
Where the averaging is performed over the quasi-steady separation time slot, namely τ ∞ > 4 and τ ∞ > 40 in the liquid/solid and gas/solid regimes, respectively. Although quite simple, our approach eliminates the influence of large-scale harmonic motions on the estimation of the pseudo-thermal fluctuations time. The fluctuations integral time varies with the dimensionless parameters of the system. In the liquid/solid regime, Fig. 13 shows the dependence of this value on Re in . A linear fit is used to correlate the fluctuations integral time to Re in in the liquid/solid regime.

Fluctuating drag as a stochastic event

We have obtained a rather complete picture of the drag force fluctuations in the range of dimensionless parameters of our PRS. We have shown that despite some differences in the functional dependence of the mean drag on Re p and ε, the average value for the given range of dimensionless parameters is almost in accordance with PRS. We now focus on a description of the drag coefficient which represents sub-grid fluctuations by a stochastic event, namely C s d :

C s d = C d + C d ( 32
)
Where C d and C d represent the fluctuating and mean components, respectively. Recall that the term mean is defined with respect to sub-grid fluctuations and simply implies that C d is a mere function of locally averaged variables. However, both C d and C d are indeed time-dependent variables in a freely-moving particles system. We seek a stochastic signal with the following mathematical features: (i) its long-run mean should be zero, (ii) it should enjoy a smooth power spectrum without any preference toward a given frequency (iii) long-run variance and time-correlation should be easily adjustable. With that in mind and inspired by time-series analysis methods [START_REF] Box | Time series analysis: forecasting and control[END_REF]), we employ a simple first order Auto-Regressive (AR-1) signal:

C d (t + ∆t) = aC d (t) + b (1 -a 2 )w(t) (33) < w(t) > t = 0, σ(w(t)) = 1 (34)
Where w(t) represents a Gaussian white noise with a zero mean and standard deviation of 1, a measures the extent to which the signal retains its memory, b controls the intensity of the randomness and ∆t is the time-step. In fact, an AR-1 process can be considered as a discrete time equivalent of the continuous Ornstein-Uhlenbeck process. Its conditional expectancy and variance can hence be written in the following form:

E[C d (t + n∆t)|C d (t)] = a n∆t C d (t) (35) Var[C d (t + n∆t)|C d (t)] = b 2 (1 -a 2n∆t ) (36

Results

We intend to compare the performance of three variants of the meso-scale model: (i) a bounding cube method for interphase coupling and a classical deterministic drag law, (ii) a Gaussian kernel method for interphase coupling and a classical deterministic drag law, and (iii) a Gaussian kernel method for interphase coupling and the proposed stochastic drag law. Results produced by these three variants of meso-scale model are compared to that of PRS which 30) and ( 31)), corresponding to the respective curve. Both plots correspond to a DEM-CFD simulation in a liquid/solid regime.

serves as a reference. We start with the liquid/solid regime. Four cases with different inlet velocities are considered in the liquid/solid regime. Next, we turn our attention to the gas/solid regime. Numerical parameters are listed in Tab. 2. We introduce an abbreviation in referring to our DEM-CFD simulations: the two first letters denote the interphase coupling method (BC for the bounding cube and GK for the Gaussian kernel) and the following number specifies the grid resolution ∆x -1 . The choice of the grid size in micro-scale simulations have been the subject of our previous study (Esteghamatian et al. [2017b]). For the meso-scale model, we performed a grid-and kernel support size-refinement analysis detailed in Appx. C. predicted by PRS. In the two lowest Re in cases, BC-0.6 with a deterministic drag law predicts large variation along z-axis which is due to the presence of nonphysical particle structures. Using smaller grid size with Gaussian kernel method attenuates the presence of such structures. In our previous study with the same physical parameters in a slightly larger configuration and using the BC-0.6 method, such effects were almost absent (Esteghamatian et al. [2017a]).

The bed height is almost well-predicted by all variants of the meso-scale model.

In general, the bed height is an indicator of the accuracy of the mean drag coefficient. In line with our findings in Section 4.1, the mean drag coefficient predicted by PRS and the deterministic drag law proposed by [START_REF] Beetstra | Drag force of intermediate reynolds number flow past mono-and bidisperse arrays of spheres[END_REF] are in agreement (see Fig. 8). Finally, there is virtually no difference between the results predicted by GK-1 det. and GK-1 sto., implying that the stochastic formulation for the drag has no effect on the mean drag force value.

Fig. 16 shows the time evolution of square root of particles granular temperature. First, using the GK method improves the prediction of the level of particles granular temperature as compared to PRS results. In spite of this improvement, the particle axial fluctuations are still not fully captured with a deterministic drag. The DEM-CFD simulations with the stochastic drag and the GK coupling method, however, correctly predict the particles fluctuation level within the uncertainty range of PRS results.

In the transverse direction as shown in Fig. 17, the general picture is similar with an improved prediction with GK-1 and stochastic drag. Nonetheless, the granular temperature is still slightly underpredicted as compared to PRS. In an effort to identify the origins of this discrepancy, we once again refer to PRS.

In Esteghamatian et al. [2017a], PRS results revealed that the fluid/particle force exerted on a particle is not necessarily co-linear with respect to its slip velocity vector. As a result, the fluid/particle force has a perpendicular component with respect to the slip velocity. Although our proposed stochastic drag force already takes into account the drag force magnitude fluctuations, it neglects the fluctuating behavior of the drag force direction. We believe that the limited deviation of transverse particles granular temperature predicted by GK-1 and stochastic drag as compared to PRS is associated with the lack of a fluctuating force perpendicular to the slip velocity vector. Nevertheless, it should be noted that particles transverse motion is not exclusively driven by the fluid/particle force. In Esteghamatian et al. [2017b], we observed that collision is an efficient mechanism of momentum transfer from mean flow to transverse particle motion. Hence, the stochastic drag (which is parallel to the slip velocity and therefore is mostly oriented in the axial direction) is implicitly projected to the transverse directions via inter-particle colli- sion, leading to the improvement of particles transverse granular temperature.

