Atomization Modelling of Liquid Jets using a Two ...

Atomization Modelling of Liquid Jets using a Two-Surface Density Approach Bejoy Mandumpala Devassy1, Chawki Habchi1*, Eric Daniel2

1: IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France [email protected] & [email protected] 2: Aix-Marseille Université, IUSTI, 5 rue E. Fermi 13453 Marseille cedex 13, France [email protected]

Abstract In internal combustion engines, the liquid fuel injection is an essential step for the air/fuel mixture preparation and the combustion process. Indeed, the structure of the liquid jet coming out from the injector plays a key role in the proper mixing of the fuel with the gas in the combustion chamber. The present work focuses on the liquid jet atomization phenomena under Diesel engine conditions. Under these conditions, liquid jet morphology is assumed including a separate liquid phase (i.e. a liquid core) and a dispersed liquid phase (i.e. a spray). This article describes the development stages of a new atomization model, for a high speed liquid jet, based on an EulerianEulerian two-fluid approach. The atomization phenomena are modelled by defining different surface density equations, for the liquid core and the spray droplets. This new model has been coupled with a turbulent and highly compressible two-fluid system of equations. The process of ligament breakup and its subsequent breakup into droplets are handled by acquiring knowledge from the available high fidelity experiments and numerical simulations. In the dense region of the liquid jet, the atomization is modelled as a dispersion process due to the stretching of the interface, from the liquid side in addition to the gas side. The model results have been compared to

1

*corresponding author: [email protected]

1

the recently published DNS results under typical direct injection Diesel engine conditions. In particular, it has been shown that the interface instabilities and the turbulence in the leading tip of the liquid core play a major role in the primary atomization of Diesel jets.

Key words Atomization, Breakup, Turbulence, Two-Surface Density, Two-fluid model, compressible liquid, Stiffened gas EOS, Eulerian-Eulerian model, Diffuse Interface method.

1. Introduction In reciprocating engines, fuel injection and the air-fuel mixture distribution plays a key role in determining the controlled combustion and the type of pollutants emitted. These factors are closely related to the degree of atomization of the injected liquid jet. 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. This disintegration or breakup occurs when the disruptive forces exceed the liquid surface tension and viscous forces. The most important factor which affects the liquid flow and its atomization is the interaction with the gas in the combustion chamber. This interaction may be affected by the gas density and temperature, the injection velocity magnitude and turbulence. In addition, the effect of cavitation and the induced secondary flows inside the injector orifices are recognized among the important parameters which directly influence the process of liquid jet atomization (Giannadakis et al., 2008 [16], Habchi et al., 2008 [19], and Tamaki et al., 2000 [39]). The final droplets size formed greatly depends on the primary breakup process of the liquid jet (Herrmann, 2011 [20], Shinjo and Umemura, 2010 [38]). But, research in two phase flow is not mature enough to predict the type of phenomena governing the primary atomization process. This is because of the range of time and length scales involved in the process. The breaking of ligaments into large primary droplets and their subsequent breakup into child droplets is a cascade process which is well accepted in the open literature for sub-critical conditions. 2

NOMENCLATURE Ck Ck = 0.006 is a constant Greek Symbols Cs Cs = 0.2 is a constant in Eq. (14)  Volume fraction Cµ=0.09 is a constant in Eq. (13)  Turbulent kinetic dissipation rate C Cv Liquid specific heat at constant Cp/Cv in SG-EOS Eq. (7) where Cp is the  volume coefficient in Eq. (9) liquid specific heat at constant pressure D Diffusion coefficient in Eq. (2) Liquid surface density  e Specific internal energy Kronecker delta function ij h Specific enthalpy Ligament diameter b k Turbulent kinetic energy Interfacial temperature parameter in Eq.(6)  q Heat Flux Conductivity coefficient  l Length scale Viscosity  P Pressure Liquid surface tension  P∞,ql SG-EOS Eq. (9) fluid parameters  Interfacial velocity, pressure and temperature relaxation to equilibrium characteristic and frequency  Q Specific heat flux Indicator function  R Perfect gas universal constant Density  Sij, S Strain tensor and strain intensity Sub and Superscripts t Time i Index of the Cartesian components T Temperature I Interfacial parameter V Velocity l Liquid phase x Cartesian coordinate g Gas phase Lb Length of a ligament m Mixture (Gas + Liquid) phase L Laminar condition Abbreviations RANS Reynolds Averaged NavierT Turbulent condition Stokes RHS Right Hand Side p,q q=3-p, index of the phases in Eqs. (3) to (6) SG-EOS Stiffened Gas Equation Of s Liquid core (Separate phase) State, Eq. (9) IFP-C3D Commercial 3D two-phase d Spray droplets (Dispersed phase) CFD and combustion software developed by IFPEN and distributed by LMS/Siemens co. 3

The ligaments start growing from the liquid core due to several reasons. It can occur due to the collapse of the cavitation pockets which have reached the exit of the injector orifice. Also, the flow of gas entrained by the high speed liquid jet can shear the liquid-gas interface, and thus be a source of interfacial instabilities and turbulence which may lead to ligaments generation. At a critical condition, these ligaments can pinch off from its parent liquid core. The ligaments thus formed are unstable and undergo breakup producing droplets. These primary drops may subsequently undergo secondary breakup in order to reach their final stable sizes. This changeover from large liquid structures to small droplets is the issue addressed in this work (Figure 1). This work attempts to improve the numerical simulation of the primary atomization process. An Eulerian-Eulerian formulation is used since it is better adapted for the numerical simulation of the liquid core in the dense region than the Lagrangian-Eulerian approach, currently used for spray modelling. This paper is organized as follows: Next sub-section 1.1 is the literature review based on the current state of art of primary and secondary atomization. Details of the two-phase system are explained in Section 2. Next, in Section 3, the main idea of this article: Modeling the atomization phenomena using two separate equations of surface density for the primary and secondary breakup processes is described along with its implementation into the software, IFP-C3D (see Bohbot et al., 2009 [6] and Velghe et al., 2010 [44]). In Section 4, the suggested atomization model is then applied for the simulation of liquid jets injected under actual diesel engine conditions. The aim of these computations is twofold: (1) To perform numerical investigations in order to identify the best suitable parameter values for the suggested atomization model. (2) To perform qualitative and quantitative comparisons between the results of the new atomization model and the DNS observations of Herrmann, 2011 [20], since the present state of art has no clear experimental evidences available close to the nozzle. Finally, the conclusions with future work are presented in Section 5.

1.1 Literature Review This section primarily aims to recall the main experimental observations and the recent numerical investigations from the literature. Indeed, we are going to focus here on the primary atomiza4

tion investigations in order to gather most of the information on the primary atomization processes. The reader interested in the secondary atomization processes may for instance refer to the recent review of Habchi, 2011 [17] and to the articles referenced inside. Firstly, experimental study to view the liquid core region is a challenging task. There exist several reasons why there are no sufficient experimental results for the primary atomization. First and the most important reason is the presence of a droplet cloud surrounding the liquid core region blocking the access to the optical rays during experiments. Besides, the phenomena happening at the liquid core region is difficult to view because the length and time scales involved are very small. With the recent advancement in the optical and X-ray imaging techniques, the dense region and the liquid core may be investigated more precisely. Several works have been carried out recently in order to visualize the primary breakup process and thereby to understand the physics happening near and at the gas/liquid interface. Marmottant and Villermaux, 2004 [28] completed a detailed analysis on the gas/liquid interface by studying in detail the ligament dynamics and its characteristics in coaxial injection regimes. Particularly, they highlighted the presence of longitudinal and transversal instabilities and concluded that the vorticity thickness controls the development of these instabilities. In addition, they performed a linear stability analysis in order to obtain the wavelength of the transversal instability. Furthermore, Marmottant and Villermaux, 2004 [28] performed an image processing analysis, to determine the size of the ligaments. They established that the diameter of the ligaments was around one fourth the transversal wavelength determined by the flow configuration. Moreover, they observed how the length of these ligaments decreased with the decrease in the shear velocity. From Figure 2, it is visible that the ligaments are formed mainly due to the shearing force acting over the liquid surface. Besides they also found out that a critical breakup Weber number Web (based on ligament diameter b ) equal to four fits well their experimental data. Also, the recent effort by Osta et al., 2012 [31] to capture the ligament dynamics using high energetic and penetrating X-ray beam is quite promising. Here water is the liquid chosen and was made to inject with a velocity of 30m/s. In the experimental setup, Osta and his co-workers tried to characterize the ligament sizes and formulated its distributions on a turbulent liquid jet injected into still air. The results reveal the 5

