Jean-Luc Estivalèzes
email: jean-luc.estivalezes@onera.fr

Davide Zuzio
email: davide.zuzio@onera.fr

A multi-scale Eulerian-Lagrangian method for efficient DNS simulation of the assisted atomization of a planar liquid sheet

Keywords: Atomization, VoF, Level-set, AMR, Euler-Lagrange coupling, DNS

The physics of assisted atomization process involves many spatial scales, where a wide variety of liquid inclusions of different sizes are produced. Large density and viscosity ratios associated with strong shearing flows make the process difficult to reproduce numerically. To correctly capture the dynamics of these spatial scales, each one should be resolved with an appropriate method, each one preserving the physical quantities and the correct interface topology. The simulation should reproduce enough physical time to allow reliable statistics. To address these problems, an original multi-scale methodology has been developed. It consists of three main items. The first is a coupled Level Set/Volume of Fluid method (CLSVOF) for accurate capture of the primary atomization process. The second an adaptive mesh refinement technique (oct-tree AMR) to dynamically optimize the structured Cartesian mesh. The third consists of a particle tracking algorithm to capture droplet dynamics. An improved Eulerian-Lagrangian coupling has been developed to assure a smooth transition between the Eulerian and the Lagrangian modelling of the droplets, where both models approach their design limits. The overall procedure is tested on simplified numerical tests and validated on a planar liquid sheet assisted atomization case. Results show its ability to reproduce the whole atomization process, from large scale instabilities to small droplet dynamics. A preliminary statistical spray analysis is performed as well, showing a realistic distribution of droplet sizes and average droplet diameter.

Introduction

The physics of atomization involves a wide range of phenomena and a strongly multi-scale dynamics, where many order of magnitude span between larger and smaller structures. Very different topologies such as drop, ligament, liquid sheet, may appear due to the many instabilities arising ( [START_REF] Stapper | An experimental study of the effects of liquid properties on the breakup of a two-dimensional liquid sheet[END_REF], [START_REF] Mansoura | Disintegration of liquid sheets[END_REF], [START_REF] Gutierrez Fernandez | Dynamic Primary Atomization Characteristics in an Airblast Atomizer[END_REF]). As the detailed simulation of all the spatial scales remains very expensive, the particle tracking method appear interesting for the smallest particles. Following this idea, the problem can be numerically split in two "moments" following the two stages of atomization, in order to adapt the local two-phase modelling. A separated phases model as the VOF or Level-Set could describe the primary atomization, where the two phases are clearly distinguished and the interface in subject to low frequency instabilities, while the dispersed phase model would be better adapted to describe the evolution of the sprays in terms of secondary atomization, transport and eventually evaporation. Indeed, if one consider a medium liquid drop, i.e. a quasi-spherical inclusion whose radius is equivalent to a few mesh cells, the Eulerian formulation is not able to correctly predict its dynamics. Actually, between 6 and 10 grid points per radius are necessary to described the evolution of such drop to avoid disappearance of liquid inclusion due numerical diffusion of level-set method, or less accurate interface representation of the piecewise linear reconstruction of VOF method. Moreover, the Lagrangian tracking could not be applied, the assumption of dispersed phase is not fulfilled any more. Hence, a third modelling scale have to be introduced so that the spatial gap between the two models is filled. In order to tackle these problems, the present work proposes an approach to model this third scale between pure Eulerian and Lagrangian representations: medium drops are supposed to have a rigid spherical shape, and they are followed by rigid translation while their mass and momentum properties are projected on the Eulerian field. This formulation results in a more conservative representation of the initial characteristics of the droplet. The hypothesis of spherical shape allows an efficient coupling with an AMR technique, where allowing a progressive mesh coarsening around the droplet makes the transition between purely Eulerian and Lagrangian formulations consistent. The reciprocal transformation has been adopted to treat the collision between Lagrangian particle with the interface of a large liquid inclusion. A CLSVOF method is used (from [START_REF] Menard | Coupling Level set/VOF/Ghost fluid methods: Validation and application to 3D simulation of the primary break-up of a liquid jet[END_REF] and [START_REF] Sussman | A coupled Level Set and Volume of Fluid method for computing 3D and axisymmetric incompressible two-phase flow[END_REF]) to accurately predict the interface shape with the Level-Set description and to ensure local mass conservation with the VOF formulation. An adaptive mesh refinement (given by the PARAMESH package [START_REF] Macneice | PARAMESH : A parallel adaptive mesh refinement community toolkit[END_REF]) is dynamically performed around the resolved CLSVOF interface throughout the computation, so that jump conditions are accurately satisfied. The smallest particles are tracked with a Lagrangian formulation and a classical drag force, while medium ones are treated using the mixed Eulerian-Lagrangian method previously described. The whole procedure has been paralleized to allow relatively large computational domains. The numerical code is an evolution of the one described in [START_REF] Zuzio | An efficient block parallel amr method for two phase flow simulations[END_REF] and [START_REF] Sarthou | Multiscale Euler-Lagrange method for parallel simulation of atomization induced by air-blast planar injectors[END_REF].

Numerical method

The numerical methodology proposed here aims to solve twophase flows under the incompressibility hypothesis. Hence in each phase we consider the Navier-Stokes equations under free divergence velocity field constraint. Interface tracking between both phases is performed thanks to a CLSVOF ( Coupled Level-Set Volume Of fluid) method [START_REF] Sussman | A coupled Level Set and Volume of Fluid method for computing 3D and axisymmetric incompressible two-phase flow[END_REF] with some further improvements described in [START_REF] Sarthou | Multiscale Euler-Lagrange method for parallel simulation of atomization induced by air-blast planar injectors[END_REF]. Surface tension forces are taken into account by a ghost-fluid method (GFM) as well as viscous jump [START_REF] Kang | A boundary condition capturing method for multiphase incompressible flow[END_REF]. Five order accurate WENO schemes are used to discretize convective terms. AMR (Adaptive Mesh Refinement) is used to refine the grid near the interface. It is bases on the Paramesh library [START_REF] Macneice | PARAMESH : A parallel adaptive mesh refinement community toolkit[END_REF]. Finally pressure Poisson equation is solved by a Krylov BiCG-Stab precondionned by a multilevel algorithm built on the AMR tree. Detailed informations on the full algorithm can be found elsewhere [START_REF] Zuzio | An efficient block parallel amr method for two phase flow simulations[END_REF], [START_REF] Sarthou | Multiscale Euler-Lagrange method for parallel simulation of atomization induced by air-blast planar injectors[END_REF].

