LES MODELING WITH A MULTIFIELD APPROACH

4th international conference on Turbulence and Interactions, TI2015 November 2-6, 2015, INSTITUT d'ÉTUDES SCIENTIFIQUES de CARGÈSE, Corsica, FRANCE

LES MODELING WITH A MULTIFIELD APPROACH Solène Fleau*,**,§, Stéphane Vincent**, Stéphane Mimouni* *

Electricité de France, R&D Division, Chatou, France,**Laboratoire de Modélisation et Simulation Multi-Echelle, Université Paris-Est, France. § Correspondence author. Email: [email protected]

ABSTRACT In nuclear power plants, flow studies with a Computational Fluid Dynamics (CFD) approach require the ability of dealing with inclusions of different sizes and shapes and turbulence effects. For this purpose, multifield methods have been developed to simulate separately the small spherical bubbles and the large deformable ones. In this article, we consider an approach in which the first range of bubbles are followed by an Eulerian dispersed method and the second structures are tracked by interface tracking methods within a two-fluid model. To deal with these large inclusions, we present and validate, in this paper, a model, called the Large Bubble Model, introduced for the simulation of large deformable interfaces between two continuous fields. The Large Bubble Model includes a surface tension model, a new drag force expression to couple the velocity of the two fields at the interface and the resolution of an interface sharpening equation to limit the numerical smearing induced by the two-fluid model. To take into account the turbulence effects, an a priori two-phase LES filtering is proposed with the two-fluid equations and the interfacial forces of the Large Bubble Model. This filtering highlights new subgrid terms compared to previous works done on the single-fluid model. Finally, DNS simulations are performed with a phase inversion test case to evaluate the order of magnitude of these terms and to compare five different turbulence models. INTRODUCTION The simulation of flows occurring in nuclear power plants remains a scientific challenge due to the complexity induced by the presence of bubbles with a large range of sizes and shapes and the necessity of dealing with the turbulence effects induced by the bubbles and the tubes topology. Therefore, in the code NEPTUNE_CFD, a multifield approach [Lahey and Drew 2001] based on the two-fluid model of Ishii [1975] has been developed [Mimouni et al. 2014 and Denèfle et al. 2015]. In this method, the large deformable bubbles are resolved whereas the small spherical bubbles are considered as a dispersed field. The required models for the interfacial transfers related to this dispersed field have been well studied and validated [Mimouni et al. 2011]. However, for the large deformable bubbles, a special treatment is necessary to locate the interfaces precisely and to take into account the velocity jump at the interface. Thus, in this article, we present a model, called the Large Bubble Model, developed specifically for the simulation of these interfaces between two continuous fields. This method includes the definition of a surface tension model, a drag force expression to couple the velocity of the two fields at the interface and the resolution of an interface sharpening equation to limit the numerical interface smearing induced by the two-fluid model. A validation test case with an elongated bubble is then proposed with a mesh refinement study. In the previous publications related to the Large Bubble Model [Mimouni et al. 2014, Denèfle et al. 2015 and Fleau et al. (a) 2015], the studies considered only laminar flows. Therefore, in this paper, we propose a first analysis of the turbulent effects. This work is based on an a priori two-phase LES filtering with the simulation of the inversed phase benchmark [Labourasse et al. 2007, Vincent et al. 2008 and Larocque et al. 2010]. The two-fluid model equations are

filtered to exhibit the specific subgrid terms of the model. Their order of magnitude is then compared for three mesh refinements and different turbulence models are applied to find the more appropriate for each subgrid term.

LARGE BUBBLE MODEL Two-fluid model The code NEPTUNE_CFD is based on an Eulerian approach with a finite volume discretization. The flow motion is followed using the two-fluid model of Ishii [1975] extended to n-phase. In this model, the density, the viscosity and the local velocity are defined for each field in each cell. This study is restricted to incompressible and isothermal cases with 𝜌𝑘 = 𝑐𝑠𝑡. No mass transfers are considered at the large interfaces. The following governing equations are solved for each field k: 

The mass balance equation: 𝜕𝑡 (αk 𝜌𝑘 ) + ∇. (αk 𝜌𝑘 𝑢𝑖,𝑘 ) = 0

(1)

With 𝛼𝑘 the volume fraction of field k, 𝜌𝑘 its density and 𝑢𝑖,𝑘 the ith component of the velocity. 

