Numerical Investigation of Eulerian Atomization Models based on a Diffuse-Interface Two-Phase Flow Approach coupled with Surface Density Equation

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1 ILASS Europe 2011, 24th European Conference on Liquid Atomization and Spray Systems, Estoril, Portugal, September 2011 Numerical Investigation of Eulerian Atomization Models based on a Diffuse-Interface Two-Phase Flow Approach coupled with Surface Density Equation B. Mandumpala Devassy 1, C. Habchi 2, E. Daniel 3 1 & 2: IFP Énergies nouvelles, 1-4 av. Bois-Préau Rueil Malmaison, France, md.bejoy@ifpenergiesnouvelles.fr & chawki.habchi@ifpenergiesnouvelles.fr 3: Université de Provence, IUSTI - Aix-Marseille, France, eric.daniel@polytech.univ-mrs.fr Abstract Atomization is a phenomenon of high importance for many practical devices involving turbulent mixing of compressible flows having significant density and velocity gradients. This study focuses on the modeling of liquid jets in the aim of improving the Diesel spray simulations. Since the liquid jet cone angle and atomization have been shown by previous simulations [1] and experiments [2] to be closely related to the flow features inside the injector, the main idea of this work is to build a new methodology for internal combustion engine simulations that simultaneously includes both the injector cavitating flow and the atomization computations. Different models need to be developed for this aim. In the first part of this paper, a compressible two-phase flow model is described. It relies on an Eulerian-Eulerian (E-E) diffused interface approach [3] implemented in the three dimensional CFD code IFP-C3D [4]. The model involves different sets of conservation equations; one for each phase : liquid fuel, its vapor or gas in the combustion chamber. The perfect gas equation of state (EOS) is used for the gas phase in the combustion chamber while the thermodynamics of the fuel phases (liquid or vapor) are governed by the Stiffened Gas EOS [5]. The previous study of Vessiller [6] has been devoted to dilute dispersed two-phase flows and the present paper describes a new formulation of the governing equations dedicated to primary atomization studies and dense or separate two-phase turbulent flow problems. In IFP-C3D the E-E atomization model is expressed by a transport equation for the liquid surface density (LSD). This quantity represents the area of gas-liquid interface and will allow the calculation of the interfacial exchange terms like heat transfer and phase changes. In the second part of this paper, a new LSD transport equation is suggested. It is based on the review of Delhaye [7] and the work of Vallet et al.[8] inspired by the flame surface density equation [9, 10]. The dynamic initialization procedure of injected LSD is first described. Then, the newly modeled term for the surface stretching process (having precise mathematical origin) has been implemented in IFP-C3D. The numerical results have shown the importance of the newly developed stretch term. Introduction The flow of single phase fluids has occupied the attention of scientists and engineers for many years. The equation of motion and thermal properties of single phase fluids are well accepted (Navier-Stokes equations) and closed form solutions for specific cases are well documented. The state-of-the-art for multi-phase flow is considerably more primitive in that the correct formulation of the governing equations is still subject to debate. For this reason, the study of multi-phase flows represents a challenging and potentially fruitful area of endeavor for scientist and engineers. Atomization of a liquid jet generally refers to a process in which a bulk liquid is disintegrated into small drops or droplets by internal and/or external forces as a result of the interaction between the liquid and surrounding medium. The disintegration or breakup occurs when the disruptive forces exceed the liquid surface tension and viscous forces. The main challenges for a typical numerical simulation of an atomization process comes from the fact that the interface location must be calculated as part of the solution process and because discontinuities in materials properties (density, viscosity e.t.c.) across the interface must be preserved. For studying the interface topology, one needs to describe the location and motion of the phase interface with high accuracy [11]. For these, mainly two types of methodology are adopted, interface tracking and interface capturing methods. For interface tracking, the surface motion is solved for a collection of marked particles on the phase interface. In the case of interface capturing methodology, the phase interface is embedded into a fixed grid via the use of an additional scalar. Mainly, three principle methods have emerged in this class: a) Volume of Fluid (VOF)[12], b) the level-set method (LSM) [13] and c) Diffuse interface method (DIM) [14]. The present study is more focused on two-phase flows using DIM. For a classical two-phase flow approaches, separate equations for mass, momentum, energy are written. These two phases are strongly coupled by the mass, Corresponding author: md.bejoy@ifpenergiesnouvelles.fr 1