Multi-scale approach

The approach proposed in this paper proposes the separate treatment of the liquid inclusions in two main scenarios, depending on the to-be-converted particle size:

1. the droplet size is lesser than the local mesh cell size. This kind of droplet will be called "small inclusion" within this paper;

2. the droplet size is greater than the local mesh size. This kind of droplet will be called "medium inclusion" within this paper.

In the first case, the dispersed phase approach hypothesis are considered as satisfied, so that a classical treatment is done, either in one or two ways coupling1 ; in the second, a kind of penalty method has been developed to allow the more complex interaction of the medium droplets and the underlying flow. The proposed multi-scale approach includes the Lagrangian to Eulerian inverse conversion. This allows in the primary atomization process the droplets to collide and merge with a resolved liquid body. The collision is handled differently for the small and medium particles.

Treatment of medium structures

Handling the medium droplets involves three main items: their detection and subsequent conversion to Lagrangian droplets, their advection and their potential liquid collision.

Detection and transformation

The first step of the multi-scale algorithm is to detect the liquid inclusion to be transformed. A tag-propagation algorithm is employed. As illustrated on figure 1, each liquid inclusion is detected by propagation of a tag over each liquid-cell, i.e. a cell where the volume fraction C i > 0.5. At first, each liquid-cell is tagged with an unique number. Then, the smallest tag is iteratively propagated among each neighbouring liquid-cell. When the tag is entirely propagated among all liquid cells which constitute the liquid inclusion, the tag is then extended to all surrounding cells where 0 < C i < 0.5. Considering a liquid inclusion l, one can evaluate the volume of this inclusion ν l , its center of mass x l and average velocity v l over all cells i l which contains the liquid inclusion through the VOF function:

ν l = V cell ∑ i∈i l C i x l = V cell 1 ν l ∑ i∈i l x i C i v l = V cell 1 ν l ∑ i∈i l v i C i (1)
where v i is the eulerian velocity interpolated at the center of cell i and V cell the volume of the cell.

As it is quite difficult to determine a physical parameter (like Weber number) to determine the stability of any drop and allow the transition between Eulerian and Lagrangian resolution, a possible transition is determined here based on geometrical considerations. Let N l be the total number of cell contained in a liquid inclusion l. For very small inclusions where N l < 2 m , with m the dimension of the considered problem (m = 2 for bidimensional problems, m = 3 for three dimensional cases), the inclusion is automatically treated in a Lagrangian point particle way, without any other arguments. For medium inclusion 2 m ≤ N l ≤ 6 m , a sphericity criterion is used, as illustrated on figure 2. To this way, a sphere of equivalent volume is superimposed on the center of mass of the liquid inclusion, for which the radius is

r p = ν l π 1/m 3 4 (m-2)/m (2)
and the difference of caracteristic length δ l and volume δ V (dashed part on the right of figure 2 are computed :

δ l = r p -L δ V = V cell ∑ i∈i l |C i -χ i | (3)
where χ i = 1 if ||x i -x l || ≤ r p , 0 otherwise and L is the smallest length between the center of mass and the interface, i.e. L = φ (x l ). The inclusion is assumed as quasi-spherical if

δ l r p < α δ V ν l < β . (4)
After some numerical test, α and β have been set between 0.5 and 1 (these values can be increased to force more conversions).

If those two criteria are satisfied, the inclusion becomes a Lagrangian particle of volume ν l , radius r p and initial velocity v p = v l at the location x p = x l . If they are not satisfied, the inclusion is still treated by the Eulerian CLSVOF resolution until a possible stabilization or fragmentation occurs. An original tracking method has been developed for the medium drop, in a way of a penalty method. Indeed, the idea of this method is to project the mass and momentum particle on the Eulerian field. Hence, for a given particle :

(ρ n+1 , µ n+1 , u n ) i = (ρ l , µ l , v p ) if x i -x p ≤ r p (ρ g , µ g , u n i ) otherwise (5)
Then, the velocity prediction equation is solved on the Eulerian grid with the new values of density, viscosity and velocity updated with respect to particle positions. The pressure inside the drops is corrected by Laplace relation:

p n+1 i → p n+1 i + σ r p if ||x i -x p | ≤ r p (6)
Afterwards, the pressure correction equation is solved with the corrected pressure. Then, the new particle velocity is obtained by averaging the Eulerian velocity inside the considered drop

v p = 1 ν p ∑ i V cell,i C i u n+1 i if |x i -x p | ≤ r p (7)
and finally equation 13 is solved with this new particle velocity. This formulation take implicitly into account the drag force on the particle and lead to a more conservative form than the method presented into previous section, while allowing the de-refinement process as the particle moves away from the interface.

Drop-interface collision

In the present work, a re-impact algorithm is implemented to take into account of a possible collision between particles and the liquid-gas interface. The re-impact is allowed if a given particle is moving throw the interface and if this particle is sufficiently closed to it (see figure 3) i.e.:

C=1 C=0 φ>0 φ<0 φ p v C=1 C=0 φ>0 φ<0
Figure 3: Illustration of the re-impact algorithm. Fat continuous line : interface (φ = 0, C = 0.5), discontinuous line : iso-value of φ . Left : before collision, right : after collision. When a particle is closed to the interface moving throw it, the re-impact is allowed : both VOF and Level Set functions are reconstruct into the drop, and the Lagrangian particle is removed.

v p • ∇φ > 0, |φ p | < r p + ∆ x,min (8)
where φ p is the interpolation of φ at a location x = x p and ∆ x,min the grid size. If those criterion are satisfied, both Level Set and VOF functions are reconstruct inside the drop:

φ n i = r p -||x p -x i ||,C n i = 1 (9)
for all cells i inside the drop (i.e.||x p -x i || < r p ). To the drop periphery, the Level Set function is fixed to 0 and the difference between the drop volume and the VOF newly added is equally distributed to the VOF function on all cells at the drop periphery, to ensure mass conservation. Afterwards, the velocity inside de drop is imposed on the Eulerian field with momentum conservation consideration :

u n i → C n i ρ l v p + (1 -C n i )ρ g u n i C n i ρ l + (1 -C n i )ρ g (10)
which is exactly equal to v p for full cells (i.e. C n i = 1) and is a mass weighted for mixed cells or small droplet (r p ≤ ∆ x,min ). Finally, the redistance algorithm is used to regularize the function φ n into the gas phase.

