Assessment and validation of liquid breakup models for high-pressure dense diesel sprays

Abstract

Liquid breakup in fuel spray and atomization significantly affects the consequent mixture formation, combustion behavior, and emission formation processes in a direct injection diesel engine. In this paper, different models for liquid breakup processes in high-pressure dense diesel sprays and its impact on multi-dimensional diesel engine simulation have been evaluated against experimental observations, along with the influence of the liquid breakup models and the sensitivity of model parameters on diesel sprays and diesel engine simulations. It is found that the modified Kelvin-Helmholtz (KH)–Rayleigh-Taylor (RT) breakup model gives the most reasonable predicted results in both engine simulation and high-pressure diesel spray simulation. For the standard KH-RT model, the model constant C_bl for the breakup length has a significant effect on the predictability of the model, and a fixed value of the constant C_bl cannot provide a satisfactory result for different operation conditions. The Taylor-analogy-breakup (TAB) based models and the RT model do not provide reasonable predictions for the characteristics of high-pressure sprays and simulated engine performance and emissions.

Keywords

breakup model / diesel engine / high-pressure injection / simulations

Introduction

High-pressure direct injection diesel engine is becoming popular for high-performance low emission automotive applications. It has been demonstrated that liquid fuel atomization and spray formation is one of the key in-cylinder processes affecting combustion and emission characteristics, and the thermal efficiency can be improved and exhaust emissions can be reduced by optimization of fuel spray characteristics and injection strategy. Badami et al. [ 1] studied the impact of injection pressure on the performance of a direct injection (DI) diesel engine with a high-pressure common rail system. They found that high-pressure injection results in an increase in maximum power, and a reduction in soot formation and fuel consumption. The influence of the geometry of the discharge nozzle hole of a diesel injector was studied by Baumgarten [ 2]. They found that the optimized injector geometry would produce a better exhaust performance. Felice et al. [ 3] investigated the potential of the multiple injection strategy for the achievement of low emissions in a high-pressure direct injection diesel engine. It was demonstrated that the peak heat release rate, the NO_x, and the smoke exhaust emissions could be reduced by using the multiple injection strategy.

In recent years, multi-dimensional computational fluid dynamics (CFD) simulation of in-cylinder processes has become the tool for engine design and optimization, in response to the enforcement of stringent emission regulations. In DI diesel engines, liquid fuel is injected directly into the combustion chamber where it is broken up into individual droplets, and eventually vaporized and ignited. Spray droplets may undergo a number of processes from the time they are injected until the time of complete vaporization. Thus, a series of spray sub-models need to be implemented to simulate the diesel dense spray processes including drop breakup, collision, evaporation, and so forth. Significant efforts have been made to develop different spray sub-models for incorporation into CFD simulation. Reitz et al. [ 4, 5] presented the wave breakup theory using the development of Kelvin-Helmholtz (KH) instabilities on a jet surface. The Rayleigh-Taylor (RT) breakup model was developed based on the theoretical considerations of Taylor [ 6, 7]. The KH-RT hybrid breakup model consists of both the Kelvin-Helmholz and Rayleigh-Taylor instability theories, and it is expected to have a greater potential than previous models to provide enhanced simulation results [ 8]. O’Rourke and Amsden [ 9] presented the Taylor-Analogy-Breakup (TAB) model based on the assumption that droplet distortion could be described as a spring-mass system. They also developed the extensions of a three-dimensional computational model for the liquid wall films formed in port-injected engines, and the computed film locations agree qualitatively with those observed in laser-induced fluorescence measurements [ 10]. Moreover, Schmidt and Rutland [ 11] presented a numerical collision scheme named the no time counter (NTC) method.

However, in spite of these many efforts, the spray simulation is not sufficiently accurate to advance fuel injection strategies and spray characteristics to the point that engines can be developed solely based on the CFD simulation that can comply with ever-stricter emission standards. On one hand, simulation results may vary considerably, depending on the details of the submodels implemented. On the other hand, the complexities of the spray processes must be included and dealt with in the spray submodels. Therefore, it is essential to assess the validity and accuracy of the previously developed models for a variety of conditions that may be encountered in high-pressure direct injection diesel engines. Sone and Menon [ 12] investigated the effect of sub-grid modeling on an in-cylinder unsteady mixing process in a direct injection engine. In their study, the predicted results of an in-cylinder turbulent fuel-air mixing process were found to be significantly sensitive to their turbulence model. The large eddy simulation for both non-evaporative and evaporative diesel spray was implemented by Hori et al. [ 13] in a constant volume vessel. Larmi and Tiainen [ 14] conducted a medium speed diesel engine simulation, and found that the fuel viscosity effect on drop sizes was well predicted by a KH-RT breakup model. Moreover, Fujimoto et al. [ 15] studied the predictive capability of different spray breakup models on a non-evaporative diesel spray. Their results showed that for a non-evaporative diesel spray simulation, the breakup model significantly affected the calculated spray shape.