position and the existence of ligaments having different sizes. If the ligaments diameters are large enough, they are separated far from its neighbors suggesting that these ligaments thus reduce the available kinetic energy to create more ligaments nearby. Besides, these experiments are also able to show the effects of injector length to diameter ratio on the breakup rates. Another interesting work has recently been performed by Crua et al., 2010 [11] using ultra high speed camera. They were able to explain the formation of oblate spheroidal cap at the tip of the jet just at the exit of the nozzle. Indeed, they found out that this is due to the transverse expansion of the jet and the physical properties of the fuel. Besides, the experiments were also able to capture the formation of fuel ligaments close to the orifice. Using Ultrafast X-ray technique, Wang et al., 2008 [48] succeeded in visualizing and proving the existence of liquid core regions for high speed jets. The details of the gas/liquid interface in the micro scale is still found difficult using their 0.47µs exposure time and 5-30 µm spatial resolution scales. Another set of experimental observations using pulsed shadowgraph by Sallam et al., 2002 [34] and Faeth et al., 1995 [14] showed that aerodynamic force has little impact on liquid structures for high liquid to gas density ratio (in the order of 1000). Moreover, they were able to provide models for the breakup length and the resulting droplet velocity. But for low density ratio flows like diesel injection (density ratio in the order of 40) aerodynamic forces plays a significant role in the atomization of liquid jet. This was suggested by Wu and Faeth, 1993 [49]. All the previous experiments are useful but they are insufficient for atomization models validations. Capturing the liquid core and breakup of ligaments from its parent liquid core and its subsequent breakup into droplets using X-ray and Ballistic imaging techniques are getting importance. In near future, spatial resolution in the submicrometer level may offer valuable databases for the validation of the primary atomization models. On the other hand, the recent developments in the computational resources improved the ability to view the atomization phenomena more accurately than before. Then, Direct Numerical Simulations (DNS) could be used along with the available experimental data to validate primary atomization models. Interfacial dynamics can be computed more precisely with resolution scales reaching to the order of one micron and temporal scales to 2ns and below (Fuster et al., 2009 [15]). The studies by Shinjo and Umemura, 2010 [38] on the liquid jets injected into a stagnant gas domain are among the very recent high fidelity simulations (HFS). Their investigation of the 6

ligament dynamics and its breakup is quite promising. For example, Figure 3 shows the ligaments and droplets formation from the liquid core in their computations. The analysis predicts the drop size distribution after primary atomization. The Sauter Mean Diameter (SMD) obtained is in the range of 4.3 µm for a Weber number of 14000 and Reynolds number of 1470. The DNS of Herrmann, 2011 [20] are also very interesting. In one of his most recent article, Herrmann, 2011 [20] demonstrates that grid independent drops may be obtained for the liquid structures resolved by at least six grid points. For this, he used a refined level set grid method (RLSG). The study also tries to compare the scaled number frequency distribution of the atomized droplets with a log-normal distribution and SMD is estimated to be 2.14µm. Desjardins and Pitsch, 2010 [12] applied the combined level set/ghost fluid method to a straight jet configuration to visualize the type of atomization happening at the phase interface. The results expose that the aerodynamic forces play a key role in the turbulent atomization of liquid jets. Besides, they are also able to explain the formation of ligaments due to the bursting of air bubbles at the liquid surface. In addition, their bibliographic review has highlighted that the surface density models could be a good candidate for the modeling of the primary atomization in the context of Large Eddy Simulations (LES) or Reynolds Average Navier-Stokes (RANS) simulations. Compared to DNS and LES studies in the literature, the present investigation uses RANS approach for predicting the behavior of primary jet atomization in actual diesel engine conditions. As mentioned by Hermann, 2011 [20], DNS methods require very refined meshes. This in turn requires huge computational resource, making it less applicable for industrial applications. Besides, the implementation of models for the fluctuating scales using RANS approach is preferred more in industries, compared to its counterpart LES or DNS approaches. The present investigation is the first of this kind carried out using diffuse interface two-fluid model to predict the primary breakup of Diesel like liquid jets. In this work, the Surface density is defined as the mean interfacial area per unit volume,  . Equation (1) is the basic equation for  adopted from Morel, 2007 [29] when the phase change is not considered.

7

I   Vi    D   AT  aT   t xi

 

I  VI s V j  i   ; AT   nn :  xi xi    D

s "I   "I  V  V j  aT   i  nn :  xi xi   

(1)

 V j"I   xi

  s

In Equation (1),  denote the average operator at the interface and n is the normal acting at the surface density. The interface velocity is decomposed as follows: V I  VI  V "I . Thereby, the terms AT and aT on the RHS arise due to this Favre averaging and represents the stretch of the interface due to the mean (resolved) and fluctuating (unresolved) turbulent velocity, respectively. It is worth to note that Equation (1) may also represent the basic equation of surface density adopted for the modeling of turbulent diffusion flames (see for instance Pope, 1988 [33], Candel et al., 1990 [8], Trouvé and Poinsot, 1994 [41] and Veynante and Vervisch, 2002 [46] ). In this case, V I is the flame speed and V "I is its fluctuation due to turbulence. In the context of two-phase flows, the interfacial “turbulent” fluctuation V "I may be due to many contributions including, in addition to the turbulent fluctuations, the interfacial instabilities such as KelvinHelmholtz and Rayleigh-Taylor instabilities, etc… . The first application of Equation (1) to the atomization process has been suggested by Vallet and Borghi, 1999 [42] and Vallet et al., 2001 [43]. The  equation proposed by them using RANS approach is the first break-through into the detailed analysis of atomization using surface density. This methodology has been inspired by several researchers (see for instance Lebas et al., 2005 [22], Ning et al., 2009 [30], Luca et al., 2009 [24], Beau et al., 2005 [5], Blokkeel et al., 2003 [7] and Jay et al., 2006 [21]) and the trend to model the atomization using surface density equations is getting importance. Vallet et al., 2001 [43] and Lebas et al., 2009 [23] modeled the first term D on the RHS of Equation (1) using

8

a first gradient approach by considering an assumption of turbulent diffusion of gas in the liquid and vice versa and thus their transport equation of  is written as follows:    Vm ,i          Ds  t xi xi  xi 





(2)

where Ds is a turbulent diffusion coefficient and Vm,i is the liquid gas mixture velocity. The sec

ond term  on RHS of Equation (2) is the source term related to the production and destruction of the interfacial area, corresponding to (AT+aT ) in Equation (1). In addition, Vallet et al., 2001 [43] 

derived a model for  , but they could only apply it for spherical droplets and thus it is not suitable to predict the atomization process near to the nozzle exit, where the droplets are not yet formed. 

In the ELSA model (Lebas et al., 2009 [23]),  includes the production source terms from dense and dilute regions by differentiating them with an indicator function ( ) as 





    dense  (1   )  dilute . But for performing this, they assumed a linear transition of volume

fraction while moving from the dense region to the dilute region, which seems to be a very rough assumption. Moreover, the determination of mass, momentum and energy exchange terms from the liquid core to the droplets is very difficult to estimate with only one equation of surface density. In addition, Chesnel et al., 2011 [9] and 2009 [10] used a similar modelling as Equation 

(2) for determining the liquid jet atomization in his LES computations. The production term  is considered as the sum of a minimum surface density ( min ) and the surface production happening at the sub-grid scale (   ). Besides, the scale similarity method is found appropriate for the computation of Ds in Equation (2).

