1 VALIDATION OF A MULTIFIELD APPROACH FOR

VALIDATION OF A MULTIFIELD APPROACH FOR THE SIMULATIONS OF TWOPHASE FLOWS Solène Fleau*,**,1, Stéphane Mimouni*,2, Nicolas Mérigoux*,3, Stéphane Vincent**,4 * Electricité de France, R&D Division, Chatou, France. ** Laboratoire de Modélisations et Simulations Multi-Echelle, Université Paris-Est, France. 1 Correspondence author. Email: [email protected] 2 Email: [email protected], 3Email: [email protected], 4Email: [email protected]

ABSTRACT Safety issues in nuclear power plant involve complex bubbly flows. To predict the behavior of these flows, the two-fluid approach is often used. Nevertheless, this model induces a numerical diffusion of interfaces, which results in a poor accuracy in the calculation of the local parameters. Therefore, to simulate large interfaces such as slugs or free surfaces, interface tracking methods have been developed using the single-fluid model. In this paper, the two approaches have been coupled in the CMFD code NEPTUNE_CFD to simulate adiabatic separated flows. The averaged momentum balance equations are solved for each field and are followed by an artificial compression step, which fixes the interface thickness and ensures mass conservation. Moreover, since the two-fluid model defines a velocity per field in the whole computational domain, a drag force is used to couple the velocity of each field at the interface. This article proposes also a new formulation for this force, to take into account the physical properties of the flow. To validate this approach, an analytical test case with a static bubble has been simulated with a mesh refinement test. Then, the simulations of a rising bubble, an oscillating bubble and the Kelvin-Helmholtz instability have been performed to highlight the effect of the modification of the drag force. Finally, model comparisons are proposed with the KelvinHelmholtz and the Rayleigh-Taylor instabilities. KEYWORDS: Two-phase flows, Multifield approach, Drag force, Interface sharpening, Capillary effects, Large bubble test cases, Free surface.

NOMENCLATURE

Greek letters: θ ∆𝑥

m

𝜖̃

Subscripts and superscripts: Angular coordinate

𝑖𝑛𝑡

Interface

Cube root of the cell volume

i

Space direction

Deformation rate

I

Cell index

𝜌

kg.m-3 Density

k

Phase index

𝜅

m−1

𝑛𝑐𝑒𝑙

Total number of cells

Γ

W.m-3 Interfacial mass transfer

𝜔

s−1

Pulsation

σ

N.m-1

Surface tension coefficient

𝜏

s

Characteristic time scale

Interface curvature

1

𝜇

Pa.s

Viscosity

𝛺

m3

Volume element

𝜀∗

Volume fraction

Roman letters: C

-

Circularity

𝐶𝑑

-

Drag coefficient

𝑑𝑝

m

Characteristic length scale

𝑑𝑝𝑑𝑖𝑠𝑝

m

Diameter of the dispersed bubbles/droplets

g

m.s-2

Gravitational constant

I

kg. m-2.s-2

Interfacial momentum transfer

k

m−1

Wavenumber

L

m

Perimeter

n

-

Unit interface normal vector

P

Pa

Pressure

R

m

Bubble radius

S

m2

Surface

t

s

Time

T

Pa

Viscous stress tensor

U

m.s-1

Average interface velocity

𝑥

-

Convergence order

1. INTRODUCTION Many situations in nuclear power plant are characterized by liquid vapor interfaces. Whereas these flows are well controlled at normal conditions, they could threaten the integrity of the reactor pressure vessel and conduct to a contamination of the environment with radioactive nuclei in case of hypothetical accidents. Therefore, important investigations are carried out to understand these complex flows. Bubbly flows occurring in nuclear power plant are often modelled with an Eulerian dispersed description within the two-fluid model of Ishii (1975). In this method, the bubbles are small enough to be considered spherical. Thus, interfacial forces such as drag force, lift force, wall lubrication, virtual mass and turbulent dispersion force are implemented (Mimouni et al., 2011). This approach seems then inappropriate to simulate larger and deformable bubbles. In fact, the numerical diffusion induced by this method does not allow an accurate interface location. Therefore, large interfaces are usually simulated through located approaches using single-fluid models such as front tracking (Unverdi and Tryggvason, 1992), level-set (Sussman et al., 1994) or Volume Of Fluid (VOF) (Hirt and Nichols, 1981). The VOF approach has been extensively used for the simulation

2

of two-phase flows because mass conservation is ensured. Nevertheless, its low order of accuracy conducted the researchers to find alternatives. Thus, Unverdi and Tryggvason (1992) demonstrated the usefulness of the front tracking method on film boiling simulations. But, the algorithm has been criticized for its difficulty to follow topology changes at vapor-liquid interfaces. More recently, the level set methods have become interesting for their high order of accuracy. Nevertheless, Enright et al. (2002) highlighted some mass conservation issues. Thus, to deal with this problem, Olsson and Kreiss (2005) proposed a conservative scheme to solve the advection equation of the level set function. In recent works, these two approaches have been combined to take advantages of both of them. The fluid motion is simulated within the two-fluid model whereas interfaces are located with a conservative level set method. The balance equations are solved for each field using the interfacial properties, accurately evaluated thanks to the level set method. As an example, Štrubelj (2009) implemented this method in his in-house code to simulate stratified flows. His comparison with other codes highlighted the effect of the method on the interface smearing. Lately, Zuzio et al. (2013) applied this approach to a primary atomization process. This method has become crucial over the last decades with the development of the multifield approach to simulate more accurately complex flows, which can contain at the same time bubbles with a large range of sizes and free surfaces. This approach is based on the concept of a four field and a two-fluid model (Lahey and Drew, 2001). Each phase is splitted into a continuous and a dispersed field. Therefore, the interfaces between two continuous fields need to be accurately located since the flow properties are highly influenced by these structures. In this paper, we adapt the above method in the CMFD code NEPTUNE_CFD (Mimouni et al., 2014). For this purpose, the interface sharpening equation developed by Olsson and Kreiss (2005) is implemented. Furthermore, a new drag force expression is proposed to take more into account the flow properties such as the fluid viscosity. In what follows, these new developments are referred as the Large Bubble Model (LBM). In this article, we will first describe the equations of the two-fluid model in the case of two separated phases. The different relevant forces necessary for the simulation of this kind of flow will be also exposed. To validate this approach, different test cases will be simulated to cover a large range of configurations, from the large bubble test cases to the free surface test cases. Thus, a mesh refinement study will be proposed to evaluate the spurious velocities induced by the method. A comparison of different drag force laws will be also presented. Finally, the results obtained with the LBM will be compared with another available model devoted to large interfaces, called the Large Interface Model (LIM). All these comparisons will be done according to experimental, theoretical and simulation results obtained with other CFD codes. 2. COMPUTATIONAL MODEL 2.1. 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 nphase. 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, laminar and adiabatic cases. Thus, no thermal energy is exchanged and no turbulence models are used. The following governing equations are solved for each phase k:

