ILASS Americas 27th Annual Conference on Liquid Atomization and Spray Systems, Raleigh, NC, May 2015

Eulerian Two-Phase Flow CFD Simulation Using a Compressible and Equilibrium Eight-Equation Model

Y. Wang1 and R. D. Reitz

Engine Research Center, Department of Mechanical Engineering

University of Wisconsin-Madison Madison, WI 53706, USA

Abstract

This study focuses on Computational Fluid Dynamics (CFD) simulation of liquid-gas two-phase flows with applica-tions to high-speed fuel injection processes in automotive engines. Assuming both liquid and gas phases to be con-tinua, Eulerian transport equations are used to describe the mass, momentum and energy conservation laws for each phase. Thermodynamic properties of the fluids are modeled with compressible Stiffened Gas Equations of State that are coupled with the conservation laws. It is assumed that the interactions between the two phases result in mechani-cal, thermal and phase equilibrium controlled by relaxation processes. Numerical methods for the governing equa-tions are presented, and critical numerical issues, such as the positivity of the liquid and gas phase volume fractions, are discussed in detail. Robustness of the CFD code is examined using several test problems involving very steep pressure gradients. These include a shock tube problem, a shock-bubble interaction problem, and submerged and non-submerged injected liquid jet problems.

1 Corresponding author: wang46@uwalumni.com (currently affiliated with ANSYS, Inc.)

ILASS Americas 27th Annual Conference on Liquid Atomization and Spray Systems, Raleigh, NC, May 2015

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Introduction Many modern internal combustion engines inject

liquid fuel directly into the combustion chamber where the fuel atomizes, vaporizes and mixes with air before combustion. The amount of fuel injected per engine cycle determines the load condition of the engine. The injection strategy in terms of single- or multiple-injections and the injection timings significantly affect engine performance. As engineers rely more-and-more on simulation for engine design, Computational Fluid Dynamics (CFD) simulations of fuel injection processes have become increasingly helpful due to the flexibility for parametric study and relatively low cost.

The fuel injection process involves both the inter-nal flow within the fuel injector nozzle passage and the external flow outside the nozzle and inside the combus-tion chamber. In this study, we focus on an “Eulerian” CFD model, which treats both the fuel and combustion chamber gas as continuous fluids. This is the most straightforward way to simulate liquid-gas, two-phase flows for practical engineering applications, since the very refined CFD grids needed for DNS (Direct Numer-ical Simulation) studies are avoided [1-4]. In DNS sim-ulations challenges arise when the grid resolution is not sufficient to resolve the droplets from atomization, the bubbles from cavitation, and small-scale turbulent flow structures. However, the Eulerian approach can de-scribe the macroscopic flow behavior if an appropriate turbulence model is applied. Also, sub-grid models, such as the interfacial area-density approach suggested by Vallet et al. [5] can be introduced to account for the physics of droplets and bubbles.

In general, there are two types of Eulerian model for liquid-gas two-phase flows: those that exactly track the interface and are suitable for DNS simulations, such as the level-set method [2, 3], and the so-called “dif-fused interface method” [4] that uses the mass or vol-ume fraction of a certain phase to indicate phase boundaries. The diffused interface method is more flex-ible in choosing CFD grid resolution, because the phase “boundary” on a coarse grid simulation could be a mix-ing zone of liquid and gas where the small structures of the interface are considered by sub-grid models.

Recent years have seen growing interest of the dif-fused interface method applied to fuel injection simula-tions [6-10]. The present work contributes to this area. Specifically, our model simulates both the internal and external nozzle flows simultaneously, starting upstream of the injector passage and extending downstream to the low-pressure combustion chamber. We use thermody-namic Equations of State (EOS) to model the properties of the fluid in both gas and liquid phases, including compressibility effects, with one component being the injected fluid, and the other being non-condensable air within the low-pressure combustion chamber. The in-

jected fluid is normally in liquid phase, but might have phase change due to cavitation effects inside the nozzle. We focus on the model’s robustness such that it can handle flows with large pressure and density gradients, as is the case in many practical fuel injection applica-tions.