9

2 Models Formulation 2.1 Eulerian-Eulerian two-phase flow governing equations The starting point for the two-fluid model proposed here is adopted directly from the work of Saurel and Abgrall, 1999 [36]. Their method is based on a compressible model involving seven partial differential equations, three for the gas phase and three for the liquid phase, in addition to an equation for the liquid volume fraction transport. In this work, these equations have been generalized for the RANS simulations of turbulent two-phase flows in a way similar to the averaging method of Drew and Passman, 1999 [13]. In addition, the specific internal energy equation is used instead of the total specific energy adopted in the work of Saurel and Abgrall, 1999 [36]. Also, phase change at the interface is not considered in this paper. For twophase flows, the model consists of the following governing equations:

 p  p t



 p  p V p ,i xi

0

(3)

 p Pp  p  p  pL,,ijT   I      p  p V p ,i   p  p V pi Vp , j   P    V q ,i  V p ,i t xi xi xi xi













L ,T        P V p ,i    L ,T V p ,i   p q p ,i + VI V  V  p  p ep   p  p V e p ,i p p p p p ,ij q ,i p ,i t xi xi x j xi











(4)



(5)

  P I Pq  Pp   (Tq  Tp )



 p



 I  p  Vi   Pq  Pp  (Tq  Tp ) t xi 





(6)

Here p={1,2} and q=3-p; say p=1 for the liquid phase (index l) and p=2 for the gas phase (index g). Equation (6) is solved only for p=1 (i.e. for the liquid volume fraction,  l   1 ). The saturation constraint  2  1  1 is then used for the calculation of the gas volume fraction,

 g   2 . The (  ) and ( ) designate the Reynolds and Favre average operation, respectively.  pL,,ijT is the shear stress tensor which has the contribution from both the laminar and turbulent flow 10

regimes.  pL,,ijT can be written as,  pL,,ijT   pL,ij   Tp,ij , where the superscripts L and T stand for laminar and turbulent flow regimes, respectively. Its modelling is given below in Section 2.2. In the energy Equation (5), ep is the Favre averaged specific internal energy and q Lp ,,iT the mean heat flux. For the latter, the laminar contribution is neglected since both the liquid and gas flows are considered here in the turbulent regime. Since the liquid and gas phase are considered in this work as single component, the mean heat flux q Lp ,,iT is mainly due to the conduction (Fourier's Law). It is modeled as follows, T q LT p ,i    p

Tp xi

(7)

where  Tp is the conductivity coefficient computed from a specified turbulent Prandtl number equal to 0.9. For the sake of simplicity, the liquid and the gas phases are assumed single-component in this paper. However, this is not a limitation of the proposed model because Equation (3) may be readily replaced by mass fraction equations for both phases. In the system of Equations (3) to (6) also intervenes an interfacial velocity and an interfacial pressure. They are assumed according to Baer and Nunziato model, 1986 [4] ( V I  Vl and P I  Pg ). In addition, the model equations include relaxation terms to restore pressure, velocity and temperature at the interfaces. Here the velocities are relaxed at the interface with an infinite relaxation by making  tending to infinity. This means that for any arbitrary small time increment, the velocities must be in equilibrium at the interfaces. Similar to the velocity relaxation, pressure at the interface is also relaxed by tending the pressure relaxation coefficient  to infinity. The description of these relaxation parameters are taken from Saurel and Abgrall, 1999 [36] and Saurel and Le metayer, 2001 [37]. Similarly, an infinite relaxation of the temperature at the interfaces is applied by tending the relaxation parameter,  to infinity (see Zein et al., 2010 [50]). It is important to note that the pressure and temperature equilibrium are accomplished by computing the exact volume fraction variations (see the right hand side (RHS) of Equation (6)). These volume fraction changes are happening without any phase change since the RHS of the mass conservation Equation (3) is zero. 11

Equations of state are used to close the above system of equations. Since the two phases considered here are compressible the present study requires two equations of state (EOS). Each fluid possesses separate EOS. In the present investigation, terms arising in the gas phase are closed by the perfect gas EOS where Rg is the gas constant:

Pg   g Rg Tg

(8)

However, the Stiffened Gas EOS (SG-EOS) is used in the liquid phase. Two different expression of the SG-EOS are given as follows: Pl  l   l  1 el  ql    l P ,l Tl 

Pl  P ,l

(9)

Cvl l   l  1

Table 1 summarizes the values of the SG-EOS parameters of n-Dodecane in liquid state. Finally, the parameter appearing in the RHS of Equation (6) is given by the following expression: Pl   l P,l Pg  l g   l 1  g 1  l g

(10)

2.2 Turbulent and Laminar Stress Tensors The stress tensor  pL,,ijT described in Section (2.1) requires modeling in order to close the

  system of momentum and energy equations. The Reynolds Stress Tensor (  Tp ,ij    pV p ,i V p , j ) is modeled using turbulent viscosity concept and is done by standard Boussinesq approximation. According to Boussinesq theory, shear stress tensor can be written as

12

 V V  2  V   pL,,ijT   p  p ,i  p , j     p p ,i  ij   x j  xi  3  xi  

(11)

where  p is the sum of the laminar and turbulent dynamic viscosities (  p   pL   Tp ). For the laminar viscosity of gas phase,  gL , which is a function of temperature of the gas and is calculated according to Sutherland's formula (Alexander and Jean-Paul, 2006 [1]). While, since the liquid is injected with high pressure, Kant's law (Aquing et al., 2012 [3]) is chosen for finding the laminar dynamic viscosity of liquid ( lL ) as function of pressure and temperature. On the other hand, turbulent kinematic viscosity  Tp of each phase (liquid or gas) has been first calculated using standard k p   p models (see for instance Amsden et al., 1989 [2]) by,

 Tp  C  p

k p2

p

(13)

with Cµ=0.09. Indeed, standard kl   l model has been initially implemented for the liquid phase in addition to the gas phase. This approach assumes a fully developed turbulence in the gas and liquid phases. However, since liquid injector flow is not fully turbulent but it experiences a laminar/turbulent flow transition in the orifice of the injector, the liquid kl   l model has proved to be inappropriate for transient injector flows. Indeed, unpublished results obtained using kl   l models in terms of turbulent viscosity in the liquid phase has proved to be overestimated and has prevented the appearance of the cavitation, especially at entrance of the nozzle orifice where the inception of the cavitation is observed experimentally. The reason for that behavior is that kl   l model is only valid for high Reynolds number conditions. In order to avoid this issue, a simple algebraic k l  l model, based on a characteristic length, l taken of the order of the grid cell, 1

l   Vol cell  3 has been used, 2  lT   l  Cs l  S

13

(14)

where Cs is a modeling constant taken equal to 0.2 and S  2 Sij Sij is the mean strain rate tensor 1  V V  intensity with Sij   l ,i  l , j  . In addition, kl is modeled according to the relation 2  x j xi  2 kl  Ck l 2 S where C k is equal to 0.006. The suggested turbulent viscosity based model ( k l  l

type model) is a first attempt to compute nozzle flows including transition from laminar flow in the sac towards turbulent flow at the exit of the nozzle. But, further improvements in the turbulence modeling are required in future work, may be using LES approach.

Further detailed analyses of each source term of the governing equations are also available in Mandumpala Devassy, 2013a [26] and Mandumpala Devassy et al., 2013b [27].

3 The new TwoSD atomization model This section aims in proposing and studying a new two-surface density (TwoSD) atomization model for primary and secondary breakup. The phenomenon of initial breakup of the bulk liquid jet into primary droplets is the primary atomization. This process occurs at the near-end of the nozzle exit while the further disintegration of the primary droplets into child droplets is called as secondary breakup and it generally occurs downstream from the nozzle. For primary atomization, the surface of the liquid (interface) behaves differently for different injection conditions. The internal flow features of the nozzle like cavitation (Habchi, 2013 [18]), pressure fluctuations, nozzle geometrical features, environmental conditions, play a major role in determining the spray structure and the type of breakup. The present investigation is concentrating mainly on the behavior of high speed liquid jets. In these conditions, the surface instabilities are so high and end up in forming cluster of ligaments. Later, the instabilities on the ligaments start growing by producing droplets. The details of this phenomenon have already been studied numerically by Shinjo and Umemura, 2010 [38] where they succeeded in producing the droplets from ligaments. In their studies, it is found out that the instabilities growing along the ligament surface produces compression waves. These compression waves, originating from the ligament tip, produce an 14

imbalance of pressure between the tip and core of the ligament which leads to the necking at the ligament tip. These ligaments later undergo disintegration producing droplets of different diameters. The present investigation also focuses on this ligament detachment and its breakup scenario, because the process of atomization requires to follow this breakup cascade which is still an ongoing debate and not completely understood, although numerous previous theoretical (Herrmann, 2011[20] and Shinjo and Umemura, 2010 [38]) and experimental (Sallam and Faeth, 2003[35] and Osta et al., 2012 [31]) studies have been performed.

