ILASS-Americas 30th Annual Conference on Liquid Atomization and Spray Systems, Tempe, AZ, May 2019

_____________________________________

*Corresponding author: ctsang@dow.com

A Stochastic Kelvin-Helmholtz/Rayleigh-Taylor Breakup Model and a Syn-

thetic Eddy Injection Model for Large-Eddy Simulations of Diesel Sprays

Chi-Wei Tsang1,2*, Christopher J. Rutland1

1University of Wisconsin-Madison

Madison, Wisconsin, 53706, United States 2The Dow Chemical Company

Lake Jackson, TX, 77566, United States

Abstract

The goal of this study is to develop and improve physical models for large-eddy simulations of

Diesel sprays. Particularly, the synthetic eddy injection model and the stochastic Kelvin-Helm-

holtz/Rayleigh-Taylor (KH-RT) atomization and breakup model were developed and tested. The

synthetic eddy injection model determines fluctuations of velocities of liquid fuel at the spray

nozzle exit. The model attempts to simulate turbulence at the nozzle exit without the need of in-

ternal nozzle flow simulations by superimposing a number of virtual coherent structures. The idea

of the stochastic KH-RT model is to stochastically and dynamically determine several model pa-

rameters, specifically the KH and RT length and time scales. This is an attempt to reduce the

sensitivity of the model parameters in the standard KH-RT model (Beale, J. and Reitz, R.D.,

1999)). The performance of these two newly developed models was compared to the classical

models, namely the standard KH-RT and the cone angle injection models (Reitz, R., 1987). Two

experimental databases, the Engine Combustion Network constant-volume sprays (Pickett, L. M.

et al., 2010) and the Engine Research Center optical engine sprays (Neal, N. & Rothamer, D.,

2016) with a range of operating conditions were used to validate the models. The stochastic KH-

RT model improved the prediction of the spatial distribution of liquid fuel in the region where

secondary breakup occurs. The stochastic KH-RT model was able to better predict liquid penetra-

tions in a wider range of operating conditions (errors within 5 %). The synthetic eddy injection

model improved the predictions of the spatial distribution of liquid mass in the near-nozzle region

and vapor penetrations at early stage of injection. The results suggest that development of insta-

bility modes and turbulent transport in the near-nozzle region were better represented by the syn-

thetic eddy injection model. The model also showed less computational grid sensitivity. Overall,

using these two new models overcomes several limitations in the original models and makes large

eddy simulations (LES) as a more predictive tool for Diesel sprays.

Beale, J. C., & Reitz, R. D. (1999). Modeling spray atomization with the Kelvin-Helmholtz/Rayleigh-Taylor

hybrid model. Atomization and sprays, 9(6). Neal, N., & Rothamer, D. (2016). Measurement and characterization of fully transient diesel fuel jet processes in

an optical engine with production injectors. Experiments in Fluids, 57(10), 155.

Pickett, L. M., Genzale, C. L., Bruneaux, G., Malbec, L. M., Hermant, L., Christiansen, C., & Schramm, J. (2010).

Comparison of diesel spray combustion in different high-temperature, high-pressure facilities. SAE International

Journal of Engines, 3(2), 156-181.

Reitz, R. (1987). Modeling atomization processes in high-pressure vaporizing sprays. Atomisation and Spray

Technology, 3(4), 309-337.

.

ILASS-Americas 30th Annual Conference on Liquid Atomization and Spray Systems, Tempe, AZ, May 2019

2

Introduction

In recent years, large-eddy simulation (LES) has be-

come a promising approach to simulate unsteady fuel

spray dynamics [1]. In LES, filtered turbulent flow fields

are directly solved on computational grids. The charac-

teristic of the filtered flow fields is that large-scale struc-

tures of turbulent flow are retained, while small-scale

structures are filtered out. Mathematically, the filtered

turbulent flow field is obtained by a convolution opera-

tion [2]. As pointed out by Rutland [1], there are several

expectations of LES applied in engine simulations. The

primary expectation is that more vortices, eddies, and

flow structures can be resolved. That is, LES provides

the “large eddy” flow field. More predicted flow struc-

tures might imply that LES is more suitable to be used to

study more complex phenomena due to turbulence, such

as cycle-to-cycle variability and engine knock. To com-

putationally study these phenomena, accurate prediction

of spatially and temporally evolving turbulent flows is

required. Generally, LES is a more appropriate approach

than RANS for these types of studies. However, LES for

engine sprays is still a relatively new research field com-

pared to RANS and a major issue is discussed in the fol-

lowing.