Since compressibility of each phase is considered, thermal properties have to be coupled with the flow parameters. The best-known approach from Baer and Nunziato [11] utilizes transport equations for mass, momentum and energy for each phase, supplemented by a transport equation for the volume fraction. Know-ing the volume fraction is particularly important, since without it the thermodynamic states of each phase are not in closure. For single-component liquid-gas flows, this is the “seven-equation model” introduced by Saurel et al. [12, 13] and Petitpas et al. [14]. In the following sections, we first present the governing equations and their extension to the two-component flows of interest in this study, and then we discuss the numerical meth-ods and present the results. Governing Equations

The gas phase is considered to be a mixture of the air and the vapor phase of the injected fluid. The mix-ing of vapor and air is assumed to be ideal, in the sense that each component behaves as if it were an ideal gas alone and occupies the entire volume of the gas mix-ture. Two partial densities are defined: 휌 and 휌 , as the mass of vapor and air divided by their shared volume. Due to the ideal mixing assumptions, we have: 휌 +휌 = 휌 where 휌 is the gas phase density. The seven-equation model of Saurel et al. [12, 13] and Petitpas et al. [14] is thus extended to an eight-equation model, written as:

휕훼휕푡 + 퐮 ∙ ∇훼 = 휇 푝 − 푝 + 훼̇

휕 훼 휌휕푡 + ∇ ∙ 훼 휌 퐮 = ∇ ∙ 훼 휌 퐷 ∇

휌휌 + 푚̇

휕 훼 휌휕푡 + ∇ ∙ 훼 휌 퐮 = ∇ ∙ 훼 휌 퐷 ∇

휌휌

퐮 + ∇ ∙ 훼 휌 퐮 ⨂퐮 + ∇ 훼 푝 = 푝 ∇훼 +∇ ∙ 훔 + 휆 퐮 − 퐮 + 푚̇

+ ∇ ∙ 훼 휌 퐸 + 푝 퐮 = 푝 퐮 ∙ ∇훼 + ∇ ∙

퐉 + 휆퐮 ∙ 퐮 − 퐮 + 푝 휇 푝 − 푝 + 푚̇|퐮|ퟐ + 푞̇ 휕(훼 휌 )휕푡 + ∇ ∙ (훼 휌 퐮 ) = −푚̇

( 퐮 ) + ∇ ∙ (훼 휌 퐮 ⨂퐮 ) + ∇(훼 푝 ) = 푝 ∇훼 + ∇ ∙훔 − 휆 퐮 − 퐮 − 푚̇퐮

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( ) + ∇ ∙ [훼 (휌 퐸 + 푝 )퐮 ] = 푝 퐮 ∙ ∇훼 + ∇ ∙ 퐉 −

휆퐮 ∙ 퐮 − 퐮 − 푝 휇 푝 − 푝 − 푚̇|퐮|ퟐ − 푞̇ (1)

In the above equations, subscript I denotes the in-terface between the two phases. 훼 (푘 = 푙or푔) is the volume fraction of phase k, which can be related to its corresponding mass fraction 푦 with 훼 = 휌푦 휌⁄ where 휌 is the density of phase k and ρ is the two-phase mixture density. It is assumed that the liquid and gas are immiscible, and the saturation condition 훼 + 훼 = 1 is applied. The specific total energy of phase k, 퐸 , is the summation of specific internal ener-gy 푒 and kinetic energy: 퐸 = 푒 + 0.5|퐮 | . 퐷 is the gas phase molecular diffusivity from Fick’s Law. σ and J denote the molecular viscous stress tensor and heat conduction vectors. Modeling of turbulent transport is not considered in the present work, and the viscous dissipation term in the energy equation is also neglected for simplicity. On the right-hand side of (1), there are terms with relaxation coefficients µ and λ. These terms account for the local momentum and ener-gy transfer between the two phases due to their velocity and pressure differences. Source terms 푚̇ and 푞̇ account for local mass and energy transfer from the liquid to the gas phase due to phase change and 훼̇ accounts for the corresponding change in the gas phase volume fraction.