3



The mass balance equation: 𝜕𝑡 (𝜀𝑘∗ 𝜌𝑘 ) + 𝜕𝑥𝑖 (𝜀𝑘∗ 𝜌𝑘 𝑢𝑖,𝑘 ) = 𝛤𝑘



(1)

The momentum equation in each space direction i: 𝜕𝑡 (𝜀𝑘∗ 𝜌𝑘 𝑢𝑖,𝑘 ) + 𝜕𝑥𝑗 (𝜀𝑘∗ 𝜌𝑘 𝑢𝑖,𝑘 𝑢𝑗,𝑘 ) = 𝜕𝑥𝑗 (𝜀𝑘∗ 𝑇𝑖𝑗,𝑘 ) − 𝜀𝑘∗ 𝜕𝑥𝑗 𝑃 + 𝜀𝑘∗ 𝜌𝑘 𝑔𝑖 + 𝐼𝑖,𝑘 + 𝐹𝑖,𝑘

(2)

where 𝐹𝑖,𝑘 denotes the extra source terms such as surface tension or drag models, which will be introduced further. Conservation of volume, mass and momentum leads to three others equations: ∑ 𝜀𝑘∗ = 1

(3)

𝑘

∑ 𝛤𝑘 = 0

(4)

𝑘

∑ 𝐼𝑘 = 𝑘

1 ∫ 𝜎 𝜅 𝑖𝑛𝑡 𝑛𝑖𝑛𝑡 𝑑𝑆 𝛺 𝑖𝑛𝑡

(5)

These equations do not allow to close the system. Therefore, in the code NEPTUNE_CFD, the assumption of a common pressure for all fields is made: ∀k, Pk = P

(6)

Finally, models are used for the interfacial transfers. 2.2. Large Bubble Model (LBM) This model has been specifically developed to simulate large interfaces within the multifield approach. It contains three main items detailed above: a surface tension model, a drag force law and an interface sharpening equation. Surface tension Clift et al. (1978) highlighted the necessity of surface tension forces to deal with flows containing large interfaces. Because the interface has a finite thickness in our approach, the choice is made to use the Continuum Surface Force (CSF) model proposed by Brackbill et al. (1992): 𝑭𝑪𝑺𝑭 = 𝜎κ𝐧

(7)

with n the interface normal vector: 𝒏=

∇𝜀𝑘∗ ||∇𝜀𝑘∗ ||

4

(8)

and κ the local curvature: ∇𝜀𝑘∗ κ = − ∇. ( ) ||∇𝜀𝑘∗ ||

(9)

However, since the code NEPTUNE_CFD is based on the two-fluid model, the Continuum Surface Force has to be splitted between the two fields. Therefore, the expression of the volumetric force becomes (Bartosiewicz et al., 2008): 𝑭𝑪𝑺𝑭 = βk 𝜎κ∇𝜀𝑘∗ Ω

(10)

The coefficient βk is chosen equal to 𝜀𝑘∗ . In fact, Štrubelj (2009) compared this formulation to a mass formulation by simulating a pressure jump over a droplet interface. In this test case, the density ratio is large and surface tension plays a dominant role. He showed that the differences between the two models were minimal but the volume averaging still gave better results. Drag force In the two-fluid model, the drag force is crucial. Contrary to the single-fluid approach, two different velocities are defined, one for each phase. Therefore, at the interface, these two velocities can have two different values. The role of this force is then to couple these velocities at the resolved interface. Different models have been developed to deal with large interfaces. Here, we present three of them, which will be compared in section 3 with different test cases. 

Law 1: This drag force is used in the Large Interface Method (LIM) developed and extensively validated in Coste et al. (2012). 𝐅𝐃𝐫𝐚𝐠 = 𝜀𝑙∗ 𝜀𝑔∗ 𝐶𝑑 (𝒖𝒍 − 𝒖𝒈 )

(11)

1 𝐶𝑑 = 103 + (1 − cos(𝜋𝜀𝑙∗ ))(2. 104 − 103 ) 2

(12)

With 𝐶𝑑 the drag force coefficient:



Law 2: This drag force was used by Štrubelj (2009) and Denèfle et al. (2015) for the simulation of large interfaces within the two-fluid model. The value of the parameter τ, proportional to the time step in the two papers (Štrubelj, 2009 and Denèfle et al., 2015), is replaced here by a constant value equal to 1. 10−7 𝑠: 𝐅𝐃𝐫𝐚𝐠 = 𝜀𝑙∗ 𝜀𝑔∗

𝜀𝑙∗ 𝜌𝑙 + 𝜀𝑔∗ 𝜌𝑔 (𝒖𝒍 − 𝒖𝒈 ) 𝜏

(13)

Nevertheless, Denèfle et al. (2015) simulated the experiment of Raymond and Rosant (2000) with this drag force. He showed that the results were in better agreement with the experimental data for cases with higher viscosities. 

Law 3: Therefore, we developed a new expression. The new formulation is based on the drag force expression applied to the dispersed fields in the approximation of spherical bubbles/droplets. The use of the drag force expression devoted to the dispersed fields as an initial step to establish a new drag force expression for the large interfaces is totally integrated in the concept of the multifield approach. In fact, in a region containing large interfaces, a specific drag

5

force will be applied. This drag force will be consistent with the drag force applied in another region of the domain with only dispersed inclusions. Thus, this ensures a smooth transition of the drag force in the whole computational domain whatever the type of inclusions. To obtain the new drag force expression for the large interfaces, let us consider the example of a bubbly flow: 𝜌𝑔 1 (𝒖 − 𝒖𝒈 ) 𝜀𝑙∗ 𝜏 𝒍

(14)

1 3𝜌𝑙 𝐶𝑑 = ||𝒖𝒍 − 𝒖𝒈 || τ 4𝜌𝑔 𝑑𝑝

(15)

𝐅𝐃𝐫𝐚𝐠 = The Ishii’s definition of τ (Ishii, 1975) is used:

The drag coefficient 𝐶𝑑 is then defined by the Schiller and Nauman equation (Schiller and Nauman, 1935). The obtained formulation is extended to large interfaces using a continuous approximation. The large interfaces are considered as a serie of aligned dispersed structures, which formed a continuous boundary between two continuous fields, as shown in Figure 1. Therefore, the drag force is multiplied by 𝜀𝑙∗ 𝜀𝑔∗ to restrict its application at the interface when the value of 𝜀𝑙∗ 𝜀𝑔∗ is non zero: 𝜀𝑔∗ < 0.3 ∶