There are few sub-grid spray models specifically for

LES. Spray droplets are usually several orders of magni-

tude smaller than the computational cell sizes. In LES,

even in direct numerical simulations (DNS) of engine

sprays, it is unlikely to resolve gas phase flow field near

droplet surfaces [3]. Therefore, sub-grid models ac-

counting for interactions of liquid droplets, sub-grid gas

motion, and resolved gas motion may be critical. When

moving towards LES of sprays, the most common ap-

proach is to use LES sub-grid models developed for sin-

gle-phase gas flow, accompanied with spray sub-models

developed and tuned in the RANS framework [1]. Often

only simple extensions are made to the RANS models,

including replacing turbulent kinetic energy and ensem-

ble-averaged velocity by sub-grid kinetic energy and fil-

tered velocity, respectively. This approach can provide

reasonable predictions, e.g., [4], [5], in terms of global

quantities such as spray tip penetrations in some particu-

lar operating conditions. However, as shown in later sec-

tions in this paper, these simple extensions may fail in

many other operating conditions and are not able to give

reasonable predictions of all quantities of interest. A pos-

sible cause is that the RANS Lagrangian spray models

were developed to interact with ensemble-averaged gas

phase models, without considerations of unsteady fluc-

tuations and interactions of different flow length scales.

In addition, the RANS-based spray models have a num-

ber of model constants. These model constants have been

optimized with RANS simulations. However, these opti-

mized constants may not be suitable for LES. Spray

models taking the unique properties of LES into account

are few. To meet the LES expectations mentioned above,

we should continue to explore possible next-generation

LES spray approaches.

The objective of this study is to address this issue by

developing spray submodels consistent with the LES

framework. Particularly, a new atomization and breakup

model, the stochastic KH-RT model, and a new injection

model, the synthetic eddy injection model, for LES of

Diesel sprays were developed. The synthetic eddy injec-

tion model dynamically determines initial velocities of

Lagrangian parcels by superposition of virtual coherent

structures. The stochastic KH-RT model is based on the

classical KH-RT hybrid model but with stochastic time

and length scales of the two breakup mechanisms.

Lagrangian-Eulerian approach

The present study uses the discrete parcel Lagran-

gian-Eulerian approach to simulate two-phase flow

problems [6]. Droplets having the same properties are

grouped into a parcel. Based on the assumption that the

parcel does not occupy any volume with respect to the

gas phase, the parcel can be treated as point sources of

mass, momentum, and energy for the gas phase.

The present study employs the dynamic structure

sub-grid stress tensor model [7] with the addition of a

near nozzle viscosity model to simulate gas phase turbu-

lence. The purpose of adding the near nozzle viscosity

model is to enhance sub-grid mixing in the near-nozzle

region where a portion of energy-containing motions

may not be adequately resolved under moderate grid res-

olution [8].

Injection

In this study, the blob injection method is used [9].

This method assumes that large “blobs” exit the nozzle

position with a diameter equal to the nozzle hole diame-

ter. In addition to the blob size and position, velocity of

a blob must be given as initial conditions to solve the

Lagrangian governing equations. In this section, two dif-

ferent methods to determine the initial velocity are intro-

duced.

Conventional Cone Angle Injection -- In RANS

spray modeling, the spray injection cone angle is speci-

fied. If the injector axis is parallel to the z-axis, the three

components of the injection velocity, 𝑉𝑖𝑛𝑗, are given by

𝑉𝑖𝑛𝑗,𝑧 = 𝑉0,

𝑉𝑖𝑛𝑗,𝑥 = 𝑉0𝑡𝑎𝑛𝜃

2𝑐𝑜𝑠𝜙,

𝑉𝑖𝑛𝑗,𝑦 = 𝑉0𝑡𝑎𝑛𝜃

2𝑠𝑖𝑛𝜙,

(1)

where 𝑉0 is given by a measured rate of injection profile, 𝜃 is assumed to be uniformly distributed between 0 and 𝜃𝑚𝑎𝑥, and 𝜙 is assumed to be uniformly distributed be-tween 0 and 2𝜋. In this study, 𝜃𝑚𝑎𝑥 is the cone angle and is chosen as 15 (deg). This value is within the range of

3

Diesel spray angle measurements in different operating

conditions [10].

The parcel distribution predicted by this specified

spray cone angle method with LES is shown in Error!