Model (1) is not closed until a thermodynamic Equation of State (EOS) is provided. In this study, the Stiffened Gas EOS [15] is used. For each phase, the property relations include a pressure law (Eq. (2)) and a caloric law (Eq. (3)):

푝 + 휋 = 휌 퐶 (훾 − 1)푇 (푘 = 푔, 푙) (2)

푒 = 퐶 푇 + 푄 + (푘 = 푔, 푙) (3) π is called ‘‘stiffness parameter’’. It is zero for the gas phase, for which the pressure law Eq. (3) reduces to the ideal gas law. For the liquid phase it is non-zero, and it significantly increases the liquid density if compared to the gas under the same temperature and pressure.

In general, the inter-phase relaxation terms are dif-ficult to model if a coarse CFD grid is used, due to lack of sub-grid information, such as the topology of the interface. However, for nozzle flow and primary atomi-zation problems with high Weber number, the pressure relaxation is assumed to be infinitely fast by neglecting surface tension effects [16]. With drag force being its mechanism, the velocity relaxation is dependent on the interfacial area density, which is orders-of-magnitude different due to atomization. However, modeling of interfacial area density can be included (e.g., Vallet et al. [5]) but is beyond the scope of this paper. Instead, the stiff mechanical relaxation suggested by Saurel and

Abgrall [12] and Murrone and Guillard [17] is imple-mented by assuming infinitely large µ and λ, such that the velocities and pressures of the two phases are equal. Eqs. (1) are then reduced to the following model:

휕훼휕푡 + 퐮 ∙ ∇훼 =

휌 푐 − 휌 푐휌 푐훼 +

휌 푐훼

∇ ∙ 퐮

휕 훼 휌휕푡 + ∇ ∙ 훼 휌 퐮 = ∇ ∙ 훼 휌 퐷 ∇

휌휌 + 푚̇

휕 훼 휌휕푡 + ∇ ∙ 훼 휌 퐮 = ∇ ∙ 훼 휌 퐷 ∇

휌휌

휕(훼 휌 )휕푡 + ∇ ∙ (훼 휌 퐮) = −푚̇

휕(휌퐮)휕푡 + ∇ ∙ (휌퐮⨂퐮) = −∇푝 + ∇ ∙ 훔

( ) + ∇ ∙ (휌퐮퐸) = −∇ ∙ (퐮푝) + ∇ ∙ 퐉 (4)

in which 푐 ,푘 = 푙,푔 is the isentropic sound speed. The Stiffened Gas Equation of State is written for the two-phase mixture as:

푝 = (5)

Although the number of governing equations is re-

duced, numerical solution is challenging due to the non-conservative terms in the gas phase volume fraction equation. Thus, an additional commonly adopted as-sumption is that the temperatures of the two phases are always equal, which further reduces the transport equa-tions to:

휕(휌푦 )휕푡 + ∇ ∙ (휌푦 퐮) = ∇ ∙ 휌푦 퐷 ∇

푦푦 + 푚̇

휕(휌푦 )휕푡 + ∇ ∙ (휌푦 퐮) = ∇ ∙ 휌푦 퐷 ∇

푦푦

휕(휌푦 )휕푡 + ∇ ∙ (휌푦 퐮) = −푚̇

휕(휌퐮)휕푡 + ∇ ∙ (휌퐮⨂퐮) = −∇푝 + ∇ ∙ 훔

( ) + ∇ ∙ (휌퐮퐸) = −∇ ∙ (퐮푝) + ∇ ∙ 퐉 (6)

This limit model does not require a transport equa-tion for 훼 . With the pressure and temperature equilib-rium constraints having been imposed to determine its value, a transport equation for 훼 would make the prob-lem over-determined. In this case, the Stiffened Gas Equation of State could still apply, but its explicit form for the two-phase mixture is not available because 훼 cannot be expressed explicitly. As a result, it is difficult

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to understand the mathematical nature of the set of par-tial differential equations in (6), although its parent models (1) and (4) have been shown as hyperbolic without applying the temperature equilibrium [12, 17].