Liquid breakup models play a key role in a spray CFD simulation. Therefore, in this paper, numerical studies on the evaporative diesel-like fuel spray have been conducted in an attempt to assess the accuracy of the existing spray breakup models which are widely used in diesel engine simulations. The effect of liquid breakup models and the sensitivity of the model parameters on the simulation of diesel fuel spray characteristics are highlighted by comparing the numerical results against experimental data available in literature. Meanwhile, the performance of the spray breakup models is analyzed for high-pressure diesel spray simulations. In addition, a diesel engine simulation is also implemented and compared with experimental measurements to enhance the understanding of the effect of spray breakup models on engine CFD simulations for high-pressure direct injection diesel engines.

Experimental

Spray

In this paper, the experimental data used for the assessment and validation of spray breakup models were taken from [ 16]. The spray was injected under high pressure into a constant volume vessel, which was used to create the high pressure and high temperature ambient condition. Inert gas, sulfur hexafluoride (SF₆), was charged into the vessel as the working medium and was heated by burning the mixture of H₂ and O₂ inside the vessel. The temperature and density of the gas in the vessel ranged from 800 to 1100 K and from 20 to 100 kg/m³, respectively. A mixture of n-decane, naphthalene (NP) and tetramethyl-p-phenylenediamine (TMPD) by a mass proportion of 90€:€9€:€1 was used to substitute for the diesel fuel and was injected into the vessel by an electronically controlled single-hole injector, with an injection pressure of 180 MPa, an injection duration of 1.0 mm and an injector diameter D_n of 0.1 mm. The equivalence ratio of the vapor-phase spray in the vessel was measured quantitatively using the planar laser induced exciplex fluorescence (PLIEF) technique. Before fuel injection, SF₆, H₂, and O₂ were charged into the vessel and ignited by a spark plug to create a high temperature and high pressure environment. The injection started when the ambient temperature dropped to the pre-set value. During the experiment, the ambient pressure was measured using a Kistler 6125B pressure sensor. Further details of the experiment can be referred to in Ref. [ 16].

Engine

In order to further increase the understanding of the predictive capability of the breakup models, a 3D engine simulation was also performed, and the simulated results were compared to another experimental measurement of engine tests conducted by Klingbeil et al. [ 17] on a Caterpillar 3401E single cylinder oil test engine (SCOTE). The engine specifications are listed in Table 1. Its fuel injector was a production style Caterpillar electronic unit injector. The characteristics of the injection system are given in Table 2.

Model formulation

Governing equations

In the numerical simulation, the dynamics of the fluid flow within the cylinder of a direct injection diesel engine and the constant-volume vessel were governed by the compressible equations for the conservation of mass, momentum, energy, and species. In these equations, the Einstein’s tensor notation was utilized for multi-dimensional flow. Considering the turbulent flow, the flow property N (u_i, h, e, T and Y_m)was decomposed by the Reynolds averaging as [ 18]

(1) N=\bar{N}+N^{\prime },

(2) \overline{N^{\prime }}\equiv \mathrm{0},

where \bar{N} is a time-averaged component, N^{\prime } is a fluctuating component, and the flow property M (p, q and ρ) is decomposed by the Favre averaging as

(3) M=\tilde{M}+M^{″},

(4) \tilde{M}\equiv {\displaystyle \frac{\overline{\rho M}}{\overline{\rho }}}.

For the Favre averaging, \tilde{N} is a mass-averaged component and N^{″} is a fluctuating component.

The variables given above mean that u_i is the instantaneous velocity in the direction, x_i, e is the specific internal energy, T is the temperature, h is the specific enthalpy, Y_m is the mass fraction of the species m, p is the pressure, q is the heat-flux vector, and ρ is the density.