Reference source not found.. It can be seen that the

cone angle method predicts the dispersion of spray in an

ensemble-averaged sense, although the gas-phase was

predicted by LES. This is not consistent with the ap-

proach of LES. The goal of LES is to capture unsteady

and three-dimensional structures of the spray. In experi-

mental [11], [12] and high-fidelity CFD images [13] of

sprays, large-scale oscillations of sprays are clearly ob-

served in the near-nozzle region, suggesting that flow

with different wavenumber modes are developed due to

shear instability and/or turbulence inside the nozzle. The

spray angle method perturbs the injection velocity ran-

domly without any time or space correlation. This white-

noise perturbation helps little in terms of developing a

range of wavenumber modes which nonlinearly interact

with each other to enhance turbulent mixing.

Figure 1. Liquid parcels distribution predicted by

the cone angle injection method with LES.

Synthetic Eddy Injection -- The objective of de-veloping a new injection model is to simulate turbulence

at the nozzle outlet and to help LES develop instability

modes not only through the basic equations solved on the

computational mesh but also by artificially perturbing

the injection velocity.

The method adopted here is called the synthetic

eddy method originally developed by Jarrin et al. [14].

This method was used extensively in generating bound-

ary conditions for LES or DNS of single-phase flows.

Here, we employ this method to generate initial velocity

data for Lagrangian parcels. The injection velocity, 𝑉𝑖𝑛𝑗,

is decomposed into mean and fluctuating parts,

𝑉𝑖𝑛𝑗,𝑖 = �̅�𝑖 + 𝑎𝑖𝑗𝑣𝑗′ , (2)

where the tensor 𝑎𝑖𝑗 is obtained from a prescribed Reyn-

olds stress tensor. Here, the Reynolds stresses are for the

flow inside the injector. If the injector axis is parallel to

the z-axis, and the injection position is fixed at the center

of the nozzle outlet, and axisymmetric flow is assumed,

then only the isotropic components of the Reynolds

stress tensor are non-zero at the centerline of the injector.

Thus, the injection velocity can be written as

𝑉𝑖𝑛𝑗,𝑧 = 𝑉0 + √𝑅𝑧𝑧𝑣𝑧′ ,

𝑉𝑖𝑛𝑗,𝑥 = √𝑅𝑥𝑥𝑣𝑥′ ,

(3)

𝑉𝑖𝑛𝑗,𝑦 = √𝑅𝑦𝑦𝑣𝑦′ ,

where 𝑉0 is given by a measured rate of injection profile. The fluctuating signal, 𝑣′, is determined by superposition of coherent structures. Error! Reference source not

found. helps explain the idea of the model for 𝑣′. There are N eddies at positions

𝑝1(𝑡), 𝑝2(𝑡), … , 𝑝𝑘(𝑡), … , 𝑝𝑁−1(𝑡), 𝑝𝑁(𝑡) in a cylinder whose center is located at the injection position, 𝑝𝑖𝑛𝑗 =

(𝑥𝑖𝑛𝑗 , 𝑦𝑖𝑛𝑗 , 𝑧𝑖𝑛𝑗), length is 2𝑙𝑠, and diameter is 2(𝑙𝑠 +

𝑟𝑛𝑜𝑧) where 𝑙𝑠 is a prescribed length scale. The velocity at 𝑝𝑖𝑛𝑗 is the sum of the contributions from the N eddies.

Mathematically, this is written as

𝑣𝑗′(𝑡)

=1

√𝑁∑ 𝜀𝑘𝑗

𝑁

𝑘=1

√𝑉𝐶𝑙𝑠

3𝑓 (

𝑥𝑖𝑛𝑗 − 𝑥𝑘(𝑡)

𝑙𝑠) 𝑓 (

𝑦𝑖𝑛𝑗 − 𝑦𝑘(𝑡)

𝑙𝑠)

∗ 𝑓 (𝑧𝑖𝑛𝑗 − 𝑧𝑘(𝑡)

𝑙𝑠)

(4)

where the index 𝑗 can be for x, y, or z, 𝑉𝐶 is the cylinder volume, and the shape function 𝑓 is given by

𝑓(𝑏) = {√3

2(1 − |𝑏|) 𝑖𝑓 |𝑏| < 1

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

(5)

The function 𝑓 satisfies the normalized condition

∫ 𝑓2(𝑏)𝑑𝑏1

−1

= 1 (6)

so that 𝑣𝑗′ has zero mean and unit variance. Positions of

the 𝑁 turbulent eddies are chosen randomly at 𝑡 = 0 within the cylinder, and these eddies are transported with

the constant speed of 𝑉0 in the z-direction. At time 𝑡 +∆𝑡, the position of an eddy is thus updated by