3.1 Outline of the TwoSD atomization model In the previous atomization models (Vallet and Borghi, 1999 [42] and Vallet et al., 2001 [43], Lebas et al., 2009 [23], Vessiller, 2008 [45], etc…) using surface density approach, both the dispersed and separate (liquid core) phases are treated using a single surface density equation coupled with an equation of the liquid mass fraction. But at the nozzle premises, the liquid jet comprises a liquid core, ligaments and droplets. Lebas et al., 2009 [23] have already developed a model for the dense and dilute regions by employing one equation of surface density. But, the method followed here is more straightforward by distinguishing the primary and secondary atomization processes using separate equations of surface density: One for the liquid core and the other for the dispersed phase (the spray droplets). As the phenomenon of primary atomization is different from droplet breakup, a two surface density (TwoSD) model may facilitate the modelling stages for both the dispersed and separate (liquid core) phases. With regards to this view, an idea of differentiating the liquid phase into dispersed and separate phase focuses attention. Thereby, separating the liquid phase into two sub-phases (dispersed and liquid core phases), it is possible to deal with the atomization and breakup phenomena more precisely. The present study is addressing this scenario by considering the initiation of the atomization as the separation of ligaments from the liquid core, which has already been proven experimentally (Sallam and Faeth, 2003 [35] and Osta et al., 2012 [31]) and is also shown in Figure 4. Besides, with this approach the droplets generated due to the primary atomization and secondary breakup can be filtered and studied separately for determining the droplets probability density function (PDF) and can later be vaporized using adequate evaporation models. Moreover, the complexity of express15

ing the complete system of primary atomization and secondary breakup processes can thus be reduced. This features the prime characteristic of the new TwoSD atomization model proposing and is discussed in the following sections.

3.2 The novel surface density equations Figure 5 shows a complex interface during the time of primary atomization of a high speed liquid jet (Sallam and Faeth, 2003 [35]). As previously indicated, two equations of surface density may be employed for determining this complex interface; one for the liquid core or separate phase ( s ) and the other for the dispersed phase ( d ). The notations, s stands for the separate phase and d stands for the dispersed phase or droplets. The development of the proposed atomization model is carried out in three different steps. Firstly, the liquid core surface is mapped as shown in Figure 6. On the one hand, mean of surface area, mean , is considered at undisturbed conditions. This reference surface density is somewhat similar to the minimum surface density ( min ) used by Chesnel et al., 2011 [9] and 2009 [10]. On the other hand, s is defined as the liquid core surface. Due to the hydrodynamic instabilities along with the effect of turbulent eddies, s surface begins deforming and producing irregularities, which later end up in the formation of ligaments as depicted in Figure 6. The quantity of surface, ( s  mean ) thus accounts for the ligament surfaces. These ligaments later undergo breakup if they satisfy a critical breakup condition, which will be discussed later. This also shows the importance and the necessity of a transport equation for the mean surface density,

mean . The second step mainly addresses the breaking of the matured ligaments from the liquid core surface. Figure 7 shows the diagrammatic representation of this ligament breakup process. In the figure, the ligament length at the breakup time is labeled as Lb and its diameter as b . Besides, Figure 7 also describes the method by which the model treats the phenomena of primary atomization. At a critical breakup condition, the matured ligaments will pinch off and the quantity of surface ( s  mean ) will detach and move away from the liquid core surface. 16

In the third and the final step, the ligaments thus detached from the liquid core become unstable and assumed to breakup as shown in Figure 7a, which is an experimental observation from Marmottant and Villermaux, 2004 [28] of the breakup of detached ligaments. The model assumes that the ligament pinching off from the surface and the droplets formation happen simultaneously. As such, under a critical ligament thickness, the ligament pinches off and at the same time it itself breaks up into primary droplets or blobs. In addition, it is assumed that the ligament breakup follows the experimental observation of Villermaux, 2004 [47]. This analysis thus concludes the requirement of three surface density equations; one for liquid core, the second for the dispersed phase and the third for the reference mean area. The next motivation in this section implies the formulation of the different surface density equations employed for defining the primary and secondary atomization phenomena. The model is built in such a manner that mean represents the undisturbed surface density (at the initial condition we have mean  s . The mean interface stretching and turbulent surface production initiates the production of ligaments over the liquid core surface ( s ). Breakup only occurs when it satisfies the critical breakup condition described in the following section. Till that moment mean is just convected by the multidimensional (3D) liquid velocity and serves as the reference for the production of ligaments by defining the quantity ( s  mean ). This is the reason why mean is simply convected and does not have any source terms and can be written as follows:

 mean  .(V l , i  mean )  0 t

(15)

This equation requires a dynamic initialization procedure for determining the initial position of the liquid jet interface close to the nozzle exit. This initialization procedure is described in Appendix A. Similarly, the transport equations of surface density for the separate liquid core and dispersed droplets phase can be written respectively as follows:

17

  •  s   n n : V    s  S s  2  relax ,l   .(V  )   . V prod l ,i s l ,i s ,i s , j l ,i s s t 3 l

(16)

   • d  Vl ,i d 2 Vl ,i 2  relax ,l    d  d prod  S d  d t xi 3 xi 3 l

(17)









The first term on the RHS of Equation (17) is the simplified stretch term for droplets compared to the similar term in Equation (16) by considering the isotropic behavior of the normal nd on the droplet surface. This term in Equation (16) and (17) is thus the stretch due to the mean velocity field (Mandumpala Devassy et al., 2011 [25]). The quantity ns in Equation (16) is the mean normal acting at the liquid core interface, and can be modeled by ns 

 s , where s  s



is the liquid core volume fraction and is discussed in next subsection.  relax,l in Equations (16) and (17) is the quantity associated with the change in the volume fraction occurring due to pressure and temperature relaxation at the gas-liquid interface and is given as follows:      relax,l   Pg  Pl  (Tg  Tl )    





(18)

Thus, Equation (16) represents the evolution of the surface due to primary atomization and Equa•

tion (17) the evolution of the surface due to secondary breakup. S s 

( s   mean ) is the rate of t

•

surface which is detached from the liquid core, see Figure 6. S d is the rate of surface which is 

added to the dispersed phase due to the ligament breakup.  s prod is the turbulent production term 

for the liquid core.  d prod is the turbulent secondary breakup production source term for the dispersed phase. The term “turbulent” used here is intended to include all the sub-grid fluctuating processes (interfacial instabilities, turbulence eddies, etc…). The models for the unknown quanti•





ties S d ,  s prod and  s prod are described in the following sections.

18

3.3 Significance of the Volume Fraction Equation With respect to the modelling description in the last section, the transfer of ligaments surface must be accompanied by a volume fraction transfer. According to this principle, two equations for liquid volume fraction can be written, one for the total liquid volume fraction (  l ) and the second for the liquid core volume fraction (  s ). I I  V l l Vl ,i l ,i   l   relax,l t xi xi

(19)

I I • V     s  s Vl ,i l ,i   s  S    s   relax,l t xi xi  l 

(20)

d  l   s

(21)

The last terms on the RHS of Equation (19) and (20) represent the relaxation of pressure and 

temperature. Besides, in Equation (20) the relaxation parameter  relax,l is multiplied by a factor  s    in order to ensure that only the liquid core is relaxed. By computing l and s , d can be  l 

computed from Equation (21). This method can allow some computational easiness by not computing d separately. This is only possible by assuming that both the liquid core and dispersed I phase regions are transported with the same liquid interface velocity V l . In Equation (20) also •

appears the S  source term. It is the transfer rate of liquid volume fraction due to the ligament(s) separation. From Figure 7, this quantity can be found out by



S 

Vollig Volume of the ligaments  Volume of the cell  t Volcell  t

19

(22)

Where t is the computational time step and Vollig can be calculated from Figure 7 as Vollig  N lig

 2 b Lb 4

(23)

and, Nlig is the total number of ligaments under breakup in the sub-grid scale.