𝐅𝐛𝐮𝐛𝐛𝐥𝐞 = 𝜀𝑙∗ 𝜀𝑔∗

18𝜇𝑙 (𝒖 − 𝒖𝒈 ) 𝜀𝑙∗ 𝑑𝑝2 𝒍

𝜀𝑔∗ > 0.7 ∶

𝐅𝐝𝐫𝐨𝐩𝐥𝐞𝐭 = 𝜀𝑙∗ 𝜀𝑔∗

18𝜇𝑔 (𝒖 − 𝒖𝒈 ) 𝜀𝑔∗ 𝑑𝑝2 𝒍

0.3 ≤ 𝜀𝑔∗ ≤ 0.7 ∶

𝐅𝐦𝐢𝐱

(16)

0.7 − 𝜀𝑔∗ 𝜀𝑔∗ − 0.3 = 𝐅 + 𝐅 0.7 − 0.3 𝐛𝐮𝐛𝐛𝐥𝐞 0.7 − 0.3 𝐝𝐫𝐨𝐩𝐥𝐞𝐭

The peculiarity of this new definition is that the viscosity becomes a parameter of the drag force intensity. Moreover, the new drag force anticipates the concept of the multifield approach, recently developed for the simulation of complex flows (Lahey and Drew, 2001). In fact, the expression depends on the properties of the continuous phase, which is determined in each cell by the value of 𝜀𝑔∗ :   

𝜀𝑔∗ < 0.3 : continuous liquid phase, 𝜀𝑔∗ > 0.7 : continuous gas phase, 0.3 ≤ 𝜀𝑔∗ ≤ 0.7 : mixture of gas and liquid.

Therefore, with this drag force, each phase can be considered as a continuous and a dispersed field in the same flow. Smooth transitions between these fields are ensured by the definition of Fmix . The multifield model is also taken into account in the definition of the characteristic distance 𝑑𝑝 . The large interfaces and the small spherical bubbles, which belong to a dispersed field, are distinguished according to the value of 𝜀𝑙∗ 𝜀𝑔∗ . Thus, in the region of the large interfaces, after resolution of the interface sharpening equation, 0.1 ≤ 𝜀𝑙∗ 𝜀𝑔∗ ≤ 0.25. In this domain, 𝑑𝑝 is evaluated by a local quantity

𝜀𝑘∗

||∇𝜀𝑘∗ ||

𝜇𝑔

corrected by the factor √ 𝜇 . This factor is necessary since the drag force model has been initially 𝑙

developed for isolated spherical particles and is now applied to large interfaces without any assumptions

6

on their shape. Then, in cells containing a dispersed field, like dispersed bubbles for instance, 𝜀𝑔∗ < 𝜀𝑙∗ with 𝜀𝑙∗ > 0.98. Therefore, for 𝜀𝑙∗ 𝜀𝑔∗ < 0.02, 𝑑𝑝 is chosen equal to the diameter of the dispersed bubbles/droplets. Between these two regions, 𝑑𝑝 is defined by interpolation of its two extreme values: 𝑑𝑝 =

𝜀𝑙∗ 𝜀𝑔∗ − 𝑏 𝜀𝑙∗ 𝜀𝑔∗ − 𝑎 𝜀𝑘∗ 𝜇𝑔 𝑑𝑝𝑑𝑖𝑠𝑝 + ∗ √ 𝑎−𝑏 𝑏 − 𝑎 ||∇𝜀𝑘 || 𝜇𝑙

(17)

with a = 0.02 and b = 0.1. An illustration of the method chosen to define 𝑑𝑝 is proposed in Figure 2. This distribution of the characteristic distance 𝑑𝑝 ensures a smooth transition between this drag force, applied to the interfaces separating two continuous fields, and the drag force between a dispersed and a continuous field. Interface sharpening In the code NEPTUNE_CFD, the interface is located using a color function. Nevertheless, in the two-fluid model, the numerical diffusion of this interface does not allow an accurate calculation of the interface properties such as the local curvature and the interface normal vector. Therefore, we locally solve the artificial compression equation proposed by Olsson and Kreiss (2005): 𝜕𝜏 𝜀𝑘∗ + ∇. (𝜀𝑘∗ (1 − 𝜀𝑘∗ )𝒏) = 𝜖∆𝜀𝑘∗

(18)

The viscosity term 𝜖∆𝜀𝑘∗ was added by Olsson and Kreiss (2005) to prevent discontinuities at the interface. The values of the parameters ∆𝜏 and 𝜖 are chosen to obtain a final interface thickness always equal to 5 cells whatever the initial interface diffusion, as shown in Figure 3. Štrubelj (2009) proposed: ∆𝜏 =

∆𝑥 32

and

𝜖=

∆𝑥 2

(19)

The interface sharpening equation is solved until convergence is obtained. The convergence criterion is based on the ratio between the variation of the volume occupied by the interface and its initial volume. This ratio has to be higher than a certain threshold 𝛽, fixed at 5.10−4 according to the best results obtained on single bubble dynamics validations (Bhaga and Weber, 1981). The criterion is evaluated at the interface, where 𝜀𝑙∗ 𝜀𝑔∗ > 0.02 and has the following expression: 𝑛𝑐𝑒𝑙

𝑛𝑐𝑒𝑙

∑ 𝛿𝜀𝑘𝐼 (1 − 𝜀𝑘𝐼 )𝛺𝐼 < 𝛽 ∑ 𝜀1𝐼 𝜀2𝐼 𝛺 𝐼 𝐼

(20)

𝐼

Therefore, in Equation (20), the term on the left-hand side corresponds to the volume variation after each resolution of the interface sharpening equation and the sum of the right-hand side evaluates the initial interface volume, before compression. The resolution of the interface sharpening equation is activated only if the interface is diffused. Thus, at the interface (for 𝜀𝑙∗ 𝜀𝑔∗ > 0.02), the term ∇𝜀𝑘𝐼 . 𝒏, which is equal to ||∇𝜀𝑘𝐼 ||, is calculated and compared 1

to 5∆𝑥. If the interface is diffused, then its thickness is higher than the thickness obtained after the resolution of the interface sharpening equation, which is fixed at 5 cells. Therefore, ||∇𝜀𝑘𝐼 || is smaller 1 than 5∆𝑥. Further details and validations of the interface sharpening equation implementation can be found in Fleau et al. (2015).