𝑧𝑘(𝑡 + ∆𝑡) = 𝑧𝑘(𝑡) + 𝑉0∆𝑡 𝑥𝑘(𝑡 + ∆𝑡) = 𝑥𝑘(𝑡) 𝑦𝑘(𝑡 + ∆𝑡) = 𝑦𝑘(𝑡),

(7)

if 𝑧𝑘(𝑡) < 𝑧𝑖𝑛𝑗 + 𝑙𝑠, i.e. the eddy at time 𝑡 is still inside

the cylinder. If 𝑧𝑘(𝑡) ≥ 𝑧𝑖𝑛𝑗 + 𝑙𝑠, the eddy is recycled

back to the inlet face of the cylinder, i.e., 𝑧𝑘(𝑡 + ∆𝑡) =𝑧𝑖𝑛𝑗 − 𝑙𝑠, and 𝑥𝑘(𝑡 + ∆𝑡) and 𝑦𝑘(𝑡 + 𝑑𝑡) are chosen ran-

domly within the inlet face. The symbol 𝜀𝑘𝑗 in Eq. (4) is

the sign of eddy k on component j. Its value is either 1 or

-1, and it is chosen randomly with equal probability

when 𝑡 = 0 and when the eddy 𝑘 is regenerated on the inlet face of the cylinder. The number of eddies, N,

should be chosen to ensure that inlet plane is statistically

covered with synthetic eddies. In the current study, N is

taken to be 100.

The two prescribed model parameters, the Reynolds

stresses tensor, 𝑅𝑖𝑗, and the length scale, 𝑙𝑠, were deter-

mined using data from high-fidelity VoF simulations

[13], [15]. Values of these two parameters are

√𝑅𝑧𝑧 = 0.024𝑉0,

√𝑅𝑥𝑥 = √𝑅𝑦𝑦 = 0.019𝑉0, (8)

4

and

𝑙𝑠 = 11𝑟𝑛𝑜𝑧 . (9) Note that the value of the length scale, 11𝑟𝑛𝑜𝑧 , is

larger than the grid size in practical Diesel spray simula-

tions. For example, in ECN Spray-A simulations, the

typical grid size is 0.25 mm, and the length scale 𝑙𝑠 =11𝑟𝑛𝑜𝑧 = 11 ∗ 0.045 = 0.495 𝑚𝑚. This means that the synthetic structures can be discretized by the LES grid.

Jarrin et al. [16] also suggested that one of the criteria for

choosing the length scale is that the length scale should

be at least larger than the grid size.

Using the synthetic eddy injection method in an LES

simulation, the parcels distribution as shown in Error!

Reference source not found. are significantly different

from that predicted by the spray angle method as shown

in Error! Reference source not found.. Large-scale liq-

uid structures can be seen indicating that this new injec-

tion model is more consistent with LES than the original

RANS based cone-angle model. More quantitative com-

parisons including gas-phase flow structures are in later

sections.

Figure 2. Schematic concept of the synthetic eddy

model.

Figure 3. Liquid parcels distribution predicted by

the synthetic eddy injection method.

Atomization and Breakup

The hybrid Kelvin-Helmholtz/Rayleigh-Taylor

(KH-RT) breakup model developed two decades ago

[17] has been widely used in industry and implemented

into a number of commercial CFD codes. Formulation

and limitations of this model and an improved version

are discussed in the following sections.

Standard KH-RT -- The KH breakup and the RT

breakup stand for two different breakup mechanisms.

The KH breakup predicts the wave growth at a gas-liquid

interphase due to the interaction of aerodynamic drag,

surface tension force, and liquid inertia. The RT breakup

mechanism accounts for wave on a droplet induced by

acceleration normal to the interface between two fluids

of different densities. The most unstable wave length and

frequency of each of these two mechanisms can be de-

rived analytically by performing linear stability analysis

[9], [18]. Then the length and the time scales are incor-

porated into the model parameters used to determine the

droplet size after breakup and the breakup time. Detailed

procedure of the model can be found in (Beale and Reitz

1999).