Small droplets

The detection of droplets smaller than the finest mesh size is performed by detecting any cells characterized by a value of VOF 0 < C i < 0.5 not attached to any larger structure, i.e. surrounded by cells with 0 < C i < 0.5. In this case the Level-Set function does not see any liquid mass, so that for the CLSVOF this liquid would be effectively lost. For these small Lagrangian droplet (i.e. r p ≤ ∆ x,min ), a classical drag coefficient for spherical point particle is used, from Schiller and Neumann:

C D (R e,p ) =
24R -1 e,p (1 + 0.15R 0.687 e,p )if R e,p < 1000 0.424 otherwise [START_REF] Déjean | Experimental study on the influence of liquid and air boundary conditions on a planar air-blasted liquid sheet, Part I: Liquid and air thicknesses[END_REF] with R e,p = 2r p ρ g ||v p -v p,e ||/µ g and where v p,e the Eulerian velocity interpolated at the particle location. These values give a relaxation time in the form τ p = 8ρ l r p /(3ρ g C D ||v p -v p,e ||).Then, the particle velocity and position follow the equations

dv p dt = τ -1 p ||v p -v p,e || (12)
and its position

dx p dt = v p (13)
Equations 12 and 13 are integrated using a classical second order Runge-Kutta scheme. The potential collision of these droplets with an interface can happen in two different ways. In the first case, the droplet is considered as small even on the finest mesh, so that its mass and momentum can be directly injected into the local cell. In the second, the particle approaching the interface crosses one or more mesh refinement jumps: once its size exceeds the cell size, it is detected by the medium droplets search algorithm and switches in this category, so that it is projected on the Eulerian field before the actual impact (see paragraph 3.1). Medium droplets entering a coarse enough region are transferred to this category too.

Results and validation

Drop in a uniform flow

In this first example, a three dimensional liquid drop at rest (ρ l = 2 kg.m -3 , µ l = 0.001 kg.m -1 .s -1 ) is immersed into an uniform gas stream (ρ g = 1 kg.m -3 , µ g = 0.001 kg.m -1 .s -1 ) with constant velocity in the x direction u(t = 0) = [U 0 , 0, 0] T with U 0 = 0.1 m.s -1 and a surface tension σ = 0.001 Nm -1 . The computational domain consists of a square box of size L = 0.25 m with slip conditions on the walls tangent to the gas stream and respectively uniform inlet and outlet on the two walls orthogonal to the stream. This lead to following non dimensional numbers, based on the drop radius R = 0.05 m :

W e = ρ l U 2 0 R σ = 1, R e = ρ l U 0 R µ l = 100 (14)
In order to avoid droplet deformation during its motion, the inlet gas stream in injected with a time ramp U in j = min(U 0 ,U 0 × t/10) so that the acceleration is smooth and the shape stable. The test consists into a study of the acceleration of the droplet up to a physical time of t end = 2 s done with the CLSVOF with increasing refined meshes. The coarsest mesh is composed by 32x32 (M32) cells, which correspond to 7 cells per droplet diameter.

The droplet can still be considered (but barely) as resolved. The more refined meshes are given by 64x64 and 128x128 cells (M64 and M128). The medium Lagrangian droplet computation is performed on the coarsest (M32) of the meshes. The particle is initialized as in the pure Eulerian case, but the multi-scale algorithm

x drop = Ω l x dv Ω l dv = ∑ i C i x i ∑ i C i v drop = Ω l u dv Ω l dv = ∑ i C i u i ∑ i C i (15)
A Lagrangian computation with a very coarse mesh (M16) has been plotted as well. The plots clearly show how the Lagrangian medium droplet modelling is able to reach almost the same accuracy as the Eulerian modelling with a mesh size twice as refined.

It is also visible how the inaccurate prediction of the increasing velocity for the M32 CLSVOF solution is affected by an initial plateau between t = 0 s and t = 0.5 s: the initial acceleration being delayed, the following evolution is shifted towards lower values.

Binary drops collision

In this section, the computation of a collision between two drops of the same size and opposite velocities is performed. Those drops are constitued of water (ρ l = 10 3 kg.m -3 , µ l = 1.137 10 -3 kg.m -1 .s -1 ) evolving into air (ρ g = 1.226, µ g = 1.78 10 -5 kg.m -1 .s -1 ), with surface tension σ = 0.072. The drop velocity and radius are respectively v d = 7m.s -1 and R = 0.4mm, leading to a Weber number W e = 272 (based on drop radius). Following Pan et al. [START_REF] Pan | Binary droplet collision at high Weber number[END_REF], the coalescence or separation regime do not occur and a breakup of initial drop is observable, leading to scattering of multiple secondary droplets (Figure 3.c and 4.a of this reference). This scenario is also visible in the present simulation in figures 6 and 7. The simulation are performed as follow : firstable, a classical eulerian resolution is perfomed on a fine mesh (equivalent resolution [128, 256, 256]) until a time t = 0.0045 s (see figure 6).

From this time, two simulations were performed : an Eulerian one with the same resolution and a Lagrangian particle tracking one with a twice coarser resolution. This is shown on figure 7 for later times. One can see that the position, size and global evolution of all 'large' drops are correctly reproduce by the present particle tracking method, whereas the second ring around the central disc is fragmented into a cloud of very small droplet, due to the lack of resolution. However, this cloud of droplet has a good description of the topological evolution of this last ring. We have check, for this simulation that mass is totally conserved throughout the simulation.

Assisted atomization of a liquid sheet

The whole multi-scale algorithm has been applied to the simulation of the primary atomization of a liquid sheet sheared by two parallel air streams. The considered numerical configuration aims to reproduce a simple atomization device as the one experimentally investigated in [START_REF] Déjean | Experimental study on the influence of liquid and air boundary conditions on a planar air-blasted liquid sheet, Part I: Liquid and air thicknesses[END_REF]. The injector, depicted in figure 8(a), consists of a NACA-shaped injector immersed into a channel gas flow, discharging the liquid by a rectangular fence 300 µm thick (visible in the middle of the device nozzle). Different geometrical configurations as well as liquid and gas flow regimes are considered by the authors. The parameters are summarized in table 1.

The simulation domain, with reference to figure 8 A comparison of some large scale features of the atomization process has been made with respect to the experimental results found in [START_REF] Déjean | Experimental study on the influence of liquid and air boundary conditions on a planar air-blasted liquid sheet, Part I: Liquid and air thicknesses[END_REF]. A comparative visualization of the behaviour of the sheet is given in figure 9. The simulation correctly captures the growing longitudinal instability of the sheet, as well as the transversal modulation. The thin ligaments formed by the primary atomization at the break-up point are fully captured by the CLSVOF eulerian solution. A first wave of droplets is formed at this location. The liquid detached from the main sheet body undergoes further deformation and break-up into a fully threedimensional flow: it can be seen in both images that large liquid parcels coexist with smaller stable droplets. A preliminary analysis has been performed on the droplet spray, the results compared to the experience by laser diffraction system allowing measurement of spray droplet size distributions. The statistics involve the droplets in the whole numerical domain; their characteristics are averaged in time, the samplings taken with a frequency corresponding to an average convection time in order to avoid repeated registrations of the same droplet. The particles of diameter inferior to the finest mesh cell (40 µm) have been discarded, as no physics is involved in their creation. A total of 18457 droplets (40 < D < 320 µm) have been registered. Figure 10 shows the corresponding distribution in volume.

