Numerical simulation of phase separation and a priori two-phase LES filtering

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1 Available online at Computers & Fluids 37 (2008) Numerical simulation of phase separation and a priori two-phase LES filtering S. Vincent a, *, J. Larocque b, D. Lacanette a, A. Toutant c, P. Lubin a, P. Sagaut d a Université Bordeaux 1, Laboratoire TREFLE, UMR CNRS 8508, 16 Avenue Pey-Berland, F Pessac Cedex, France b CEA-CESTA, Le Barp, France c LMDL, CEA-Grenoble, Grenoble, France d LMM, UMPC/CNRS, Paris, France Received 1 October 2006; received in revised form 22 December 2006; accepted 6 February 2007 Available online 9 October 2007 Abstract This work reports on the potential application of Large Eddy Simulation (LES) in the calculation of turbulent isothermal two-phase flows, in the case where the large scales of each phase are resolved and small interface structures can be smaller than the mesh size. In comparison with single phase flows, application of LES to two-phase flow problems should account for the complex interaction between the interface and the turbulent motion. The complete filtered two-phase flow equations are formulated to deal with turbulence at the interface. Explicit filtering of 3D direct numerical simulations of a phase separation problem has been employed to evaluate the order of magnitude of the specific subgrid contributions. Analyses of the numerical results have been conducted to derive conclusions on the relative importance of the different subgrid scale contributions. Modeling issues and turbulent energy transfer across the interface are discussed. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction Many industrial and environmental applications involve high-reynolds number turbulent multiphase flows. Examples include spray formations, fuel injectors, oil transportation, sea aerosol formation and many others. A great deal of research effort has been oriented towards the numerical modeling of multiphase turbulent flows. The great majority of numerical computations have adopted Reynolds Averaged Navier Stokes (RANS) modeling in which shear and multiphase induced turbulence is considered (Homescu and Panday [1]). Large Eddy Simulation (LES) has been also utilized for multiphase flow calculations of particles smaller than the Kolmogorov scale (Boivin et al. [2]), undeformable interfaces (Calmet and Magnaudet [3]), bubble formation and break-up (Liovic et al. [4]), turbulent plane jet and liquid film interaction * Corresponding author. Tel.: ; fax: address: vincent@enscpb.fr (S. Vincent). (Lacanette et al. [5] and [6]) or wave breaking (Christensen and Deigaard [7], Lubin et al. [8]). However, in the abovementioned models, the impact of the multiphase topology of the flow on the turbulent characteristics has never been carefully studied, and the modeling is mainly based on analytical considerations. More precise studies are required to validate and formulate subgrid scale modeling of turbulence interface interaction. Recent studies of Liovic and Lakehal [9] and [10] investigate for the first time the LES of turbulent two-phase flows and discuss the formulation and modeling of specific LES subgrid terms linked to the two-phase character of the flow. The objectives of the present work are to present the filtered conservation equations in the framework of multiphase flows with large deformable interface and to lead one-field Direct Numerical Simulation (DNS) of phase separation without explicit turbulence modeling. A convergence study is achieved and a macroscopic investigation of the multiphase flow is presented. Finally, a priori tests dedicated to LES modeling are investigated. A low-pass /$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi: /j.compfluid

2 S. Vincent et al. / Computers & Fluids 37 (2008) frequency filter is applied to the one-field DNS simulations in order to estimate the order of magnitude of the specific LES subgrid terms appearing in the filtered conservation equations. 