The KH-RT model is widely used in RANS simula-

tions, and in recent years it has been employed in LES

spray simulations, e.g. [19]–[21]. A common issue of the

KH-RT model used in both RANS and LES is that sim-

ulation results heavily depend on the choice of the model

constants. As pointed out by Reitz [9] and Beale and

Reitz [17], the choice of the value of the KH time con-

stant, 𝐵1, depends on nozzle flow conditions, but there is no guidance of how to relate the value of the constant to

the nozzle flow conditions. The model constants may

also depend on fuel. Brakora et al. [22] showed that to

improve the prediction of biodiesel sprays, the values of

𝐵1, the RT size constant, 𝐶𝑅𝑇, and the breakup length constant, 𝐶𝑏, need to be adjusted from the standard by a factor of 2. The choice of the value of 𝐵1 may also vary with grid resolution. Senecal et al. [4] used a value of 7

for 𝐵1 in a very refined mesh (∆𝑥 = 0.0625 𝑚𝑚). Fu-jimoto et al. [23] studied the effect of 𝐵1 on the predic-tion of spray tip penetrations and droplet size, and found

that larger value of 𝐵1 predicts longer liquid penetration and larger Sauter mean diameter (SMD). Tsang and Rut-

land [24] studied the effect of KH size constant, 𝐵0, Kh time constant, 𝐵1, and breakup length constant, 𝐶𝑏. It was found that the momentum exchange between liquid

and gas is sensitive to 𝐵1. Smaller value of 𝐵1 predicts larger momentum transfer from liquid to gas, resulting in

longer vapor penetration. It was also found that the pre-

diction of liquid penetration is sensitive to the constant,

𝐶𝑏. Smaller 𝐶𝑏 predicts shorter liquid penetration. Perini and Reitz [25] performed the multi-objective optimiza-

tion to find the best set of the model constants which can

give reasonable predictions of liquid penetration, vapor

penetration, and vapor mass distribution under the En-

gine Combustion Network “Spray-A” conditions [26]. It

5

was found that a value giving good prediction of an ob-

jective is not an optimal value for another objective. For

instance, an optimal value of the RT time constant, 𝐶𝜏, to predict vapor penetration and fuel vapor distribution well

is ~0.1, but the optimal value for liquid penetration is

2.39.

These studies suggest that the KH-RT model con-

stants are not universal. An optimal set of the model con-

stants may only be effective for a specific flow configu-

ration, a specific mesh resolution, or a specific simula-

tion output. This statement is further justified in the sim-

ulation results shown in this paper. Thus, the model pre-

dictive capability is limited. The KH-RT model only pre-

dicts atomization and breakup due to wave instability.

Other possible breakup mechanisms such as turbulence

[27] and nozzle cavitation [28], [29] are neglected. This

may be one of the reasons why predictions are highly

sensitive to the model constants.

Stochastic KH-RT -- The central idea of the sto-

chastic KH-RT model is that the model parameters are

determined stochastically and dynamically according to

local flow conditions. The model parameters include i)

KH breakup time, 𝜏𝐾𝐻, ii) Breakup length, 𝐿𝑏, iii) RT breakup time, 𝜏𝑅𝑇 , and iv) RT wavelength, 𝜆𝑅𝑇 . That is, even if two parcels have the exact same conditions (We-

ber number, Ohnesorge number, normal acceleration,

etc.), they may still have different breakup characteris-

tics due to the stochastic model. This is an attempt to re-

duce the sensitivity of the model constants partly result-

ing from the incapability of predicting breakup mecha-

nisms other than wave instabilities

The four parameters mentioned above are all func-

tions of the droplet radius. Instead of using a fixed value

of the droplet radius, the droplet radius in the equations

of the four parameters are determined stochastically by

time-dependent probability density functions (PDFs).

The KH breakup time is written as

𝜏𝐾𝐻,𝑠𝑡𝑜 = 3.726𝐵1𝑟𝑑,𝑠𝑡𝑜𝐾𝐻 𝛬𝐾𝐻𝛺𝐾𝐻⁄ , (10) where 𝛺𝐾𝐻 is the fastest KH growth rate, and 𝛬𝐾𝐻 is the corresponding wavelength, derived from linear stability

analysis. The rate equation for the parent droplet size, 𝑟𝑑, becomes

𝑑𝑟𝑑𝑑𝑡

= −(𝑟𝑑 − 𝑎) 𝜏𝐾𝐻,𝑠𝑡𝑜⁄ , (11)

where 𝑎 is the child droplet radius assumed to be linearly proportional to 𝛬𝐾𝐻. By assuming inviscid flow and in-finite Weber number, the breakup length can be written

as [17]