Nevertheless model (6) has been widely used, for example, in [6-9]. There are some differences in the assumptions of the mixing state of the vapor and air and in the Equation of State of liquid, but a general chal-lenge is to maintain the positivity of the volume frac-tions, i.e., 0 < 훼 , < 1 when both phases are present. Violation may result in negative gas phase density and pressure, which are unphysical. This positivity condi-tion seems obvious, but is not guaranteed by the equa-tions of (6). In particular, the imposed pressure and temperature equilibrium conditions turn the volume fractions from primitive variables into derived variables that have no control over the solution process. Howev-er, even for model (4) which solves for 훼 as a primi-tive variable, positivity is still difficult to maintain due to the complexity of its transport equation.

Therefore, instead of solving the limit models (4) or (6), we have adopted the method of Saurel and Abgrall [12] by solving the parent model (1) with a relaxation approach. We split the solution into a pre-relaxation step, a relaxation step which imposes pres-sure and velocity equilibrium, and finally a phase change step, as detailed in the next section. It will be shown that this relaxation approach ensures the posi-tivity condition for 훼 .

Numerical Method

In any time step, the transport equations of (1) are first solved in the pre-relaxation step without consider-ing relaxation and phase change between the two phas-es:

휕훼휕푡 + 퐮 ∙ ∇훼 = 0

휕 훼 휌휕푡 + ∇ ∙ 훼 휌 퐮 = ∇ ∙ 훼 휌 퐷 ∇

휌휌

휕 훼 휌휕푡 + ∇ ∙ 훼 휌 퐮 = ∇ ∙ 훼 휌 퐷 ∇

휌휌

퐮 + ∇ ∙ 훼 휌 퐮 ⨂퐮 + ∇ 훼 푝 = 푝 ∇훼 +∇ ∙ 훔

휕 훼 휌 퐸휕푡 + ∇ ∙ 훼 휌 퐸 + 푝 퐮

= 푝 퐮 ∙ ∇훼 + ∇ ∙ 퐉 휕(훼 휌 )휕푡 + ∇ ∙ (훼 휌 퐮 ) = 0

휕(훼 휌 퐮 )휕푡 + ∇ ∙ (훼 휌 퐮 ⨂퐮 ) + ∇(훼 푝 )

= 푝 ∇훼 + ∇ ∙ 훔

휕(훼 휌 퐸 )휕푡 + ∇ ∙ [훼 (휌 퐸 + 푝 )퐮 ] = 푝 퐮 ∙ ∇훼 + ∇ ∙ 퐉

(7)

Consider the numerical solution from time step n to n+1. Note that both velocity and pressure are in equilib-rium at time step n due to the relaxation procedure ap-plied in the last time step, therefore, 퐮 = 퐮 = 퐮 =퐮 and 푝 = 푝 = 푝 = 푝 . After Eqs. (7) are solved, we apply the stiff relaxation terms in Eqs. (1):

휕훼휕푡 = 휇 푝 − 푝

휕 훼 휌휕푡 = 0

휕 훼 휌휕푡 = 0

휕 훼 휌 퐮휕푡 = 휆 퐮 − 퐮

휕 훼 휌 퐸휕푡 = 휆퐮 ∙ 퐮 − 퐮 + 푝 휇 푝 − 푝

휕(훼 휌 )휕푡 = 0

휕(훼 휌 퐮 )휕푡 = −휆 퐮 − 퐮

휕(훼 휌 퐸 )휕푡 = −휆퐮 ∙ 퐮 − 퐮 − 푝 휇 푝 − 푝

(8)

The end states of the relaxation process are known to provide equilibrium in the flow velocities and pres-sures, and algebraic relations can be derived to calcu-late the equilibrium states [12]. Momentum relaxation is straightforward because the mass of each phase is conserved during the relaxation process. As a result of the stiff momentum relaxation (휆 → +∞), the two-phase mixture’s velocity is readily computed as:

퐮 =훼 휌 퐮 + 훼 휌 퐮

훼 휌 + 훼 휌

in which the “~” sign refers to the flow states after the pre-relaxation step. Superscript “n+1” refers to flow states at the new time step. For the subsequent pressure relaxation, we apply 휇 → +∞ to Eqs. (8). After some algebraic manipulation, we derive:

휕푒휕푡 + 푝

휕푣휕푡 = 0푘 = 푔, 푙

in which 푣 = 1 휌⁄ is the specific volume of phase k. Integrating from 푡 = 0to휏 in which 휏 is the relaxation time scale (휏 → 0), and applying the EOS (2)(3), we have:

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1휌 =

1휌

푝 + (훾 − 1)푝̂ + 훾 휋푝 + (훾 − 1)푝̂ + 훾 휋 푘 = 푔, 푙

(9)

In which 푝̂ = ∫ 푝 푑푡, which can be ap-

proximately calculated as: 푝̂ = 푝 . We then use the saturation condition ∑ 훼, = 1 to derive a quad-ratic equation for the equilibrium pressure 푝 :

(푝 ) + 퐵푝 + 퐶 = 0 (10) Where the coefficients B and C are: 퐵 = 푝̂ 훼 (훾 − 1) + 훼 훾 − 1 − 훼 푝 −

훼 푝 + 훼 훾 휋 + 훼 훾 휋 퐶 = −훼 푝 [(훾 − 1)푝̂ + 훾 휋 ]− 훼 푝 훾 −

1 푝̂ + 훾 휋 The coefficient C is guaranteed to be negative if 푝 > 0 and 푝 > 0, which suggests that one of the roots of Eq. (10) is guaranteed to be positive, and the positive root is taken as the solution for 푝 . The two phases’ density and specific internal energy are then found by Eq. (9) and the EOS (2) and (3), respectively. At this stage, the relaxation step is complete. Note that the total mass and energy of the two-phase mixture are conserved during the relaxation.

Let us examine the positivity of the gas phase den-sity 휌 and volume fraction 훼 . From the EOS (2) and (3), we know that positive gas phase density depends on both positive pressure and positive specific internal energy. Usually, positivity of specific internal energy is easily ensured by a proper numerical solution of the energy equation. It has been shown that during the re-laxation step, a positive equilibrium pressure 푝 is ensured if 푝 > 0 and 푝 > 0. Given this, one only need focus on the pre-relaxation step and make sure that 푝 > 0 and 푝 > 0 are always satisfied. This is not difficult to achieve if the volume fraction 훼 is solved appropriately by its convection equation in (7).

Due to the hyperbolic nature of the governing equations, we adopt an explicit, hybrid HLLC-Rusanov scheme [18]. Note that 훼 varies significantly across the material interface, and thus it requires the convec-tion scheme to be strictly monotone.

The last step is to simulate mass and energy trans-fer between the two phases:

휕훼휕푡 = 훼̇

휕 훼 휌휕푡 = 푚̇

휕 훼 휌휕푡 = 0

휕 훼 휌 퐮휕푡 = 푚̇퐮

휕 훼 휌 퐸휕푡 =

12 푚̇

|퐮| + 푞̇

휕(훼 휌 )휕푡 = −푚̇

휕(훼 휌 퐮 )휕푡 = −푚̇퐮

( ) = − 푚̇|퐮| − 푞̇ (11) A phase equilibrium model based on the entropy

maximization principle was formulated and implement-ed, as described in [18]. Results and Discussion

We first validate the present fluid solver for a one-dimensional, two-phase shock tube problem. This prob-lem is a benchmark test for compressible two-phase flows that has been studied by many authors [12-14, 17]. We consider a 1-D duct 4 meters long where the region [-2 m, 0.7 m] is initially filled with pure liquid and [0.7 m, 2 m] initially filled with pure gas. For nu-merical reasons, a trivial amount (α=10-8) of the other phase is mixed with the pure phase on each side. This is because from the definition of the volume fraction α, it must satisfy 0<α<1, otherwise the liquid and gas densi-ties cannot be defined. The added trivial substances could be considered as impurities, e.g., dissolved gas in liquid and humidity in gas. The duct is discretized with 1000 grid points in the CFD simulation.

The initial pressure of the liquid phase is 10,000 bar, and the gas pressure is 1 bar. The working fluid is water as the liquid phase, and steam vapor as the gas phase. The Stiffened Gas EOS parameters for water and steam have been tuned to match their measured ther-modynamic properties, such as density and vapor pres-sure, and the parameters are summarized in Table 1. For completeness, the specific heat C and reference energy Q are also included, but we note they are im-portant only when heat transfer and phase change are considered. For the current “frozen” shock tube flow their effects can be neglected.