The governing equations considering the compressible turbulent flow are expressed as

Conservation of mass:

(5) {\displaystyle \frac{\partial \bar{\rho }}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\bar{\rho }{\tilde{u}}_j\right)=S_n,

where S_n is the mass source term derived from the evaporation of the injected fuel.

Conservation of momentum:

(6) {\displaystyle \frac{\partial \bar{\rho }{\tilde{u}}_i}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\bar{\rho }{\tilde{u}}_j{\tilde{u}}_i\right)=-{\displaystyle \frac{\partial \bar{p}}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[{\bar{\sigma }}_{ji}-{\tau }_{ji}\right]+S_j,

where σ_ji is the stress tensor, τ_ji is the Reynolds stress tensor, and S_j is the source term arises from fuel spray and gravitational acceleration.

Conservation of energy:

(7) \begin{array}{l}{\displaystyle \frac{\partial }{\partial t}}\left[\overline{\rho }\right(\tilde{e}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\tilde{u}}_i{\tilde{u}}_i\left)\right]+{\displaystyle \frac{\partial }{\partial x_j}}\left[\overline{\rho }{\tilde{u}}_j\right(\tilde{h}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\tilde{u}}_i{\tilde{u}}_i\left)\right]={\displaystyle \frac{\partial }{\partial x_j}}(-{\overline{q}}_j+{\tilde{u}}_i{\overline{\sigma }}_{ji})-{\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\overline{\rho }\widetilde{{u^{″}}_i{u^{″}}_i}\right)+\\ {\displaystyle \frac{\partial }{\partial x_j}}(-{\tilde{u}}_j{\tau }_{ji}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\tilde{u}}_j{\overline{{u^{″}}_i{u}^{″}}}_i-{\displaystyle \frac{{\mu }_t{c}_{\mathrm{p}}}{P{r}_t}}{\displaystyle \frac{\partial \tilde{T}}{\partial x_j}})+S_{\mathrm{e}},\end{array}

where e=C_{\mathrm{v}}T , q_j=-K{\displaystyle \frac{\partial T}{\partial x_j}}=-{\displaystyle \frac{\mu }{Pr}}{\displaystyle \frac{\partial h}{\partial x_j}} , μ_t is the turbulent viscosity; c_p is the specific heat at constant pressure; C_v is the specific heat at constant volume; Pr_t is the turbulent Prandtl number, P{r}_{\mathrm{t}}={\displaystyle \frac{c_{\mathrm{p}}{\mu }_{\mathrm{t}}}{k}} , k is the thermal conductivity; and S_e represents the source from chemical reactions and turbulent dissipation.

Conservation of species:

(8) {\displaystyle \frac{\partial \bar{\rho }{\tilde{Y}}_m}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\bar{\rho }{\tilde{u}}_j{\tilde{Y}}_m\right)=-{\displaystyle \frac{\partial {\bar{J}}_j^m}{\partial x_j}}-{\displaystyle \frac{\partial \bar{\rho }\overline{{u^{″}}_j{Y^{″}}_m}}{\partial x_i}}+S_m,

where m represents the individual chemical species in the fluid mixture, and S_m is the species source term derived from the chemical reactions and fuel evaporation. The diffusion flux is given by

(9) {\bar{J}}_j^m=-\bar{\rho }{D}_m{\displaystyle \frac{\partial {\tilde{Y}}_m}{\partial x_j}},

in which D_m is the mass diffusion coefficient for the species m,

(10) \overline{\rho }\overline{{u^{″}}_j{Y^{″}}_m}=-{\displaystyle \frac{{\mu }_t}{S{c}_{\mathrm{t}}}}{\displaystyle \frac{\partial {\tilde{Y}}_m}{\partial x_j}}.

The stress tensor, σ_ji, is given as

(11) {\overline{\sigma }}_{ji}={\mu }_{\mathrm{t}}\left({\displaystyle \frac{\partial {\tilde{u}}_i}{\partial x_j}}+{\displaystyle \frac{\partial {\tilde{u}}_j}{\partial x_i}}\right)-{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{\mu }_{\mathrm{t}}{\displaystyle \frac{\partial {\tilde{u}}_k}{\partial x_k}}{\delta }_{ij.}

The Reynolds stress tensor, τ_ji, is given as

(12) {\tau }_{ji}=-\bar{\rho }\overline{{u^{″}}_i{u^{″}}_j}={\mu }_t({\displaystyle \frac{\partial \widetilde{u_i}}{\partial x_j}}+{\displaystyle \frac{\partial \widetilde{u_j}}{\partial x_i}})-{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}(\bar{\rho }\kappa +{\mu }_t{\displaystyle \frac{\partial \widetilde{u_i}}{\partial x_i}}){\delta }_{ij},

where δ_ij is the Kronecker delta, and the turbulent kinetic energy, k , is given by k={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\overline{{u^{″}}_i{u^{″}}_i} . The turbulent kinetic energy is obtained using the turbulence model which is described in Section 3.3.