2. Numerical modeling 2.1. Direct numerical simulation of two-phase flows In each phase, the flow is incompressible and isothermal. Let u be the velocity, t the time, q the density, l the dynamic viscosity, p the pressure, g the gravity vector, S D ¼ 1 2 ðru þrt uþ the deformation rate, c the surface tension coefficient, j the curvature of interface, n i the normal to the interface and d i a Dirac function indicating the interface. If k = 0 refers to one phase and k = 1 to the other, the Navier Stokes equations in each fluid can be written as ru ¼ 0 ð1þ ou q k þrðuuþ þrðpid 2l ot k S D Þ q k g ¼ 0 At the interface, mass and momentum conservation is ensured thanks to jump relations (Delhaye [11]) u 1 u 0 ¼ 0 ðp 1 Id 2l 1 S D;1 Þn i ðp 0 Id 2l 0 S D;0 Þn i cjn i ¼ 0 where subscripts 0 and 1 refer to the characteristics in each phase. A unique set of equations can be obtained to allow a one-field description of multiphase flows by introducing a phase function v equal to 1 in one phase and 0 elsewhere and by convolving and summing (1) with v. A one-fluid model is thus obtained (Scardovelli and Zaleski [12]), in which an advection equation for v is introduced to describe the interface evolutions ru¼0 q ou þrðuuþ þrðpid 2lS D Þ qg cjn i d i ¼ 0 ð3þ ot ov ot þ u rv¼0 In this model, q, l and j depend on v. They are defined, for example, as q ¼ vq 1 þð1 vþq 0 l ¼ vl 1 þð1 vþl 0 rv j ¼r krv k where v * corresponds to a smooth approximation, in the sense of continuous, of v (see Brackbill et al. [13]) Filtered Navier Stokes equations for two-phase flows ð2þ ð4þ Let v k = C be the phase function indicating phase k(v k = 1 in phase k and 0 elsewhere), G the low-pass frequency filtering operator defining the spatial filtering of r by U ¼ G U; eu ¼ P k q kv k U k = P k q kv k the phase-weighted filtering of r and U ¼ P k v ku k the filtered phase average of r. As a definition, v 1 = v and v 0 =1 v. The commutation between the spatial filter and the derivative is widely assumed in single-phase LES modeling (Sagaut P. [14] and Vasilyev et al. [15]). The same assumption is considered to apply for two-phase flows concerning U. Such assumption has to be verified, even if the commutation error depends more on the topology of the mesh than on the characteristic of the flow (Dakhoul and Bedford [16] and [17]). Under this assumption, the spatial filtering and averaging over phases of the incompressible Navier Stokes equations (3) leads to the following set of equations r~u ¼ r 0 q o~u ot þrðq ~u ~u þr 2le S D þ pid qg cjn i d i ¼ X3 oc ot þ ~u rc ¼ r 1 Three specific tensors s 0, s 1, s 2 and s 3, associated, respectively, to acceleration, inertia, viscous effects and interface presence, appear due to the filtering and averaging operations of the momentum equations (Labourasse et al. [18]). These subgrid terms are expressed with the following definitions s 0 ¼ q ou o~u þ q ot ot s 1 ¼ rðquu q ~u ~uþ s 2 ¼ r l½ru þr T uš l½r~u þr T ~uš rc rc s 3 ¼ crcr þ crcr krck krck i¼0 s i ð5þ ð6þ In the same way, the scalar terms r 0 and r 1 stem from the treatment of the divergence and advection equations r 0 ¼ r:ðu ~uþ r 1 ¼ ðurc ~u:rcþ The terms (6) and (7) must be modeled to close the LES equations of two-phase flows, as they depend on unsolved variables. They result from the Favre and filtered phase averages of velocity, density, viscosity and phase function. Other formulations can be derived with conservative formulations or U unknown filtering. For example, Liovic and Lakehal [9] and [10] use a conservative formulation and eu unknown filtering to obtain a LES two-phase model of macroscopic interfaces, equivalent to (5 6 7). In their simulations, they assume in a first approach that the subgrid terms, which correspond to r 0 and r 1 in our model, are equal to zero. ð7þ

3 900 S. Vincent et al. / Computers & Fluids 37 (2008) Discretization and solvers The DNS of two-phase flows is based on the previous equations system in which the velocity, pressure and phase function are not filtered and so tensors s 1, s 2 and s 3 and scalar terms r 0 and r 1 are zero. This system is approximated by implicit finite volumes of second order in time and space on a fixed Cartesian staggered grid. An implicit augmented Lagrangian procedure (Fortin and Glowinski [19] and Vincent et al. [20]) is implemented for the treatment of the velocity pressure coupling and the divergence free constraint. The non-linear inertia term is linearized in time by considering u n Æ $u n+1, where n is the time indice corresponding to time ndt. Concerning the space derivatives in the momentum equations, centered schemes are used to discretize the inertia and viscous terms. An iterative BiCGSTAB II solver (Van Der Vorst [21]), preconditioned under a modified and incomplete LU approach (Gustafsson [22]), is used to solve the linear system. The free surface evolutions are simulated thanks to a Piecewise Linear Interface Construction (PLIC) volume of luid (VOF) interface tracking method of Youngs [23]. Concerning surface tension modeling and discretization, the Continuum Surface Force (CSF) of Brackbill et al. [13] is utilized to include this interfacial force as a volume force in the momentum equations. More details on the methods and physical and numerical validations can be found in Vincent and Caltagirone [24]. 