The simulation seems able to capture the predominant diameters around 255 microns, but no information has been extracted from structures larger than 320 microns in diameter, so that the distribution is truncated after this value. It would seem that the evolution of the distribution with decreasing diameters is underestimated, a possible consequence of the considered volume of interest close to the injector and the absence of a secondary atomization model. The numerical diameter is D 32 = 158 µm, where the experimental one is D 32 = 165 µm showing a pretty good agreement.

Conclusion

An innovative multi-scale methodology for the direct numerical simulation of atomization has been presented. It consists of three main items. The first is a coupled Level Set/Volume of Fluid method (CLSVOF) for accurate capture of the primary atomization process. The second an adaptive mesh refinement technique (oct-tree AMR) to dynamically optimize the structured Cartesian mesh. The third consists of a particle tracking algorithm to capture droplet dynamics. An improved Eulerian-Lagrangian coupling has been developed to assure a smooth transition between the Eulerian and the Lagrangian modelling of the droplets, where both models approach their design limits. Results show its ability to reproduce the whole atomization process, from large scale instabilities to small droplet dynamics. A preliminary statistical spray analysis is performed as well, showing a realistic distribution of droplet sizes and average droplet diameter.

Acknowledgements

The PARAMESH software used in this work was developed under NASA's HPCC and ESTO/CT projects and under grant NNG04GP79G from the NASA/AISR project. The present study has been granted by the Foundation STAE-RTRA-research program COFFECI. This work was granted access to the HPC resources of CINES under the allocation 2014-x20142b7264 made by GENCI.

Figure 1 :

 1 Figure 1: Illustration of liquid inclusion detection.

Figure 2 :

 2 Figure 2: Sphericity criterion for medium inclusion

Figure 4 :

 4 Figure 4: (a) x(t = t end ) position of the droplet (b) x velocity component of the droplet versus time. Continuous lines : Eulerian computation, dotted lines: Eulerian/Lagragian.

Figure 5

 5 shows a direct comparison between the mesh converging Eulerian results and the Lagrange medium droplet formulation. The visualization clearly shows that the final position of the droplet (x(t = t end ) is shifted forward when a more accurate resolution is performed. The M32 mesh droplet is clearly late in comparison with the reference result of the M128 mesh. Conversely, the Lagrangian result (on M32 mesh) shows a good agreement with the most accurate M64 and M128 meshes. The velocity field obtained with the mixed formulation is close to the one obtained with the Eulerian, whereas a classical Lagrangian dragforce model fails to correctly reproduced the flow topology.

Figure 5 :

 5 Figure 5: Advection of the liquid droplet at t end = 2 s. From left to right, three CLSVOF results with increasing mesh refinement (M32, M64, M128) and the result of the multi-scale computation. The colour field is the z component of the velocity u. The vertical dotted white line represents the x position of the droplet center of mass, as calculated with the CLSVOF M128 (the finest) mesh.

Figure 6 :

 6 Figure 6: Binary drop collision at W e = 272. Eulerian simulation at time t = 0, 3, 3.5, 4 ms on the finest grid Time increase from left to right.

Figure 7 :

 7 Figure 7: Binary drop collision at W e = 272. Top eulerian simulation at time t = 5, 5.64 ms, bottom : mixed Eulerian/Lagrangian formulation at time t = 5, 5.64 ms. Time increase from left to right. yellow : interface, green : Lagrangian particle.

Figure 8 :

 8 Figure 8: (a) Actual liquid sheet injector device. (b) Numerical computation set-up.

Figure 9 :

 9 Figure 9: Snapshots of both experimental visualization and simulation, arbitrary time step.

Figure 10 :

 10 Figure 10: Droplet volume distribution obtained from the simulation dispersed phase, comparison with the experience.

Table 1 :

 1 Dimensionless parameters of the 3D atomization simulation.

May 22nd -27th 2016, Firenze, Italy ICMF-2016 -9th International Conference on Multiphase Flow May 22nd -27th 2016, Firenze, Italy

In the application foreseen in this paper, the one way approach is considered as sufficient, even if the small droplets can in practice have almost the size of the cell


F Gand

P Grenson

I Mary

H Deniau

P Reulet

B Aupoix

Experimental and numerical investigation of a heated impinging jet for a small nozzle-to-plate distance and high Reynolds number

, which has been selected for its high Reynolds number Re D = 60,000, small nozzle-to-plate distance H/D = 3 and temperature difference between the heated jet and the ambient. First, the experiments are described and analysed, then numerical simulations (DNS, LES, ZDES and RANS) are assessed to evaluate the resolution level required to capture the salient features of this configuration.

Introduction

The impinging jet configuration is of widespread use in industrial applications due to the high heat transfer experienced by the impinged surface. Nevertheless, it remains an attractive and challenging configuration for turbulence modeling and numerical simulation assessment. A large number of published work deals with numerical simulations of impinging jets configurations [START_REF] Dewan | Recent Trends in Computation of Turbulent Jet Impingement Heat Transfer[END_REF] [3] [4] [START_REF] Hadziabdic | Vortical structures and heat transfer in a round impinging jet[END_REF] [START_REF] Uddin | LES simulations of an impinging jet: On the origin of the second peak in the Nusselt number distribution[END_REF], however at rather low Reynolds number and different nozzle-to-plate distances and based on often incomplete and scattered experiments.

The goal of the present paper is to assess the accuracy of a wide range of numerical methods -RANS, ZDES, LES and DNS-for the prediction of an impinging jet configuration using a single, well-documented experimental dataset created in the framework of an ONERA joint project for numerical validation purposes by Grenson et al. [START_REF] Grenson | Investigation of an impinging heated jet for a small nozzle-to-plate distance and high Reynolds number: An extensive experimental approach[END_REF]. The configuration considered has been selected for its high Reynolds number Re D = 60,000, small nozzle-to-plate distance H/D = 3 and temperature difference between the heated jet and the ambient.