Releasing the “diaphragm” separating the two phases at t=0, the analysis expects a shock wave to propagate into the gas phase, followed by the original contact surface between liquid and gas. Rarefaction waves propagate into the liquid phase and accelerate the liquid behind the contact surface. In Figure 1 the CFD simulation results are compared to the analytical solu-tions at 0.9 ms. Excellent agreement is observed on the strength of the shockwave, as seen by the velocity mag-nitude. The density field between the contact surface and the rarefaction waves is also precisely predicted. Although there is evidence of slight oscillation in the predicted solution behind the shock, in general the pres-sure field is predicted well. The initial pressure gradient in this case is very steep (10,000:1), and is covers the typical operating range of modern diesel and gasoline engine injection systems.

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analytical CFD

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analytical CFD

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Figure 1: CFD results at 0.9 ms compared with analyti-

cal solutions of the water shock tube problem.

Figure 2: Initial conditions of the shock-bubble

interaction problem. High pressure liquid (region 1) propagates into low pressure liquid (region 2) that

contains a cylindrical bubble (region 3).

Next, we validate the flow solver’s capability to solve two-dimensional problems. We consider a shock-interface interaction problem for the set up shown in Fig. 2 [19]. The domain is 1.2 m wide and 1.0 m high and is discretized with a 240×200 uniform Cartesian grid. On the left side of the domain high pressure liquid (region 1) propagates towards the right and into a low pressure liquid region (region 2). The shock wave speed is 452.7 m/s. The pressure difference is 10,000:1 and the density difference is about 1,200 kg/m3 versus 1,000 kg/m3. A cylindrical “bubble” (region 3) is placed qui-escently within the low pressure liquid. The bubble contains gas phase that has a density of 1.2 kg/m3, and it is in pressure equilibrium with the ambient region 2

liquid. We examine how the flow field develops after the shock wave hits the liquid-gas interface.

Figure 3: Simulation results of the shock-bubble inter-

action problem at 0.1, 0.2, 0.3, 0.4 ms.

Figure 3 shows the simulation results of the density field (left) and the gas phase volume fraction (right) at 0.1, 0.2, 0.3 and 0.4 ms after the start of the calcula-tions. The bubble is seen to be compressed and acceler-ated upon impact by the shock. Acceleration at the cen-ter of the bubble starts earlier because the impact from shock wave occurs earlier. Given the large density ratio between the liquid and gas, this acceleration leads to the deformation of the interface, which eventually breaks

EOS parameters Fluid

훾 휋 (Pa) C (J/kg-K) Q (J/kg)

Water (liquid phase) 3.0 9×108 1400 -1.084e6 Steam (gas phase) 1.3 0 1615 1.892e6

Table 1: Equation of State parameters for water and steam

1 23

432.7 m/su

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up the bubble. Associated with the bubble breakup is the rise of pressure and density in the surrounding liq-uid. Eventually a cylindrical shock wave is generated, and it propagates to the ambient regions. The current CFD results are found to agree well with the results reported in the literature [19].

Figure 4: Initial conditions for the liquid jet problem. The high pressure chamber is the left (red) region; the nozzle and the low pressure chamber are the blue re-

gions.

Figure 5: Simulation results of vapor mass fraction for

the submerged liquid jet problem at 0.06, 0.10, 0.14, 0.18 ms.

Figure 6: Simulation results of vapor mass fraction for

the non-submerged liquid jet problem at 0.04, 0.08, 0.12 ms.