3.4 Primary atomization model 3.4.1 Ligament breakup from the liquid core The model deals with simultaneous breakup of ligaments and droplets from the liquid core and ligaments, respectively. This section describes these two breakup processes. Firstly the breakup of ligaments occurs at a critical Weber number, Web , based on ligament diameter ( b ). The work of Marmottant and Villermaux, 2004 [28] is quite promising in finding out the breakup condition of ligaments. The present work adopts their experimental observation which says that the ligaments will breakup at a critical value of ligament aspect ratio and Weber number based on ligament diameter ( b ). This gives

 gVrel2 b Web  

(24)

where  is the liquid surface tension and Vrel is the relative velocity between gas and liquid. The critical aspect ratio of the ligament is checked during the ligament break-up process. Besides, it is also understood that the viscosity of the liquid (i.e. Ohnesorge number) plays necessarily a certain role in breakup and at the present state its effect has been considered along with other 

physical effects through the modeling constant, C prod s , in the production term ( aT   s prod , Section 3.5). In near future, the model will be more refined by considering these effects separately by including different source terms in the s and d equations. The model considers that the ligament length at the time of breakup is directly proportional to the ligament diameter, b . This gives,

Lb  Cb  b 20

(25)

Web and C b are the modeling constants which need to be investigated. During the time of liga

ment pinch off, the quantity of surface density S s leaving the liquid core surface density ( s ) is computed as follows,



S s 

 s   mean Surface Area of ligaments Nlig  b  Lb   t Volcell  t Volcell  t

(26)

And gives, N lig 

  s   mean  Volcell Cb  b2

(27)



Breakup will occur if Nlig  1, and S given by Equation (22) may be rewritten as follows:

 b3



S  Cb N lig

4Volcell  t

(28)

3.4.2 Ligament breakup into droplets The moment when the ligaments detach from the surface, it undergoes further breakup. Figure 7 describes this process. According to the experimental observation of Villermaux, 2004 [47], the size of the droplets formed by this ligament breakup is comparatively bigger than that can be produced by the Rayleigh theory. Besides, the DNS studies of Shinjo and Umemura, 2010 [38] on primary atomization also detects the same observation. With regards to these works, the present study invokes the findings of Villermaux and assumes that the ligaments are breaking into droplets with mean diameter's, d  2.5b (Marmottant and Villermaux, 2004 [28] and Villermaux, 2004 [47]). This assumption is also meant to simplify the model and to reduce the number of modeling constants. In future, this will be modeled more precisely. Let us denote by 21

•

Ndrop the number of droplets formed from the breakup of the ligaments. If S   t is the volume fraction of these droplets then its number is given according to the relation.



N drop  S

Volcell t  d3 6

(29)

From d and Ndrop, the total surface of droplets added to the dispersed phase can be calculated as



Sd  Ndrop

d2 1 Volcell t

(30) 

3.5 Modeling Turbulent Surface Production Source term,  s prod 

The source term  s prod is the most important term of the primary atomization model (Equation 16). According to the work of Morel (Equation (1)), it corresponds to the term aT for the production of the surface density due to turbulent velocity fluctuations. This production term accounts for the formation of ligaments on the liquid core surface. Indeed, in addition to the turbulent eddies fluctuations, this term may include the interfacial instabilities such as KelvinHelmholtz (K-H) and Rayleigh-Taylor (R-T) instabilities, etc… . Even if the relation between interfacial instabilities and local turbulent eddies is still not well understood, the modeling of 

 s prod with respect to the work of Vallet et al., 2001 [43] is found promising and is given as, 

 s prod  C prods

'' ''  V p ,i V p , j V p ,i s x j kp

(31)

'' ''   V p ,iV p , j In the above equation, the normalized turbulent shear stress    gives the directions of  kp   

interface stretching in the fluctuating scale. More precisely, the diagonal components of this tensor are assumed to be responsible of the ligaments formation at the liquid core surface. Yet, 22

Shinjo and Umemura, 2010 [38] have observed that ligament creation is strongly correlated with its gas flow field and turbulence as the strong shear near the liquid surface deforms the liquid surface. In addition, the main source of K-H and R-T instabilities are induced by the local shear which is the source of turbulence as well. The present state of the model considers one modeling constant, C prods for the combination of these different physical phenomena at the liquid-gas interface. These are the prime reasons why the ligament formation is modeled in this work with respect to the gas and liquid turbulence in Equation (31). Thus, present investigation involves the 

turbulence in both the gas and the liquid, a numerical investigation of the production term  s prod is required and is carried out in Section 4 in order to check the turbulence of which phase (gas, liquid or both) dominates the surface density production. This is the reason why the turbulent term in the above Equation (31) is denoted generally with phase index p. Thus, Equation (31) can be considered as a good candidate for the production of ligaments by stretching of the liquid core surface, s . However, future works should introduce the K-H and R-T interface instabilities in a 

more explicit manner for the  s prod modelling.

3.6 Secondary Atomization model This section aims to describe the secondary atomization model employed in Equation (17) for the dispersed phase surface density, d , for further breaking of the blobs generated from •

the primary atomization. In Equation (17), S d is the quantity of surface which is added to the dispersed phase. It acts as a source term for performing the further disintegration of these blobs into child droplets. The model which has been used in this article for performing the secondary breakup, has already been studied by Vessiller, 2008 [45], according to the work of Vallet et al., 2001 [43]. The present investigation recalls this model for the computation of the surface pro

duction source term,  d prod . This model is recalled here for completeness and is written as;

23



d prod 

aC7 2 a d     k d g g production l 

(32)

destruction 1 2 t





Where a  C5 Re  g / k g with Ret  k g 2 /  g  g . The constants C5 and C 7 are kept as 0.1 and 0.16, respectively. The Equation (32) adopted for this study considers that the primary atomized droplets or blobs are unstable and can face further disintegration. The process of this secondary breakup thus assumes that there exists an equilibrium condition for these child droplets formed by defining a production and destruction terms. Although this model seems to be physically limited and cannot take into account the different secondary breakup regimes already experimentally identified (Pilch and Erdman, 1987 [32], Habchi, 2011 [17]), a preliminary analysis of the comprehensive atomization model described above is carried out in the next section.

4 Results and Discussion The TwoSD atomization model proposed in Section 3 is analysed here. Due to the lack of experimental results close to the injector, as discussed in the literature review Section 1.1, DNS results of Herrmann, 2011 [20] are used for the assessments and calibration of the constants of the suggested atomization model. Besides, it is added that the simulations presented here are the first tests of a model which is dedicated to primary atomization of high speed liquid jets. These results will allow, thereby highlighting the different methods applied for modelling the physics associated with the liquid jet atomization.

4.1 Computational conditions. This section focuses mainly to figure out the details of the computational domain considered for the analysis. A typical 3D finite volume hexahedron grid (Figure 8) has been generated. The configuration has an inflow boundary at the injector pipe inlet and an outflow at the right boundary. The diameter value of the inflow injector is 100  m . Since most of the comparison is carried out with the DNS results of Herrmann, 2011 [20], the computational domain is also set with respect to that. Initially, the injector hole is filled by a liquid fuel (n-dodecane) and the 24

chamber is filled by gas (nitrogen). The initial pressure and temperature in the liquid and gas phases are shown in the Table 2. The simulation is carried out with Injection Reynolds and Weber number at 5000 and 17,000 respectively. For the initial conditions, all velocities inside the liquid are set to (0,0,100)T m/s, whereas all velocities inside the gas phase are set to zero. The injected liquid velocity is specified to (0,0,100)T m/s. It is assumed having a uniform profile throughout the orifice section. Also, wall-slip boundary condition is assumed. For numerical reasons, each phase contains a weak volume fraction  of the other phase (typically  = 10-6). Since this test case has stagnant gas inside the computational domain, the initial turbulent kinetic energy kg and its dissipation rate  g values considered for this study is taken as very small. In addition, weak turbulent kinetic energy (10-3 m2/s2) and dissipation rate (0.08 m2/s3) values have been specified at the inlet. The present study primarily focuses on the surface perturbations occurring on the liquid surface due to the effect of gas entrainment similar to the work of Shinjo and Umemura, 2010 [38]. Indeed, in this work, they observed that ligament creation is strongly correlated with its gas flow and turbulence as the strong shear near the liquid surface deforms the liquid surface. The computations are carried out for 20  s and the total computational CPU time required is 3.2hrs running in 8 AMD Barcelona 2.3Ghz processors. As a first results, Figure 9 shows the liquid volume fraction and velocity distribution at t=20  s .