Gas/solid

The type of fluidization in the gas/solid regime is pretty different from a hydrodynamic viewpoint as compared to the liquid/solid regime Esteghamatian et al. [2017b]. Based on PRS, we clearly observe the emergence, growth and burst of a bubble in a periodic fashion. predict the porosity profile within 5 % error with respect to PRS, excluding the values near the bed upper limit and the entry. Deviations with respect to PRS in the range of z ∈ [20 : 28] is associated with the slug burst event near the upper surface of the bed and is mildly improved by employing the GK method with stochastic drag. In the vicinity of the entry, all variants of the meso-scale model underestimate porosity, as is also the case in the liquid/solid regime (Fig. 15).

As shown in Fig. 20, while all DEM-CFD variants are in good agreement with PRS results, the particles granular temperature is rather insensitive to DEM-CFD modifications. Figs. 20(a) and 20(b) show that employing a GK method improves DEM-CFD predictions of the mean axial granular temperature by about 10 %. In the transverse direction, employing the GK method and a finer fluid grid cell slightly overestimates the transverse granular temperature:

5 % and 10% overestimation as compared to PRS, with a deterministic and stochastic drag, respectively.

Discussion and perspectives

In this study, we examined two directions of improvement of the meso-scale model. First, we employed a more sophisticated interphase coupling scheme to decouple the averaging length scale from the fluid grid cell size. This socalled Gaussian kernel method, introduced by Capecelatro and Desjardins

[2013], provides us with the flexibility of using a full range of grid sizes while guaranteeing a numerically stable solution. Second, we suggested an ad hoc remedy based on PRS results to capture the fluctuating nature of the drag force. We meticulously investigated the drag coefficient PDFs for given pairs of (Re p , ε) and quantified their properties with respect to the locally filtered dimensionless variables. While there is no fundamental basis on the randomness of these sub-grid fluctuations, our numerical experiments based on PRS revealed that the shape of these PDFs is close to Gaussian particularly in the homogeneous liquid/solid regime. Thus, we proposed as a tentative solution a stochastic description for the drag force in the meso-scale model, assuming that sub-grid convective gradients can be simply represented by a localized random process. The characteristic time of the proposed stochastic drag is also extracted from PRS results. We adopted a first-order auto-regressive signal to introduce a fluctuating component in the drag force in the direction of the slip velocity. The magnitude of this fluctuating component is a function of the local Re p . We intentionally avoided to tackle the famous issue of the inconsistency between mean drag laws, as described by [START_REF] Kriebitzsch | Fully resolved simulation of a gas-fluidized bed: a critical test of dem models[END_REF]:

"the appropriateness of these static correlations for fluidized systems is a bit of a hit and miss affair". Among recent efforts in the literature to clarify these ambiguities, [START_REF] Rubinstein | Lattice Boltzmann simulations of low-Reynolds-number flow past fluidized spheres: effect of Stokes number on drag force[END_REF] highlighted the impact of Stokes number on drag force, while [START_REF] Tang | Direct numerical simulations and experiments of a pseudo-2D gas-fluidized bed[END_REF] proposed a modified drag law taking into account particles mobility. In the present work, we turned our attention to a different question: assuming that we already have access to a mean drag correlation which is on average in accordance with PRS, can we capture the correct level of particles fluctuations with a meso-scale model? The present study has shown that the answer highly depends on the extent to which the already-resolved scales prevail.

We evaluated the contribution of the proposed improvements in two different regimes. We showed that in the homogeneously bubbling liquid/solid regime, both modifications have positively contributed to an enhanced DEM-CFD prediction of the particles granular temperature. By using a GK method and a stochastic drag, DEM-CFD simulations fully capture the axial particles granular temperature level as compared to PRS. In the transverse direction, we still observe minor discrepancies between DEM-CFD simulations and PRS.

Our impression is that these minor differences stem from the direction of the fluctuating component. Recalling that the stochastic drag model mimics fluctuations originating from sub-grid scale convective gradients, it might not be necessarily collinear with the mean slip velocity. In a more realistic approach, one may consider a stochastic drag component which is not only fluctuating in magnitude, but also in direction.

In the gas/solid regime, we observed a highly heterogeneous fluidization with the dominance of meso-scale structures. For a given drag law (on average in agreement with PRS), the main characteristic phenomena of the system were well captured by a standard DEM-CFD model. Hence, we did not observe any favorable contribution of the stochastic drag to the DEM-CFD simulations in this system. In the presence of large and dominant meso-scale structures as, e.g., a slugging mechanism, the dynamics of particles is already well resolved by DEM-CFD simulations with GK (Gaussian kernel) and ∆x -1 in the range A similar formalism is already adopted in the literature to improve the coarsegrid DEM-CFD simulations based on highly-resolved DEM-CFD simulations.

In [START_REF] Andrews | Coarse-grid simulation of gas-particle flows in vertical risers[END_REF], the authors have already shown that although the main characteristics of gas/particle flows in a vertical riser are captured by the mean drag and a coarse-grid DEM-CFD simulation, a quantitative estimation requires a stochastic sub-grid correction.

In general, we believe that the present stochastic description of the drag force is particularly promising owing to its conceptual features: (i) all parameters can be extracted from PRS without any artificial fitting, and (ii) it takes into account the influence of already-resolved scales. Nonetheless, our approach is only an ad hoc remedy and is far from being a general solution to the classical problem of particle/fluid momentum transfer. As an engineering approach, one can perform various PRS with different dimensionless parameters in order to find the functional dependence of the stochastic model parameters on relevant dimensionless variables. In doing so, one may move to a tri-periodic box configuration in order to (i) decouple the dimensionless numbers of the system such as Re p , ε and ρ r , and (ii) eliminate the inlet boundary condition effects. Another parameter which needs to be taken into account in such an analysis is the size of the filter with respect to the particle size. The present study has shown that the filter size is an important parameter in the estimation of sub-grid scale fluctuations. This has been also reported by Capecelatro Finally, another potential direction in improving drag laws (and in turn mesoscale models) is to better understand the physical origins of sub-grid fluctuations. We already know that an important source of drag modulation on a particle in a dense regime is the local fluctuations induced by neighboring particles. Thanks to the direct handling of particles motion in DEM-CFD models, detailed information about the local pore geometry is available. Hence, rather than merely computing the fluid volume fraction, more detailed information can be exploited by DEM-CFD models. For static particles, [START_REF] Akiki | Pairwise interaction extended point-particle model for a random array of monodisperse spheres[END_REF] has recently proposed a Faxén correction to quantify the fluid/particle force perturbation due to the presence of neighbors. The authors assumed that the pairwise interaction of particles can be linearly superposed even in fairly inertial regimes Re p = 170. A major advantage of such approach is that due to its deterministic nature, it is applicable to a wide range of configurations.

