Florence Drui 1, 2 , Alexandru Fikl 2 , Pierre Kestener 2 , Samuel Kokh 2, 3 , Adam Larat 1, 4 , Vincent Le Chenadec 1 and Marc Massot 1,4
Abstract. Many physical problems involve spatial and temporal inhomogeneities that require a very fine discretization in order to be accurately simulated. Using an adaptive mesh, a high level of resolution is used in the appropriate areas while keeping a coarse mesh elsewhere. This idea allows to save **time** and computations, but represents a challenge **for** distributed-memory environments. The MARS project (**for** Multiphase Adaptative Refinement Solver) intends to assess the **parallel** library p4est **for** adaptive mesh, in a case of a finite volume scheme applied to **two**-**phase** **flows**. Besides testing the library’s performances, particularly **for** load balancing, its user-friendliness in use and implementation are also exhibited here. First promising 3D simulations are even presented.

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The purpose of the paper is to implement the bi-projection scheme analyzed in [4] in the context of the numerical study of **two**-**phase** immiscible, icompressible and isothermal **flows** of viscoplastic medium. A level set formulation is used to track the interface of the fluids. In this approach (see [26, 25]), the interface is the set of points where the level set function vanishes and the latter is transported by the velocity field satisfying the Navier-Stokes equations over the whole domain. **For** obvious reasons, it is desirable to transport a smooth function rather than a discontinuous one. The level set function is therefore initialized and maintained, with the help of a redistancing procedure, as the normal signed distance function from the interface (see [19]). The reinitialization step consists in solving a Hamilton-Jacobi equation with an artificial **time**: the stationary solution of which being a distance function (see [25, 22]). It has been observed in [22] that, during iterations of the reinitialization procedure the zeros off the level set function have the tendency to move towards the closest grid point and a subcell fix method has been proposed to remedy this numerical artifact. In [17], the effect of the temporal discretization of the Hamilton-Jacobi coupled with the subcell fix **algorithm** was studied. These methods have been implemented in the present study.

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6 Conclusion
In this paper, we have assessed the capability of a scheme issued from the incompressible flow context, namely a pressure correction scheme, to compute discontinuous solutions of hyperbolic systems. This scheme was already known to be unconditionally stable [8, 10]. Numerical tests performed here show that, provided that a sufficient numerical dissipation is introduced in the scheme, it converges to the (weak and entropic) solution to the continuous problem; in addition, it shows a satisfactory behaviour up to large CFL numbers. Since the scheme boils down to a usual projection scheme when the density is constant, this approach yields an **algorithm** which is robust with respect to the flow Mach number, and the present solver is indeed now routinely used to compute viscous **two**-**phase** low Mach number **flows**, as bubble columns **for** instance. The present work may be extended in various ways. First, the observed convergence can be comforted by theoretical arguments; even if a complete convergence proof seems difficult at this **time**, because of the lack of compactness of sequences of discrete solutions due, in particular, to the absence of diffusion terms, it is possible to show, **for** one-**phase** **flows**, by passing to the limit in the scheme, that any limit of a convergent sequence is an entropy weak solution to the continuous problem [17].

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7. Computational **time**, efficiency and simplicity
All computational examples confsidered in this paper led to the same observation: the present relaxation solver is much faster than the root-finding iterative **algorithm** given in Appendix. In our previous work [1] CPU savings from 5 to 50% were obtained, depending on the conditions. Due to the increased complexity of the iterative **algorithm** **for** the multi-component version, the CPU saving is now 50% in all reported cases. The main argument **for** this **time** gain is related to its simplicity: the relaxation solver is direct whereas the iterative method requires solving a non-linear-algebraic system that may cause difficulties as a result of non-linearities. Besides, the root-finding iterative method requires the calculation of the saturation pressure via Eq. (2.4) at each iterative step, which itself requires an iterative method.

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Figure 7. Pressure-based Riemann solver applied to test 1 of Table 1: Pressure-based solver (red and blue lines), Roe solver (sky blue line) and exact (dash) solutions compared at **time** 0.2
Test 2 consists of **two** symmetric rarefaction waves and a trivial contact wave, with the star region between the non-linear waves close to vacuum. This problem is a good assessment of the performance of numerical methods **for** low-density **flows**. The results **for** this test are shown in Figure 8 against the exact results and we can see that the accuracy of the numerical results is nearly from those of the exact Riemann solver unlike Roe scheme which fails near low-density **flows** [13].