2 momentum and energy exchanges between them and their existence is defined separately by defining a separate volume fraction equation. The general studies of governing equations and its derivation for diffused interface is done from thermodynamically consistent theories of continuum phase transitions [14]. Jacqmin[15] used thermodynamic principles to obtain a diffuse interface model for two-phase flows. From the atomization point of view, previous analysis, for example Habchi [16] has shown that the disintegration of liquid core into ligaments can occur due to the stringing of cavitation pockets and transient secondary flows. Besides, study on the interface between the liquid surface and the air, its stretching, deformation and emergence of minute droplets plays a major role in the atomization phenomena. At the present time, there are many attempts to simulate these processes using VOF and LSM approaches [17]. But the current trend to track the liquid-gas interface by defining a surface density Σ (mean interfacial area per unit volume in the turbulent two-phase flows) equation is getting more importance. Its origin comes from turbulent diffusion flame equation [18, 10, 19, 20]. An application of similar equation has been done on two-phase flow regimes. The first of this kind has been done by Vallet et al.[8]. The work of Truchot[21] and later by Vessiller [6] provides foundation of a two-phase Eulerian modeling with a new separate equation for interfacial surface density Σ, based on the grounds of Vallet et al.[8]. Finally the work of Bayoro [22] made it possible to calculate the cavitation inside Diesel injector nozzles using a Baer and Nanziato DIM model. In this work, the idea is that the atomization process can be modeled by defining a separate equation for Σ coupled with the equation of volume fraction and also with suitable break-up conditions. This paper is organized as follows. First, the system of Eulerian two-phase flow equations are explained in detail. Later, the new formulation of surface density equation with stretch term is presented and the procedures carried out for doing the dynamic initialization of surface density for liquid jets computation is given. Next the results obtained from a two-dimensional liquid injection test case are discussed especially with the new transport equations. Finally a brief conclusion along with future work is presented. Two-Phase Flow Governing Equations In this section, a detailed review of averaged governing equations is carried out. Besides the governing equations, the representation of liquid gas interface is also described by formulating an instantaneous surface area density equation. To obtain the multiphase flow models we use the averaging method of Drew and Passman [23] applied to compressible Navier-Stokes equations. Here we neglect the heat flux and mass transfer terms and turbulent fluctuations at the interface. The interfacial pressure and velocity is assumed according to Baer and Nunziato model [24] (Vl,i I = V l and Pl I = P g ). We write here the final averaged system of equations for two-phase flows (liquid and gas). For two-phase flows, the model consists of seven governing equation and an equation for interfacial area density (Σ). Continuity Equations: α l ρ l t + α lρ l Ṽ l,i = 0 (1) α g ρ g t + α gρ g Ṽ g,i = 0 (2) Momentum Equations: ) (α l ρ l Ṽ l,i + ( ) α l ρ l Ṽ l,i Ṽ l,j t = α lp l + Pl I α l + α lτ l,ij (3) ) (α g ρ g Ṽ g,i + ( ) α g ρ g Ṽ g,i Ṽ g,j t = α gp g + P I g α g + α gτ g,ij (4) where τ p,ij = τp,ij T + τp,ij. L The term τp,ij T = ρ pv p,i V p,j is the Reynolds Stress Tensor (superscript T denotes turbulence) and τp,ij L the laminar shear (superscript L denotes laminar), p denotes the phases either liquid (l) or gas (g). 2