Appendices A Description of drag correlations

For the sake of consistency with the literature, all drag law formulations are given in dimensional variables and the Reynolds number is computed with the superficial slip velocity |u * -U * |ε = V * c as a characteristic velocity and the particle diameter d * = L * c as a characteristic length:

Re p = ρ * f |u * -U * |εd * µ * (A.1)
Based on experiments for a wide range of Reynolds numbers and porosity, Di Felice [1994] proposed the following formulation:

F * drag = 0.5C d ρ * f πd * 2 4 |u * -U * |(u * -U * )ε -χ (A.2) χ = 3.7 -0.65 exp(- (1.5 -log 10 Re p ) 2 2 ) (A.3) C d = 24 Re p (1 + 0.15Re 0.687 p ) (A.4)
where C d is the drag coefficient for a single unhindered particle for Re p less than 1000 [START_REF] Schiller | A drag coefficient correlation[END_REF]) and the pre-factor ε -χ accounts for the presence of neighboring particles. This type of drag law was first proposed by [START_REF] Wen | Mechanics of fluidization[END_REF], with a pre-factor ε -3.7 . However, the drag law was not applicable for ε < 0.8 . Thanks to the modifications of Di Felice [1994], the drag law is valid on a wide range of porosity.

The second drag law is based on the numerical simulations of Beetstra et al.

[2007]:

F * drag = 3πd * µ * (u * -U * )g(ε, Re p ) (A.5) g(ε, Re p ) = 10(1 -ε) ε 2 + ε 2 (1 + 1.5 √ 1 -ε) + 0.413Re p 24ε 2 ( ε -1 + 3ε(1 -ε) + 8.4Re -0.343 p 1 + 10 3(1-ε) Re -0.5-2(1-ε) p ) (A.6)

B Mathematical description of the Gaussian Kernel interphase coupling method

We first show how any Eulerian quantity and its derivatives can be reconstructed at a Lagrangian position. Starting with a Taylor expansion of an

Eulerian quantity φ about the particle position: In a 1-D system, B.6 directly yields to an expression for the derivative:

φ(x) = φ(r i ) + m (x -r i ) ∂φ(r i ) ∂x m + ... (
dφ(r i ) dx m = Ω [φ(x) -φ(r i )] dW(x -r i ) dx m dx Ω (x -r i ) dW(x -r i ) dx m dx , m = 1 (B.
A =               Ω (x 1 -r i,1 ) ∂W(x -r i ) ∂x 1 dx Ω (x 1 -r i,1 ) ∂W(x -r i ) ∂x 2 dx Ω (x 1 -r i,1 ) ∂W(x -r i ) ∂x 3 dx Ω (x 2 -r i,2 ) ∂W(x -r i ) ∂x 1 dx Ω (x 2 -r i,2 ) ∂W(x -r i ) ∂x 2 dx Ω (x 2 -r i,2 ) ∂W(x -r i ) ∂x 3 dx Ω (x 3 -r i,3 ) ∂W(x -r i ) ∂x 1 dx Ω (x 3 -r i,3 ) ∂W(x -r i ) ∂x 2 dx Ω (x 3 -r i,3 ) ∂W(x -r i ) ∂x 3 dx               (B.9) X =               ∂φ(r i ) ∂x 1 ∂φ(r i ) ∂x 2 ∂φ(r i ) ∂x 3               , B =               Ω [φ(x) -φ(r i )] ∂W(x -r i ) ∂x 1 dx Ω [φ(x) -φ(r i )] ∂W(x -r i ) ∂x 2 dx Ω [φ(x) -φ(r i )] ∂W(x -r i ) ∂x 3 dx               (B.10)
In practice, the matrix inversion is done only once and the solution is explicitly derived. Similarly, higher-order derivatives can be achieved by replacing the kernel with its high-order derivatives in equation B.2 and following the same procedure. In our formulations, however, we do not need such expressions.

So far, we have shown how to reconstruct a particle property from the background fluid field. For the reverse operation, i.e., projecting the particles information on the Eulerian field, the procedure is essentially similar. We start with a Taylor expansion of a particle quantity φ about an arbitrary Eulerian position φ(x j ): .11) on the predicted solution is subsequently investigated. For the sake of brevity, we limit our analysis only to one set of parameters: Re in = 6, ρ r = 10.

φ(x) = φ(x j ) + m (x -x j ) ∂φ(x i ) ∂x m + ... (B
We study the sensitivity of particle fluctuations to grid resolution in DEM-CFD simulations in The definition of abbreviations is given in Tab. 2. 40

  the particle volume, M * p the particle mass, d the dimension of the system, I p = I * p /ρ * s L * d+2 c the particle inertia tensor, r the position vector and R the vector connecting the gravity center and the contact point. The following dimensionless numbers are also introduced:

( a )

 a Micro-scale model: (I) A classical L2-projection scheme is employed to solve the Navier-Stokes equations. An intermediate fluid velocity and pressure field is obtained.

Figure 2 .

 2 Figure 2. Gaussian Kernel averaging method: Illustration of (a) distribution of a particle property over the neighboring cells and (b) performance of the projection kernel close to a boundary. In (b), the real and image particles are in solid green, and red hashed pattern, respectively. The solid lines represent the projection kernel of the real and image particles, while the red dashed line is the superposition of both kernels.

Fig. 2

 2 Fig. 2(b)). While the true choice of the boundary condition for particle variables is the subject of many studies particularly in TFM approaches Benyahia et al. [2005]Li et al. [2010]Jenkins and Louge [1997], a Neumann boundary condition avoids the over-prediction of fluid volume fraction near walls as claimed by Capecelatro and Desjardins [2013]. According to our PRS numerical experiments, a slight increase of fluid volume fraction near walls is physical and related to the local geometric constraint. However, a non-conservative treatment of the projection operation over-predicts this increase and results in an unphysical behavior of the flow. Generally, a Neumann boundary condition along with a mass-conserving projection operation is deemed to properly pre-

Figure 4 .