Breakup models

To model the breakup of the injected liquid bulk, the Kelvin-Helmholtz (KH) model, the Rayleigh-Taylor (RT) and the Taylor-analogy-breakup (TAB) models are widely used in 3D engine simulations.

Kelvin-Helmholtz (KH) breakup model

The Kelvin-Helmholtz instability is based on a liquid jet stability analysis that is described in detail by Reitz [ 4, 5]. The analysis considers the stability of a cylindrical, viscous, liquid jet with an initial radius r₀ which is penetrating into an incompressible, inviscid gas with a relative velocity u_rel. It is also assumed that the turbulence generated inside the nozzle hole results in the presence of a spectrum of sinusoidal waves on the liquid jet surface. These surface waves have an infinitesimal axisymmetric displacement initially, and grow due to the aerodynamic forces derived from the relative velocity between the liquid and gas. As shown in Fig. 1(a), it is assumed that the new droplet size is proportional to the maximum wavelength, Λ_KH, and the change rate of the droplet size is given as

(13) {\displaystyle \frac{\mathrm{d}r}{\mathrm{d}t}}={\displaystyle \frac{r-r_{\mathrm{new}}}{{\tau }_{\mathrm{KH}}}},

where τ_KH is the KH model breakup time and r_new is the radius of the new droplet, which are described as

(14) {\tau }_{\mathrm{KH}}={\displaystyle \frac{\mathrm{3.726}{B}_{\mathrm{1}}r}{{\mathrm{\Lambda }}_{\mathrm{KH}}{\mathrm{\Omega }}_{\mathrm{KH}}}},

(15) r_{\mathrm{new}}=B_{\mathrm{0}}{\mathrm{\Lambda }}_K,

where B₀= 0.61, B₁ is an adjustable model constant, and {\mathrm{\Omega }}_{\mathrm{KH}} is the maximum growth rate.

Rayleigh-Taylor (RT) breakup model

The Rayleigh-Taylor instability mechanism as shown in Fig. 1(b) considers that the unstable RT waves occur because of the rapid deceleration of the drops which arises from the aerodynamic drag force F_aero.

(16) F_{\mathrm{aero}}=\mathrm{\pi }{r}^{\mathrm{2}}{c}_D{\displaystyle \frac{{\rho }_g{u}_{\mathrm{rel}}^{\mathrm{2}}}{\mathrm{2}}}.

Dividing the drag force by the mass of the drop, the acceleration of the interface can be expressed as

(17) a={\displaystyle \frac{\mathrm{3}}{\mathrm{8}}}{c}_D{\displaystyle \frac{{\rho }_g{u}_{\mathrm{rel}}^{\mathrm{2}}}{{\rho }_{l}r}},

where c_D is the drag coefficient of the drop and r is the radius of the drop.

The new droplet radius, r_RT, and the model breakup time, τ_RT, are given as

(18) r_{\mathrm{RT}}=\mathrm{0.5}{\mathrm{\Lambda }}_{\mathrm{KH}},

(19) {\tau }_{\mathrm{RT}}={\mathrm{\Omega }}_{\mathrm{RT}}^{-\mathrm{1}}.

KH-RT hybrid breakup model

A single model is usually unable to sufficiently describe the whole breakup process of the engine sprays. Hence, in this paper, the KH-RT hybrid breakup model was implemented to simulate the diesel breakup, as in Ref. [ 7]. The droplet breakup modeled by the RT model is too fast if such model is implemented at the nozzle hole [ 2]. Therefore, in this model, the KH mechanism is responsible for drop breakup before the breakup length L_b, while both KH and RT mechanisms are activated beyond L_b as shown in Fig. 1(c). First, it was checked if the RT mechanism could break up the droplet. If not, the KH mechanism would be responsible for the breakup.