From a technical point of view, a key quantity to predict is the heat transfer on the impinged plate, which presents a dual-shaped profile for the nozzle-to-plate distance considered. This is attributed to the dynamics of the plate boundary layer re-attachment which is challenging for turbulence modelling. The use of different levels of resolution in this paper aims at highlighting some trade-offs between accuracy and computational cost for the prediction of heat transfer.

Experimental dataset

The main features of an impinging jet configuration are illustrated in Fig. 1. Three areas of interest are identified, namely the free jet issued from the nozzle (in the present case, the nozzle is actually a fully developed pipe), the stagnation area and the wall jet. The characteristics of the configuration studied in the present paper are given in Table 1. The nozzle-to-plate distance is equal to H/D=3, the flow in the pipe is heated up to 400 K so that the free jet is hot. The Reynolds number based on the pipe exhaust diameter is equal to Re D = 60, 000. A complete description of the experiments and instrumentation can be found in [START_REF] Grenson | Investigation of an impinging heated jet for a small nozzle-to-plate distance and high Reynolds number: An extensive experimental approach[END_REF] as well as the analysis of the results. The Nusselt number distribution on the impinged plate obtained by an inverse method displays a secondary maximum typical of such small nozzle-to-plate distance configurations (Fig. 2). Besides, PIV measurements enable to detect large coherent vortices created in the free jet area and track them during the impact, as depicted in Fig. 3. A detailed analysis of these vortex regions using time-accurate LDV measurements enabled to link the dynamics of the free jet and that of the wall jet. At last, the secondary maximum of the Nusselt number distribution was attributed to a possible separation in the decelerating flow region, which is a shared interpretation with previous works [4] [6].

Simulations

Parameters of the simulations performed

Four simulations of increasing turbulence resolution level are evaluated in the present paper as summarized in Table 2: • A RANS computation using the k-ω SST model is used as a reference of the industrial capabilities, although it is known that eddy-viscosity RANS models are not suited for this kind of configuration [START_REF] Zuckerman | Jet impingement heat transfer: physics, correlations, and numerical modeling[END_REF]; • A hybrid RANS/LES simulation using the ZDES mode 2 [START_REF] Deck | Recent Improvements of the Zonal Detached Eddy Simulation (ZDES) Formulation[END_REF] is performed in order to assess the compromise of this approach in terms of CPU cost and resolution level since the attached flow on the impinged plate is treated in RANS mode. The use of mode 2 of ZDES makes it straightforward for the user since the RANS-to-LES switch is automatic [START_REF] Deck | Recent Improvements of the Zonal Detached Eddy Simulation (ZDES) Formulation[END_REF], which makes this approach useful in an industrial context. The pipe is not included in this simulation. Instead, the pipe exhaust flow is generated using the Synthetic Eddy Method [START_REF] Jarrin | Reconstruction of Turbulent Fluctuations for Hybrid RANS/LES Simulations using a Synthetic-Eddy Method[END_REF] to provide realistic unsteady turbulent content. This simulation is presented in [START_REF] Gand | Investigation of turbulence development in incompressible jets with zonal detached eddy simulation (ZDES) and synthetic turbulent inflow[END_REF], which focuses on the free jet area; • A wall-resolved LES simulation provides a first assessment of the increased near-wall turbulence resolution level on the impinged plate. The pipe is included in the computational domain, and the inflow turbulence necessary to compute the pipe in wall resolved LES is generated using a combination of vortex rings method near the wall and isotropic homogeneous turbulence in the center of the pipe. This simulation is presented and analyzed in details in [START_REF] Grenson | LES of a impinging heated jet for a small nozzle-to-plate distance[END_REF];

•
A DNS simulation is finally used as an even further resolved numerical dataset. The pipe is included in the computational domain, the Lund method is used to generate resolved turbulence in the pipe.

The RANS, ZDES and LES simulations involve an isothermal boundary condition at T=298.15 K for the impinged plate. On the other hand, the experimental temperature profile was used as a boundary condition for the DNS simulation in order to match exactly the conditions during the PIV measurements.

The RANS, ZDES and LES simulations were performed with the elsA software of ONERA [START_REF] Cambier | The Onera elsA CFD software: input from research and feedback from industry[END_REF]. The DNS simulation was carried out with the LES/DNS dedicated code Funk of ONERA. The difference in resolution on the impinged plate between the three eddyresolving simulations assessed in this paper is illustrated in Table 3. This is also an indicator of the cost of the simulations. Note that the ZDES grid is 20 times smaller than the LES one and 30 times smaller than the DNS one.

Name

Simulation

Pipe exhaust conditions

Mean and fluctuating velocity profiles close to the pipe exhaust are plotted in Fig. 4. All simulations match the experimental velocity profile, and satisfactory turbulent statistics are achieved. The ZDES simulation which involves SEM to generate these statistics does not reproduce the near-wall turbulence intensity peak due to the setup of the SEM which was only aimed at reproducing isotropic core jet turbulence [START_REF] Gand | Investigation of turbulence development in incompressible jets with zonal detached eddy simulation (ZDES) and synthetic turbulent inflow[END_REF] (for user-friendliness purposes). The time-averaged fields depicted in Fig. 6 provide qualitative insights on the accuracy of the simulations. All simulations agree fairly well with the experiment for the velocity magnitude. Conversely, significant discrepancies appear for second order statistics in the mixing layer. The DNS and ZDES simulations seem to underestimate the thickness and intensity of the shear layer, while the LES simulation displays the opposite trend. The turbulence levels also seem to be overestimated in the wall jet area for the LES simulation, which is further studied in section 3.4. The profiles extracted in the free jet at x/D = 1.5 presented in Fig. 7 confirm the previous observations. Overall, the simulations are in fair agreement with the experiment. The ZDES simulation provides arguably the best agreement with the streamwise and radial experimental velocity fluctuations, which is consistent with several previous works with ZDES and jets and free shear flows [START_REF] Deck | Recent Improvements of the Zonal Detached Eddy Simulation (ZDES) Formulation[END_REF] [9] [START_REF] Verrière | Zonal Detached-Eddy Simulations of a Dual-Stream Jet[END_REF]. Regarding the RMS temperature fluctuations, the DNS is closer to the experiment. The shear layer in the LES simulation is too large and intense which may be related to a travelling spurious pressure wave inside the pipe [START_REF] Grenson | LES of a impinging heated jet for a small nozzle-to-plate distance[END_REF].