4. DNS of phase separation 4.1. Description of the problem and macroscopic behavior of turbulence In a 1 m 3 cubic cavity full of water, a cubic oil drop is initially placed in a bottom corner of the cavity (Fig. 1). The motion of the cubic oil inclusion is driven by gravity and induces a turbulent flow behavior (Vincent et al. [25]). The water density and dynamic viscosity are 1000 kg.m 3 and PaÆs, respectively, while the oil characteristics are 900 kbæm 3 and 0.1 PaÆs. The constant surface tension coefficient is equal to NÆm 1. On three different grids, Fig. 2 presents the first 12 s of the inversion flow where the large and small scale deformations of the initial oil drop are predominant. Three stages are observed: an overturning motion during the first 4 s, strong interface deformation and rupture for 4 6 t 6 8 s, and the beginning of the separation of oil and water with coalescence of numerous droplets for 8 6 t 6 12 s. Towards the end, a sloshing motion is observed while water lies in the bottom part of the cavity and oil in the upper part (see Fig. 3 right). In all the simulations, a time step of s is utilized and 100,000 calculation steps are calculated. An example of the velocity field structure in a 2D x z slice at y = 0.5 m is proposed in Fig. 3 for t = 12 s. The phase separation is clearly well established at this time while the flow remains inertial in each fluid. Fig. 1. Definition sketch of the water/oil phase inversion problem. Let X be the calculation domain. The macroscopic behavior of the potential energy R qkgkzdv, normalized X by the theoretical potential energy after phase separation, and the kinetic energy R 1 X 2 u2 dv, normalized by the maximum kinetic time energy in water, is presented in Fig. 4. The potential energy demonstrates that the phase separation is established after 20 s even if the flow remains inertial in water and oil. After t = 20 s, the oil phase is located in the upper part of the cavity and the interface remains almost horizontal. From t = 0 to 3 s, the gravity accelerates the oil drop and increases inertia in an overturning flow. Then, the two-fluid layers are almost stable (the lighter fluid lies on the heavier one) and inertia in each phase decreases under the action of the viscous effects. This decreasing turbulent flow is modulated by a sloshing motion of a period of 3.33 seconds whatever the grid size. This surface wave characteristic is measured by applying a Fast Fourier Transform to the time evolution of the kinetic energy in water or oil after t =20s. The DNS requires the independency of the simulations with respect to the grid. A convergence study is led in Vincent et al. [26] and is illustrated in Fig. 2. As for the interface description, it is observed that for 85 3 and grids, the macroscopic interfacial structures are similar. However, the grid is required to track the small oil droplets which are generated under the shearing and surface tension actions at the interface. A finer grid would certainly be required in this case to reach convergence with respect to the interfacial description. As is well known from numerous publications, the VOF PLIC method induces artificial droplet generation (Rider and Kothe [27]) if the density of the mesh points is larger than 2 3 times the thinner interfacial structures. Concerning hydrodynamic characteristics, near convergence is observed with respect to potential

4 S. Vincent et al. / Computers & Fluids 37 (2008) Fig. 2. 3D macroscopic evolution of the water oil interface for 43 3 (left), 85 3 (centered) and (right) grids x, y and z are, respectively, the longitudinal, transverse and vertical axis the gravity is oriented in the z-direction. and kinetic energies (see Vincent et al. [26]) since a grid is used for the simulations. Concerning enstrophy, differences are observed between 85 3 and simulations, indicating that the present results remain under-resolved DNS (see Fig. 7). However, the grid is considered to be a good compromise for dealing with DNS of phase sep-

5 902 S. Vincent et al. / Computers & Fluids 37 (2008) Fig. 3. x z slice at y = 0.5 m presenting the interface (C = 0.5) and the corresponding unsteady velocity field after 12 s (left) and 3D topology of the interface after 100 s (right). Fig. 4. Time evolution of the dimensionless potential (left) and kinetic (right) energy in water and oil for converged simulations 85 3 and grid. aration, in the sense that no explicit modeling of turbulence is accounted for, while keeping reasonable calculation costs and converged first order moments of turbulence (see Figs. 4 and 6). Numerous micro-droplets are formed after the dislocation of the initial oil drop (6 < t < 20 s). Scardovelli and Zaleski [12] have demonstrated that spurious parasitic currents can be generated in the interfacial area of droplets when the