7

2.3. Large Interface Model (LIM) This model has been developed to deal with large interfaces in a two-field code and implemented in NEPTUNE_CFD (Coste, 2013). Validations of this model can be found in Coste et al. (2012). In this section, we propose to detail this model which will be used to validate the Large Bubble Model. In order to simulate large interfaces, the first step of the LIM is to recognize which cells are crossed by them: here comes the interface recognition. The second step is to calculate the necessary inputs of the specific models applied to the marge interfaces: velocities, turbulent quantities and the distance between the computational point where these inputs are calculated and the interface. Here comes the three-cell thick approach. Notice that these two steps are independent. One should not mistake one for the other. At each time step, starting from the computed gas volume fraction for a given grid, a large interface recognition algorithm gives the cells (ST) crossed by the large interface. The standard NEPTUNE_CFD LIM interface recognition algorithm is described in (Laviéville and Coste, 2008). It gives a correct recognition in many flow configurations which means that a great majority of cells is correctly recognized as crossed by a large interface (as displayed in Figure 4) but not all cells in any complex situation. Under the action of gravity, the interface thickness is ensured and the recognition algorithm provides satisfactory results. However, the interface recognition is intrinsically intended neither to detect large interfaces when the interface is too much diffused nor to correct the interface smearing when the mesh refinement is not sufficient to follow the interface location or when some transient effects may disturb the interface in the CFD. In other words, on the contrary to reconstruction methods, it is not intended to compensate some CFD limits with the use of an additional numerical equation which controls the mass position, at the risk of violating the basic mass and momentum equations results. However, in some CFD cases which a priori deal mainly with large interfaces, such reconstruction techniques may lead to interesting results. In the view to better master the interface phenomena, it is necessary, on the one hand, to carefully handle the residual phases in the closure laws, i.e. gas variables on the liquid side of the large interface when the gas volume fraction is negligible there, or liquid variables on the gas side of the large interface when the liquid volume fraction is negligible there. On the other hand, the variables input in the physical closure laws should be physically relevant. Moreover, the results should not be dependent on the relative position of the large interface with regard to the computational cells. These are reasons why the large interface models were written within a three-cell stencil: one cell on the liquid side (STL), one cell in the intermediate region (ST) and one cell on the gas side (STG), both sides being detected with the help of the local ∇𝜀𝑙∗ which gives the surface normal 𝒏 (Coste et al. , 2007). The group of these cells triples makes a three-cell thick line in a 2D calculation or a three-cell thick surface in a 3D calculation. The main specific large interface physical models are as follows. An anisotropic two-phase friction is assumed close to the free surfaces. The friction of bubbles with liquid, in the case of bubbles coming up to a free horizontal stratified flow surface, for example, is different from the friction of the gas over the free surface. Then, the drag coefficient in the direction normal to the large interface is derived from classical bubbles or drops drags whereas the drag coefficient in the large interface plane is specific to large interfaces, with the hypothesis that the free surface is a wall for the gas, a wall moving at the interface velocity, a wall that can be rough when there are waves. 2.4. Numerical scheme A Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) solver is used in the code NEPTUNE_CFD (Patankar and Spalding, 1972). This algorithm can be summarized as followed in adiabatic conditions:   

Set the boundary conditions, Evaluate the gradients of velocity and pressure, Solve the discretized momentum equation to estimate an intermediate velocity field,

8

     

Compute the intermediate mass fluxes at cell faces, Solve the pressure correction equation, Update the pressure field by taking into account the pressure correction, Solve the interface sharpening equation, Correct the face mass fluxes, Correct the velocity field using the pressure correction.

An iterative coupling of the equations is applied to ensure mass conservation in each cell and in the whole domain. The data structure is face-based to allow simulations on arbitrary-shaped cells including nonconforming meshes. 3. SIMULATION RESULTS In this part, we will validate the approach developed above on three different test cases. Beginning with a validation test case including a mesh refinement analysis, we will then compare the effect of the modification of the drag force expression in the LBM with the simulation of three different test cases: the Bhaga’s rising bubble test case (Bhaga and Weber, 1981), an oscillating bubble and the Thorpe’s experiment (Thorpe, 1969). A model comparison is also proposed with the second and the fourth cases. Finally, the results of the simulation of the Rayleigh-Taylor instability will be compared with other codes. Thus, the approach will be validated on three large bubble test cases and two free surface configurations. 3.1. Static bubble For this test case, an air bubble is simulated in still water with an initial round shape and without gravity forces. Its initial diameter is taken equal to 2 cm. The mesh is a square, with 5-cm sides. Seven different mesh refinements are tested: 45 x 45 cells, 64 x 64 cells, 91 x 91 cells, 128 x 128 cells, 181 x 181 cells, 256 x 256 cells and 512 x 512 cells. The time step is constant and equal to 0.1 ms. Therefore, the CFL number is kept under 0.9. The simulations are performed for 1 s. At this time, we assume that the bubble has reached its equilibrium state. For all the simulations, the LBM with the third drag force law is used. In this case only, the criterion used to solve the interface sharpening equation only on diffused interface is not activated to see the effect of this compression. The quantities observed are the relative error for the circularity C, evaluated using Equation (21), for the pressure, defined by the Laplace equation (24), and the average bubble velocity. 𝐶 =

2𝜋𝑅𝑛 𝐿

(21)

where L denotes the bubble perimeter at the end of the simulation. 𝑅𝑛 , the final bubble radius, has the following expression: 𝑆𝑛 𝑅𝑛 = √ 𝜋

(22)

with 𝑆𝑛 the estimated bubble surface: 𝑛𝑐𝑒𝑙

𝑆𝑛 = ∑ 𝜀𝑔𝐼 𝐼

9

(23)

Finally, the Laplace equation is given by: 𝑝𝑖𝑛 − 𝑝𝑜𝑢𝑡 =

𝜎 𝑅𝑛

(24)

where 𝑝𝑖𝑛 is the pressure in the bubble and 𝑝𝑜𝑢𝑡 out of the bubble, 𝜎 corresponds to the surface tension coefficient equal to 0.08 N.m-1. To calculate 𝑝𝑖𝑛 and 𝑝𝑜𝑢𝑡 , the following expressions are used : 𝑝𝑖𝑛 =

∑𝑛𝑐𝑒𝑙 𝜀𝑔𝐼 𝑝𝛺 𝐼 𝐼

and

∑𝑛𝑐𝑒𝑙 𝜀𝑔𝐼 𝛺 𝐼 𝐼

𝑝𝑜𝑢𝑡

∑𝑛𝑐𝑒𝑙 𝜀𝑙𝐼 𝑝𝛺 𝐼 𝐼 = 𝑛𝑐𝑒𝑙 𝐼 𝐼 ∑𝐼 𝜀𝑙 𝛺

(25)

This test case is particularly interesting to quantify the spurious velocities induced by a model since the motion of the bubble depends entirely on these velocities. The mesh refinement study highlights also the order of convergence of the code for different quantities. In Figures 5 and 6, the relative error for the circularity and the pressure and the average bubble velocity are plotted for the seven mesh refinements. The X axis of the six graphs (Figures 5 and 6) corresponds to the dimensionless quantity obtained by dividing the bubble diameter by the cell length. In Figure 5, we note a convergence of the three quantities, especially for the average bubble velocity. Therefore, the intensity of spurious velocities decreases with the mesh refinement and remains low even with coarse meshes. To evaluate the order of convergence x for the quantity X, we apply the following expression (Roache, 1998), based on the Richardson’s extrapolation:

𝑥=

𝑋𝑚 − 𝑋𝑚 ln (𝑋 3 − 𝑋 2 ) 𝑚2

1 ln (2)

𝑚1

(26)

with 𝑚1 , 𝑚2 and 𝑚3 three mesh refinements, such as: ∆𝑥𝑚3 =

∆𝑥𝑚2 ∆𝑥𝑚1 = 2 4

(27)

To determine the order of convergence, the meshes with 128 x 128 cells, 256 x 256 cells and 512 x 512 cells are used. In fact, as we can see in Figure 6, these meshes belong to the asymptotic regions, where the Richardson’s extrapolation is valid. Therefore, for the circularity, the order of convergence is equal to 1.7, for the pressure 1.4 and for the average velocity to 0.4. 3.2. Bhaga’s rising bubble test case In this test case, an air bubble is rising up in viscous water. For this simulation, the properties of the two fluids are: 𝜌𝑙 = 1350 kg. m−3 , 𝜇𝑙 = 0.77 Pa. s, 𝜌𝑔 = 1.35 kg. m−3 and 𝜇𝑔 = 1.8. 10−5 Pa. s. Surface tension is equal to 0.0785 N.m-1. An hydrostatic pressure is imposed in the column: P = 𝑃𝑜 + 𝜌𝑙 𝑔(𝑧𝑚𝑎𝑥 − 𝑧) With 𝑃𝑜 the atmospheric pressure.

10

(28)

The bubble radius is initialized at 1.3 cm in our simulations. Moreover, the bubble is initially located at 3.9 cm from the top of the mesh, which corresponds to three radii. The dimensions of the computational domain are chosen large enough to avoid wall effects on the bubble and high enough to reach the limit velocity. Therefore, to limit CPU consumption, a 2D axisymmetric mesh, whose definition sketch is given in Figure 7, is used. The mesh contains 179 x 540 cells. The time step is kept constant, equal to 1.10-4 s to have a maximum CFL number equal to 0.9. The bubble shape and final velocity is extracted at 0.6 s before inducing bubble deformations due to the outlet on top of the domain. Comparison of the drag force expression The simulation is performed with the three drag force laws presented in section 2.2. The predicted bubble shapes are displayed in Figure 8. No results are shown for the first drag force law because the model induces a fragmentation of the bubble as soon as it begins to rise. For the two other models, a reasonable agreement with the experimental data is obtained. The second law (Figure 8 left) elongates the bubble extremities whereas the third law causes a lateral elongation. In terms of final velocity (obtained at 0.6 s), the second law predicts 29.3 cm.s-1 corresponding to a relative error of 1 % (according to the experimental results, Bhaga and Weber, 1981) and the third law 28.9 cm.s-1, 0.3 % of relative error. Therefore, the second and the third drag laws allow an accurate simulation of the Bhaga’s bubble. In what follows, we do not consider the first drag force law in the comparisons since it has not been able to keep the integrity of the Bhaga’s rising bubble. Comparison with the LIM The simulation is now performed with the LIM. The bubbles shapes are compared with the experimental results in Figure 8. We see that the two methods (LBM and LIM) overpredict the bubble width. Concerning the final velocity of the bubble, the LIM predicts 27.7 cm.s-1 corresponding to a relative error of 4.5 %. Thus, the results are very close between the two models. 3.3. Oscillating bubble An oscillating air bubble in a still liquid has been simulated without gravity (Caltagirone et al., 2011) to validate the new drag force expression developed for the LBM. The fluid properties are: 𝜌𝑙 = 7000 kg. m−3 , 𝜇𝑙 = 4.0. 10−2 Pa. s, 𝜌𝑔 = 1.17683 kg. m−3 and 𝜇𝑔 = 1.8. 10−5 Pa. s. Surface tension coefficient is equal to 1.5 N.m-1. The mesh is a square with 5 cm side length and 512 x 512 cells. The bubble is initialized with an ellipsoidal shape with a semi-minor axis equal to 0.95 cm and a semi-major axis of 1.05 cm. The interfacial position of the bubble is given in polar coordinates by Lamb (1932): 𝑅(𝜃, 𝑡) = 𝑅0 (1 + 𝜖̃ cos(2𝜃) cos(𝜔𝑑 𝑡) exp (−

𝑡 )) 𝜏𝑑

(29)

With 𝑅0 the final bubble radius equal to 1 cm, 𝜖̃ the initial perturbation of the bubble equal here to 0.05, 𝜃 the angular coordinate, 𝜔𝑑 the oscillation frequency and 𝜏𝑑 the characteristic time of decay due to viscous damping: 𝜔𝑑 2 =

6𝜎 𝑅0 3 (𝜌𝑙 + 𝜌𝑔 )

and

𝜏𝑑 =

𝑅0 2 (𝜌𝑙 + 𝜌𝑔 ) 4(𝜇𝑙 + 3𝜇𝑔 )

(30)

With the fluid properties, the expected bubble frequency is equal to 5.71 s-1 and the characteristic time of decay to 4.37 s.

11

The results of the simulations obtained with the second and the third drag forces are presented in Table 1. We see that the new drag force model predicts better the frequency and the characteristic time of decay of the bubble. As a comparison, Caltagirone et al. (2011) with a VOF-PLIC method found a relative error for the oscillation frequency equal to 12 % and for the characteristic time of decay of 9 %. 3.4. Kelvin-Helmholtz instability In this study, the Kelvin-Helmholtz instability is observed in the Thorpe’s experiment configuration (Thorpe, 1969). With this test case, we propose different levels of validation. First, as previously done with the Bhaga’s rising bubble and the oscillating bubbles, we will compare the two last drag forces. Then, the simulation will be performed with the LIM, which is devoted to the simulation of large interfaces. Theory Two immiscible fluids are contained in a rectangular box, which is tilted for a small angle, sin(γ) = 0.072, as displayed in Figure 9. The Kelvin-Helmholtz instability is observed when the relative velocity between the two fluids exceeds a critical velocity. In this test case, the velocity difference between the two fluids can be high in the bulk. Therefore, the two-fluid model can predict high relative velocities at the sheared interface. Thus, in this context, the drag force plays a crucial role. In fact, if the intensity of the drag force is inadequate, the simulated flow behavior can be dramatically affected. The sheared interface highlights the role of the drag force to predict the flow behavior with a high level of accuracy. We consider here an inviscid fluid flow with: 𝜌={ with ℎ = ℎ1 = ℎ2 =

𝐻 2

𝜌2 for 0 < 𝑧 < ℎ 𝜌1 𝑓𝑜𝑟 ℎ < 𝑧 < 𝐻

(31)

.