The breakup length, L_b, of the injected diesel fuel jet is calculated by

(20) L_b=d_{\mathrm{0}}{C}_{\mathrm{bl}}\sqrt{{\displaystyle \frac{{\rho }_l}{{\rho }_g}}},

where d₀ is the nozzel diameter, ρ₁ and ρ_g are the ambient gas density and droplet density, respectively. The breakup length constant C_bl can be tuned from 0 to 50. In this paper, C_bl was set to 0, 10 and 40, respectively, in order to assess its impact on the results of the spray simulation.

Further, as an alternative to the KH-RT breakup model, the modified KH-RT model was also implemented in this paper. In this model, the specific breakup length L_b was removed. Instead, the KH model was responsible for the primary breakup of the injected “Parent” liquid blobs, during which “Child” drops were created. Thus, the secondary breakup of these drops was modeled by examining the competing effects of the KH and RT mechanisms.

Taylor-Analogy-Breakup (TAB) model

The TAB breakup model is a classic method of calculating drop distortion and breakup. This method was developed based on Taylor’s analogy between an oscillating and distorting droplet and a spring-mass system [ 9]. In the TAB model, the breakup drop radius r is able to be calculated both with and without a drop size distribution. For the model without the drop size distribution, the new droplet radius r^{\prime } is determined as

(21) r^{\prime }={\displaystyle \frac{r_{\mathrm{0}}}{\mathrm{1}+\mathrm{8}K/\mathrm{20}+({\rho }_l{r}_{\mathrm{0}}^{\mathrm{3}}/\sigma ){\dot{y}}^{\mathrm{2}}(\mathrm{6}K-\mathrm{5})/\mathrm{120}}},

where r₀ is the particle radius before breakup; \dot{y} is the velocity of the parameter y, y=2x/r₀, which is the non-dimensional displacement of the particle surface; x is the displacement of the drop equator from its equilibrium position; K is the ratio of the distorting energy of a particle to its total energy; and σ is the surface tension of the particle.

In the TAB model with a drop size distribution, Eq. (21) provides the Sauter mean radius r₃₂ ( r^{\prime }=r_{\mathrm{32}} ). The chi-squared and the Rosin-Rammler distribution may be used in the TAB model, respectively.

For the chi-squared distribution, the probability density function is given by

(22) C(r)={\displaystyle \frac{\mathrm{1}}{\bar{r}}}\exp \left(-{\displaystyle \frac{r}{\bar{r}}}\right),

where r is the drop radius and \bar{r} is the number averaged drop radius given by

(23) \bar{r}={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{r}_{\mathrm{32}}.

For the Rosin-Rammler distribution, the probability density function is described as

(24) R(r)=\mathrm{1}-\exp \left(-{\displaystyle \frac{r^a}{q}}\right),

where a and q are empirical model constants.

In this paper, the above breakup models, including the modified KH model, the KH-RT hybrid models with different breakup lengths, the TAB model (without the drop size distribution), TAB-CHI model (with the chi-squared drop size distribution), and the TAB-RR model (with the Rosin-Rammler drop size distribution) were implemented and their impact on the spray characteristics were investigated.

Turbulence modeling

Turbulence directly influences fuel injection and atomization processes, spray characteristics, mixing and combustion processes in an internal combustion engine, thereby, the turbulence model plays an important role in internal combustion engine simulations. The rapid distortion RNG k-ε model was developed by Han and Reitz [ 19] based on the standard k-ε [ 20] and RNG k-ε models [ 21]. The superiority of the rapid distortion RNG k-ε model was demonstrated earlier [ 22]. Hence, the rapid distortion RNG k-ε turbulence model was applied in this paper.

The transport equation for the turbulent kinetic energy, k, of the rapid distortion RNG k-ε model developed by Han and Reitz et al. [ 19] is given as

(25) {\displaystyle \frac{\partial \overline{\rho }k}{\partial t}}+{\displaystyle \frac{\partial \left(\overline{\rho }{\tilde{u}}_{j}k\right)}{\partial x_j}}={\tau }_{ij}{\displaystyle \frac{\partial {\tilde{u}}_i}{\partial x_j}}+{\displaystyle \frac{\partial }{\partial x_j}}\left({\displaystyle \frac{{\mu }_t}{P_{\mathrm{r}}}}{\displaystyle \frac{\partial k}{\partial x_j}}\right)-\overline{\rho }\epsilon +S_s,

where ε is the dissipation of turbulent kinetic energy, S_s is the source term, μ_t is the turbulent viscosity, Pr is the Prandtl number, and τ_ij is the Reynolds stress.