4.2 Grid Sensitivity Study The proposed atomization model requires a grid sensitivity study for the main following two reasons. On one hand, the mesh needs to be refined enough in order to resolve correctly the flow fields especially at the liquid-gas interfaces. On the other hand, the grid size should be larger than the diameters of the produced ligaments. Indeed, this requirement may be viewed as a numerical accuracy condition, which is similar to one required for classical spray droplets modelling where the droplet needs to be much smaller than the mesh spacing. This section explains this grid sensitivity study performed in order to find a trade-off regarding the previous constraints on the mesh and hence to check the best possible range of cell size with which the new atomization model is performing better. 25

The droplet diameter results in terms of probability density functions (PDF) are considered in this study. Since present investigation is proposing a new primary and secondary atomization models, PDFs may be considered as the most crucial parameter for the grid sensitivity study. Meshes with four cell sizes, 10, 15, 20 and 25  m are considered. As shown in Figure 10, increasing the mesh size more than 20  m seems not appropriate and the cell size in the range of 10 - 20  m shows a fairly good comparison with the DNS results. In the rest of this paper, a mesh with 20  m cell size (32,000 hexahedral cells) is used in order to minimize the computational cost.

4.3 Numerical investigation of modelling constants Since both the present work and the DNS results of Herrmann, 2011 [20] focus mainly on the primary atomization process, the present examination is done without the secondary breakup model. This section handles the investigation of the modelling constants of the new primary atomization model by comparing our numerical results with the DNS of Herrmann, 2011 [20] in 

terms of the total mass of the atomized droplets, mD , and the drop size PDFs. There are mainly three modelling constants which demand the inquiry. They are Web , C b and C prod s appearing in Equations (24), (25) and (31), respectively.

4.3.1 Determination of the model constant C b The present atomization modelling methodology considers that the production of surface occurs by stretching from the fluctuating (aT) and mean (AT) scales. Marmottant and Villermaux, 2004 [28] have shown experimentally that the average droplet size resulting from the breakup of a ligament can be obtained as, d  0.4d0 (I), where d0 is the diameter of the equivalent sphere containing the ligament volume. Besides, in Villermaux, 2004 [47], a relation can be derived between the average size of the droplets resulting from the ligament breakup and ligament diameter as d  2.5b (II). Using (I) and (II) experimental correlations, a relation can be formulated

26

for the ligament length ( Lb ) during the time of breakup as, Lb  6b which implies Cb  6 . Of course, this value needs to be checked in the future work, for the case of Diesel injection conditions.

4.3.2 Determination of the model constant C prod s C prod s is the constant in Equation (31) for the modelling of the stretching production term 

( aT   s prod ). This subsection describes the effect of C prod s on the disintegration of the liquid jet. It is obvious that an increase in its value indicates an increase in production rate of the liquid 

core surface density,  s prod ; and subsequently, an increase of the amount of atomized droplets. 





The quantity ( mD / min ) computed by Herrmann, 2011 [20] is used for determining C prod s . mD is 

the rate of droplet mass generation due to primary atomization and min is the total injected liquid mass flow rate. 



Figure 11 shows the comparison of ( mD / min ) for different C prod s values from 45 to 205. From the figure, it is clear that the sensitiveness of this parameter is small. Indeed, a slight increase in C prod s will not lead to a significant increased atomization. Besides, all the curves in Figure 11

follow the same trend compared to the DNS results of Herrmann and out of which the value of C prod s  65 provides the best comparison. However, more refined value could be determined

from future experimental and DNS results.

4.3.3 Determination of the model constant Web The Weber number Web is an important parameter which determines the primary atomization mechanism. It is defined by Equation (24). The knowledge of this parameter is obtained from the experimental study of Marmottant and Villermaux, 2004 [28]. Besides, it has also been employed by Shinjo and Umemura, 2010 [38] in their DNS study of liquid jet disintegration. It is 27

worth to note that an increase in the Web leads to an increase in the critical ligament diameter b which later corresponds to an increase in the droplet diameter formed from the ligament breakup. As such, small critical Weber number leads to small atomized droplets. From preliminary numerical tests, the results have proved to be very sensitive to the value of the critical Weber number, Web . Then, Web was tested with four different values of 5, 6, 7 and 8. For this analysis the value

of C prod s is considered with a value of 65 and Cb  6 . Figure 12 shows the Web effect on droplet distribution. From the figure, it is found that the behaviour of droplet size distribution is sensitive to the small change in the value of Web . For a higher value of Web the trend of the distribution is to shift towards the region of maximum number of droplets with higher diameters. Present investigation chooses the value Web = 6 for which the droplet distribution is near to the DNS results and not too far from the value obtained experimentally by Marmottant and Villermaux, 2004 [28]. Nevertheless, this value needs to be refined based on future experiments and DNS results.



4.3.4 Effect of gas and liquid turbulence on surface production,  s prod The present atomization model involves in Equation (31) the turbulence in both the gas and the liquid in order to model the production of the liquid core surface density due to interfacial instabilities and small energetic turbulence eddies. A numerical investigation of this production term is then required in order to check the turbulence of which phase (gas, liquid or both) dominates the ligaments production. Figure 13(a) shows the gas turbulent kinetic energy distribution. From the velocity plot shown in Figure 9, the maximum value of k g will be in the region between the spray tip and the nozzle end, where the turbulent eddies are well developed. For a well-developed liquid jet, this turbulent shear enables the generation of ligaments from the liquid core surface. From Figure 13(c), it is also clear that liquid turbulence ( kl ) is particularly concentrated on the cap (in the head of the jet). The reason behind this may be due to the effect of liquid impingement on the compressed gas. 28



The term  s prod accounts for this production which is arising in s equation. The main feature of this term corresponds to the formation of surplus surface above mean which represents the growth of ligaments on the liquid core at the fluctuating scale level. The main motivation here is thus to identify the role of gas, liquid or mixture turbulence on the stretching and surface production. 

 s prod  C prods

'' ''  V p ,i V p , j V p ,i s x j kp

(33)

For this analysis, three different simulations have been performed with the values of p = l (liq

uid), g (gas) & m (mixture (gas + liquid)) in the  s prod source term re-written above for the sake of clarity of the discussion. The simulation results for these different test cases are plotted in Figure 14 at 20  s after start of injection. The values of C prod s and Web are kept constant to values of 6 and 65, respectively. From Figure (14), the turbulence in the liquid and the turbulence in the gas seem to play complementary roles for the ligaments formation and the primary atomization. The liquid turbulence seems to dominate the atomization of the leading tip of the liquid jet, while the gas turbulence helps in the atomization of the entire liquid core. On the one hand, from the Figure 14(a), one can observe that a high surface density is preferentially produced at the leading tip of the liquid core. This result is due to the high level of turbulence in this region, as depicted in Figure 13. On the other hand, Figure 14(a) shows that gas turbulence seems more effective than liquid turbulence for the atomization from the liquid core side. Moreover from Figure 14(b), the few droplets produced at the side of the liquid core are due to the gas turbulence. In addition, it clear than much more droplets is produced due to liquid turbulence. Besides, these droplets are mainly produced at the tip of the liquid core. During the injection period, it is the jet cap which is confronting the gas phase that undergoes severe disintegration with high rate of production of droplets. A mushroom-shape tip is created due to the lateral liquid spread by impingement against the stagnant gas and roll-up. This kind of phenomenon has already been studied extensively by Shinjo and Umemura, 2010 [38] and similar kind of observations is noted by Trinh, Huu Phuoc, 2004 [40]. 29

Present investigation recommends the using of the mixture turbulence for surface density production in Equation (33). The reason is that from the above study, it is found that liquid turbulence has an effect on the spray cap and gas turbulence mainly involves the shearing of the liquid core surface. This study considers these two factors important and thus takes the mixture turbulence as the right choice for the production of ligaments and primary atomization.