The steady velocity distribution along the rectangular tube is: ∆𝑢 𝑓𝑜𝑟 0 < 𝑧 < ℎ 2 𝑢={ ∆𝑢 𝑓𝑜𝑟 ℎ < 𝑧 < 𝐻 2 −

(32)

This parallel flow is assumed to be a solution of Euler equations upon which is superposed a small perturbation proportional to exp(𝑖(𝑘𝑥 + 𝜔𝑡)). The linearization of the Euler equations gives the following dispersion relation: ∆𝑢(𝜌2 − 𝜌1 ) 𝜎𝑘 3 + 𝑔𝑘(𝜌2 − 𝜌1 ) 𝑘 2 ∆𝑢2 𝜌1 𝜌2 𝜔=𝑘 ±√ th(𝑘ℎ) − 2(𝜌1 + 𝜌2 ) (𝜌1 + 𝜌2 ) (𝜌1 + 𝜌2 )2

(33)

The system becomes unstable when the complex part of ω is non-zero, which provides the condition for the minimum critical velocity difference: ∆𝑢2 >

(𝜌1 + 𝜌2 ) 𝑔(𝜌2 − 𝜌1 ) (𝜎𝑘 + ) th(𝑘ℎ) 𝜌1 𝜌2 𝑘

12

(34)

The critical wavenumber is then obtained by calculating the minimum of the right-hand side of Equation (34): 𝑘𝑐 = √

𝑔(𝜌2 − 𝜌1 ) 𝜎

(35)

This corresponds to a theoretical value of 232 m−1. Experimentally, the measured values were equal to 𝑘𝑐 = 197 ± 58 m−1. Moreover, if viscosity and closed-end effects are neglected, the velocity distribution at the beginning of the simulation is: (𝜌2 − 𝜌1 )ℎ1 𝑔sin(𝛾) 𝑡 𝑓𝑜𝑟 0 < 𝑧 < ℎ (𝜌1 ℎ2 + 𝜌2 ℎ1 ) (𝜌2 − 𝜌1 )ℎ2 𝑔sin(𝛾) 𝑢1 = 𝑡 𝑓𝑜𝑟 ℎ < 𝑧 < 𝐻 (𝜌1 ℎ2 + 𝜌2 ℎ1 ) { 𝑢2 = −

(36)

From this equation, Thorpe (1969) predicted the time of the instability onset between 1.5 s and 1.7 s and observed experimentally tonset = 1.88 ± 0.07 s. Finally, he evaluated the wave velocity at uwaves = 2.38 cm.s-1 and observed uwaves = 2.6 cm.s-1. Both fluid layers have the same initial height h1 = h2 =1.5 cm. The properties of the two fluids are: ρ1 = 780 kg.m-3, μ1 = 1.5.10-3 Pa.s, ρ2 = 1000 kg.m-3 and μ2 = 1.10-3 Pa.s. Surface tension is equal to σ = 0.04 N.m-1. The dimensions of the computational domain are L = 1.83 m and H = 3 cm (see Figure 9). The mesh contains 80 x 4880 cells. A wall boundary condition is imposed everywhere except in front and behind, where symmetry boundary planes are defined. The simulation is performed with a constant time step equal to 0.5 ms, which ensures that the CFL number stays under 0.9. Comparison of the drag force expression With this test case, we first observe the effect of the modification of the drag force. The results are compared in terms of interface shape, critical wavenumber, wave velocity, time of the instability onset and evolution of the maximum value of the average interface velocity at the beginning of the simulation. The following expression is used to evaluate U: U=

𝜀1∗ 𝜌1 𝑢1 + 𝜀2∗ 𝜌2 𝑢2 𝜀1∗ 𝜌1 + 𝜀2∗ 𝜌2

(37)

In Figure 10, an example of the evolution of the Kelvin Helmholtz instability over time is displayed for the two drag forces. We can notice that the waves appeared later with the second drag force with τ = 1.10−7 s. To determine the critical wavenumber, the interface profile is extracted at 3 s in Figure 11 and the waves distance is evaluated. Therefore, the critical wavenumber obtained with the second drag force with τ = 1.10−7 s is equal to 222 m−1 and to 219 m−1 with the third drag force. These results are in good agreement with the experimental and theoretical data. As a comparison, Bartosiewicz et al. (2008) found 143 m−1 and Štrubelj (2009) 157 m−1. The interface is also examined in terms of amplitude growth in order to evaluate the time of the instability onset. For this purpose, the standard deviation of the interface is evaluated every 0.2 s between 1 s and 3.4 s. The results are shown in Figure 12. With the second drag force with τ = 1.10−7 s, tonset is found equal to 2.4 s and 2.1 s with the third drag force. These results agree well with the simulations of

13

Bartosiewicz et al. (2008) with tonset = 1.9 s, Štrubelj (2009) with tonset = 2 s and the theoretical and experimental data. Moreover, Figure 13 presents the maximum value of the average interface velocity U over time. The results are compared with Equation (36), which is valid at short times, where the linear approximation can be applied. Therefore, we see that the two drag forces ensure an accurate prediction of the interface velocity. In Figure 14, the velocity profiles are displayed for the two drag forces. We can notice that, at the interface, the velocities of the two fluids are equal, which confirms that the new drag force expression ensures the cancelation of the velocity differences between the two continuous fields. Moreover, we observe that the profiles are symmetrical. In fact, in our case: ℎ1 = ℎ2 = 1.5 𝑐𝑚. Thus, Equation (36) predicts that, for short times, the velocity of each field has the same magnitude and evolves in an opposite direction. In Figure 14, we see that, from 2 s, the symmetry begins to disappear. Finally, the wave speed uwaves is evaluated by calculating the crest-to-crest distance at different positions in the tube. Figure 15 gives an example of the waves used for this calculation. We find uwaves = 3.3 cm.s1 with the second drag force with τ = 1.10−7 s and uwaves = 3.1 cm.s-1 with the third drag force. These results agree well with the simulations of Bartosiewicz et al. (2008) with uwaves = 2.5 cm.s-1 and Štrubelj (2009) with uwaves = 3 cm.s-1. In Table 2, all the parameters evaluated in this simulation are summarized and compared with the theoretical and experimental data and other simulation results. Thus, we see that the flow behavior is well predicted in our simulation. Moreover, this test case highlights that the third drag force improves significantly the results. Comparison with the LIM The same study is then performed with the LIM. The critical wavenumber, time of the instability onset and the wave speed are displayed in Table 2. We see that the three parameters are higher than the results predicted by the LBM. Compared to the experimental and theoretical data, it seems that the critical wavenumber and the wave velocity are slightly overpredicted by the LIM. Nevertheless, the results remain with the same order of magnitude with the two models. Then, in Figure 13, the maximum value of the average interface velocity U over time is displayed. The evolution at the beginning of the simulation, where the linear approximation can be applied, is predicted with the same accuracy for the LBM and LIM. Thus, the LBM allows to obtain accurate results compared to other methods dedicated to the simulations of large interfaces. For some quantities, the error is even reduced by using the LBM. 3.5. Rayleigh-Taylor instability The Rayleigh-Taylor instability occurs in a system with two immiscible fluids of different densities in the presence of a gravity field, perpendicular to the interface. The fluid with higher density (ρ1 = 3 kg.m-3, μ1 = 0.03 Pa.s) is initially located above the fluid with lower density (ρ2 = 1,78.10−3 kg.m-3, μ2 = 0.01 Pa.s). The aim of this test case is to compare our results to other codes. Theory We define the Atwood number At , which is equal to 0.5 in our case: At =