The momentum equation: 𝜕𝑡 (αk 𝜌𝑘 𝑢𝑖,𝑘 ) + ∇. (αk 𝜌𝑘 𝑢𝑖,𝑘 𝑢𝑗,𝑘 ) = ∇. (αk 𝜇𝑘 𝑆𝑘 ) − αk ∇𝑃 + αk 𝜌𝑘 𝑔𝑖 + 𝐹𝑖,𝑘

(2)

With 𝜇𝑘 the dynamic viscosity of field k, 𝑆 the viscous stress tensor, 𝑃 the pressure field, 𝑔 the gravitational constant and 𝐹 extra source terms due to the pressure of large scale interfaces (surface tension) or coupling terms between the continuous fields (drag forces). In the code NEPTUNE_CFD, the assumption of a common pressure for all fields is made. The solver SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) is used [Patankar and Spalding 1972] with a collocated arrangement for all variables. An iterative coupling of the equations is implemented to ensure mass conservation. Surface tension The surface tension model used in the code NEPTUNE_CFD for the simulation of the large interfaces corresponds to the volumetric formulation [Bartosiewicz et al. 2008, Denèfle et al. 2015, Fleau et al. (a) 2015] of the Continuum Surface Force (CSF) model proposed by Brackbill et al. [1992]: 𝑭𝑪𝑺𝑭 = αk 𝜎 κ ∇αk

(3)

∇α

With 𝜎 the surface tension coefficient and κ = − ∇. (||∇αk ||) the local curvature. k

Drag force In the two-fluid model, the drag force is crucial since it allows to couple the velocity of the two continuous fields at the interface. Thus, a new drag force expression has been developed to deal with large interfaces within the multifield approach. Details can be found in Mimouni et al. [2014] and Fleau et al. (a) [2015]:

𝛼2 < 0.3:

𝐅𝐛𝐮𝐛𝐛𝐥𝐞 = 𝛼1 𝛼2

18𝜇1 (𝒖 − 𝒖𝟐 ) 𝛼1 𝑑𝑝2 𝟏

𝛼2 > 0.7:

𝐅𝐝𝐫𝐨𝐩𝐥𝐞𝐭 = 𝛼1 𝛼2

18𝜇2 (𝒖 − 𝒖𝟐 ) 𝛼2 𝑑𝑝2 𝟏

0.3 ≤ 𝛼2 ≤ 0.7:

𝐅𝐦𝐢𝐱 =

(4)

0.7 − 𝛼2 𝛼2 − 0.3 𝐅𝐛𝐮𝐛𝐛𝐥𝐞 + 𝐅 0.7 − 0.3 0.7 − 0.3 𝐝𝐫𝐨𝐩𝐥𝐞𝐭 𝜇

𝛼

The characteristic length scale 𝑑𝑝 is equal to √𝜇2 ||∇𝛼𝑘 || close to the large interfaces, for 0.1 ≤ 1

𝑘

𝛼1 𝛼2 ≤ 0.25, and to the diameter of the dispersed bubbles/droplets far from them, for 𝛼1 𝛼2 ≤ 0.02. Between these two regions, a linear extrapolation of these two values is defined. Interface sharpening To limit the interface smearing induced by the resolution of the twofluid equations, the interface sharpening equation proposed by Olsson and Kreiss [2005] and adapted to the two-fluid formulation is solved for each continuous field [Fleau et al. (b) 2015]: 𝜕𝜏 αk + ∇. (αk (1 − αk )𝒏) = 𝜖∆αk ∇α

(5)

∆𝑥

With 𝒏 = ||∇αk || the interface normal vector. The values of the parameters ∆𝜏 = 32 and 𝜖 = ∆𝑥

k

(∆𝑥 denots the cell size) are chosen to obtain a final interface thickness always equal to 5 2 cells whatever the initial interface diffusion [Štrubelj 2009 and Denèfle et al. 2015].