Next, the flow solver is applied to a submerged

liquid jet issuing from a 2-D planar nozzle with 2 mm width and 8 mm length (Figure 4). The high-pressure chamber is connected to the nozzle with a sharp rectan-gular inlet. Water at room temperature fills the whole simulation domain (the EOS parameters are summa-rized in Table 1). The injection pressure (푝 ) is 1,000 bar and the back pressure (푝 ) is 1 bar. The Reynolds number is estimated to be 106, and the cavitation num-ber, defined as (푝 − 푝 )/(푝 − 푝 ) by Nurick [20] in which 푝 is vapor pressure, is 1.001. From the vapor mass fraction plots in Figure 5, cavitation is seen to occur by 0.06 ms after the start of the simulation. A pair of vapor bubbles is seen to be generated near the nozzle exit and two small spots of cavitation are seen near the nozzle inlet. As the simulation proceeds, both the in-nozzle cavitation and the wake bubbles grow in size. Eventually, the entire nozzle wall is covered by cavitat-ing flow, and a large region of cavitated fluid (a “cloud”) appears in the chamber. The in-nozzle cavita-tion is due to flow expansion and separation from the nozzle walls as it flows past the sharp nozzle inlet. The downstream cavitation is due to the pressure drop at the center of the vortices in the flow structures. These two cavitation regions develop independently, but the cavi-tated flow inside the nozzle propagates downstream and

8mm2mm

푝 = 1bar Low pressure chamber

푝 = 1000bar High pressure chamber

8

contributes to the growth of the bubble cloud. This is clearly seen in the plots at 0.14 ms.

Finally, the same setup is used to consider liquid water at 1,000 bar injected into air at 1 bar. The high-pressure chamber is initially filled with liquid, and both the nozzle and low-pressure chamber are initially filled with non-condensable air. Viscous flow was assumed and no-slip boundary conditions were applied at the walls, but the mass diffusion terms in Eqs. (1) were neglected due to large (~30,000) Péclet number. From the vapor mass fraction results in Figure 6, it is seen that cavitation is generated immediately after the liquid enters the nozzle (0.04 ms) and by t = 0.14 ms the cavi-tation regions extend over the entire nozzle length. Al-most no vapor mass is seen outside the nozzle, indicat-ing condensation of the vapor as fuel is injected out of the nozzle. This is in contrast to the results of the sub-merged jet, where large amounts of vapor were gener-ated in the chamber. Summary

An Eulerian CFD model was developed to simulate high-pressure liquid injection processes through a noz-zle into a low-pressure ambient environment, and both the internal- and external-nozzle flows were considered. This modeling approach is consistent at the nozzle exit because the same Eulerian conservation laws and ther-modynamic EOS are applied throughout the simulation domain. The Stiffened Gas EOS was used to model the thermodynamic states of the compressible two-phase mixture, and mixing of non-condensable air with the vapor phase was modeled with an ideal mixing ap-proach. Positivity of the gas phase volume fraction, which is critical for numerical stability, was shown to be preserved by the proposed stiff relaxation approach used to solve the transport equations. The CFD model was found to be robust for flows with large pressure and density differences.

For future studies, use of an advanced Equation of State to replace the current Stiffened Gas EOS is sug-gested. This will allow different fuel blends to be mod-eled for realistic injector flow simulations. Other areas of future study include removing the stiff relaxation assumptions by including resolved or modeled-sub-grid interfacial information, and by considering surface ten-sion effects.

Acknowledgement

The authors thank Drs. Ramachandra Diwakar and Tang-Wei Kuo at the Propulsion Systems Research Lab of General Motors Company for financial support. Dr. Chawki Habchi at IFP Energies nouvelles (France) and Prof. Richard Saurel at Polytech Marseille (France) made many helpful comments. We also thank Drs. Lu

Qiu, Won-Geun Lee, Wei Ning, Chang Wook Lee and Mr. Ben A. Cantrell, for their help. Nomenclature α volume fraction 훼̇ source term in the volume fraction equation B coefficient in the pressure equation (10) c isentropic sound speed C specific heat (Equation of State) or coefficient in the pressure equation (10) D molecular mass diffusivity e specific internal energy E specific total energy γ specific heat ratio (Equation of State) J heat transfer (vector) λ momentum relaxation coefficient 푚̇ source term for mass transfer µ pressure relaxation coefficient p pressure π stiffness parameter (Equation of State) Q reference energy state (Equation of State) 푞̇ source term for heat transfer density σ shear stress (tensor) t time T temperature τ relaxation time u flow velocity (vector) v specific volume y mass fraction Subscripts a air b back g gas I interface inj injection l liquid v vapor Superscripts n previous time step n+1 new time step References

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