3 Energy Equations: t (α lρ l ẽ l ) + ( α l ρ l Ṽ l,i ẽ l ) V l,i = P l α l α lq l,i V l,i + α l τ l,ij (5) x j t (α gρ g ẽ g ) + ( α g ρ g Ṽ g,i ẽ g ) = P g α g V g,i α gq g,i + α g τ g,ij V g,i x j (6) where e p is the internal energy and q p,i the heat flux. For gas phase, which is multi component, heat flux transfer will be mainly due to the conduction (Fourier s Law) and diffusion flux. While, for fuel, which is single component, only conduction flux plays the role. For a multi-component fuel medium this parameter can be written as, N T p p q p,i = λ p + ρ p h k Y k V Dk,i (7) where p is the phase, k is the species, N p is the number of species in phase p, Y k the mass fraction, h k the enthalpy and V Dk,i the diffusion velocity [25]. Then a liquid volume fraction equation to close the system: where α l = 1 α g k=1 α l t + Ṽ l,i I α l = 0 (8) Turbulent and Laminar Viscosity and Stress Tensors The stress tensor modeling (see Equation (3) and (4)) is done by standard Boussinesq expression and is given by: τ p,ij = ν p ( Vp,i x j + V ) p,j 2 ( ) V p,i ν p δ ij 3 where ν p is the kinematic viscosity which is the sum of laminar and turbulent kinematic viscosities (that is ν p = ν L p + ν T p ). Laminar viscosity ν L p, which is a function of temperature and is calculated according to Sutherland s formula[26]. In case (p = g), for gas phase, ν T g is given using a k g ε g model [27] by : ν T g = C µ k 2 g ε g (10) In case (p = l) for liquid phase, Smagorinsky model [28] is taken for the simplicity for modeling νl T, given by = (C s ) 2 S (11) ν T l where S = 2S ij S ij and the Smagorinsky constant C s usually has the value of and = (cell volume) (1/3). The Smagorinsky model is a very popular subgrid model that originated from the research in meteorological field. The Smagorinsky model states that the subgrid stress tensor is a scalar multiple of the resolved rate of strain tensor. This choice need to be justified in future work. Equations of State Equation of state are used to close this systems. In order to circumvent the difficulty of models having negative squared speed of sound in the two-phase region, the present study uses two equations of state (EOS). Each fluid possesses separate EOS. In the present investigation source terms arising in the gas phase are closed by perfect gas EOS and for the liquid phase by the Stiffened Gas EOS (SG EOS) and generally for both phases we can write, P p = ρ p (γ p 1)(e p q p ) γ p P,p (12) Table (1) summarizes the values of the equations of state parameters (γ p,q p,p,p ) of each phase: 3 (9)

4 Table 1. Parameters of the Equation of State. Parameters Gas Liquid(dodecane) γ p q p 0.0 J/kg J/kg P,p 0.0 Pa Pa Boundary Conditions [4] In an injection test case, the content of the ghost cells at the boundary is read in a file that is created in the initialization of the run. In the present study specific flags are set for the inlet and outlet nodes that allow building inlet and outlet face arrays. The simulations are carried out by giving the time dependent mass flow rate at the inlet. From the entrance area and the mass density a velocity is deduced. This velocity is set at the inlet nodes. At the outlet, a fixed pressure is set at a given capacity distance. Then all quantities like velocity at output nodes and scalar fluxes through output faces, can be computed. Surface Density Equation As we all know the two phases do not evolve independently since they are strongly coupled through the mass, momentum and energy exchanges between them. Most of this exchanges are proportional to the available contact area between the two phases, per unit volume of the mixture. This interfacial area per unit volume (surface density - Σ) is therefore a fundamental quantity in two-phase flow studies. So by defining a separate equation for the quantity Σ, is the best way of describing the development of surface near to the nozzle exit for a typical jet flow, which can thus describes primary atomization. The final averaged equation of Σ proposed is given below, the details of its derivation and its averaging procedures is explained in detail [29] Σ t +.( V l Σ ) ( +. V l s Σ ) = ) ( (.V l (nn) s : V l Σ +.V l nn : V s ) l Σ (13) }{{}}{{} A T a T The left hand side of Equation (13) contains two convection terms : convection by the mean velocity V l and convection by the fluctuating velocity V l. The later is generally modeled using a first gradient method [30]. Following Vessiller [6], this term is assumed negligible in order to be coherent with the volume fraction equation. The right hand terms represent the stretch of the surface density Σ due to the velocity divergence. Due to Reynolds decomposition two stretch terms appear: one due to the mean velocity field (A T ) and the other due to fluctuating velocity filed (a T ). Here the stretch due to the mean velocity field (term A T ) involves the orientation tensor (nn) s which is the surface average of the dyadic product of the normal vector n by itself. This paper investigates first the effect of the stretch term due to mean velocity field (term A T ) in the Σ equation. The modeling work of stretch due to fluctuation velocity field term (term a T ) will be considered later in future work. First, the new Σ equation (only with term A T ) is given below and a diagrammatic representation of the model with the normal acting at the interface is shown in Figure (1) Σ t +.