Jet flow development

Flow on the impinged plate

Fig. 8 presents profiles in the wall jet area. The overall agreement between experiments and simulations is satisfactory. Contrary to what has been observed in the free jet area, the LES and DNS results for the velocity magnitude and fluctuations are closer to the experiments than the ZDES ones. This is attributed to the progressive switch to RANS mode of the ZDES as the wall jet reattaches, whereas it is fully resolved in the LES and DNS simulations. These discrepancies are likely to affect the heat transfer since it is a shared opinion [START_REF] Dairay | Direct numerical simulation of a turbulent jet impinging on a heated wall[END_REF] that the near-wall dynamics of the wall jet is responsible for the Nusselt number distribution -in particular the secondary maximum observed in Fig. 2.

Near-wall quantities on the impinged plate

Fig. 9 illustrates the dispersion of the numerical results in terms of heat transfer. On the left part of Fig. 9, the Nusselt number based on the pipe exhaust diameter is computed as 𝑁𝑁𝑁𝑁 𝐷𝐷 = 𝑞𝑞 𝑐𝑐 (𝑇𝑇 𝑤𝑤 -𝑇𝑇 𝑎𝑎𝑤𝑤 )𝐷𝐷/𝑘𝑘 𝑝𝑝 where q c is the heat transfer, T w is the plate temperature, T aw is the adiabatic wall temperature taken from the experiments and k p is the thermal conductivity of air at T w . The LES simulation matches the experiment in the stagnation region whereas the k-ω SST and ZDES overestimate the heat transfer. All simulations predict the existence of a secondary maximum but its intensity and location is not accurately captured. The LES is closer to the experimental data than the ZDES and k-ω SST simulations. An additional ZDES simulation was performed without injecting turbulence at the pipe exhaust (ZDES (no SEM) in Fig. 9). It is interesting to note that the results of this simulation do not exhibit a secondary peak which is attributed to the fact that the free jet is not correctly developed if the turbulence coming from the inside of the pipe is not taken into account (see [START_REF] Gand | Investigation of turbulence development in incompressible jets with zonal detached eddy simulation (ZDES) and synthetic turbulent inflow[END_REF]).

Since the DNS simulation was performed at the thermal equilibrium (conditions identical to the ones when the PIV measurements were carried out), the Nusselt distribution is not relevant for this simulation. To assess the DNS nearwall data, the wall friction is plotted on the right of Fig. 9. This quantity was not measured during the experiments, therefore the data from the experiments of [START_REF] Alekseenko | Local characteristics of impinging round jet[END_REF] for a similar configuration (H/D=3 and Re D =44,000) are shown. The secondary maximum for this quantity appears challenging to capture and only the DNS is able to predict it quantitatively. An inflexion point can also be seen in the LES curve, which is not the case for the ZDES simulation. The additional ZDES without SEM at the pipe exhaust provides a slightly lower friction coefficient than the one with injected turbulence near r/D=2, which is consistent with the results for the Nusselt number and illustrates the link between these quantities.

Summary and conclusion

The comparison of different levels of turbulence modellings ranging from RANS to DNS with an extensive experimental dataset has enabled to identify the flow features that can be captured with each approach. Since the cost of the simulations roughly scale with the grid size and refinement of the method, this helps identify the best compromise between the accuracy required for the desired quantity and the computational cost / turnaround time.

It was shown that for the free jet area, the ZDES mode 2 approach -which acts as LES in the free jet -provides very satisfactory results insomuch as the pipe exhaust conditions are correctly taken into account (using SEM here, also refer to [START_REF] Gand | Investigation of turbulence development in incompressible jets with zonal detached eddy simulation (ZDES) and synthetic turbulent inflow[END_REF]). Some discrepancies have been observed between the ZDES, LES and DNS simulations regarding the shear layer intensity and growth rate, which is probably linked to the turbulence generation and resolution in the pipe, as well as grid stretching near the pipe exhaust.

Conversely, it was observed that some near wall turbulence resolution is required in the wall jet area in order to accurately capture the heat transfer and friction coefficient (this is consistent with previous studies [START_REF] Hadziabdic | Vortical structures and heat transfer in a round impinging jet[END_REF] [6] [START_REF] Zuckerman | Jet impingement heat transfer: physics, correlations, and numerical modeling[END_REF]). The LES data tend to indicate that the heat transfer can be predicted with this approach, while DNS seems to be necessary to capture the friction coefficient profile. ZDES mode 2 reproduces some features of the heat transfer distribution but the switching back to RANS mode in the attached wall jet seems to be counterproductive in this regards.

In order to combine the good features of ZDES mode 2 in the free jet area and provide some near-wall turbulence resolution in the wall jet area, it would be interesting to implement ZDES mode 3 (i.e. Wall Modelled LES) in the wall jet area. This requires an increased grid density near the plate, which can be achieved effortlessly with the chimera approach while maintaining a computational cost compatible with industrial constraints. More accurate conclusions could also be reached by using the same numerical methods (turbulence generation, space and time schemes, boundary conditions, CFD software, etc.) and only refining the grid when the turbulence resolution approach requires it.

Fig. 1

 1 Fig. 1 Schematic description of a strong flow deviation

Fig. 2

 2 Fig. 2 Measured Nusselt number distribution

Fig. 3

 3 Fig. 3 Location and size of the vortex regions

Fig. 4

 4 Fig. 4 Pipe exhaust conditions

Fig. 5

 5 Fig. 5 presents instantaneous visualizations of the ZDES, LES and DNS simulations. The difference in resolution level is self-evident, especially on the impinged plate where fine scale turbulence is resolved in the LES and DNS simulations whereas only larger detached eddies are captured in the ZDES simulation.

Fig. 5

 5 Fig. 5 Instantaneous visualizations of the simulations (Q criterion)

Fig. 6

 6 Fig. 6 Mean velocity (first row), streamwise velocity fluctuations (second row) and radial velocity fluctuations (third row)

Fig. 7

 7 Fig. 7 Mean velocity, velocity fluctuations and temperature fluctuations in the jet at x/D=1.5

Fig. 8

 8 Fig. 8 Profiles of streamwise, radial and crossed velocity fluctuations and temperature fluctuations near the impinged plate

Fig. 9

 9 Fig.[START_REF] Gand | Investigation of turbulence development in incompressible jets with zonal detached eddy simulation (ZDES) and synthetic turbulent inflow[END_REF] Nusselt number and friction coefficient on the impinged plate. Experimental data for friction taken from[START_REF] Alekseenko | Local characteristics of impinging round jet[END_REF]

Table 1

 1 Parameters of the configuration investigated

	Quantity	Value
	Bulk velocity U j	25.6 m/s
	Jet temperature T j	403.15 K
	Ambient temperature T a	298.15 K
	Exhaust diameter D	60 mm
	Reynolds number Re D	60,000
	Nozzle-to-plate distance H/D	3

Table 2

 2 Simulations performed

Table 3

 3 Grid sizes on the plate (x is the wall normal direction relatively to the plate)

Acknowledgments

The DNS and LES simulations presented in this paper were performed using HPC resources from GENCI-TGCC (Grants n° c20142a7253 and x20152a7373).