VALIDATION OF THE LARGE BUBBLE MODEL In this section, we study a large bubble validation test case. For this purpose, an oscillating bubble in a still liquid (𝜌𝑎𝑖𝑟 = 1.17683 kg. m−3, 𝜇𝑎𝑖𝑟 = 1.85. 10−5 Pa. s, 𝜌𝑙𝑖𝑞 = 7000 kg. m−3 and 𝜇𝑙𝑖𝑞 = 4. 10−3 Pa. s, 𝜎 = 1.5 N. m) [Caltagirone et al. 2011] is simulated. With these fluid properties, the expected bubble oscillation frequency is equal to 5.71 s−1 and the characteristic time of decay to 4.37 s [Lamb 1932]. For this simulation, four different square meshes (5 cm side length) are used: 64 x 64 cells, 128 x 128 cells, 256 x 256 cells and 512 x 512 cells. The time step is kept constant and is respectively equal to 0.1 ms, 0.05 ms, 0.025 ms and 0.0125 ms. The bubble is initialized with an ellipsoidal shape with a deformation rate equal to 0.05 and a theoretical final radius of 1 cm. The results in terms of oscillation frequency are presented in Table 1. With the coarser mesh, spurious oscillations in the diagonal directions are observed. Therefore, we have to be cautious with the values displayed in this Table. The frequency converges well with the mesh refinement. Concerning the characteristic time of decay, for the mesh 512 x 512 cells, we obtain 3.16 s, which corresponds to a relative error of 28%. As a comparison, Caltagirone et al. [2011] predicts this value with a relative error of 9%. To conclude, the code is able to simulate accurately the bubble motion.

Table 1 Bubble oscillation frequency (s-1) according to the mesh refinement for an initial deformation rate of 0.05, the corresponding relative errors are given in brackets and the results are compared with Caltagirone et al. [2011] Mesh refinement NEPTUNE_CFD Caltagirone et al. [2011]

64 x 64 cells 5.71 (-) 4.79 (16%)

128 x 128 cells 5.56 (2.6%) 4.95 (13%)

256 x 256 cells 5.68 (0.5%) 4.99 (13%)

512 x 512 cells 5.67 (0.7%) 5.04 (12%)

Further validations with free surface test cases, convergence studies, evaluation of spurious velocities and comparison with other models can be found in Fleau et al. (a,b) [2015].

STUDY OF A TURBULENT TWO-PHASE FLOW: THE INVERSED PHASE BENCHMARK The inversed phase benchmark is a phase separation test case, which have been previously used in different publications to study the role and to compare the order of magnitude of the two-phase subgrid terms [Labourasse et al. 2007, Vincent et al. 2008 and Larocque et al. 2010]. Contrary to the previous studies, the simulation is performed with a two-fluid model using the Large Bubble Model presented and validated above. In this test case, an oil drop with a cubic shape (size of L/2) is initially placed in a cubic box (size L = 0.1 m) containing liquid water (see Figure 1). The evolution of the system is Figure 1. Initial conditions of the inversed driven by the gravity forces. At the end of phase benchmark. the simulation, the oil phase is supposed to be located in the top part of the box with the liquid water beneath. The fluid properties are: 𝜌𝑜𝑖𝑙 = 900 kg. m−3 , 𝜇𝑜𝑖𝑙 = 0.1 Pa. s, 𝜌𝑤𝑎𝑡𝑒𝑟 = 1000 kg. m−3 and 𝜇𝑤𝑎𝑡𝑒𝑟 = 1. 10−3 Pa. s, 𝜎 = 0.045 N. m. The test case is simulated with three different mesh refinements: 1283 cells, 2563 cells and 5123 cells. The time steps are kept constant and are respectively equal to 0.8 ms, 0.2 ms and 0.05 ms. The simulations have been performed with 144 cores for the first mesh and 1152 for the two others during respectively 7 hours, 47 hours and 2 months to reach 13 physical seconds. Macroscopic behavior First, we study different macroscopic quantities to validate the Large Bubble Model in this configuration. Thus, Figure 2 shows the evolution of the normalized 1 kinetics energy (E𝑐,𝑘 = ∑𝐼 𝛼𝑘 𝜌𝑘 𝑢𝑘2 Ω, 𝐼 denoted the cell index and Ω the cell volume), 2

1

potential energy (E𝑝,𝑘 = ∑𝐼 𝛼𝑘 𝜌𝑘 gzΩ) and enstrophy (E𝑠,𝑘 = 2 ∑𝐼 𝛼𝑘 𝑟𝑜𝑡(𝑢𝑘 )Ω, 𝑟𝑜𝑡(𝑢𝑘 ) denoted the rotational of the velocity 𝑢𝑘 ). These quantities are normalized using Table 2. The results obtained with the Large Bubble Model reproduce the same trends observed with other CFD codes [Vincent et al. 2008]. Oscillations are observed for the potential and kinetics energies due to the sloshing motion of oil when it reaches the top of the box. Concerning the

enstrophy, the location of the peak is also found to occur at around 3 in dimensionless time. Mesh convergence is obtained with the oil phase but not with the water phase. In fact, contrary to the two other quantities, enstrophy is particularly affected by the small scale motions, especially vorticity. Thus, finest grids are required to reach convergence [Vincent et al. 2015]. Table 2 Macroscopic quantities normalization for the inversed phase benchmark, 𝑈𝑔 =