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CONCLUSIONS
In the first part of our research, new variational stabilised formulations are proposed to solve the level set transport equation without using the reinitialisation method. These formulations are obtained by adding new terms that depend on the residual of the Eikonal equation to the basic SUPG formulation of the level set equation. These methods were compared with a modified variant or alternative of the penalty method proposed by Li et al. (Li et al., 2005) and the geometric reinitialisation through a “brute force search **algorithm**”. The temporal discretisation was made using a semi-implicit Crank-Nicolson scheme, while the space discretisation is carried out by approximation using quadratic elements. The numerical comparisons were made using standard validation tests such as a periodic reversed vortex and a rotational movement of the Zalesak disk, which is a rigid body. The proposed stabilisation methods improve the numerical behaviour of the SUPG method. They allow the effective of capture the interfaces between the phases, which are weakly or strongly deformed, and provide at the same **time** a retention of the acceptable mass. These methods are relatively easy to implement in existing finite element solvers in 2D and in 3D.

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Fig. 14. Interface location, pressure (Pa) and velocity (m s − 1 ) ﬁelds at different times.
strengthening the ability of the **time** splitting projection method **for** compressible ﬂows to perform an eﬃcient coupling between strong interface deformation and volume oscillation. In the Fig. 22 and in the Fig. 23 , the temporal evolution of the equivalent radius and the average vertical velocity of the bubble are plotted **for** different grids in order to check the spatial convergence of the overall **algorithm**. Although the **simulation** performed on the thinnest grid seems to be well converged in the ﬁrst **time** of the **simulation**, more important deviations appear in the end of the simulations. It suggests that a more reﬁned grid could be used to improve the numerical results. However, we must also notice that our numerical model is based on the Euler equations which do not include the viscous dissipation effects. As a rotational ﬂow and a deformed bubble are considered in this situation, the computations could not converge since the model does not account **for** any spatial cut-off due to the low scale viscous dissipation. Indeed, in such a situation the Reynolds number is inﬁnite due to the zero viscosity in the **two** phases. Therefore, hydrodynamic instabilities in the wake of the bubble or shape instabilities can develop in a way which could depend on the mesh size. This is why the spatial convergence can sometimes be diﬃcult to achieve **for** inﬁnite Reynolds number simulations involving rotational ﬂows. Moreover, as stronger deformations of the bubble are involved in this case than in the previous section, we can assume that a more reﬁned grid could be required to obtain a better spatial convergence in this situation.

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Table 1: Number of **time**-iterations **for** each test case
the second one (top right) and the third one (bottom) involve several distinct waves. The scheme we propose here is denoted LP implicit and is compared with its explicit version (which amounts to replacing (9) by its **time**-explicit version) and the well-known Rusanov scheme (see [8]). We observe that our approach is clearly less diffusive around the contact discontinuities since the CFL condition is well-adapted to the corresponding speed of propagation, but more diffusive around the acoustic waves since it is implicit. Table 1 gives **for** each test case the number of iterations needed to perform the computations. As expected, the gain is important when using the proposed implicit-explicit **algorithm** and the corresponding CFL restriction based on the material waves (instead of the acoustic waves as **for** the explicit scheme). A careful evaluation of the CPU cost necessitates an additional programming effort that has not been implemented yet.

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ically arise.
The main advantage of this formulation is to use the natural set of unknowns **for** the hydrodynamical and thermodynamical laws and to extend to a large class of compositional Darcy flow models ranging from immiscibility to full miscibility (see [11]). On the other hand, its main drawbacks are an additional complexity to deal with sets of unknowns and equations depending on the set Q, and the use of a fixed point **algorithm** to compute the set of present phases Q at each point of the space **time** domain. The efficiency of this formulation has mainly been shown **for** reservoir **simulation** test cases with complex thermodynamics, **two** and tri **phase** Darcy **flows**, but with usually small capillary effects and the use of a reference pressure in the thermodynamical state laws rather than the **phase** pressures. In the next section it will be assessed and compared with the **two** other formulations on test cases with both strong or weak capillary effects.