𝐿𝑏,𝑠𝑡𝑜 = 𝑈𝑟𝑒𝑙𝜏𝐾𝐻,𝑠𝑡𝑜

=1

√𝜋𝐵1

𝑟𝑑,𝑠𝑡𝑜𝐾𝐻𝑟𝑑

√𝐴𝑛𝑜𝑧𝜌𝑑𝜌𝑐

, (12)

where 𝑈𝑟𝑒𝑙 is the relative velocity magnitude between liquid and gas, 𝐴𝑛𝑜𝑧 the nozzle cross-sectional area, 𝜌𝑑 the droplet density, and 𝜌𝑐 the gas density. The form of

the PDF for 𝑟𝑑,𝑠𝑡𝑜𝐾𝐻 refers to the stochastic breakup model developed by Apte et al. [30]. It follows Kolmo-

gorov’s hypothesis, which is that the probability to break

each parent particle into a given number of fragments is

independent of the parent particle size, and a differential

Fokker-Planck equation for the PDF of droplet radii can

be derived. The solution of the differential equation is a

log-normal distribution function in the form of, 𝑓𝐾𝐻(𝑟𝑑,𝑠𝑡𝑜𝐾𝐻 , 𝑡𝑟,𝐾𝐻)＝

1

𝑟𝑑,𝑠𝑡𝑜𝐾𝐻√2𝜋〈𝜉𝐾𝐻2 〉𝜈𝐾𝐻𝑡𝑟,𝐾𝐻

∗

𝑒𝑥𝑝 [−(𝑙𝑛𝑟𝑑,𝑠𝑡𝑜𝐾𝐻 − 𝑙𝑛𝑟𝑑,0 − 〈𝜉𝐾𝐻〉𝜈𝐾𝐻𝑡𝑟,𝐾𝐻)

2

2〈𝜉𝐾𝐻2 〉𝜈𝐾𝐻𝑡𝑟,𝐾𝐻

],

(13)

if 𝑡𝑟,𝐾𝐻 > 0. If 𝑡𝑟,𝐾𝐻 = 0, 𝑟𝑑,𝑠𝑡𝑜𝐾𝐻 = 𝑟𝑑. The KH breakup frequency, 𝜈𝐾𝐻 , is the reciprocal of the stochas-tic KH time, 𝜏𝐾𝐻,𝑠𝑡𝑜. The characteristic time, 𝑡𝑟,𝐾𝐻, for a parcel is tracked and is set or reset to zero if certain con-

ditions are met. More details of the conditions can be

found in [31].

The two shape parameters,〈𝜉𝐾𝐻2 〉𝜈𝐾𝐻𝑡𝑟,𝐾𝐻 and

〈𝜉𝐾𝐻〉𝜈𝐾𝐻𝑡𝑟,𝐾𝐻 in Eq. (13), determine the shape of the distribution. By definition, 〈𝜉𝐾𝐻〉 and 〈𝜉𝐾𝐻

2 〉 are the first two moments of 𝜉𝐾𝐻 where 𝜉𝐾𝐻 is the logarithm of the ratio of the size of a child droplet to the size of a parent

droplet. These two parameters are determined dynami-

cally according to local flow conditions and fuel proper-

ties. Apte et al. came up with an assumption and related

the droplets breakup process to Einstein’s theory of

Brownian motion to determine these two parameters dy-

namically. The assumption is that, in the intermediate

range of scales between large droplets with large Weber

numbers and the maximum stable droplets with a critical

Weber number, there is no preferred length scale. This is

analogous to the inertial range of homogenous turbu-

lence. Thus, by assuming the relationship of

𝑈𝑟𝑒𝑙3

Λ𝐾𝐻 ~

𝑈𝑟𝑒𝑙,𝑐𝑟3

Λ𝐾𝐻,𝑐𝑟, (14)

one can obtain

𝑟𝑑,𝑐𝑟𝑟𝑑

~ (Λ𝐾𝐻

Λ𝐾𝐻,𝑐𝑟)

2/3

(𝑊𝑒𝑐𝑟𝑊𝑒𝑔

), (15)

where 𝑈𝑟𝑒𝑙,𝑐𝑟 is the magnitude of relative velocity at which disruptive forces are balanced by capillary forces

(similar to the turbulent velocity scale of the smallest ed-

dies), 𝑟𝑑,𝑐𝑟 = 𝑊𝑒𝑐𝑟𝜎 𝜌𝑐𝑈𝑟𝑒𝑙2⁄ is the maximum stable

droplet radius where 𝑊𝑒𝑐𝑟 is the critical Weber number taken to be a value of 6, 𝜎 the surface tension coefficient, Λ𝐾𝐻,𝑐𝑟 is the critical KH wavelength written as