surface tension force dominates the viscous effects. The Laplace number La ¼ qlr, with L the characteristic interfacial lengthscale of the droplets (L is chosen l 2 equal to Dx), is used to estimate whether these parasitic currents can alter the velocity field and the subsequent fluctuating turbulent characteristics. A limit value of La =10 6 is admitted in many publications. In oil and water, La is equal to 52 and , respectively, in the phase separation problem. Therefore, the flow is not affected by this numerical effect. In order to characterize the inertial effects of the turbulent flow, we introduce the Reynolds number in water, based on the size of the larger eddy structures (integral lengthscale), the density of water, the maximum velocity magnitude measured in this phase and the water viscosity at each time step (see Fig. 5). From time t = 0 to 2 s, the oil drop rises and the flow becomes more and more dynamic due to the high acceleration brought by the buoyancy forces. From t = 2 to 7 s, the flow is unsteady, obviously turbulent, and the Reynolds number is between 50,000 and 60,000. Until time t = 7 s to the end of the DNS, the flow inertia decreases under the combined actions of turbulent dissipation in early times and viscous dissipation in later times. The Reynolds number strongly decreases from t = 7 to 20 s (Fig. 5), corresponding to the beginning of a decaying of the kinetic energy as a function of t 2 (Fig. 6). On a numerical point of view, it is observed in Figs. 4 and 6 that the macroscopic quantities such as the

6 S. Vincent et al. / Computers & Fluids 37 (2008) Fig. 5. Time evolution of the Reynolds number in water based on the integral scale and the maximum velocity magnitude. Fig. 7. Time evolution of the enstrophy in water and oil for various simulations. t = 6.5 s in oil. As previously explained, the flow is mainly turbulent as soon as gravity has induced strong deformations of the initial oil drop (2 6 t 6 20 s). Later, a dissipative behavior is observed, due to viscosity and confinement effects, which leads to near equilibrium and phase separation at t = 100 s. Concerning the DNS and the resolution of the small turbulent scale, Fig. 7 demonstrates that, even if the potential and kinetic energy are converged on a grid and the enstrophy admits the same behavior in time on the two considered grids in oil and water, enstrophy remains 40 % less in water and 10 % less in oil on a 85 3 grid compared to a grid. The proposed simulations are under-resolved DNS if the small vortical structures of the flow are considered and a grid would certainly be required in order to obtain a real DNS. 5. A priori tests Fig. 6. Time evolution of the dimensionless kinetic energy in water (log log scale) compared to a fitting curve f(t) =At and simulations from t =1stot = 100 s. potential and the kinetic energies are converged in oil and water on a grid. Moreover, the same t 2 decay of the kinetic energy and sloshing motion of period 3.33 s are verified for the 85 3 and grids. The characterization of the turbulent flow is difficult. Indeed, no Fourier analysis can be driven on the space variations of velocity or phase function as the calculation domain is not periodic and the flow is not statistically stationary nor homogeneous. In addition, no average and fluctuating quantities can be directly calculated since a two-phase flow is considered. Let w be the local vorticity of the flow depending on u. The enstrophy R 1 X 2 w2 dv gives relevant information in the present problem. Fig. 7 shows that the maximum enstrophy, normalized by the maximum time enstrophy in water, is obtained for t = 7 s in water and This section is devoted to estimating the relative order of magnitude of the specific subgrid terms which appear in Eq. (5) according to the filtered resolved inertial, viscous and capillary contributions of the flow. In this work, all the tensors have been investigated, except s 0 which will be considered in future works. A top hat filter has been applied to the DNS results for sizes of the filter corresponding to 2 (filter F2) and 4 (filter F4) times the size of the DNS control volumes in each direction. If i, j and k are the grid indices and r the filtered variable, the filters are defined by F 2ðU ijk Þ¼ F 4ðU ijk Þ¼ Z xi 1 Z yj 1 Z zk 1 x i 1 y j 1 z k 1 Z xi 2 Z yj 2 Z zk 2 x i 2 y j 2 z k 2 Uðx; y; zþdxdy dz Uðx; y; zþdxdy dz Fig. 8 presents the time evolution of the components of the averaged filtered terms ^s 1 ; ^s 2 ; ^s 3 and ^r 1, normalized by the maximum at each time step of the filtered resolved

7 904 S. Vincent et al. / Computers & Fluids 37 (2008) Fig. 8. Time evolution of subgrid filtered terms ^r 0, ^s 1, ^s 2, ^s 3 and ^r simulation.