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the Navier–Stokes equation. The reason is the convection term and its numerical treatment with the characteristic method. In fact, if the velocity extension step is placed after the fluid equation resolution, the velocity considered **for** the flow when the characteristic curve crosses the free surface and goes to the other side of the computational domain, will be 0 . This is not coherent with the physics of the system nor with the continuous mathematical model which represents it. Indeed, one of the features of this work is to compare the free surface **simulation** with the bi-fluid **simulation** where air represents one of the **two** phases. In other words, in the case of free-surface **flows**, the action of the second fluid is describe by an atmospheric pressure. On the other hand, according to the interface condition, the flow velocity is continuous through the bifluid interface. This is why the consideration of a 0 value **for** velocity when the characteristic curve crosses the free surface is not convenient. Of course, in these latter cases, we could also take the value of the last point of the advection but since we may need to extend the velocity **for** the advection equation, it would be more elegant to use the extended velocity to solve the Navier–Stokes equation in the fluid domain. This means that even though the Navier–Stokes equation is solved only on the fluid domain, the velocity that the Navier–Stokes solver takes as entry is the extended velocity defined on the whole computational mesh T n . However, only values of the extended velocity on the vicinity of free surface and in the fluid domain which are used to construct the right hand side of the linear system ( 49 ). As a confirmation, we can check by numerical simulations that the behavior of the free surface is different if we give the non extended velocity to the Navier–Stokes solver. Let us also mention that if we consider a Stokes flow, the placement of the velocity extension step before the resolution of the fluid equation is not necessary. This is due to the absence of the convection term and the absence of the characteristic function on the right hand side of the linear system in the case of Stokes flow. The **algorithm** summarizes as follows:

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Our final goal is to build an efficient numerical tool **for** EHD problems, which can be applied to arbitrary complex geometry configurations. In recent years, the unstructured grid methodology is quickly evolving in CFD field mainly due to its inherent flexibility **for** discretization of complicated geometries [89][90]. However, most HR schemes cannot be applied to unstructured grids directly. The main difficulty is the lack of far-upwind cell information [91]. To circumvent this difficulty various methods have evolved [71][91-93]. There are **two** main thoughts to apply the HR schemes on unstructured meshes. The first is to avoid explicitly using far-upwind points, like GAMMA scheme [71] and Bruner’s approach [92]. GAMMA scheme uses an ‘unbounded indicator’ without far-upwind information, whereas it uses the CD scheme as the high order base scheme. However, GAMMA scheme requires that the nodal point locating in the centre of the control volume, which is not always the case **for** unstructured meshes. The second is to construct the virtual far-upwind points, like Darwish’s r-factor **algorithm** [91] and Lian-xia Li’s new r-factor **algorithm** [93]. Both algorithms need to calculate nodal gradients and to use extra memory **for** virtual points. The most serious problem of all above mentioned methods is all of them show minor oscillations, even if the grid size is sufficiently small.

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The ignition **phase** of a jet engine cycle is by definition a critical **phase** and is part of the certification of the jet engines which needs **for** security reasons to ignite or re-ignite in specific conditions. To fullfil such requirements, fundamental ignition study receives great interest from industry. However this fully transient **phase** is a complex phenomenon, not yet fully understood and controlled. Experiments show that ignition may be successfull or fail in the same operating conditions, even in simple geometry lab-scale congurations [ 142 , 143 , 144 ]. As a consequence, ignition is usually described in terms of probability of success [ 37 ]. This probability is the result of the space and **time** fluctuations of the fluid state at the igniter location, and is therefore con- ditioned by local phenomena. To understand the process of ignition, previous numerical studies have been devoted to gaseous ignition [ 145 , 146 ] and in particular demonstrate the concept of minimum energy **for** successful ignition [ 40 , 147 ]. More recently, Garcia-Rosa [ 148 ] used zero and one-dimensional sub-models to describe the very first moments of ignition. Less work has been devoted to the ignition of a **two**-**phase** mixture, mainly because the **simulation** of **two**- **phase** **flows** still presents multiple challenges associated with the dispersed **phase** description (cf. Chapter 3 ). Note however that LES of **two**-**phase** reacting **flows** offer access to fully tran- sient phenomena that could not be addressed numerically with other CFD models (other than DNS). In particular, the fully transient **phase** that results from a sparking ignition **phase** can be simulated with LES which thus provide a dynamic description of the spatial and temporal evo- lution as well as the propagation of the flame front until full ignition or extinction of the burner. In the following, LES is used to reproduce the details of ignition sequences and burning of the **two**-**phase** mixture on the MERCATO bench. The objective is to validate ignition scenario and understand the probabilistic character of ignition within the limits of the LES model. In particular the main factors influencing the failure or success of ignition are identified and implications **for** ignition systems optimization are derived.