𝛬𝐾𝐻,𝑐𝑟

=9.02𝑟𝑑(1 + 0.45𝑂ℎ

0.5)(1 + 0.4𝑇𝑎𝑐𝑟0.7)

(1 + 0.87𝑊𝑒𝑐𝑟1.67)0.6

(16)

6

where 𝑇𝑎𝑐𝑟 = 𝑂ℎ𝑊𝑒𝑐𝑟0.5 where 𝑂ℎ is the Ohnesorge

number. Taking the logarithm of Eq. (15), the model for

〈𝜉𝐾𝐻〉 can be written as 〈𝜉𝐾𝐻〉

=2

3𝑙𝑛 (

Λ𝐾𝐻Λ𝐾𝐻,𝑐𝑟

) + 𝐾1,𝐾𝐻𝑙𝑛 (𝑊𝑒𝑐𝑟𝑊𝑒𝑔

), (17)

where 𝐾1,𝐾𝐻 is a model constant. The model for the sec-ond moment of 𝜉𝐾𝐻 proposed by Apte et al. based on Einstein’s theory of Brownian motion is written as

〈𝜉𝐾𝐻2 〉 = −

〈𝜉𝐾𝐻〉

𝐾2,𝐾𝐻𝑙𝑛 (𝑟𝑑

𝑟𝑑,𝑐𝑟)

(18)

where 𝐾2,𝐾𝐻 is another model constant.

The stochastic RT breakup time is written as

𝜏𝑅𝑇,𝑠𝑡𝑜 =𝐶𝜏

𝛺𝑅𝑇,𝑠𝑡𝑜 (19)

where 𝛺𝑅𝑇,𝑠𝑡𝑜 is given by

𝛺𝑅𝑇,𝑠𝑡𝑜 = √2

3√3𝜎

[−𝑔𝑡,𝑠𝑡𝑜(𝜌𝑑 − 𝜌𝑐)]3 2⁄

𝜌𝑑 + 𝜌𝑐 (20)

where

𝑔𝑡,𝑠𝑡𝑜 = 𝑔𝑡 (𝑟𝑑

𝑟𝑑,𝑠𝑡𝑜𝑅𝑇)

3

. (21)

The stochastic RT wavelength is given by

𝜆𝑅𝑇,𝑠𝑡𝑜 =2𝜋𝐶𝑅𝑇𝐾𝑅𝑇,𝑠𝑡𝑜

, (22)

where the stochastic RT wave number is

𝐾𝑅𝑇,𝑠𝑡𝑜 = √−𝑔𝑡,𝑠𝑡𝑜(𝜌𝑑 − 𝜌𝑐)

3𝜎. (23)

Analogous to the log-normal distribution for

𝑟𝑑,𝑠𝑡𝑜𝐾𝐻, the stochastic radius, 𝑟𝑑,𝑠𝑡𝑜𝑅𝑇 , is also randomly chosen from the log-normal distribution,

𝑓𝑅𝑇(𝑟𝑑,𝑠𝑡𝑜𝑅𝑇 , 𝑡𝑟,𝑅𝑇)＝ 1

𝑟𝑑,𝑠𝑡𝑜𝑅𝑇√2𝜋〈𝜉𝑅𝑇2 〉𝜈𝑅𝑇𝑡𝑟,𝑅𝑇

∗ 𝑒𝑥𝑝 [−(𝑙𝑛𝑟𝑑,𝑠𝑡𝑜𝑅𝑇 − 𝑙𝑛𝑟𝑑,0 − 〈𝜉𝑅𝑇〉𝜈𝑅𝑇𝑡𝑟,𝑅𝑇)

2

2〈𝜉𝑅𝑇2 〉𝜈𝑅𝑇𝑡𝑟,𝑅𝑇

],

(24)

if 𝑡𝑟,𝑅𝑇 > 0. If 𝑡𝑟,𝑅𝑇 = 0, 𝑟𝑑,𝑠𝑡𝑜𝑅𝑇 = 𝑟𝑑. The RT breakup frequency, 𝜈𝑅𝑇 , is the reciprocal of the stochastic RT breakup time, 𝜏𝑅𝑇,𝑠𝑡𝑜. Analogous to the KH shape pa-rameters, the RT shaper parameters, 〈𝜉𝑅𝑇〉 and 〈𝜉𝑅𝑇

2 〉 are respectively given by

〈𝜉𝑅𝑇〉 =2

3𝑙𝑛 (

𝜆𝑅𝑇𝜆𝑅𝑇,𝑐𝑟

) + 𝐾1,𝑅𝑇𝑙𝑛 (𝑊𝑒𝑐𝑟𝑊𝑒𝑔

), (25)

and

〈𝜉𝑅𝑇2 〉 = −

〈𝜉𝑅𝑇〉

𝐾2,𝑅𝑇𝑙𝑛 (𝑟𝑑

𝑟𝑑,𝑐𝑟)