 4 Figure 4. Flow domain and layout of particles at t = 0 in (a) liquid/solid configurations and (b) gas/solid configurations.

  Figure 5. (a) Reconstruction of the PRS fluid field (black cells) at the DEM-CFD grid cell centers (white cells/points). Volume-averaged values are then linearly interpolated at the particle center. (b) Decomposition of the hydrodynamic force acting on a particle. The plane P is perpendicular to the slip velocity vector (U slip = u -U ).F f p denotes the total hydrodynamic force, F drag and F drif t are the components in parallel and perpendicular directions with respect to the slip velocity vector.

Figure 6 .

 6 Figure 6. Probability Density Function (PDF) of drag coefficient, C d , for a given pair of Re p and ε computed from the PRS results and different drag laws (Beetstra et al. [2007], Huilin and Gidaspow [2003], Di Felice [1994]) in (a) liquid/solid regimes and (b) gas/solid regimes. PRS results are shown as a PDF, and the deterministic drag correlations are shown by a Dirac function. The uncertainty of the values computed by the drag correlations are shown by solid lines and corresponds to the relative uncertainty tolerance of 1% in Re p and ε.

  7(a) and 7(c), while an increase of N real smoothes the PDF curves, its general shape is only subject to mild variations from 200 and 800 realizations in the liquid/solid and gas/solid regimes, respectively. Variations of the mean and standard deviation values with the number of realizations are shown in Figs. 7(b) and 7(d).

Figure 7 .

 7 Figure 7. Variation of the drag coefficient PDF for a given pair of Re p and ε with respect to the number of realizations in (a) liquid/solid regimes and (c) gas/solid regimes. (b) and (d) show the dependence of the mean and standard deviation values to the number of realizations, in liquid/solid regimes and gas/solid regimes respectively.

  tween σ(C PRS d ) and Re p requires a parametric analysis by performing several PRS for a wide range of Re p , ε and density ratio.

Figure 9 .

 9 Figure 9. Standard deviation of the drag coefficient in liquid/solid regimes as a function of Re p and for different ε: (a) absolute value and (b) value normalized by mean drag coefficient.

4. 2

 2 Time behavior of fluctuations So far we have analyzed the drag force fluctuations based on snapshots of the PRS velocity field. In our approach PDFs are extracted from the ensemble of particles at different times: a combination of spatial and temporal fluctuations. These fluctuations which stem from sub-grid convective gradients, can be modeled by a stochastic process. Such treatment by definition assumes that

Figure 10 .Figure 11 .

 1011 Figure 10. Standard deviation of the drag coefficient in the gas/solid regime as a function of Re p and for different ε: (a) absolute value and (b) value normalized by mean drag coefficient.

Figure 12 .Figure 13 .

 1213 Figure 12. Auto-correlation function of the drag force magnitude as a function of the dimensionless separation time in (a) liquid/solid regimes and (b) gas/solid regimes.

)

  Fig. 14(b) by the integral time corresponding to the respective curve in order to compare the shape of auto-correlation functions (c.f. (30) and (31) for definition of T ). C det d corresponds to a DEM-CFD simulation with a classical deterministic drag force without any fluctuating component, viz. C det d = C d . Likewise, C sto d and C d correspond to a stochastic drag coefficient and its fluctuating component in a DEM-CFD simulation incorporating a stochastic drag

  Figure 14. (a) An example of C d signal as a function of time and (b) autocorrelation function of a stochastic drag coefficient C sto d and its fluctuating component, C d (c.f. (33)) compared to a classical drag coefficient without stochastic component, C det d . The x-axis is normalized by the integral time, T (defined in (30) and (31)), corresponding to the respective curve. Both plots correspond to a DEM-CFD simulation in a liquid/solid regime.

  Fig. 15 compares the axial fluid volume fraction profile predicted by different models. Except for the very beginning and end of the bed, a smooth profile is

Figure 15 .

 15 Figure 15. Fluid volume fraction axial profile in the liquid/solid regime with different inlet velocities. The definition of abbreviations is given in Tab. 2.

Figure 16 .Figure 17 .Figure 18 .

 161718 Figure 16. Evolution of axial granular temperature as a function of time in the liquid/solid regime with different inlet velocities. The definition of abbreviations is given in Tab. 2. Graph legends are the same in all figures and shown in figure (a) only.

Figure 19 Figure 20 .

 1920 Figure 19. (a) Bed height expansion as a function of time and (b) fluid volume fraction axial profile, in the gas/solid regime. The definition of abbreviations is given in Tab. 2.

[ 1 :

 1 3], and no stochastic drag, and is almost insensitive to local pseudo-random fluctuations. Please note that the grid resolution of our DEM-CFD simulations is by far more refined than what is used in coarse-grid simulations for industrial-scale applications. In DEM-CFD simulations of risers for instance, the fluid grid size is typically larger than the particles diameter by a factor of ten. In that situation, DEM-CFD simulations with grid resolutions equivalent to the ones we employed in the present study are considered as highly-resolved.

  et al. [2015] in fully developed cluster-induced turbulence. Furthermore, we have developed and tested our stochastic drag model in a bi-periodic fluidized bed configuration only, i.e., a flow configuration without wall effects. Beyond exploring a wider range of dimensionless parameters and thereby examining other flow configurations as, e.g., dilute turbulent flows or clustering in moderately dilute flows, investigating the influence of wall boundary conditions is certainly another milestone to achieve a full description of the stochastic drag force.