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In the present work, we are interested in moment methods, because of their com- putational efficiency with regards to other approaches. One of the main issues with moment methods is the accurate description of the velocity distribution of the par- ticulate **phase**. Actually, in turbulent **flows**, the velocity distribution can drastically change with the inertia of the particles, which can be quantified by the Stokes number based on the Kolmogorov **time** scale. **For** Stokes number smaller than one, the NDF is monokinetic, i.e. all particles at the same position have the same velocity, and such a distribution can be uniquely determined using zero and first order moments, i.e. density and momentum. **For** higher Stokes number, particles trajectories may cross, and the velocity distribution is no longer a unique Dirac δ-function, and higher order moments are needed. To handle these higher order moments, several methods can be found in the literature, and can be split into **two** categories. On one hand, Algebraic-Closure-Based Moment Methods (ACBMM) [2, 46, 55, 54] derive closures **for** the second order moments using physical and/or mathematical assumptions. On the other hand, Kinetic-Based Moment Methods (KBMM) close the system by using a presumed shape **for** the NDF [48, 56, 17, 45, 72, 70, 71], which has as many pa- rameters as the number of moments required to be controlled to describe the NDF accurately. The choice between each type of closure is motivated by the structure and the complexity of the encountered PTC, and is directly related to the number of moments.

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higher concentration of void in the upper parts of the tube bundle probably due to larger gas structure predictions. It is difficult to conclude on the gas velocity profiles coming from the Multi-field approach. Therefore, the GLIM is probably the most competitive model regarding these profiles.
According to Figure 3.12 , 3.13 and 3.17 , void fraction distributions from the three approaches are strictly different even if the profiles along NS and WE lines were similar. The conclusion **for** the dispersed approach is illustrated on Figure 3.17 given that there is no strong discrepancy between the void fraction between **two** horizontal tubes and in front of cylinders, and the inclination has no real influence on the bubble distributions. The NS **time**-averaged void fraction distribution from the dispersed approach is similar to the one from the GLIM (aside from the front of the cylinders). In Figure 3.12 , the instantaneous profiles illustrate an important dispersed contribution ponctuated by larger continuous gas structure **flows** which appears to be more concentrated higher in the tube bundle. It would explain the WE **time**-averaged void fraction distribution which predicts slightly more and more gas in the upper parts of tube when we are higher in the tube bundle. Consequently, the higher we are in the tube bundle, the larger gas structures we have, the higher concentration of void in the upper parts of the tube since large gas structure are rising along the tube depending on the inclination. In Figure 3.13 , each slice is strongly different from the GLIM ones. First, instantaneous profiles reveal a low dispersed gas contribution and a high continuous gas field contribution in mirror with the GLIM. Then, regarding the NS **time**-averaged slice, the higher we are in the tube bundle, the lower is the void fraction. This is explained by the WE **time**-averaged slice which shows a high concentration of void in the upper parts of tube, thus an important influence of the inclination and possibly large gas structures. Since the continuous gas is used when bubble are considered too large to be modeled by the dispersed approach, the transition criterion from dispersed to continuous is probably too low compared to the transition criterion from the GLIM. In the present case, the **two** main differences between them is the use of a surface tension force in the continuous gas field of the multi field approach and the transition criterion (discrepancies between both approaches are detailed in appendix).

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easily use the fast and accurate **algorithm** **for** water properties calculation proposed by the authors in [79]. Instead, with Kapila’s model, iterative algorithms would be necessary in order to guarantee
pressure equality, p l = p v = p. However, **for** the present paper, only the simple stiffened gas EoS are
used. This is in particular advantageous to derive analytical solutions to be used as references. The six-equation model has nonconservative products appearing in the phasic total energy transport equations, which express interphase total energy transfers. We can recognize **two** different nonconservative terms in the phasic total energy equations. One refers to the gradient of the volume fraction, which is typical of many **two**-**phase** flow models and is inactive across shock waves due to the character of the advection equation of the volume fraction. Another term serves to split the mixture total energy coherently with the assumption of equal velocity and is active across shock waves. The 5-equation model of Kapila et al. [139] also contains nonconservative products. In particular, a nonconservative term depending on the divergence of the velocity appears in the transport equation governing the volume fraction of one of the phases. Due to this term it is difficult to obtain discretizations of the five-equation model that ensure positivity of the volume fraction. This difficulty does not arise in discretizations based on the six-equation model since only a simple homogeneous advection equation **for** the volume fraction needs to be integrated. On the other hand, the presence of nonconservative terms in the phasic energy equations of the six- equation model poses difficulties in its numerical solution. In the work of Pelanti–Shyue [200, 201] **two** Riemann solvers were proposed **for** this **two**-**phase** flow model: a Roe-type solver accounting naturally **for** nonconservative terms through the Roe matrix associated to the quasi-linear form of the model system, and a simple HLLC-type