( V l Σ ) ) = (.V l (nn) s : V l Σ (14) This model (Equation (14)) represents the gas-liquid interface and treats them as separate phases. The key point in the above equation is the calculation of the (nn) s, orientation tensor. The different ways of modeling this orientation tensor is explained in the next section..v l is the effect of compressibility in Σ equation. Dynamic Initialization of Surface Density The present study has contributed the definition of dynamic initialization for surface density equation. This term permits to initialize the surface density calculations without artificially injecting liquid surface density in to 4

5 Figure 1. The diagrammatic representation of the model (Equation (14)) and the normal n acting at the wrinkled interface in a computational cell. the computational domain. At each time step, the initialization is done on the cells separating liquid and gas by taking Σ = max(σ, 2 α ) Models for normal n computation Model 1 : The first suggested model for the normal (n s ) is the basic way of taking n s as the α α. This accounts for taking the interfaces as straight as the one did during the initialization. So generally we can say that this model accounts only for stretching and do not take into account possible wrinkling of the interface, because there can be only one average normal acting over the surface in a cell. Model 2: Hawkes and Cant method [31] This work is concerned with the formulation of a transport equation for flame surface density for premixed turbulent combustion. Here the value of n takes care the effect of wrinkling of the interfaces. The normal n is treated as per the Hawkes and Cant model[31] given below A T = (.V l n i n j s : V l ) (15) where n i n j s = n i s n j s rδ ij,r = 1 n s 2 The presence of the term r is designed to melt down the isotropic part of tensor orientation when the flow becomes laminar or very well resolved, while it becomes important when the flow is turbulent or very poorly resolved. This term thus bring the concept of wrinkling of the interfaces by taking the values of n s as α Σ. First suggested model for primary atomization From the detailed literature review and from the new stretch term, the final surface density equation selected for the present study is discussed in this section. Σ t +.( V l Σ ) ) = (.V l (nn) s : V l Σ + aσ(1 C 7σ l Σ) (16) α l ρ g kg This model has its turbulent counter part which is the last term of the above equation other than the stretch term. This term is the production and destruction terms from Vallet et al.[8]. This equation (Equation (16)) typically contains a separated phase model (stretch term) and the dispersed phase model (turbulent production term). Even though no criterion has been modeled for the breakup of the wrinkled interface, this equation will help in future for defining the same and its work is under progress. So presently the actual interface will be the overall average of the interfaces due to the stretching and the production terms which is well presented in Figure (2). The normal in Equation (16) is modeled as per the Hawkes and Cant model[31] and the complete equation (Equation (16)) is implemented in IFP-C3D for further validation. 5

6 Figure 2. The diagrammatic representation of the final suggested model (Equation (16)) for the present study Injection Test Case This section presents the test cases carried out for primary atomization. The first part of this section explains the computational domain, initial and boundary conditions. The second part describes the analysis carried out on the newly implemented surface density equation. Computational Conditions For the present test case a typical 2D finite volume hexahedron grid shown in Figure (3) has been generated. The domain consists of an injector hole filled by a liquid fuel (n-dodecane) and a chamber initially filled by gas (nitrogen). The initial pressure and temperature in the liquid and gas phases are 60 bar and 300K respectively. Fuel is injected at the same temperature with an inlet velocity equal to 100 m/s or 300 m/s. This injected velocity has a uniform profile through out the injector section. The computational mesh has hexahedral cells. The diameter of the injector is taken as a typical value of 200µm. Figure 3. Injection test case computational domain for primary atomization modeling Results and Discussion The analysis done here explains the effect of new surface density equation with