  B.1)We multiply both sides by a symmetric kernel function centered at the particle's position, W(xr i ), and integrate over the flow domain Ω :Ω φ(x)W(x-r i )dx = φ(r i ) Ω W(x-r i )dx+ m ∂φ(r i ) ∂x m Ω (xr i )W(xr i )dx +... (B.2)Neglecting all the derivative terms, we can have an estimation of φ(r i ):φ(r i ) = Ω φ(x)W(xr i )dx Ω W(xr i )dx (B.3)Due to the symmetry of the kernel, integral of the terms with odd-order derivatives are zero far from the boundaries. Hence the error produced by the truncation of the derivative terms is of order |xr i | 2 for the interior particles, and of order |xr i | for particles near the boundaries. Now we estimate the first derivative of φ(x i ). We replace W(xr i ) with the derivative of the kernel∂W(x-r i ) ∂xm in equation B.2: i ) ∂x m Ω (xr i ) ∂W(xr i ) ∂x m dx + ... (B.5)By neglecting second-and higher-order derivatives and rearranging equation B.5 we have:Ω [φ(x)φ(r i )] ∂W(xr i ) ∂x m dx = m ∂φ(r i ) ∂x m Ω (xr i ) ∂W(xr i ) ∂x m dx (B.6)

  7)In order to avoid singular values in the denominator, the dW(x-r i ) dxm needs to be an anti-symmetric function. Similar to 17, the truncation error is of order |xr i | 2 for the interior particles, and of order |xr i | for particles near the boundaries. In a 3-D system, B.6 represents a set of linear equations for different components of the derivative:

  Figure B.1. Grid refinement analysis for DEM-CFD simulations. Evolution of (a) axial and (b) transverse granular temperature as a function of time in the liquid/solid regime.

Table 1 Physical

 1

parameters of the system Case Re p,in Fr Stk ρ r u * in,z /u * mf,z

Table 2

 2 Numerical parameters of DEM-CFD and PR simulations.

In fact, unsteady contributions of the relative velocity to the fluid/particle interaction force are lumped into added mass effects and Basset force which are virtually insignificant for large particle-fluid density ratios/non-oscillating flows[START_REF] Fan | Principles of gas-solid flows[END_REF]).

We multiply both sides by a symmetric kernel function centered at the position of the selected Eulerian point, W(xx j ), and integrate over Ω. V total p = N p i=1 V i . By neglecting all the derivative terms we have:

Given that Ω = Ω f ∪ V total p where V total p = N p i=1 V i , B.12 expands as:

Given that any particle variable is constant over the respective particle domain and is zero over the fluid domain, B.13 can be transformed to:

Although not required in our formulations, all the derivatives of a particle property at an Eulerian position can also be achieved in a similar fashion.

C Grid and kernel size refinement analysis in DEM-CFD simulations

We choose a range of 0.5 ≤ ∆x -1 ≤ 3 to investigate the impact of grid resolution on computed solutions. We denote δ sup = 2δ 0.5 as the kernel support size. In our grid-refinement analysis, we set δ sup = 2, while the influence of δ sup resolution. This was predictable since DEM-CFD simulations are intrinsically controlled by the averaging operation. When the BC method is adopted, the grid refinement implicitly influences the local averaging length scale and in turn the level of fluctuations. With the GK method, however, the local averaging is decoupled from the grid size thanks to a smooth filtering kernel and its length scale is hence adjusted by the kernel support.

Finally, we study the impact of the kernel support size δ sup on the predicted solution. We choose the case with 2 grid points per diameter, GK-2, and perform different simulations for a range of kernel support sizes. A kernel support size of less than two diameters is not acceptable since it violates the basic principle of locally averaged Navier-Stokes equations on which our model is based: the local averaging volume should be about an order of magnitude larger than the particle diameter. The upper bound is also limited by the transverse domain size which is 8 diameters. We hence choose 2 ≤ δ sup ≤ 4 as a reasonable range for the kernel support size analysis.


InterMedia Eiliv Hauge

J Enkenberg

M-B Kentz

Björn Skaar

Yngve Refseth

Jan Dolonen

Turi Digernes

MA students)at InterMedia Silje Andresen

Andreas Lund

Alf Sjuls Hansen

Jon Magne Vestöl

Heid Leganger-Krogstad

Torgeir Christiansen

Shane Colvin

Digital case methodology: A study of student teachers' co-construction of professional knowledge 1

Introduction

Case pedagogy has a growing position in inquiry oriented teacher education programmes as means of bridging gaps between every-day and professional conceptions of teaching and learning and enhancing contextual and collaborative learning. Although research in the field still might be in its earlier stages, studies seem to reveal that case pedagogy has the potential to foster reflective and critical thinking, challenge technical-skill orientated teaching and stimulate situated learning [START_REF] Zeichner | The teaching portfolio in US teacher education programs: What we know and what we need to know[END_REF][START_REF] Lunderberg | Who learns what from cases and how? The research base for teaching and learning with cases[END_REF][START_REF] Morine-Dershimer | What's in a case -and what comes out?[END_REF][START_REF] Lunderberg | Looking twice means seeing more: Developing pedagogical knowledge through case analysis[END_REF][START_REF] Harrington | Written case analysis and critical reflection[END_REF][START_REF] Barnett | Fostering critical analysis and reflection through mathematics case discussion[END_REF]. While most of the studies are focusing on salient dimensions of professional learning, there is a need to deepen the understanding of collective and collaborative knowledge building when implementing case pedagogy in such a context.

In the field of computer-supported collaborative learning (CSCL) (cf. Koschman, 1996;Wasson, Ludvigsen & Hoppe, 2003) there is a growing body of research exploiting the possibilities of collaborative learning by computer technology. However, providing students with communication technology does not automatically result in collaborative learning. Like any other educational approach, CSCL has to provide a sound pedagogical context to support students' learning (Strijbos, Krischner & Martens, 2004).

In the present paper we will look at how a CSCL environment may enforce and improve collective thinking and professional reflection by the use of a cased based learning model in pre-service teacher education. The study is based on computer-supported learning resources developed by InterMedia, Unversity of Oslo, in the Digital Learning Case Project (DLC) 2 . These learning resources are based on video productions from classroom settings integrated in digital environments and framed by a set of complementary resources supporting individual and collective activities. One of these resource units, the classroom leadership case, are presented and analysed in light of the student teachers' perceptions and first hand experiences in their study. In particular we are looking at:

• how do student teachers perceive the digital case learning resource as a means of understanding classroom leadership?

• to what extent does the digital case learning resource support collaborative learning and collective reflection?