5 Conclusions and future work This paper started by highlighting the importance of liquid jet atomization and its effect on the mixture preparation and combustion in internal combustion engines. From the literature study it was found that for sub-critical and trans-critical injection conditions there exists always a well-developed liquid core surrounded by a droplet cloud. Present study has suggested a novel atomization model using a two-surface density approach within the framework of an Eulerian two-phase system. The new TwoSD atomization model follows a breakup cascade process containing, first the production of ligament like structures, which subsequently undergo further breakup to form droplets of different sizes. The results also signify the formation of a spray cap at the jet tip where the disintegration seems to occur due to liquid turbulence in addition to the gas turbulence. The model is performing better with a mesh size ranging from 10-20 microns. For the production of surface density, it has been shown that both the gas and liquid turbulence is necessary. Moreover, the new TwoSD atomization model suggested in this work requires further improvement which will be studied in the near future. The turbulence modelling seems easier using LES, particularly when laminar-turbulent transition needs to be computed. The consideration of separate source terms of the evaporation for the liquid core and the spray droplets. Besides, in future, if new precise experimental observations from the near nozzle (first 20D from the nozzle hole) of the spray are obtained then the proposed TwoSD atomization model can be further refined. Future works will also include investigations of the cavitation effects on the atomization of liquid jets (Habchi, 2013 [18]) and studies of more physical secondary breakup models such as the energy spectrum analogy breakup (SAB) model (Habchi, 2011 [17]) recently developed by the authors for the numerical simulation of sprays. 30

Reference [1]

Alexander, J.S. and Jean-Paul, D., Turbulent shear layers in supersonic flows, Birkhäuser, Germany, 2006.

[2]

Amsden, B., O'Roukre, P.J., Butler, T.D., KIVA-II: A computer program for chemically reactive flows with sprays, Los Alamos National Lab Report, No. LA-11560MS, 1989.

[3]

Aquing, M., Ciotta, F., Creton, B., Féjean, C., Pina, A., Dartiguelongue, C., Trusler, J.P.M., Vignais, R., Lugo, R., Ungerer, P., Nieto-Draghi, C., Composition analysis and viscosity prediction of complex fuel mixtures using molecular-based approach, Energy & Fuels., 26(4), 2220-2230, 2012.

[4]

Baer, M.R. and Nunziato, J.W., A two-phase mixture theory for the deflagration to detonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flow., 12, 861-889, 1986.

[5]

Beau, P.A., Funk, M., Lebas, R., Demoulin, F.X., Applying quasi-multiphase model to simulate atomization processes in diesel engines: Modeling of the slip velocity, SAE, 2005-01-0220, 2005.

[6]

Bohbot, J., Gillet, N., Benkenida, A., IFP-C3D: An unstructured parallel solver for reactive compressible gas flow with spray, Oil Gas Sci. Technol., 64, 309-335, 2009.

[7]

Blokkeel, G., Barbeau, B., Borghi, R., A 3d Eulerian model to improve the primary breakup of atomizing jet, SAE, 2003-01-0005, 2003.

[8]

Candel, S.M. and Poinsot, T.J., Flame stretch and the balance equation for the flame area, Combust. Sci. Tech., 70, 1-15, 1990.

[9]

Chesnel, J., Reveillon, J., Ménard, T., Demoulin, F.X., Large eddy simulation of liquid jet atomization, Atom. Sprays., 21(9), 711-736, 2011.

[10]

Chesnel, J., Réveillon, J., Ménard, T., Doring, M., Berlemont, A., Demoulin, F.X., LES of atomization: From the resolved liquid surface to the sub grid scale spray, ICLASS Paper ICLASS2009-250, 2009. 31

[11]

Crua, C., Shoba, T., Heikal, M., Gold, M., Higham, C., High-speed microscopic imaging of the initial stage of diesel spray formation and primary breakup, SAE., 201001-2247, 2010.

[12]

Desjardins, A. and Pitsch, H., Detailed numerical investigation of turbulent atomization of liquid jets, Atom. Sprays, 20(4), 311-336, 2010.

[13]

Drew, D.A. and Passman, S.L., Theory of multi component fluids, In: Applied Mathematical Sciences, vol. 135. Springer, 1999.

[14]

Faeth, G.M., Hsiang, L.P., Wu, P.K., Structure and breakup properties of sprays, Int. J. Multiphase Flow., 21, 99-127, 1995.

[15]

Fuster, D., Bague, A., Boeck, T., Le Moyne, L., Leboissetier, A., Popinet, S., Ray, P., Scardovelli, R., Zaleski, S., Simulation of primary atomization with an octree adaptive mesh refinement and VOF method, Int. J. Multiphase Flow., 35, 550565, 2009.

[16]

Giannadakis, E., Gavaises, M., Arcoumanis, C., Modelling of cavitation in diesel injector nozzles, J. Fluid Mech., 616, 153-193, 2008.

[17]

Habchi, C., The energy Spectrum Analogy Breakup (SAB) model for the numerical simulation of sprays, Atom. Sprays, 21(12), 1033-1057, 2011.

[18]

Habchi, C., A Gibbs free Energy Relaxation Model for Cavitation Simulation in Diesel injectors, 25th Annual Conference ILASS-Europe, 2013.

[19]

Habchi, C., Dumont, N., Simonin, O., Multidimensional Simulation of Cavitating Flows in Diesel Injectors by a Homogeneous Mixture Modeling Approach, Atom. Sprays, 18(2), 129-162, 2008.

[20]

Herrmann, M., On simulating primary atomization using the refines level set grid method, Atom. Sprays, 21(4), 283-301, 2011.

[21]

Jay, S., Lacas, F., Candel, S., Combined surface density concepts for dense spray combustion. Combust. Flame, 144(3), 558-577, 2006.

[22]

Lebas, R., Blokkeel, G., Beau, P.A., Demoulin F.X., Coupling vapourization model with the Eulerian–Lagrangian Spray Atomization (ELSA) model in diesel engine conditions, SAE, 2005-01-0213, 2005. 32

[23]

Lebas, R., Ménard, T., Beau, P.A., Berlemont, A., Demoulin, F.X., Numerical simulation of primary break-up and atomization: DNS and modelling study, Int. J. Multiphase Flow, 35, 247-260, 2009.

[24]

Luca, M.D., Vallet, A., Borghi, R., Pesticide atomization model for hollow-cone nozzle, Atom. Sprays, 19(8), 741-753, 2009.

[25]

Mandumpala Devassy, B., Habchi, C., Daniel, E., Numerical investigation of Eulerian atomization models based on a diffused-interface two-phase flow approach coupled with surface density equation, In: Proceedings of the 24th Annual Conference ILASS-Europe, 2011.

[26]

Mandumpala Devassy, B., Atomization Modelling of Liquid Jets using an EulerianEulerian Model and a Surface Density Approach, PhD Thesis, Aix-Marseille University, France, 2013a.

[27]

Mandumpala Devassy, B., Habchi, C., Daniel, E., A new Atomization Model for High Speed Liquid Jets using a Turbulent, Compressible, Two-Phase Flow model and a Surface Density Approach, 25th Annual Conference ILASS-Europe, 2013b.

[28]

Marmottant, P. and Villermaux, E., On spray formation, J. Fluid Mech., 498, 73-111, 2004.

[29]

Morel, C., On the surface equations in two-phase flows and reacting single-phase flows, Int. J. Multiphase Flow., 33, 1045-1073, 2007.

[30]

Ning, W., Reitz, R.D., Diwakar, R., Lippert, A.M., An Eulerian-Lagrangian spray and atomization model with improved turbulence modeling, Atom. Sprays, 19(8), 727-739, 2009.

[31]

Osta, A.R., Lee, J., Sallam, K.A., Fezzaa, K., Study of the effects of the injector length/diameter ratio on the surface properties of turbulent liquid jets in still air using X-ray imaging, Int. J. Multiphase Flow, 38, 87-98, 2012.

[32]

Pilch, M., and Erdman, C., Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop, Int. J. Multiphase Flow, 13(6), 741-757, 1987. 33

[33]

Pope. S., The evolution of surfaces in turbulence, In. J. Engg. Sci., 26(5), 445-469, 1988.

[34]

Sallam, K.A., Dai, Z., Faeth, G.M., Liquid breakup at the surface of turbulent round liquid jets in still gases, Int. J. Multiphase Flow, 28, 427-449, 2002.