Considering the interactions of turbulence with the discrete phase, the source term S_s includes the fluctuating component of the gas-phase velocity given as

(26) S_s=-{\displaystyle \frac{{\displaystyle \sum N_p({F^{\prime }}_{\mathrm{drag},i}{u^{″}}_i}{)}_p}{V}},

where the summation is the overall parcels in a grid cell, N_p is the number of the drops in a parcel, V is the cell volume, and {F^{\prime }}_{\mathrm{drag},i} is defined by

(27) {F^{\prime }}_{\mathrm{drag},i}={\displaystyle \frac{F_{\mathrm{drag},i}}{(u_i+{u^{\prime }}_i-v_i)}}{u^{″}}_i,

where F_drag,i is the drag force on a drop and v_i is a drop velocity.

The transport equation of the dissipation of turbulent kinetic energy, ε, is given by

(28) \begin{array}{l}{\displaystyle \frac{\partial \overline{\rho }\epsilon }{\partial t}}+{\displaystyle \frac{\partial \overline{\rho }{\tilde{u}}_j\epsilon }{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left({\displaystyle \frac{{\mu }_{\mathrm{t}}}{pr}}{\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right)-[{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{C}_{\mathrm{1}}-C_{\mathrm{3}}+{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{\displaystyle \frac{C_{\mu }\eta (\mathrm{1}-\eta /{\eta }_{\mathrm{0}})}{(\mathrm{1}+\beta {\eta }^{\mathrm{3}})}}{\displaystyle \frac{k}{\epsilon }}{\displaystyle \frac{\partial {\tilde{u}}_k}{\partial x_k}}]\overline{\rho }\epsilon {\displaystyle \frac{\partial {\tilde{u}}_i}{\partial x_j}}\\ +\left[\right(C_{\mathrm{1}}-{\displaystyle \frac{\eta (\mathrm{1}-\eta /{\eta }_{\mathrm{0}})}{(\mathrm{1}+\beta {\eta }^{\mathrm{3}})}}){\displaystyle \frac{\partial {\tilde{u}}_i}{\partial x_j}}{\overline{\sigma }}_{ij}-C_{\mathrm{2}}\overline{\rho }\epsilon +C_{\mathrm{s}}{S}_{\mathrm{s}}]{\displaystyle \frac{\epsilon }{k}},\end{array}

where the model constants β, C₁, C₂, C₃,C_μ and C_s are 0.012, 1.42, 1.68, –1.0, 0.0845 and 1.5, respectively. η is given by \eta ={\displaystyle \frac{k}{s}}\left|S_{ij}\right| in which S_ij is the mean strain rate tensor.

Other sub-models used in this paper are listed in Table 3.

Computational grids

In this paper, the numerical simulation was implemented using the Converge^TM CFD code. It is well known that spray simulation is sensitive to the resolution of the numerical grids used. In this paper, in order to reduce the grid dependency, the original grid resolution of 2.0 mm×2.0 mm was refined to 1.0 mm×1.0 mm and 0.5 mm×0.5 mm. The adaptive mesh refinement (AMR) technique was also implemented to refine the grid to a minimum of 0.25 mm×0.25 mm. During the simulation, the variation of the velocity, temperature, species, and passives in a grid cell were referred to determine whether the cell was embedded or the embedding should be removed [ 7]. At present, the Eulerian-Lagrangian method is widely used to simulate the spray and atomization. As a result, the spray simulation has a high grid resolution dependency. The simulated liquid penetration increases with a decrease in the grid size, but does not converge to the experimental data. Over-small grid size will lead to an over-prediction of the liquid penetration [ 27]. AMR technique is one of methods to optimize the grid resolution and eliminate the grid resolution dependency. As depicted in Fig. 2, the spray structure is quite different for each grid resolution used. In the cases of the grid resolutions of 2.0 mm×2.0 mm and 1.0 mm×1.0 mm, the spray shape is not reasonable compared to the experimental observations. Reasonable spray shapes are obtained for both the fine grid (0.5 mm×0.5 mm) and the AMR methods, respectively. Therefore, the AMR method with a minimum grid size of 0.25 mm×0.25 mm was implemented to refine the grid and save computational costs.