Case learning ideas

The underlying philosophy of case based learning and case methodology in the DLC project is based on ideas implemented in a teacher education reform programme at the University Oslo during the years 2000 -2004. Ideas and principles of case based learning as described by [START_REF] Shulman | Teacher-written cases with commentaries: A Teacher-researcher collaboration[END_REF], [START_REF] Wasserman | Introduction to case method teaching. A guide to the galaxy[END_REF] and [START_REF] Colbert | The case for Education: Contemporary Approaches for Using Case Methods[END_REF], approaches of writing to learn [START_REF] Bereiter | The Psychology of Written Composition[END_REF][START_REF] Flower | Problem Solving Strategies for Writing[END_REF][START_REF] Dysthe | Skrive for å laere, Skriving i høyere utdanning[END_REF] and strategies of problem based learning (Koschman, Kelson, Feltovich & Barrows, 1996) have been integrated in the programme. The students are working with different types and genres of cases involving problem-based learning, individual and collective writing and discussions face to face and online. The case rationale is founded on descriptions, observations and analyses of everyday problems and dilemmas of teaching and learning in school. The cases combine use of personal beliefs and values, theoretical and practical knowledge about teaching and learning. The cases are corner stones for seminar work, group discussions and portfolio productions.

Follow-up research of the reform project, based on activity theoretical analyses (Engeström, 1987(Engeström, , 1999)), has revealed that the cases and their intertwined learning strategies are functioning as institutional artefacts integrating knowledge from different study domains in the program [START_REF] Ludvigsen | Innovasjon i laererutdanningen: Hvordan skapes endring[END_REF][START_REF] Hauge | Casearbeid og mappevurdering som integreringsverktøy i laererutdanningen[END_REF][START_REF] Hauge | Portfolio assessment and student teachers' professional development. A study of learning outcomes in co-located and digital learning environments[END_REF][START_REF] Hauge | Learning portfolio and ICT as cultural artefacts in teacher education[END_REF]. The research studies point to findings of case based learning as supporting means for student teachers and teacher educators in bridging different knowledge domains.

The DLC case learning model

The case learning model described above has been the foundation for the Digital Learning Case Project (DLC) launched in 2003 as a team work between researchers, video producers, designers and teacher educators at the University of Oslo (InterMedia & the Department of Teacher Education and School Development). The DLC project attempts to visualise everyday phenomena in teaching and learning in school and to establish a more complete and integrated digital learning environment for collaborative purposes. While the first case model in the teacher education programme relied on textual descriptions of phenomena in school, to some extent supplied by video films, followed up by writing tasks and discussions in the virtual learning environment (LMS)3 , this second model was using video films as the prime case input in combination with an indvidual and collaborative task structure organised in a common virtual environment. The video films were based on script narratives of students and classroom practices in different school subjects. Seven digital case learning units have so far been developed.

According to Strijbos, [START_REF] Kirschner | CSCL in higher education? A framework for designing multiple collaborative environments[END_REF] the creation of a CSCL design requires a careful orchestration of various components likely to facilitate the attainment of the learning objectives. In creating the DLC case learning design several aspects had to be considered simultaneously in their interrelations, such as learning objectives for each of the cases, the kind of individual and collaborative learning activities best suited to attain these objectives, the types of additional learning resources supporting discussions and reflections (e.g. internet resources, study literature references, teacher supervision) and technical environment. Intentionally the whole technical and educational architecture was created to facilitate individual and collective learning and professional reflection among the student teachers.

Based on experiences with case model one the DLC design was also created as a means for enhancing new kinds of learning scaffolds. We wanted to develop new learning resources with sharp focusing points on phenomena and problems of classroom teaching, to highlight significant concepts of pedagogical content knowledge and organise the interaction patterns more clearly. This is in line with what [START_REF] Pea | The social and technological dimensions of scaffolding and related theoretical concepts for learning, education and human activity[END_REF] recommends as a clarification of social and technological dimensions of scaffolding. The notion of scaffolding in the actual CSCL environment is extended to include physical/technological as well as cognitive artefacts for learning. This extended meaning is significant for the understanding of computer supported learning as discussed by [START_REF] Davies | Exploration of scaffolding in complex classroom systems[END_REF] and [START_REF] Pea | The social and technological dimensions of scaffolding and related theoretical concepts for learning, education and human activity[END_REF]. As a consequence the DLC -project has been created to:

• narrow and intensify situated experiences for the student teachers,

• highlight certain moments of actions in classroom teaching and learning,

• let the students work on salient and constitutive elements of teacher professionalism, • and reinforce joint problem solving and collaboration between the student teachers.

Methodological approach

One of the DLC units: Classroom management and leadership, was tried out on a large scale for all students in a part time teacher education programme in the autumn 2004. This implies that the students worked through the application distributed locally where they were living. They were introduced for the application in a common face-to-face meeting and they were required to deliver a final group assignment at the end as a part the case work. This assignment was delivered electronically in January 2005 for responses from the teacher.

The case unit was distributed through internet with restricted access by passwords. The students got access to the application through computers with certain minimum standards and by use of the Quick time software. A small group of students without necessary broadband facilities got access to the case by a special CD-version. 158 students participated in the try-out situation, whereas 129 answered an electronic questionnaire in a virtual learning environment (It's learning). The answer rate is 75%, which is quite acceptable for the purpose of the study.

The questionnaire was constructed to assess the scaffolding qualities of the application regarding the student teachers' understanding of classroom leadership. Fixed and open answers are used in the form.

The following dimensions have been analysed:

-the authenticity of the video film -technical structure of the application -concept structure -student collaboration and collegial supervision -personal perceptions of classroom leadership skills The students' answers have been analysed with regard to internal correlation and variance structures (by factor and regression analyses).

In addition to the questionnaire we have made a rough analysis of 26 student group assignments (two students in each group) forwarded to two seminar teachers for supervision and feedback. These assignments have been reviewed with regard to: 1) what extent they reflect substantial conceptual dimensions in the DLC application, 2) use of scaffolding functions in the application, 3) use of formal and practical knowledge and meta-knowledge (knowledge of practice).