[35]

Sallam, K.A. and Faeth, G., Surface properties during breakup of turbulent liquid jets in still air, AIAA, 41, 1514-1524, 2003.

[36]

Saurel. R. and Abgrall, R., A multiphase Godunov method for compressible multi-fluid and multiphase flow, J. Comp. Physics, 150, 425-467, 1999.

[37]

Saurel, R. and Le metayer, O., A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation, J. Fluid Mech., 431, 239-271, 2001.

[38]

Shinjo, J. and Umemura, A., Simulation of liquid jet primary breakup: Dynamics of ligament and droplet formation, Int. J. Multiphase Flow, 36, 513-532, 2010.

[39]

Tamaki, N., Shimizu, M., Hiroyasu, H., Enhanced atomization of a liquid jet by cavitation in a nozzle hole, In: Proceedings of the 8th Annual Conference ILASSAmerica, 2000.

[40]

Trinh, Huu Phuoc., Modeling of turbulence effect on liquid jet atomization, PhD Thesis, The University of Alabama in Huntsville, USA, 2004.

[41]

Trouvé, A. and Poinsot, T., The evolution equation for the flame surface density, J. Fluid Mech., 278, 1-31, 1994.

[42]

Vallet, A. and Borghi, R., Modélisation Eulerienne de l'atomisation d'un jet liquide, C.R. Acad. Sci. Paris Sér. II b 327, 1015-1020, 1999.

[43]

Vallet, A., Burluka, A.A., Borghi, R., Development of a Eulerian Model for the atomization of a liquid jet, Atom. Sprays, 11(6), 619-642, 2001.

[44]

Velghe, A., Gillet, N., Bohbot, J., 22nd International Conference on Parallel Computational Fluid Dynamics (ParCFD 2010), 2010.

[45]

Vessiller, C., Contribution à l'étude des bouillards denses et dilués par les approaches Eulerinennes et Lagrangiennes, PhD Thesis, Ecole Centrale Paris, France, 2008. 34

[46]

Veynante, D. and Vervisch, L., Turbulent combustion modelling, Progress in Energy and Combust Sci., 28, 193-266, 2002.

[47]

Villermaux, E., Unifying ideas on mixing and atomization, New Journal of Physics. 6:125, 2004.

[48]

Wang, Y., Liu, X., Im, K.-S., Lee, W.-K., Wang, J., Fezzaa, K., Hung, D.L.S., Winkelman, J.R., Ultrafast X-ray study of dense liquid jet flow dynamics using structure tracking velocimetry, Nat. Phys. 4, 305-309, 2008.

[49]

Wu, P.K., and Faeth, G.M., Aerodynamic effects on primary breakup of turbulent liquids, Atom. Sprays, 3(3), 265-289, 1993.

[50]

Zein, A., Hantke, M., Warnecke, G., Modeling phase transition for compressible twophase flows applied to metastable liquids, J. Comp. Physics., 229, 2694-2998, 2010.

35

Appendix A Dynamic Initialization Procedure In this study, a dynamic initialization procedure for the surface density s and mean is required especially in a region just at the exit of the nozzle as shown in Figure 15. The transport velocity is the liquid velocity which is a highly 3D velocity field. The method of performing this initialization procedure is described below in 2D for sake of simplicity.

.

Figure 15. Need of Dynamic Initialization

Figure 16. Notations for the Dynamic Initialization formulation. Dashed line indicates the liquid-gas inter-

From Figure 16, we can write,

face. 4

   1  1 1 dv   ni dS        A n ,i   Vol cell Vol  1  xi cell Vol cell xi If one assumes square cells as in Figure 16, A  A and l 

36



(A1)

A , then Equation (A1) becomes, Vol

4        l    n ,i  1  xi cell

 l 1n1,i   2 n2,i   3 n3,i   4 n4,i 



(A2)

= l 1n1,i   2 n2,i   3 n1,i   2 n2,i  = l 1   3  n1,i

From Figure 16, we have, 1 

 il   ir1  2

and  3 

 il  ir 3  2

, putting this in Equation (A2)

gives,

   l     ir1   ir 3  n1,i  xi cell 2



(A3)

In Figure 16,  ir1  0 and  ir 3  1 and substituting this in Equation (A3) gives;

   l     n1,i 2  xi cell



(A4)

Finally, modulus of Equation (A4) gives,

 l  2  Similar expression can be retained even if the interface is diagonally oriented in the computational hexahedron cell. This dynamic initialization is performed in all the cells where the two following conditions are true: 1)  s or  mean   min 2) 2  s  min with min is a critical value taken equal to 1000. Such dynamic initialization is also needed in order to be able to model surface appearance in case of cavitation, for instance.

37

Tables

Table 1: Parameters of the liquid SG-EOS Parameters

Units

Liquid (Dodecane)

l

-

2.35

ql

J/kg

-755269

P ,l

Pa

4.108

Cvl

J/kg/K

1077

Table 2: Operating Conditions 24 kg/m

Gas density ρg

3

850 kg/m

Liquid density ρl Temperature of gas in the chamber T g Temperature of liquid in the injector T l

3

300 K 325 K

Pressure of gas in the chamber P g

22.3 bar

Pressure of liquid in the nozzle P l Injector diameter D Injection Velocity Computational Domain

22.3 bar 100 µm 100 m/s 20D ҳ8D ҳ8D

38

Figures

Figure 1: Schematics of liquid jet atomization

Figure 2: Experimental observation of ligament formation from the liquid core in co-axial injection (Adapted from Marmottant and Villermaux, 2004 [28]). The non-dimensional numbers and liquid properties are Red0 = 4600, Wed0 = 88980, d0= 7.8e-3 m and g = 1.2 kg/m3.

39

Figure 3: Ligament formation from the liquid core (Adapted from Shinjo and Umemura, 2010 [38]), dominated by the fluctuating scale ligaments which are breaking up into finer droplets.

Figure 4: a) Shadowgraph and b) X-ray image of ligaments.(adapted from Sallam and Faeth, 2003 [35] and Osta et.al., 2012 [31] respectively)

40

Figure 5: Shadow graph image of the existence of ligaments and droplets Sallam and Faeth, 2003 [35].  d : Surface density of dispersed phase (droplets) and  s : Surface density of separate phase (liquid core).

Figure 6: Layout of the different surface density ( mean and s ) used in the suggested atomization model.

41

Figure 7: a) Experimental observation (Marmottant and Villermaux, 2004 [28]). b) The working layout of the new atomization model and definition of the problem.

Figure 8: The working layout of the computational configuration and definition of the problem. 42

Figure 9: Liquid jet volume fraction and velocity distribution at 20  s at the middle cross section of the configuration.

Figure 10: Grid sensitivity study of the computational domain with 10, 15, 20 and 25  m cell sizes. Scaled number frequency distribution (f(D)) on the Y-axis and Droplet Diameter’s on the X-axis. The values of Web , C b , C prod s are set to 6, 6 and 65 respectively and the turbulent pro

duction term  s prod is considered with mixture properties of both gas and liquid. 43







Figure 11: Effect of C prod s in the variation of mD min with time.  s prod is considered with mixture properties and Web is taken as 6. The results of the new atomization model are compared to DNS results of Herrmann, 2011 [20].



Figure 12: Effect of Web in the final droplet distribution.  s prod is considered with mixture properties and C prod s is taken as 65. Scaled number frequency distribution (f(D)) on the Y-axis and Droplet Diameter’s on the X-axis. The results of the new atomization model are compared to DNS validated distribution from Herrmann, 2011 [20]. 44

Figure 13: (a)-Turbulent Kinetic Energy ( k g ) and (b)-Dissipation (  g ) of gas and (c)-Turbulent Kinetic Energy ( kl ) of the liquid. The dashed circle indicates the highest liquid turbulence intensity location.

45

(a) Surface density of Liquid core ( s )

(b) Droplet diameter distribution Figure 14: (a) Surface density of the liquid core ( s ) and (b) Droplet diameter distribution for the three different test cases distinguishing the effect of gas, liquid and mixture turbulent produc-

46



tion of surface density (  s prod ). Web and C prod s are kept constant to values of 6 and 65 respectively. The DNS results of Herrmann are plotted for the sake of qualitative comparisons.

47