𝜌1 − 𝜌2 𝜌1 + 𝜌2

(38)

As long as the flow can be analyzed with linearized Navier-Stokes’ equations, the amplitude of the interface deformation can be expressed as followed: a(t) = 𝛿0 exp(𝜔𝑡)

14

(39)

with 𝛿0 the initial amplitude of the interface equal to 1 mm. The pulsation ω is given by: 𝜔2 = 𝑔𝑘At +

𝑘2𝜎 𝜌1 + 𝜌2

(40)

𝜋

with g = 10 m.s-2, 𝑘 = 𝐿 the wavenumber of the initial perturbation. The simulation is performed in a closed box (L = 1 m, H = 5 m). The mesh contains 96 x 480 cells. A wall boundary condition is imposed at the top and the bottom of the mesh and symmetry boundary planes everywhere else. A variable time step is chosen for the simulation. Its initial value is taken equal to 1 ms and the maximum CFL number is fixed at 0.9. The interface between the two fluids is initialized as a small cosine wave with amplitude equal to 1 mm. The following expression is implemented for the initialization: 2πx 𝑧 = 𝛿0 (cos ( − π) + 1) L

(41)

with 0 < x < L. The simulations are performed without surface tension. Comparison with other codes The interface position over time predicted by the LBM with the third drag force expression are compared with the theory using the analytical expression (36) and the results obtained by Štrubelj (2009). Figure 16 shows that our simulation predicts well the interface position for short times. The difference observed at the very beginning is due to the interface initialization. Figure 17 proposes a comparison of our results with other codes. We can notice that the interface smearing is well controlled by the use of interface sharpening or the geometrical interface reconstruction in code FLUENT. Moreover, the shape of the mushroom obtained with the different codes remains the same except with the code FLUENT, where the mushroom is more lately extended. A first difference can be observed with the Štrubelj’s in-house code (Štrubelj, 2009). In fact, some particles of the above fluid are early created and detached due to the mushroom extension. This is due to the implementation of a transition criterion between a stratified flow and a dispersed flow. Moreover, at t = 4 s, the mushroom keeps its symmetrical shape with CFX and with the Štrubelj’s in-house code but not with FLUENT and with our simulation. This loss of symmetry is caused by spurious velocities induced by numerical diffusion. Thus, the more diffusive the numerical method is, the less accurate the simulations are but also the less sensitive to spurious velocities they are. In CFX and the Štrubelj’s in-house code, the numerical methods are more diffusive, which prevents them from numerical instabilities. Nevertheless, this effect is well controlled in the code NEPTUNE_CFD and was analyzed in the first study with the simulation of the static bubble. Finally, we can note that, for all the test cases presented in this section, the mass balance error in the whole domain is equal to 10−17 %.

15

4. CONCLUSION A two-fluid model coupled with a conservative level set method has been implemented in the CMFD code NEPTUNE_CFD to simulate separated flows. This approach includes an artificial compression step to control the numerical interface diffusion induced by the two-fluid approach. This adding equation allows a more accurate evaluation of the local parameters such as the interface normal vector or the curvature. Moreover, because the two-fluid model can predict non zero relative velocities at large interfaces, the drag force plays an important role in the simulation of free surfaces to cancel the velocity difference between the two continuous fields. Thus, in this paper, a new expression of this force has been developed, which takes into account the physical properties of the flow. To validate this method, a mesh refinement test has been first performed on a static bubble. The simulations showed a reasonable mesh convergence especially for the intensity of the spurious velocities, which remains low even with coarse meshes. Then, four other test cases have been simulated: two with large bubbles and two free surfaces test cases. These simulations highlighted that the LBM with the new drag force expression (law 3) allowed to obtain results in good agreement with experimental, theoretical data and other code simulation results. Moreover, the comparison with the LIM showed that the LBM was able to reproduce results with the same accuracy than methods dedicated to large interfaces. The recent developments of the multifield approach have been included in the definition of the new drag force. Nevertheless, work is still in progress to validate the method on complex flows with two continuous fields, liquid and gas, and a gaseous dispersed field. Moreover, no turbulence models have been implemented yet with this approach. However, we can point out the work of some research groups (Larocque et al., 2010, Liovic and Lakehal, 2007, Vincent et al., 2008) on LES modeling of two-phase turbulence terms. 5. ACKNOWLEGMENTS 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 MSME laboratory (Université Paris-Est, Champs sur Marne, France). 6. REFERENCES 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, 29, pp. 460–478. Bhaga, D. and Weber, M.E., (1981) Bubbles in viscous liquids: shape, wakes and velocities, J. of Fluid Mech., 105, pp. 61-85. Brackbill, J. U. et al., (1992) A continuum method for modeling surface tension, J. Comput. Phys., 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. Clift, R. et al., (1978) Bubbles, drops, and particles, New York: Academic Press. Coste, P. et al., (2007) Modeling Turbulence and Friction around a Large Interface in a Three-Dimension Two-Velocity Eulerian Code, Proc. of Int. Conf. NURETH 12, Pittsburgh, USA.