In distinguishing between formal and practical knowledge we are using a conceptual framework according to Cochran-Smith & Lytle (1999) which they call images of teacher learning. This framework has been adopted in an earlier study of student teachers' learning portfolios in the reform programme [START_REF] Hauge | Portfolio assessment and student teachers' professional development. A study of learning outcomes in co-located and digital learning environments[END_REF]. The first of these images refers to what they call 'knowledge-for-practice', a term that often is applied to as formal knowledge and theory for teachers to use in order to improve practice. This includes the ways teachers organise their lessons and units of study, the activities and materials teachers use for various groups of students, the sequence of content matter teachers present, the methods they use etc. These approaches to teaching are mostly individualistic oriented. The second image of teacher learning reflects 'knowledge-in-practice'. An alternative to this concept may be practical knowledge, i.e. knowledge that is embedded in practice and in teachers' reflections on practice. It is assumed that teachers learn when they have opportunities to examine and reflect on the knowledge that is implicit in good practice. Facilitated teacher groups, teacher communities and other kinds of collaborative arrangements that support reflection in and on practice are important in this context. The knowledge-in-practice conception is very similar to Schön's (1987) concept of knowingin-action. The third image of teacher knowledge named by Cochran-Smith & Lytle is 'knowledge-of-practice'. This is a knowledge form understood as a pedagogical actconstructed in the context of use, intimately connected to the knower and the situation, and also inevitable a process of theorising. This is inquiry oriented knowledge, where teachers connect their work to a larger context and take a critical perspective on theory and research of others [START_REF] Erlbaum | Relationships of knowledge and practice: Teacher learning in communities[END_REF].

The DLC unit: Classroom leadership

The digital case learning unit Classroom Leadership is build around a rather chaotic classroom episode with a group of 15 years old students in a lower secondary school. The opening scene is a video film of about 2 ½ min. It shows how the students rush into the classroom busy talking with friends without noticing the teacher and her management. One of the students brings with her a pet rat which attracts a group of students that reinforce disorder in the classroom. Other students are opening their lap-tops without noticing the teacher.

The case unit is designed by seven steps where the student teachers start with observing a video film, then they have to reflect individually on the film episode, write an individual text, analyse the phenomenon together with a fellow student, join into a peer supervision and produce a joint group text about the case problem according to some key theoretical concepts.

Step one -introduction

This first step focuses on task clarification, aims, procedures and products. Before entering this application site the students have to be introduced for the tasks, procedures and expectations by the teacher educator in a face-to-face seminar. The teacher will not be interacting with the students while working with the application. This case work is the foundation of a group portfolio work that has to be delivered to the final exam.

Step two -watching the video film

Watching the dramatised classroom episode is the first step towards discussion and elaboration of the theme, cf. figure one. Step three -individual reflection and writing 6 In this phase the students have to write a short individual note based on some guiding questions. The note has to be stored for later use and the application directs the user to this work. The questions are open-ended and descriptive helping the students to write down their first thoughts and reflections. The questions points to problems of classroom management. The application is organised in a way that makes it possible for the students to rewind and play the film over again if they need to.

Fig. 2 The writing of an individual note of reflection

Step four -collaboration and sharing of ideas

In the fourth case phase the students have to organise themselves into a fellow group, making appointments of peer supervision and elaborate the case further by use of their personal experiences. This phase leads the students into two new sites describing working procedures for each of the students. The students have to make appointments about when, how and where to share ideas and advices. The application is constructed to store texts and send e-mails.

Step five -case elaboration

Phase five invites one of the students to identify him/herself with the classroom teacher in the video film and personalise his/her case writing and problem solving. Three constraints are now given to this process: 1) student A is asked to teach the class by the headteacher in the middle of the first semester. He/she has two years of teaching experience, 2) student A has to ask a colleague of him/her (student B) who he/her respect very much, for an advice and he/she has to write a note about the problem to him/her, 3) student A have to make it concrete what he/she want to discuss with student B and receive feedback on. Student A may add new elements to the classroom case already discussed so far.

Step six -giving advice

In step six the student adviser (B) has received an e-mail from the fellow student (A) and given him/herself the role as an experienced teacher. He/she has to answer the questions based on basic guidelines. A sub-figure on the site leads to links with guidelines and literature in the field of peer supervision. This literature is a reminder of earlier course study work in the pedagogy course.

Fig. 3 Questions used for advising the fellow student

Step seven -reflections on advices

After fulfilling their case elaborations and delivering advices the two students have to write down some individual reflections about their collaborative experiences. The students have to write these notes before moving into the last collective step of cowriting.

Step eight -the writing of a joint text

In the final phase the students have discuss class management and leadership in relation to key concepts in the course study literature. The students are given some support for this discussion by use of a topic map focusing on important concepts in the field. They have to decide how to write a collective text using the digital storing functions (LMS) in the course program. The final process of the work towards a portfolio assignment will be continued in the face-to-face seminar. Fig. 4 The final extension process of a joint portfolio document

Results and discussions

Survey results

Table 1 gives an overview of the student teachers' opinions and assessments of the digital case learning unit. Answers are given in percent and mean values.

Table 1 Student teachers' assessment of DLC features

Very little

Some extent

Great extent Table 1 describes a differentiated picture of the student teachers' experiences and opinions of the digital case application. In general the answers seem to support an impression of the application as supportive for discussions and elaborations of classroom leadership problems. This is the case for the majority of the students when assessing different aspects of the application. However, it is quite clear that some of the sub-tools or functions need to be further improved as scaffolding structures. This regards at least the concept map model and the peer supervision model. The progression of the content matter in the case as a whole also needs to be reconsidered and clarified according to responses from one fourth of the students. As a general impression the digital video case seems to be accepted by the students as a stimulating starting point and reflection basis for the work with the material.

The differences in the answer patterns in table 1 may be further clarified by looking at correlations and variances between items and individuals. Factor analysis of the answers for all items reveals one significant common factor explaining 26% of the total variance. This means that there is a common answer pattern across many of the items, i.e. students who assess the digital resource positive as a whole also assess the many sub-functions to be positive. There is reason to believe that the different components of the application support each other when students move through it.

When looking at how the students assess their own skills as classroom leaders, table 1 shows that only a small portion of the students are reporting these to be weak. The majority report that their qualifications are good or even very good. However, all of them confess that they need to learn more. Linear regression analyses based on this self-reporting reveal that students who are open for deepen their understanding of classroom leadership also are inclined to assess the digital learning resource more positive than other students. (Significant Beta values varies between 0.17 -0.26 on different aspects, explaining 19 -24% of the total variance.)

In summarising these findings we may say that the student teachers' opinions about the DLC application to some extent are depending on their classroom leadership orientation.