𝜌𝑤𝑎𝑡𝑒𝑟 − 𝜌𝑜𝑖𝑙 𝜌𝑜𝑖𝑙

𝑔𝐿

√

2

corresponds to the gravitational velocity

Phase Oil Water

∗ 𝐸𝑐,𝑘

𝜌𝑘 𝑈𝑔2 𝐿3 = 16

3.41.10-4 J 3.78.10-4 J

∗ 𝐸𝑝,𝑘

= 𝐸𝑝,𝑘 (𝑡 → +∞) 0.1035 J 0.3755 J

∗ 𝐸𝑠,𝑘 (maximum of enstrophy for code DyJeAT with 5123 grid [Vincent et al. 2015]) 7.33.10-2 m3.s-2 1.3759 m3.s-2

𝑡∗ =

𝑈𝑔 𝐿

0.643

Figure 2. Evolution of the dimensionless kinetic energy, potential energy and enstrophy, the solid lines correspond to the oil phase, the dashed lines to water, the black curves refer to the coarser mesh: 1283 cells, the red ones to the intermediate mesh: 2563 cells and the green ones to the refined mesh: 5123 cells. Filtered two-fluid model equations As previously done in Labourasse et al. [2007] and Vincent et al. [2008] with the single-fluid model and Lakehal [2004] with the two-fluid model including a dispersed field, we apply the LES filter to the two-fluid model equations for two continuous fields within the Large Bubble Model. The subgrid terms appearing are displayed in Table 3. The subgrid term corresponding to the drag force is obtained with the same method applied to the surface tension term using Equation (4). 

The mass balance equation: 𝜌𝑘 𝜕𝑡 αk + 𝜌𝑘 ∇. (αk 𝑢𝑖,𝑘 ) + 𝜏𝑖𝑛𝑡𝑒𝑟𝑓 = 0

(6)

With αk the filtered volume fraction of field k and 𝜏𝑖𝑛𝑡𝑒𝑟𝑓 a subgrid terms related to the relationship between the filtered velocity 𝑢𝑖,𝑘 and the interface topology (see Table 3). 

The momentum equation:

𝜌𝑘 𝜕𝑡 (αk 𝑢𝑖,𝑘 ) + 𝜏𝑡𝑖𝑚𝑒 + 𝜌𝑘 ∇. (αk 𝑢𝑖,𝑘 𝑢𝑗,𝑘 ) + 𝜏𝑐𝑜𝑛𝑣 = 𝜇𝑘 ∇. (αk 𝑆𝑘 ) + 𝜏𝑑𝑖𝑓𝑓 − αk ∇𝑃 − 𝜏𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒

(7)

+ αk 𝜌𝑘 𝑔𝑖 + 𝐹𝐶𝑆𝐹 + 𝜏𝑠𝑢𝑝𝑒𝑟𝑓 + 𝐹𝐷𝑟𝑎𝑔 + 𝜏𝑑𝑟𝑎𝑔

With 𝜏𝑐𝑜𝑛𝑣 , 𝜏𝑑𝑖𝑓𝑓 the convective and diffusive subgrid terms and 𝜏𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 , 𝜏𝑠𝑢𝑝𝑒𝑟𝑓 and 𝜏𝑑𝑟𝑎𝑔 three specific subgrid terms of the two-fluid model applied to two continuous fields (see Table 3 for the expressions). We can notice that, with the Favre’s averaging, three subgrid terms 𝜏𝑖𝑛𝑡𝑒𝑟𝑓 , 𝜏𝑡𝑒𝑚𝑝𝑠 and 𝜏𝑑𝑖𝑓𝑓 are equal to zero. Therefore, the modeling effort concerns less subgrid terms. Table 3 Subgrid term appearing in the filtered two-fluid equations, 𝑢̃ 𝑖,𝑘 = ∇α