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Eulerian descriptions **for** the dispersed **phase** are obtained by some kind of averaging method (**for** example by ensemble average section 1.3 or volume average 1.4 ). The different averaging procedures lead however not to exactly the same set of equations. With the filtering the Eulerian field equations in the sense of LES, a second averaging procedure is introduced. This procedure may be compared to the process **for** obtaining LES equations **for** single **phase** flow: the Navier Stokes equations are obtained from kinetic theory of gases by ensemble average of the molecular dynamics. Then they can be filtered to obtain the filtered LES equations (Fig. 5.1 ). Conceptually the approach used **for** the dispersed **phase** is similar in the case of one way coupling. The Eulerian equations of the dispersed **phase** are obtained from ensemble average of kinetic theory and the filtered equations by application of a LES filter. The major difference concerns the involved length and **time** scales and the sampling rate. Whereas in gaseous **flows** 6.0 10 23 molecules per mole contribute to the average, the corresponding particle number density in the dispersed **phase** is much smaller. Furthermore, the molecular “diameter” is of the order of some Angstr¨ om (10 −10 m) whereas the particle diameter is typically of the order of microns (10 −6 m) or larger. In the case of volume average, a filter size sufficiently larger than the typical particle diameter is required to guarantee continuous Eulerian equations. Therefore volume filtered equations can be understood as beeing already LES type equations **for** the **two** phases with explicit **phase** coupling.

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disappears, and (ii): the lack of positivity of the Roe scheme. Each of these difficulties will receive a specific treatment.
To address the first difficulty, we propose the use of the so-called polynomial schemes [10]. This choice is motivated by the asymptotic analysis of the **two**-**phase** model in the limit of vanishing volume fraction of one of the phases. In this limit, the **two** phases almost decouple and the minority **phase** obeys a pressureless gas dynamics system [6, 7]. This system is not hyperbolic because the Jacobian of the flux matrix is not diagonalizable. This implies that **two** eigenvectors of the original **two**-**phase** model collapse in the limit of small volume fraction. Therefore, most shock-capturing schemes, which require a complete basis of eigenvectors, breakdown in this limit. To overcome this problem, schemes that do not require that eigenvectors form a complete basis are needed. There are many such schemes, such as Lax-Friedrichs, or central schemes [26], but many of them are too diffusive **for** safety studies of nuclear power plants. The interesting feature with the polynomial schemes is that it is possible to tune the amount of numerical diffusion. Polynomial schemes have been used e.g. in [24, 28].

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1. Introduction
1.1. General motivations. The models **for** multiphase porous media **flows** have been widely studied in the last decades since they are of great interest in several fields of applications, like e.g. oil-engineering, carbon dioxide sequestration, or nuclear waste repository management. We refer to the monographs [5, 6] **for** an extensive discussion on the derivation of models **for** porous media **flows**, and to [4, 11, 3, 13] **for** numerical and mathematical studies.

Figure 13: Set up of the 3D simulations of the axisymetric and non-axisymetric merging of **two** bubbles
6.5. Fully 3D simulations of the axisymetric and non-axisymetric merging of **two** bubbles
To test high efficiency of the method **for** **two**-fluid **flows** involving complex topological changes of the interface, we consider the merging of **two** air bubbles with first co-axial and then oblique coalescence. This test case was studied by [49, 50] and the results suggested that due to surface tension a merging will occur. We consider **two** gas bubbles with a diameter d = 0.01 m that are initially set in a quiescent liquid. The density and viscosity ratios (gas to liquid) are respectively 0.001 and 0.01. The dimensionless parameters we use **for** this problem are the Eotvos number Eo = 16 and the Morton number M = 2 × 10 −4 . The dimensions of the computational domain are [−2d, 2d] × [0, 8d] × [−2d, 2d]. **For** the co-axial case, the center of the upper bubble is at [0, 2.5d, 0] and the lower bubble is at [0, d, 0]. As **for** the non-axisymetric case, the lower bubble is shifted to the position [0.8d, d, 0]. We use free-slip boundary conditions on all sides of the domain. It can easily be observed in figures 14 and 15 that the **two** bubbles gradually merge together and form a larger bubble under capillary force. **For** both cases, the computed evolution of the bubble shape is well captured by the dynamic mesh adaptation. The interfaces of the **two** bubbles are very well captured and the boundary layers as well as the detachments are automatically detected. Our results agree with the experimental photographs in [49] and the numerical results of [51]. We can see that the evolution of the lower bubble is completely different from the leading one. The tailing bubble catches the leading bubble later to form one single bubble. **For** the non-axisymetric case shown in figure 15, the dynamics are similar to the previous one except we can note that the flow field is clearly asymmetrical.

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