Results and discussion

Effect of breakup models on spray simulation in a constant volume vessel

The fuel spays considered were injected into a constant volume vessel having a gas medium under the conditions given in Table 4. The simulated results are compared with the experimental results in Ref. [ 16] in an attempt to assess the predictive capability of different breakup models. The fuel used in the experiment is a mixture of n-decane, naphthalene (NP) and tetramethyl-p-phenylenediamine (TMPD) at a mass proportion of 90:9:1, and the TMPD is used for its fluorescence characteristics in the PLIEF measurement technique [ 16]. In the present simulation, the mixture of 91% n-decane and 9% naphthalene in the mass fraction was used. The breakup models evaluated are listed in Table 5.

Figure 3 is a comparison of the liquid spray penetration between the present model predictions and the measured results taken from Ref. [ 16]. It is seen that all the spray breakup models have the same prediction for the liquid spray penetration in the initial injection period, but then they behave quite differently for the succeeding period in which the experimental liquid spray tip penetration remains almost constant. The three TAB based models (TAB, TAB-CHI and TAB-RR) significantly under-predict the liquid spray tip penetration. The drop size distribution in the TAB model makes only a slight difference in the prediction of the liquid spray tip penetration. For the KH-RT hybrid model, the liquid breakup length has a significant influence on the predictive capability for the liquid spray penetration because the breakup length is proportional to the breakup time τ_KH as expressed in Eq. (14). The predicted spray liquid penetration increases with an increase in the breakup time. The results indicate that the predicted penetration using the KH-RT model with a breakup length constant of C_bl= 10 in Eq. (20) agrees with the experimental data reasonably well. Also, the modified KH-RT model in which a specific breakup length is removed provides a predicted result which is in reasonable agreement with the experimental data. The RT model significantly under-predicts the liquid penetration. This suggests that such mechanism breaks up the droplets too fast if it is used for the earlier stage of the spray development. However, in the initial fuel injection period, all models have a shorter predicted penetration compared with the experimental results. This behavior may occur because the relative measurement error in the initial injection period is higher than that in the later stage of the injection period. It is also observed that the liquid spray penetration predicted by all the models increases initially, reaches a peak value, and then reduces to an asymptotic constant, except for the KH-RT model with a breakup length constant of C_bl= 20. The reason for this is that these models predict small droplets in the spray tip region, which vaporize in the high temperature gas environment, resulting in the peak value in the penetration. The long breakup length predicted by the KH-RT model with a breakup length constant of C_bl= 10 predicts unreasonable spray penetration as well as other spray characteristics as demonstrated in Figs. 4 and 5.

Shown in Fig. 6 is a comparison of the liquid phase sprays predicted by the present simulation employing the different breakup models given in Table 5 against the experimental results in Ref. [ 16]. It is clearly observed that the predicted liquid phase spray is significantly influenced by the breakup models used in the numerical simulation. The modified KH-RT model provides a reasonable simulation result. For the KH-RT model, the predicted liquid phase spray is very sensitive to the numerical value of the breakup length constant C_bl appeared in Eq. (20). The KH-RT model with C_bl=10 provides a reasonable predicted results. However, using C_bl= 20 leads to a longer breakup time τ_KH, and a longer penetration; even the overall characteristics of the liquid phase spray deviate considerably from the experimental observations. Further, it can be seen that the three TAB based models under-predict both the spray angle and spray tip penetration for the liquid phase.

A comparison of the simulation results with the experimental results is given in Fig. 5 for the simulated vapor phase spray based on the seven breakup models considered in this paper. The numerical results are presented in terms of the contour plots for the equivalence ratio, similar to the experimental results in Ref. [ 16]. The equivalence ratio is determined by the competition between the rate of local fuel evaporation and the ambient gas entrainment, and the latter is related to the liquid breakup, penetration and turbulent transport. However, in general, the vapor phase distribution is more like a turbulent jet injected into a stationary medium, as displayed in Fig. 5. It can be observed that the effect of the breakup model on the vapor phase simulation is much smaller than that on the liquid phase simulation discussed earlier. The predictions of the vapor phase shape of every model, except the KH-RT model at C_bl = 20, agree reasonably with the experimental measurements for the vapor phase. This might be attributed to two reasons. First, it can be seen from Fig. 6 that for different liquid breakup models, except the KH-RT model at C_bl = 20, the predicted liquid spray mass which remains in the constant-volume vessel is very similar for the sprays investigated, although the simulated liquid-phase shape of the injected fuel is quite different among different liquid breakup models studied. This suggests that different liquid breakup models used, except the KH-RT model at C_bl = 20, predict the similar amount of vapor-phase fuel which is derived from the injected liquid-phase fuel. Second, the numerical results of the isosurface of the turbulent kinetic energy of the ambient flow, as illustrated in Fig. 7, indicate that the model constant C_bl in the KH-RT model and the drop size distribution in the TAB model have a quite small effect on the turbulent kinetic energy of the ambient gas. Therefore, similar vapor-mass and turbulent kinetic energy are predicted using the different liquid breakup models, except the KH-RT model at C_bl = 20.