αk 𝑢𝑖,𝑘 αk

being the Favre’s

average of 𝑢𝑖,𝑘 and κ̂ = − ∇. (||∇αk ||) the filtered local curvature k

𝜏𝑑𝑖𝑓𝑓 𝜏𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝜏𝑐𝑜𝑛𝑣 𝜏𝑖𝑛𝑡𝑒𝑟𝑓 𝜏𝑡𝑖𝑚𝑒 𝜌𝑘 (∇. (αk 𝑢𝑖,𝑘 ) 𝜌𝑘 (𝜕𝑡 (αk 𝑢𝑖,𝑘 ) 𝜌𝑘 (∇. (αk 𝑢𝑖,𝑘 𝑢𝑗,𝑘 ) 𝜇𝑘 (∇. (αk 𝑆𝑘 ) αk ∇𝑃 LES filter − ∇. (αk 𝑢𝑖,𝑘 ) ) − 𝜕𝑡 (αk 𝑢𝑖,𝑘 )) − ∇. (αk 𝑢𝑖,𝑘 𝑢𝑗,𝑘 )) − ∇. (αk 𝑆𝑘 )) − αk ∇𝑃

𝜎(αk κ ∇αk − αk κ̂ ∇αk )

αk ∇ 𝑃 − αk ∇𝑃

𝜎(αk κ ∇αk − αk κ̂ ∇αk )

Filter

Normalized order of magnitude (%)

Favre’s averaging

-

𝜌𝑘 (∇. (αk 𝑢𝑖,𝑘 𝑢𝑗,𝑘 )

-

− ∇. (αk 𝑢̃ 𝑖,𝑘 𝑢̃ 𝑗,𝑘 ))

12

Conv

10

Diff

8

Superf

6

Pressure

4 Interf 2 Conv_Favre 0 128

256

512

Mesh refinement

Figure 3. Order of magnitude of the normalized subgrid terms for the oil phase, only the z component is displayed. Table 4 Classification of the subgrid terms according to their relative contribution LES filter Water 𝜏𝑠𝑢𝑝𝑒𝑟𝑓 𝜏𝑐𝑜𝑛𝑣 𝜏𝑐𝑜𝑛𝑣 𝜏𝑠𝑢𝑝𝑒𝑟𝑓 𝜏𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝜏𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝜏𝑑𝑖𝑓𝑓 𝜏𝑖𝑛𝑡𝑒𝑟𝑓 𝜏𝑖𝑛𝑡𝑒𝑟𝑓 𝜏𝑑𝑖𝑓𝑓 Oil

1 2 3 4 5

Favre’s averaging Oil Water 𝜏𝑠𝑢𝑝𝑒𝑟𝑓 𝜏𝑐𝑜𝑛𝑣 𝜏𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝜏𝑠𝑢𝑝𝑒𝑟𝑓 𝜏𝑐𝑜𝑛𝑣 𝜏𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 -

-

𝜏𝑠𝑢𝑝𝑒𝑟𝑓

These subgrid terms are compared in terms of order of magnitude to find the predominant and negligible ones. For this purpose, a top hat filter is applied to the simulation results extracted at the peak of enstrophy for the three grids. Only the first neighborhood of each cell (filter size of 2) is considered to obtain the value of the subgrid terms. Each subgrid term in the momentum equation is normalized by the convection resolved term. The interfacial subgrid term 𝜏𝑖𝑛𝑡𝑒𝑟𝑓 appearing in the mass balance equation is normalized by its corresponding resolved part. The subgrid term 𝜏𝑡𝑖𝑚𝑒 is not considered in this study since the analysis is proposed only for one time. The results are presented in Figure 3. The relative contributions of each subgrid term depends on the phase (see Table 4). Moreover, as documented in Labourasse et al. [2007], Vincent et al. [2008], Larocque et al. [2010], the subgrid term 𝜏𝑠𝑢𝑝𝑒𝑟𝑓 is predominant for