8 S. Vincent et al. / Computers & Fluids 37 (2008) contributions of the flow (inertia for the momentum equation and advection for the phase function equation). The average filtered contributions are expressed as Z ^U ¼ F ðu ijk ÞdV X where F is one of the F2 or F4 filters. From a global point of view, it is observed that, for a given filter, the magnitude of the vertical component of the subgrid terms is larger than the contributions in the x and y directions. In the z-direction or gravity direction, the flow is strongly anisotropic. This is induced by the key phenomenon of the phase separation, i.e. the buoyancy force. Moreover, it is calculated that the magnitude of the subgrid terms is comparable in the x and y directions. Whatever the filter size, ^r 0 is small but not negligible compared to the resolved r~u. In addition to the explicit modeling of ^r 0, attention will have to be paid concerning the numerical treatment of incompressibility (projection methods, for example) in Eq. (5). The estimate of the subgrid tensors demonstrates that, in the case of the phase separation flow, subgrid terms ^s 1 and ^s 3 can represent 25% and 3%, respectively, of the maximum filtered inertial resolved contribution during the most turbulent phase (5 < t < 15 s). In the viscous dissipation phase, ^s 1 keeps its relative magnitude while ^s 3 rises up to 20% as soon as the subgrid scale capillary effects becomes predominant in front of the decreasing inertia. They could not be neglected in future LES simulations based on single fluid model. Concerning ^s 2, this subgrid term appears to be three orders of magnitude less than the filtered contribution of inertia. Its modeling would not be necessary in the two-phase LES model (5). To finish with, ^r 1 is of the same order of magnitude as the filtered resolved advection term. Its modeling will be of major importance in future LES of multiphase flows. This point is reported by Toutant et al. [28] in a priori filtering of a DNS of a deformable bubble interacting with isotropic homogeneous turbulence. Concerning the effect of the filtering operations on the subgrid terms, it is observed that the magnitude of the components of ^s 1 increases with the size of the filter. In the range considered in this work, the filtering does not induce a remarkable effet on the magnitude of the components of ^s 3. This subgrid term contribution is certainly driven by the larger resolved scale of interface. Even if the interfacial flow structure looks like a disperse flow (Fig. 2 for times between 8 and 12 s) constituted by small droplets, the average integrated effect of droplets with a diameter smaller than 4 DNS grid cells is negligible in this case. This conclusion testifies the assumption that a grid is large enough to lead a DNS of the phase separation in this case. Indeed, the majority of the interfacial structures are resolved by the VOF PLIC method. As for ^s 3, ^r 1 is not sensitive to the size of the filtering. The turbulent transfers at the interface highly depend on the interfacial area. A simple approach chosen to approximate this interfacial area R int is the following a binary phase function C _ is first built by stating C _ ¼ 1ifCP0:5 and C _ ¼ 0 elsewhere R int ¼ P kr _ S cell C 6¼0k where S cell = DxDy = DxDz = DyDz is an approximation on a regular Cartesian grid of the interfacial area in the cells cut by the interface. This operation has been applied to the resolved and filtered phase function field. A more efficient method for calculating the surface area of a contour is proposed by Geurts [29]. Fig. 9 presents the time evolution of the interfacial area normalized by the value R int = 0.75 m 2 corresponding to the initial free surface area for our phase separation case. The filtering operation involves an important decrease of the resulting R int with increasing filtering size. Differences are observed between the filtered interfacial area obtained thanks to 85 3 and grids. However, the filtering operations involve the same decrease of the interfacial area whatever the simulation grid. It has to be noticed that the filtering operation is conservative with respect to mass by construction but it induces a numerical coalescence of the small drops which involves an artificial decrease of R int. This effect of the filtering could have dramatic consequences in LES of multiphase flows, as the dynamics of droplets and large drops are different, especially in their interaction with turbulence. In order to build a two-phase LES turbulent model for macroscopic interfaces, the modeling of ^s 1 ; ^s 3 and ^r 1 will require taking into account the artificial effect of the LES filtering on interfacial area. To finish, we can notice that concerning the interfacial area, the F2 filtering applied to the simulations (corresponding to a LES simulation on a 64 3 grid) provides similar results to 85 3 DNS simulations. In this way, future LES Fig. 9. Temporal evolution of interfacial length depending on the size of the filter comparison between 85 3 and simulations.