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Coste, P. et al., (2012) Validation of the Large Interface Method of NEPTUNE_CFD 1.0.8 for Pressurized Thermal Shock (PTS) Applications, NED, 253, pp. 296–310. Coste, P., (2013) A large interface model for two-phase CFD, NED, 255, pp. 38-50. Denèfle, R. et al., (2015) Multifield hybrid approach for two-phase flow modeling – Part 1: Adiabatic flows, Comput. Fluids., 113, pp. 106–111. Enright, D. et al., (2002) A hybrid particle level set method for improved interface capturing, J. Comput. Phys., 183, pp. 83–116. Fleau, S. et al., (2015) Conservative implementation of the interface sharpening equation within an incompressible isothermal multifield approach, submitted to Comput. Fluids. Hirt, C.W. and Nichols, B.D., (1981) Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39, pp. 201–225. Ishii, M., (1975) Thermo-fluid dynamic theory of two-phase flow, University of Michigan: Eyrolles. Lahey, R.T. and Drew, D.A., (2001) The analysis of two-phase flow and heat transfer using a multidimensional, four field, two-fluid model, NED, 204, pp. 29-44. Lamb, H. (1932) Hydrodynamics, 6th Edition of Cambridge University Press. Larocque, J. et al., (2010) Parametric study of LES subgrid terms in a turbulent phase separation flow, Int. J. Heat Fluid Flow, 31, pp. 536–544. Laviéville, J., and Coste, P., (2008) Numerical modelling of liquid-gas stratified flows using two phase eulerian approach, Proc. 5th Int. Symposium on Finite Volumes for Complex Applications, Aussois, France. Liovic, P. and Lakehal, D., (2007) Multi-physics treatment in the vicinity of arbitrarily deformable gas liquid interfaces, J. Comput. Phys., 222, pp. 504–535. Mimouni, S. et al., (2011) Combined evaluation of second order turbulence model and polydispersion model for two-phase boiling flow and application to fuel assembly analysis, NED, 241(11), pp. 45234536. Mimouni, S. et al., (2014) Multifield approach and interface locating method for two-phase flows in nuclear power plant, submitted to NED. Olsson, E., and Kreiss, G., (2005) A conservative level set method for two phase flow, J. Comput. Phys., 210, pp. 225–246. Patankar, S. and Spalding, D., (1972) A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Transf., 15(10), pp. 1787–1806. Raymond, F., and Rosant, J.-M., (2000) A numerical and experimental study of the terminal velocity and shape of bubbles in viscous liquids, Chem. Eng. Sc., 55, pp. 943–955.

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Roache, P.J., (1998) Verification and Validation in Computational Science and Engineering, Albuquerque, New Mexico: Hermosa Publishers. Schiller, L., and Nauman, A., (1935) A drag coefficient correlation, V.I.D. Zeitung, 77, pp. 318-320. Štrubelj, L., (2009) Numerical simulations of stratified two-phase flows with two-fluid model and interface sharpening, PhD, University of Ljublana. Sussman, M. et al., (1994) A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, pp. 146–159. Thorpe, S., (1969) Experiments on the instability of stratified shear flows: immiscible fluids, J Fluid Mech., 39, pp. 25–48. Unverdi, S.O. and Tryggvason, G., (1992) A front-tracking method for viscous, incompressible, multifluid flows,J. Comput. Phys., 10, pp. 25–37. Vincent, S. et al., (2008) Numerical simulation of phase separation and a priori two-phase LES filtering, Comput. Fluids, 37, pp. 898–906. Zuzio, D. et al., (2013) Numerical simulation of primary and secondary atomization, C. R. Mecanique, 241, pp. 15-25.

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Figure 1. Schematic view explaining the extension of the drag force expression to large interfaces based on a continuous approximation, left: dispersed approach with dispersed particles distributed statistically over the domain, right: aligned dispersed particles forming a continuous boundary between two continuous fields.

Figure 2. Repartition of 𝑑𝑝 along the domain, large interfaces are located at high values of 𝜀𝑙∗ 𝜀𝑔∗ .

Figure 3. Sensitivity to the sharpening method, left: two interfaces with two different initial thicknesses, right: final thickness of the two interfaces after resolution of the interface sharpening equation.

19

Figure 4. Example of a three-cell stencil for LIM

Figure 5. Effect of the mesh refinement on the relative error for the circularity and the pressure and the average bubble velocity, logarithmic axes.

20

Figure 6. Location of the asymptotic region for the estimation of the order of convergence of the circularity, the pressure and the average bubble velocity, linear axes.

Figure 7. Schematic view of the mesh used for the simulation of the Bhaga’s rising bubble (Bhaga and Weber, 1981).

21

Figure 8. Comparison of the simulated bubble shape with the Bhaga’s experimental observations (Bhaga and Weber, 1981), left to right: second drag force model, third drag force model and LIM results, the bubble isosurface is extracted at 0.6 s.

Table 1 Frequency and characteristic time of decay of the bubble oscillations according to the drag force model, mesh with 512 x 512 cells, the relative errors are written in brackets Drag force model Second drag force with τ = 1.10−7 s Third drag force

𝜔𝑑 (s-1) 5.56 (2.6 %) 5.67 (0.7 %)

𝜏𝑑 (s) 2.35 (46 %) 3.16 (28 %)

Figure 9. Schematic view of the Thorpe experiment at initial conditions (Thorpe, 1969).

Figure 10. Influence of the drag force expression in terms of interface shape, left: second drag force with τ = 1.10−7 s, right: third drag force given by Equation (16), only the middle 0.6 meters long section of the channel is shown.

22

Figure 11. Physical location of the interface at 3 s, left: second drag force with τ = 1.10−7 s, right: third drag force given by Equation (16), only the middle 0.6 meters long section of the channel is shown.

Figure 12. Amplitude growth obtained by evaluating the standard deviation of the interface over time, only the middle 0.6 meters long section of the channel is used for this analysis.

Figure 13. Average interface velocity U over time, U is defined in Equation (37), the theory is given by Equation (36), the black, red and green curves are superposed.

23

Figure 14. Variation of the average interface velocity U along the tube width, left: second drag force with τ = 1.10−7 s, right: third drag force given by Equation (16).

Figure 15. Physical location of the interface at different times, used to evaluate the wave speed, left: second drag force with τ = 1.10−7 s, right: third drag force given by Equation (16).

Table 2 Comparison between our simulation, the theoretical and experimental data and the simulations of Bartosiewicz et al. (2008) and Štrubelj (2009)

𝑘𝑐 (m−1) 222 219 303 232 197 ± 58 143 157

Results Second drag force, τ = 1.10−7 s Third drag force, Equation (16) LIM Theory Experiments Bartosiewicz et al. (2008) Štrubelj (2009)

24

tonset (s) 2.4 2.1 2.3 1.5 – 1.7 1.88 ± 0.07 1.9 2

uwaves (cm.s-1) 3.3 3.1 3.9 2.38 2.6 2.5 3

Figure 16. Interface location simulated with the code NEPTUNE_CFD using the new drag force expression compared with Štrubelj’s simulation results (Štrubelj 2009) and the analytical expression given by Equation (36).

Figure 17. Evolution of the Rayleigh Taylor instability obtained with various models, (a) VOF with geometrical interface reconstruction in FLUENT (Štrubelj, 2009), (b) VOF with interface sharpening in CFX (Štrubelj, 2009), (c) single-fluid model with interface sharpening with Štrubelj’s in-house code (Štrubelj, 2009), (d) two-fluid model with interface sharpening with Štrubelj’s in-house code (Štrubelj, 2009), (e) two-fluid model with interface sharpening in the NEPTUNE_CFD code, left to right: t = 2 s, t = 2.5 s and t = 4 s.

25