stretch term. Two different simulations with 100m/s and 300m/s as injection velocity are carried out. Figure (4) shows the evolution of spray and its spray structure over different time intervals. From the figure one can see an unsymmetrical spray structure while moving from 60µs to 80µs. This is due to the fact that the turbulent viscosity of the injected liquid (dodecane) is modeled with Smagorinsky theory [28] described before (Equation (11)) and the gas phase by k g ε g model [27]. The value of total surface density is computed at different time after start of injection (SOI) through the spray axis. Here surface density values are computed for three cases: one without any stretch (A T = 0), second with 6

7 Figure 4. Spray structure at different time intervals stretch (A T ) and without wrinkling (n s = α α ) and the third with wrinkled stretch term (ns = α ). From the Σ general visualization of the spray structures, Figure (5) shows the variation of Σ for the above three different cases. It is clear that an increase in Σ can be seen for the second case (with stretch and without wrinkling) compared to the other two cases proving that in each cells the normal acting at the interface is maximum given by n s = α α (straight interface). While for the case with wrinkling(third) a decrease in Σ can be seen (shown in Figure (5)) over the entire spray structure, which is the contrary that we expected, showing the inaccuracy in using the model of Hawkes and Cant[31]. Thus with stretch term there is a considerable increase in surface density compared to the one without stretching. But within stretch term the one without wrinkling has more surface density than the one with wrinkling. This is actually the contrary which was expected showing the inaccuracy in using the Hawkes and Cant model[31] for defining the wrinkling of interfaces in two-phase flows. Figure 5. Comparison of surface density for three different cases (without stretch, with stretch and without wrinkling and with stretch and with wrinkling) at 40 µs after start of injection Variation of Σ over the spray axis Figure (6(a)) shows the method of calculation of total surface density (Σ) along the spray axis. Here the entire spray is made to split to 20 different bins and the value of Σ on each cell is added together for each bins to show the cumulative progress in the surface density development. Figure (6(b)) shows the variation of Σ along the spray axis presented in Figure (6(a)) for an injection velocity of 100m/s. The values of Σ at 3 different time(10, 30 and 50 µs) are plotted. The results reveal that the second case (with stretch and without wrinkle) has an considerable increase in Σ with a high possibility of breakup. While, with the wrinkled stretch term a decreased effect of 7

8 (a) Quantitative calculation of the surface density at different locations along the spray axis (b) Calculation of Σ along the spray axis for an injection velocity of 100m/s for the three different cases shown in Figure (5) at three different times (10µs, 30µs and 50µs) Figure 6. Variation of Σ over the spray axis wrinkling can be noticed by decreasing the value of Σ which is not expected. This difference is more for higher velocities (shown in Figure (7)). Calculation is also carried out with the available production(sgs) and destruction Figure 7. Calculation of Σ along the spray axis for an injection velocity of 300m/s for the three different cases shown in Figure (5) at three different times (5µs, 10µs and 15µs) term from literature ([8]). Figure (8) shows the similar kind of variation by an increase in value of Σ. Conclusions Primary atomization represents an example of a complex gas-liquid flow. Near the nozzle, the liquid, initially introduced as a continuous jet, disintegrates into filaments and drops by interacting with the gas. This work mainly focuses on the primary atomization of an Eulerian, separate two-phase liquid jet. The objective of this work is to build a system of separate two-phase flow equations with focusing mainly on the near nozzle end. A detailed analysis of governing equations and bibliographic study for surface density equations are carried out. From the knowledge of flame modeling from combustion, a similar kind of analogy is implemented here by developing a new separate equation for surface density with a wrinkled stretch term. For the injection process, dynamic initialization of surface density is proposed. Analysis were carried out for proving the effect of stretching on liquid jet in conditions similar to diesel injection. The results reveals that with stretch term there is a considerable increase 8