Published in Nordiske

 Organisasjons-Studier, 1, 2007: 111-117.

SOME LESSONS FROM THE NORDIC WAY OF DOING SCIENCE

Jean-Claude THOENIG 1

What has happened in Northern European countries in the last part of the 20 th century shall remain as a milestone for the development of knowledge in social sciences. Examples are quite few of remarkable and lasting contributions made by a wide set of scholars: the Ecole des Annales in history, the Cowes Commission in economics, the Aston school in the 1970s, the Carnegie Mellon neo-behavioral approach in the 1950s and early 1960s in the field of sociology of organizations.

It has become common practice to characterize the contributions of French school known as Les Annaless as placing emphasis on long-term periods and socio-economic contexts, or to call the perspective developed by Herbert Simon and his disciples a behavioral revolution. To define in a few words a distinctive Nordic style is not an easy endeavor. The Nordic colleagues themselves do not feel at ease when listing the distinctive features of their style.

Many of them refer to their countries of origin. Historical and cultural backgrounds such as that of the Vikings are said to be relevant factor explaining the longevity and the intensity of Nordic studies. Reference is also made to legacies such as a common agenda about democracy, the status of the state, the nature of political consent or the style of business management (Byrkjeflot 2003). A third factor often mentioned is the fact that members of the Nordic wave share a common enemy, namely normal science. NATO, or North American Theory of Organizations, would be the challenge or the danger to fight (Czarniawska and Sevon 2003).

There is no doubt that the Nordic contributions are a unique phenomenon in the field of organization theory. Non-Nordic scholars are struck by the fact that this wave has combined several characteristics in a unique manner:

* a collective dynamics running of more than 25 years of duration * a couple of thousand publications in high-ranking journals and books2 , * close to one hundred collaborative research projects3 , * a lasting common identity shared by its many members, despite the fact that most of them show rather diversified research interests and agendas, and explore rather heterogeneous paradigms, * a high degree of collaboration and dialogue between individuals who speak different native languages and belong to academic institutions located in countries which in the past were not very close allies, to put it mildly.

While it is quite difficult to capture in a few sentences the intellectual core -if such a core exists, which is still an open question -of the Nordic way or open school, and while such a list would be quite unfair to many of its members and their contributions,given the variety of perspectives, etc.-, the next-best ambition might be to name the institutional conditions that have facilitated the existence of such a strong and lasting cooperative and communitarian scientific enterprise. They may also provide lessons for knowledge production to the scientific community at large.

The Nordic way blends local embeddedness and global visibility. It has strong roots in specific social science disciplines while being open to cross-disciplinary approaches. It avoids normal science syndromes, and has created a virtuous circle of regional identity and debate.

Global and local

The first fact relates to publication practices. The Nordic open school has been able to develop a global presence while keeping roots in its own region. For instance, note the way its authors consider publishing matters. They write very actively in prestigious top-ranked international journals. Nevertheless, they have not neglected to send manuscripts to local or regional journals. They publish in English but also write in their native languages. Publishing in your native language suggests that English is not the only ultimate sign of success and quality. And sending your most ambitious papers not only to global journals but also to Nordic journals maintains, keeps a local publishing capacity alive both economically and scientifically. Another striking fact is that journal articles are not considered the main, or the only, legitimate criterion of scientific relevance. Books also matter as scientific achievements. They still provide a major channel of information and publications (Engwall 2003).

Nordic research has, therefore, at the same time benefited from and kept alive favorable pre-conditions at a time such as the latter part of the 20 th century, when the geo-political landscape of organization science was changed in a dramatic manner. These factors have, at the same time, fueled distinctiveness and kept a local tissue of stimuli available while improving access to and visibility on the international scene. Nordic scholars probably rank among those having had the longest professional stays in foreign countries (Europe as well as the USA) among the organization theory community. They travel abroad, and they make an effort to understand how their foreign guests are doing research.

Intensive exchange with non-Nordic colleagues induces Nordic scholars to avoid becoming prisoners of a regional ghetto in terms of ways of thinking, and to refrain from abandoning their own agenda or style for the purpose of becoming more universal.

Disciplinary roots combined with interdisciplinary openness

A second fact addresses intellectual creativity and scientific rigor.

The Nordic research has maintained strong, explicit linkages with humanities areas such as philosophy, and with basic social science disciplines. This is clearly the case with regard to political science. Other disciplines such as ethnology, linguistics or cognitive psychology, just to name a few, also served as dynamic sources of inspiration and reference. Sociology as such seems to have been less involved, which may surprise some continental colleagues. The way the Nordic movement or school handles interfaces and interactions with the area of economics should also be a source of inspiration for non-Nordic colleagues. Nordic colleagues have avoided behaving passively and become over-fascinated by this discipline, borrowing its latest fads or becoming overly intrigued by this discipline, borrowing its latest fads or replicating its dominant criteria of methodological excellence.

When comparing the Nordic situation with that of other European and non-European countries, it is remarkable to see that several business administrationoriented departments and schools were established quite early on, and that they had forged rather close connections with older universities. In other terms, they were not emerging -as sometile occurred in other national settings -as parallel or even alternative models to the traditional academic institutional core. Nordic university reforms were usually handled as a balanced compromise between innovation and tradition, enabling business professors to maintain an academic ethos in daily professional life. More than in many other countries interdisciplinary pluralism and cooperation have therefore been kept alive.

Dialoguing with basic disciplines has helped the organization theory perspective to pursue a broader intellectual and societal agenda, even when rooted in an action or management perspective. But it has not neglected epistemological issues, in any case.

Pluralism

The Nordic way also deals in a unique manner with paradigmatic and empirical pluralism.

Some approaches have been a major source of inspiration for the Nordic way, at least as measured in quantitative terms. The most conspicuous illustration of this is the overwhelming influence of an agenda for institutional theory. Other sources of inspiration are linked to the sociology of translation and to critical sociology. Most ot the time the founding fathers were not Nordic scholars, but their American, French or German colleagues. At the same time, and this is a remarkable achievement, Nordic scholars have not produced clones or behaved as as passive imitators of the masters. On the contrary, they have made decisive contributions in testing the models, refining when not transforming them. Nordic research has therefore co-authored knowledge in areas that are now universally recognized as key stages in the advancement of organization theory. Therefore, it is quite difficult to refer to a typical content profile of Nordic research. There are very different perspectives covering a variety of topics. At the same time, and to a large extent, peers conduct a dialogue and cooperate on common research projects. . This openness, if not this broad coverage, may also be a condition for lasting success.