the oil phase. Moreover, a small contribution of the diffusion subgrid term 𝜏𝑑𝑖𝑓𝑓 is observed for the two phases. Nevertheless, it cannot be neglected for the oil phase. The Favre’s averaging does not affect the classification of the subgrid terms. Finally, as expected, we also observed a decrease of the magnitude of each term when the mesh is refined except for 𝜏𝑠𝑢𝑝𝑒𝑟𝑓 in the oil phase. We can note that the contribution of the drag subgrid term 𝜏𝑑𝑟𝑎𝑔 is not mentioned in Figure 3 in the interests of clarity but its order of magnitude is largely higher than the contribution of the other subgrid terms. Turbulence model Finally, we apply five different turbulence models to these subgrid terms: the Smagorinsky model [Smagorinsky 1963], the Wall-Adapting Local EddyViscosity (WALE) model [Nicoud and Ducros 1999], the Bardina’s model [Bardina et al. 1980], the mixed Smagorinsky-Bardina’s model [Bardina et al. 1983] and the Adaptative Deconvolution Model (ADM) [Adams and Stolz 2002] with an order of 6. In Figure 4, we present the relative error of each model with the DNS prediction of Figure 4. Relative error obtained by comparison each subgrid term. This figure between the modeled subgrid term and the terms highlights that the ADM is the most appropriate model for all the subgrid obtained by DNS for the oil phase, similar terms. Figure 5 displays the dispersion results are obtained with the water phase, the and the slope obtained between each Smagorinsky and WALE models applied to the model and the DNS results for the Favre’s averaged subgrid terms give the same 3 convective subgrid term. ADM is the results (relative error of 100%), mesh with 512 only model to present a slope close to 1 cells. with a limited dispersion. Nevertheless, the error for the pressure term remains high with this model. This has a limited effect since the pressure term is not predominant, as shown in the previous section. Moreover, for all the turbulence models, the error is stable when the mesh is refined. No data are given in Figure 4 for the drag subgrid term since the error is always higher than 100%. As shown in Figure 5, this is probably due to the region splitting of the drag force expression, which induces a deviation of the modeled subgrid term at the boundaries [Sagaut and Germano 2005, Carrara and DesJardin 2008]. A solution to model this subgrid term could be to adapt the phaseconditioned filtering proposed by Trontin et al. [2012] for the velocity and stress tensor jump at interfaces to the regions defined in the drag force expression. 120

Conv

Relative error (%)

100

Diff

80

Superf

60

Pressure

Interf

40 20

0

CONCLUSION A specific model, called the Large Bubble Model, has been presented and validated on an oscillating bubble test case. This model has been developed to simulate accurately a large range of two-phase flows with large deformable interfaces within the multifield approach. Work is still in progress to extend the validations to cases containing also a dispersed field to show the ability of the model to take into account the transitions between the small spherical bubbles and the larger ones. Moreover, this article has been devoted to the LES study of the two-fluid equations with large interfaces configuration. The equations filtering highlighted new subgrid terms with non negligible orders of magnitude. Finally, different models have

Figure 5. Correlation between the turbulence models and the DNS results, left: convective subgrid term for the oil phase, right: drag subrid term in the intermediate region 0.3 ≤ 𝛼2 ≤ 0.7, only the component in the z direction is considered, mesh with 1283 cells. been applied and compared. The best correlation with the DNS results has been obtained with the ADM for all the subgrid terms. The same study should be performed in the future on another test case involving large rising bubbles and the phase inversion benchmark in a more turbulent configuration, in which small spherical bubbles are created. ACKNOWLEDGEMENTS This work has been achieved in the framework of the NEPTUNE project, financially supported by CEA (Commissariat l’Énergie Atomique et aux Energies Alternatives), EDF (Électricité de France), IRSN (Institut de Radioprotection et de Sûreté Nucléaire) and AREVA NP, in collaboration with the MSME laboratory (Université Paris-Est, Champs sur Marne, France). REFERENCES Adams, N., and Stolz, S. [2002], A subgrid-scale deconvolution approach for schock capturing. J. Comput. Phys., Vol. 178, Issue 2, pp. 391–426. Bardina, J., et al. [1980], Improved subgrid scale models based on large eddy simulation, AIAA Paper, 80-1357. Bardina, J., et al. [1983], Improved turbulence models based on large eddy simulation of homogeneous, incompressible, turbulent flows, Tech. rep., Thermosciences Division, Department of Mechanical Engineering, Standford University. Bartosiewicz, Y. et al. [2008], A first assessment of the NEPTUNE_CFD code: Instabilities in a stratified flow comparison between the VOF method and a two-field approach, Int. J. Heat and Fluid Flow, Vol. 29, pp. 460–478. Brackbill, J.U. et al. [1992], A continuum method for modeling surface tension, J. Comput. Phys., Vol. 100, pp. 335–354. Caltagirone, J.P. et al. [2011], A multiphase compressible model for the simulation of multiphase flows, Comput. Fluids., Vol. 50, pp. 24-34. Carrara, M.D. and DesJardin, P.E. [2008], A filtered mass density function approach for modeling separated two-phase flows for LES II: Simulation of a droplet laden temporally developing mixing layer, Int. J. Multiphase Flow, Vol. 34, pp. 748-766.

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