9 Figure 8. Σ variation along the spray axis for an injection velocity of 100m/s with the production (SGS) and destruction terms of Vallet et al.([8]) for the three different cases shown in Figure (5) at three different times (10µs, 30µs and 50µs) in surface density compared to the one without stretching. But within stretch term the one without wrinkling has more surface density than the one with wrinkling. This is actually the contrary which was expected showing the inaccuracy in using the Hawkes and Cant model[31] for defining the wrinkling of interfaces in two-phase flows. The inclusion of the production terms of Vallet et al.[8] leads to considerable amount of increase in surface density and can be tuned further for taking into account the evolution of ligaments from the wrinkled surfaces. This will later accounts for high breakup regions along the spray axis. The future work needs to be mainly focused on modeling the new terms of conservation and surface density equations. The turbulent production term in the Σ equation of Vallet et al.[8] could be a good candidate for predicting the primary atomization on separate phase and secondary atomization on dispersed phase. Nomenclature Symbols e internal Energy [J kg -1 ] n normal P pressure [Pa] τ shear [N m -2 ] t time [s] T temperature [K] V velocity [m s -1 ] Greek Symbols α volume fraction γ stiffened gas EOS parameter ν kinematic viscosity [m 2 s -1 ] ρ density [kg s -3 ] σ surface tension [N m -1 ] Subscripts g gas l liquid value at infinity Superscripts I interface L laminar -s averaging over the surface T turbulent 9

10 References [1] Habchi, C., Dumont, N., Simonin, O., Atomization and Sprays 18: (2008). [2] Blessing, M., König, G., Krüger, C.,Michels,U., Schwarz, V., SAE transactions 112: (2003). [3] Abgrall, R., and Saurel, R., Journal of Computational Physics 186: (2003). [4] Bohbot, J., Gillet, N., Benkenida, A., Oil and Gas Science and Technology-Revue de l Institut Francais du Petrole 64: (2009). [5] Saurel, R., Petitpas, F., Abgrall, R., Journal of Fluid Mechanics 607: (2008). [6] Vessiller, C., Contribution à l étude des brouillards denses et dilués par la simulation numérique Euler / Euler et Euler / Lagrange, ECP/IFPEN thesis No [7] Delhaye, B., Etude des flammes de diffusion turbulentes. Simulations directes et modélisation, Ecole Centrale Paris Thèse, No. 94 ECAP [8] Vallet, A., Burluka, A., Borghi, R., Atomization and Sprays 11: (2001). [9] Poinsot, T., and Veynante, D., Theoretical and numerical combustion, R.T. Edwards, 2nd edition., [10] Candel, S.M., and Poinsot, T., Combustion Science and Technology 70:1-15 (1990). [11] Gorokhovski, M., and Herrmann, M., Annual Review of Fluid Mechanics 40: (2008). [12] Hirth, C.W., and Nichols, B.D., Journal of Computational Physics 39: (1981). [13] Sethian, J.A., 21 st Proceedings of National Academy of Science, 1996, pp [14] Anderson, D.M., and McFadden, G.B., Annual Review of Fluid Mechanics 30: (1998). [15] Jacqmin, D., Journal of Computational Physics 155: (1999). [16] Habchi, C., Modélisation Tridimensionnelle de l injection et de la préparation du mélange gaz/carburant dans les moteurs à combustion interne., IFPEN/INPT Report No f44b7989e skydrive.live.com/browse.aspx/HDR. [17] Desjardins, O., and Pitsch, H., Atomization and Sprays 20: (2010). [18] Pope, S., International Journal Engineering Science 26: (1988). [19] Trouvé, A., and Poinsot, T., Journal of Fluid Mechanics 278:1-31 (1994). [20] Vervisch, L., Bidaux, E., Bray, K.N.C., Kollmann, W., Physics of Fluids 7:2496 (1995). [21] Truchot, B., Développement et validation d un modèle eulérien en vue de la simulation des jets de carburants dans les moteurs à combustion interne, INPT/IFPEN thesis No [22] Bayoro, F., Habchi, C., Daniel, E., 21 st European Conference on Liquid Atomization and Spray Systems, Como Lake, Italy, September [23] Drew, D. A., and Passman, S. L., Theory of Multicomponent Fluids, Springer-Verlag, [24] Baer, M. R., and Nunziato, J. W., International Journal of Multiphase Flow 12: (1986). [25] Hirschfelder, J. O., Curtiss, C.F., Byrd, R. B., Molecular theory of gases and liquids, John Wiley & Sons, [26] Alexander, A. A., Jean-Paul, D., Turbulent shear layers in supersonic flow, Birkhäuser, [27] Amsden, B., O Roukre, P. J., Butler, T. D., KIVA II: a computer program for chemically reactive flows with sprays, Los Alamos National Laboratory report No. LA MS, [28] Smagorinsky J., Monthly Weather Review 91: (1963). [29] Mandumpala Devassy, B., Atomization Modeling of Liquid Jets by an Eulerian Approach., IFPEN Report No [30] Anand, M., and Pope, S. B., Symposium on Turbulent Shear Flows-4, Karlsruhe, West Germany, September 1983, pp [31] Hawkes, E. R., and Cant, R. S., Proceedings of the Combustion Institute 28:51-58 (2000). 10