Effect of breakup models on engine simulation

The numerical simulations were implemented to clarify the predictive capability of the breakup models for engine simulations by comparing the numerical results with the engine test data of Klingbeil et al. in Ref. [ 17] in a single cylinder high pressure direct injection diesel engine. The engine operating conditions are listed in Table 6.

The results of the present numerical simulation implementing different liquid breakup models are compared in Fig. 8 against the experimental engine test results obtained in Ref. [ 17]. For the measured and simulated in-cylinder pressure histories shown in Fig. 8(a), it is seen that the modified KH-RT model, the KH-RT model at C_bl=10, and the RT model provide the simulated results that are in good agreement with the experimental results. The KH-RT model at C_bl=20 significantly under-predicts the in-cylinder pressure. The three TAB based models have a similar prediction of the in-cylinder pressure which is lower than the experimental data. The experimental and numerical heat release rates shown in Fig. 8(b) indicate that the three kinds of TAB models and the KH-RT model at C_bl=20 predict a significantly small premixed combustion phase and a quite large diffusive combustion phase. The predicted heat release rates using the KH-RT model at C_bl=10 and the RT model are close to the experimental data, but a little lower. It also can be seen that the modified KH-RT model can predict the heat release rate reasonably, being in better agreement with the experimental results.

Figure 9 illustrates the simulated distribution of the in-cylinder equivalence ratio and spray at a crank angle of 5˚ATDC. It can be observed that the RT model and the three TAB based models predict a shorter spray penetration compared to the modified KH-RT model. As a result, their spray does not have any impact on the combustion chamber. The high equivalence ratio areas of the three TAB models are close to the injector due to their over prediction of droplet breakup. The KH-RT model at C_bl=20 significantly under-predicts the droplet breakup, leading to a long predicted spray penetration. It is found that for the KH-RT model, the breakup length constant, C_bl, have a strongly effect on the simulated results of the spray and atomization process and hence the mixture formation process as well.

A comparison of the soot and NO_x emissions is exhibited in Fig. 10 between the present simulated results using different liquid breakup models and the experimental results obtained by Klingbeil, et al. [ 17]. The soot emission predicted by the modified KH-RT model and the RT model are in a good agreement with the experimental results at the engine-out point. The three TAB models and the KH-RT model at C_bl=10 over-predict the soot emission, while the KH-RT model at C_bl=20 predicts an obviously unreasonable soot emission result compared to the experimental data at the engine-out point. For the NO_x emission, only the prediction using the modified KH-RT model matches the experimental results well at the engine-out point, while the RT model over-predicts and the TAB models and the KH-RT model with C_bl=10 and 20 under-predict the engine-out NO_x emission considerably. It is seen that some models yield good prediction for NO_x emission while others yield good prediction for soot emission. However, only the modified KH-RT model provides good prediction for both NO_x and soot emission simultaneously.

Conclusions

In this paper, the effect of different liquid breakup models was investigated on the simulated characteristics of high-pressure fuel sprays injected into a high pressure and high temperature ambience in a constant volume vessel and on the simulated engine performance for a high-pressure DI diesel engine. The results indicate that the modified KH-RT breakup model, in which the fixed breakup length is removed, gives the most reasonable predicted results in both engine simulation and constant-volume vessel spray simulation. For the standard KH-RT model, the model constant C_bl has a significant effect on the predictability of the model, and a fixed value of constant C_bl cannot provide a satisfactory result for different operation conditions. The three TAB based models predict a quite small premixed combustion phase and a large diffusive combustion phase due to their over prediction of droplet breakup. The RT model is not appropriate to be used as a single model in a diesel-like fuel spray simulation.

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