Students who look at themselves as qualified leaders seem to appreciate the application more than others. The analyses give no support to the belief that the collaborative tasks in the application can be looked upon as decisive for the students' understanding of classroom leadership.

In the open answers to the questionnaire the students have given a lot of good advices for further improvements of the application. In general the majority of the students are positive to what they have been trough, but they seem to agree on that the application need be revised with regard to some instructional parts, the structure of collaborative work, clarification of the concept mapping procedures and the users access to additional learning resources. Also technical problems are mentioned.

Student assignments

When writing the case assignments the students had to follow a common core of guidelines related to the use of formal knowledge and conceptual base in the DLC application, and they had to formulate an inquiry based question for their problem solving.

The 26 student group assignments were sorted in three main categories according to how they were constructed by the use of three different knowledge forms: formal knowledge, DLC content knowledge and practical knowledge. Most of the assignments are placed in a main group A: Heavy use of formal knowledge. A main group B comprised assignments with a mixed use of the different knowledge forms. Main group C is a group of assignments utilising practical knowledge quite extensively as the main base for writing. The amount of assignments in these three main categories are distributed as in a 5 : 3 : 2 proportion. One representative assignment in each of the groups is selected to illustrate the qualities.

Group A -assignment

The students in this group start their assignment with a short description of the case problem without any reference to DLC interactions or explanation of how they had developed their understanding of the case. Their research question is based on two premises: The classroom teacher has a weak competence in classroom management and his/her students are badly motivated for learning. The paragraphs that follow in the assignment are constructed by use of formal theory related to reference books in the education course. In many ways the rest of the assignment text is very much a paraphase of theory in their study literature combined with some alignments to the actual case problem. The student teachers' way of handling the leadership problem is quite normative and it is heavily based on advices given in classroom management literature. Meta-perspectives are missing and there are very few references to what the students have done in the DLC case.

Group B -assignment

The chosen assignment in this group has a good structure, a clear research question and combines different knowledge forms: formal, personal, practical and DLC related knowledge. The research question illuminates how two student teachers reflect metacognitively on the case problem by use of relevant theory. They are asking how a teacher can act constructively for enhancing conditions for learning and motivation, and improving student -teacher relationships. The student teachers discuss the case problem in light of one main theme -classroom leadership, and they use five categories chosen from a reference book outside the core study literature in their analyses. The research question and case problem is elaborated in a convincing way. The DLC support structure is not very visible in their writing, but the students seem to have internalised some key knowledge exposed in application.

Group C -assignment

This assignment opens up with two research questions focusing on how a teacher may create a good learning environment for his/her students, and how a teacher may establish order and confidence in his/her teaching. Then the student teachers move to the study literature to legitimate their questions. After that they move back again to the DLC case video film. They make their own interpretation of the teacher figure in the video and question what really went wrong for this particular teacher. They discuss this problem by use of their own experiences in teaching and by formal knowledge. Their personal practical knowledge seems to be their strongest resource for problem solving. The support components in the DLC application are not very visible except for video film itself. The focus point of discussions is fading away as their writing develops and the conclusions are not very clear with regard to problem solving.

Looked upon as a whole the three group assignments differ in writing style, structure and content matter, and they reflect different approaches to the understanding of professional knowledge according to [START_REF] Erlbaum | Relationships of knowledge and practice: Teacher learning in communities[END_REF]. The assignment of Group A follows in many ways a traditional academic style of writing by presenting theory and advices given from authorities in education, while the assignment of Group C is the opposite -relying very much on personal practical knowledge. The Group B assignment expresses the most autonomous approach to the task by integrating different forms of knowledge along a meta-cognitive line of discussion. The DLC content knowledge is not utilised very explicit in any of the assignments, but seems to be best internalised in Group B.

Discussion

The present follow-up study of the DLC application reflects a mixed picture of the tool qualities and the student teachers' appraisal of the learning opportunities. The students' assessments confirm a need for further revisions of conceptual as well as technical dimensions of the application. Nevertheless, the students have found the DLC application interesting and motivating for the study of classroom management and leadership considered in general.

The students have been exposed to a product still under development. The first version of it was only composed of video films that could be used as an added resource in the education course model. The second version is an integrated environment composed of various learning resources and a framework for student collaboration. This integrated educational and technological environment is not yet fully developed as a free 'plug-in solution' to other technological tools or educational environments. It has to be used as part of a planned educational course structure. In this particular case the course teacher decided to use the application as a starting point for a case writing integrated in a learning portfolio system. The study reveals that this interconnection or alignment has to be developed further to better enhance the students' learning.

When looking at the DLC application in particular, the study has revealed that the application has been supportive for students' collaboration and learning about classroom management and leadership. However, students who tell that they are confident as classroom leaders are also more inclined to appreciate the DLC qualities for further learning about leadership. In other words different groups of students with regard to leadership confidence seem to utilise the technical and social support structure differently.

The analyses of the final student assignments delivered for further feedback from the teacher clearly demonstrate different task expectations and traditions of academic writing among the students. Three categories of assignments have been found corresponding to different knowledge orientations to teaching and learning among the students. About half of the group of students has written their assignments in a traditional way by using formal knowledge. The DLC application and its conceptual structures are not very visible in their reflections. About two third of the group have been using the DLC content actively in writing their case assignments. This is done together with alternative theory input and by use of their own practical knowledge. The last part of the students, about 20%, has primarily based their case assignments on their own personal practical knowledge supplied with references to theory and concepts in the DLC application.

The intention of the DLC design has been to narrow and intensify situated learning episodes that can be used in teacher education. The application should also bring forward to the students some significant concepts concerning classroom management and leadership, and it should strengthen problem solving and collaborative activities among the students. The follow-up study seems to confirm that these intentions have been active during the students' work on DLC, however there is still a need for further revisions and clarifications of the interactive trajectories supporting the students' learning activities.

Fig. 2 A

 2 Fig. 2 A video snapshot of the classroom episode

LMS = Learning Management System (Virtual Learning Environment)

Acknowledgements

I would like to thank InterMedia and the Department for Teacher Education and School Development at University of Oslo for financial support to this research project. The research is part of the strategic research effort "Competence and Media Convergence (CMC)" at the University of Oslo. For more information about CMC, see http://cmc.uio.no/.