On should add that several Nordic scholars have not only been quite mobile in terms of specific topics, fields and theories, but also in taking part in specific ad hoc cooperative ventures. Niches that provide an income for life are not common. Scholars renew their research agendas, from the study of hypocrisy to the understanding of standardization, or from European integration to local government. Curiosity and creativity, these two challenges that most tenured faculty members do not handle easily, go hand in hand in the Nordic way. Two phenomena that often coincide with constraints on knowledge advancement, imagination, exploration and risk taking have been avoided.

One is the emergence of so-called schools or micro-sects governed by one charismatic and sometimes despotic academic figure. He or she controls resources such as career opportunities in a discretionary way, his or her clique of disciples being expected to do research in strict conformity with particular dogmas. Research becomes a way of building a monument to the master. Social and intellectual dependence are so important that any deviation from the common identity is punished by excommunication.

Another phenomenon that has been avoided is concerned with the evasion of processes that engender normal science effects. In too many countries outside the Nordic region, doctoral topics and paradigms are selected by students because they are currently considered fashionable in terms of publications by top-ranked journalists, or as opportunistic ways of entering the academic job markets. The danger lies in studying phenomena with stereotyped perspectives, and in producing a decreasing return on knowledge. Paradigmatic differentiation coupled with "publish or perish" policies foster overspecialized niches, repetitive agendas and over-formalized approaches. Technicians take over intellectuals. Nordic conditions favor the exact opposite. They allow people to take scientific risks at all ages, to explore new challenges without having to pay too high a cost for non-conformism to dominant paradigms. One may well argue that there are not many Nordic contributions that are rooted in quantitative as well as in highly formalized approaches. If we want to legitimize inductive research once again as a way of good science, we should take the Nordic tradition since the 1970s into close consideration.

Co-constructed identities

A fourth observation suggests how a common identity and scientific fate are engendered.

Nordic achievements should not be perceived as the outcome of a Prussian army type of integration. Institutional polycentrism, some forms of free riding, and strong national as well as linguistic and historical diversities, are all parts of the scene. At the same time, some processes have hindered centrifugal forces and have established socialization and cooperation processes that should not be underestimated: regional professional associations, ad hoc meetings, and ambitious cross-national projects.

Therefore, introducing regional cooperation settings and research agendas is not incompatible with building up a European scene, just as well as the establishment of the European base of EGOS is not contradictory to dialoguing at the international level with North America or Asia.

  One well-known example of the Nordic way is the Stanford consortium founded in 1988. SCANCOR (the Scandinavian Consortium for Organizational Research) generated a program of more than thirty field studies conducted on public sector organizations, especially in Norway and Sweden. Inciendtally it is worth noting that a US scholar, James March, played a major role as the founding father of a European cooperative effort, as was the case with EGOS when Michael Aiken, in 1993, convinced continental scholars to create a European forum. This program has been equally important as some other pluri-annual projects such as the Aston program on the formal structures of organization or the Centre de Sociologie des Organisations program on public administration in the 1960s and early 1970s. It has brought together a new generation of scholars and has generated many field studies. Last, but not least, it has provided a masterful demonstration of the fallacy of New Public Management, rational choice and agency theory, linking a normative approach of institutions with a powerful analytical tool kit for organizational functioning.The point to be made here is that it has created a virtuous circle of identity. Nordic scholars have published quite a few collective books devoted specifically to Nordic studies. In a way, they have become excellent marketers in the field of science. Not only do they offer original and innovative products, but they also build a brand around them. Instead of aggressively pushing a new theory with an -ism at the end, they have found seductive ways to package a series of different perspectives and agendas under a common umbrella. Reviewing the March (Brunsson and Olsen 1998) and the Olsen (Egeberg and Laegreid 1999) Festchrifte, Thoenig compares organization theory in Northern European countries to professional basketball in the USA (Thoenig 2000). The Scandinavian League, like the NBA, celebrates Most Valuable Players, identifies Rookies of the Year and shares rites. In any case, ethnographers should pay close attention to such an unusual dynamic, considering the factors that allow the participants to apparently find compromises between differentiation and integration, between the pursuit of specific sub-identities or vested interests, on the one hand, and more collective identities, on the other .Concluding remarksAny reference to the Nordic area made by a non-Nordic resident should be handled with care. Starting in the 1930s, the Nordic area was considered by the rest of the world as the region where utopias became reality. Their grass was considered greener in all kinds of aspects of social life: the emancipation of women, industrial relations, participative democracies, the humane quality of work organizations, consensus building, etc. As far as organization theory is concerned, one should not overemphasize the non-Nordic academic vices and over-idealize Nordic academic virtues. At the same time, facts show that Nordic scholars have delivered. Even the theoretical and interpretative limits of their discoveries should be considered challenges for further research, by them as well as by all scholars around the world who belong to other traditions(Thoenig 2003). The Nordic contributions have paved a new way for our understanding of organizations as social configurations, of quasi-organized action setups, and of organizing processes. All of us around the world owe them a great deal.Brunsson N. and J.P.Olsen (eds) (1998). Organizing Organizations,Stockholm, Fagbokforlaget. Byrkjeflot H. (2003). "Nordic management: Frome functional socialism to shareholder value", inCzarniawska B. and G. Sevon (eds). The Northern Lights. Organization Theory in Scandinavia, Copenhagen Business School Press. Czarniawska B. and G. Sevon (eds) (2003). The Northern Lights. Organization Theory in Scandinavia, Copenhagen Business School Press. Egeberg M. and P. Laegreid (eds) (1999). Organizing Politial Institutions, Stockholm, Scandinavian University Press. Engwall L. (2003). " On the origin of the Northern lights", in Czarniawska B. and G. Sevon (eds). The Northern Lights. Organization Theory in Scandinavia, Copenhagen Business School Press. Thoenig J.C. (2000). "New institutionalism and organizational theory: A review article", Public Administration: 455-60. Thoenig J.C. (2003). Institutional theories and public institutions: Traditions and appropriateness", Peters G. and Pierre J. (eds) Handbook of PublicAdministration, London, Sage, 2003: 127-148.

Directeur de recherche, Dauphine Recherche en Management, université Paris-Dauphine. He is a co-founder and Honorary Member of EGOS, as well as its first chairperson of the board. .jean-claude.thoenig@dauphine.fr

This number has to be considered a rough conservative estimate. It updates numbers cited byEngwall (2003)

This order of magnitude results from an informal estimate made with the help of three Nordic colleagues.