9 906 S. Vincent et al. / Computers & Fluids 37 (2008) will allow us to simulate multiphase flow with reduced calculation costs while keeping accurate results. All the results presented in this section will have to be confirmed by a parametric study on q, l and c. 6. Conclusions and perspectives The DNS and LES motion equations for incompressible and isothermal two-phase flows have been presented. Onefield simulations of water/oil phase separation were carried out thanks to an implicit finite volume approach, coupled to a volume of fluid interface tracking method. The a priori analysis of the DNS results have demonstrated that specific subgrid tensors introduced by the LES formulation of turbulent two-phase flows cannot be neglected, particularly due to the presence of an interface. Moreover, increasing the size of the filter induces a dramatic decrease of the interfacial area. A parametric study of the phase separation will have to be lead according to surface tension and density or viscosity ratios in order to verify the relative magnitudes of the various tensors, including the acceleration term. The t 2 decay of the kinetic energy will also have to be investigated. Moreover, the characterization of the turbulent flow will require defining mean and fluctuating velocities, by constructing, for example, conditional filtered phase averages. Future subgrid scale modeling of the specific tensors will have to take into account the generation or modulation of turbulence involved by the presence of the interface and the local volume fraction fluctuations in order to correct the effect of the filtering on the interfacial area for example. In addition, specific subgrid terms linked to drag, lift or buoyancy forces of small droplets or capillary effects will have to be included into the momentum and interface tracking equations, as in two-way coupling modeling of turbulent particle flows (Boivin et al. [2]). The knowledge of the accurate subgrid scale volume fraction will require interface tracking methods to include a resolved diffusion of the volume fraction which will represent droplets or bubbles smaller than the grid size. References [1] Homescu D, Panday P. Forced convection condensation on a horizontal tube: influence of turbulence in the vapor and liquid phases. J Heat Transfer 1999;174: [2] Boivin M, Simonin O, Squires K. On the prediction of gas solid flows with two-way coupling using large eddy simulation. Phys Fluids 2000;12: [3] Calmet I, Magnaudet J. Statistical structure of high-reynolds number close to the free surface of an open-channel flow. J Fluid Mech 2003;474: [4] Liovic P, Lakehal D, Liow J-L. LES of turbulent bubble formation and break-up based on interface tracking. In: Gueurts BJ, Friedrich R, Metais O, editors. Direct and large eddy simulation V. ERCOF- TAC series, vol. 10. Kluwer Academic Publishers; p [5] Lacanette D, Gosset A, Vincent S, Buchlin J-M, Arquis E. Numerical simulation of gas-jet wipping in steel strip galvanizing process. Int Steel Iron J 2005;45: [6] Lacanette D, Vincent S, Arquis E, Gardin P. Macroscopic analysis of gas-jet wipping: numerical simulation and experimental approach. Phys Fluids 2006;18:1 15. [7] Christensen E, Deigaard R. Large eddy simulation of breaking waves. Coastal Eng 2001;42: [8] Lubin P, Vincent S, Abadie S, Caltagirone J-P. Three-dimensional large eddy simulation of air-entrainment under plunging breaker waves. Coastal Eng 2006;53: [9] Liovic P, Lakehal D. Multi-physics treatment in the vicinity of arbitrarily deformable gas liquid interfaces. J Comput Phys 2007;222: [10] Liovic P, Lakehal D. 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