, (26)

where the critical RT wavelength is given by [18]

𝜆𝑅𝑇,𝑐𝑟 = 2𝜋 (𝜎

−𝑔𝑡,𝑠𝑡𝑜(𝜌𝑑 − 𝜌𝑐))

1 2⁄

, (27)

and 𝐾1,𝑅𝑇 and 𝐾2,𝑅𝑇 are model constants. The model constants used in the standard and stochas-

tic KH-RT models for the remainder of the current study

are listed in Table 1 and Table 2. Note that although there

are 4 more model constants of the stochastic model, this

set of constants is expected to be more universal and are

not changed from one operating condition to another.

Table 1. Standard KH-RT model constants.

KH size constant, 𝐵0 0.61

KH time constant, 𝐵1 40

RT size constant, 𝐶𝑅𝑇 0.1

RT time constant, 𝐶𝜏 1.0 Breakup length constant, 𝐶𝑏 2.0

Table 2. Stochastic KH-RT model constants.

KH size constant, 𝐵0 0.61

KH time constant, 𝐵1 80

RT size constant, 𝐶𝑅𝑇 0.1

RT time constant, 𝐶𝜏 1.0

𝐾1,𝐾𝐻 1.0

𝐾2,𝐾𝐻 0.007

𝐾1,𝑅𝑇 1.0

𝐾2,𝑅𝑇 0.008

Results and Discussion

The injection and atomization and breakup models

were systematically validated against two experimental

datasets, the Engine Combustion Network (ECN) sprays

[26] and the ERC optical engine sprays [32]. A range of

operating conditions are used for model validation, as

shown in Table 3. The ERC optical engine conditions are

listed in Table 4. Four cases using different combinations

of the breakup and the injection models are evaluated, as

listed in Table 5.

7

Table 3. Selected experimental conditions of ECN sprays for model validations.

Case name Temperature

(K)

Density

(kg/m3)

Nozzle diameter

(mm)

Injection

pressure

(MPa)

Fuel Oxygen (%)

Non-vap A 300 22.8 0.09 150 C12H26 0

Vap A 900 22.8 0.09 150 C12H26 0

Vap A,

High T and

low den.

1400 7.6 0.09 150 C12H26 0

Vap A, low

inj. P 900 22.8 0.09 50 C12H26 0

Vap A,

low T 700 22.8 0.09 150 C12H26 0

Vap H 1000 14.8 0.1 150 C7H16 0

Table 4. ERC optical engine specification.

Engine type Single-cylinder optically accessible

Injector Bosch Crin 2 six-hole

Fuel type #2 Diesel

Bore (mm) 82.55

Stroke (mm) 76.2

Bowl depth (mm) 6.12

Bowl diameter (mm) 65.18

Clearance (mm) 1.04

Compression ratio 14.03

Engine speed (rpm) 1200

Cylinder wall temperature (℃) 50 Start of injection (SOI) timing -5 ATDC

Injection duration (𝜇s) 1500 Injector nozzle diameter (mm) 0.11

Angle of the plume off the horizontal plane (degs) 13

Mean temperature at ROI (°C) 945.1±11.4

Table 5. Test matrix to evaluate breakup and injection

models.

Atomization and

breakup

Injection

Case 1 Standard KH-RT Cone angle

Case 2 Stochastic KH-RT Cone angle

Case 3 Standard KH-RT Synthetic eddy

Case 4 Stochastic KH-RT Synthetic eddy

Simulated spatial distributions of liquid mass are com-

pared against the X-ray ECN data [33] for non-vaporiz-

ing sprays (Non-vap A in Table 3) as shown in Figure 4

to Figure 6. Mesh size was 0.25 mm. The experimental

results were time- and ensemble-averaged. Simulation

results were time-averaged and spatial-averaged in the

azimuthal direction. The duration of the time-averaging,

0.4 ms to 1.2 ms ASOI, is the same in the simulations

and in the experiments. Comparing Case 1 and Case 2,

the standard KH-RT and the stochastic KH-RT models

have similar predictions in the upstream region. Both

cases under predict the PMDs at y = 0 before z = 6 mm.

Moving further downstream, a notable difference be-

tween the PMD profiles at y = 0 predicted by the stand-

ard KH-RT and those predicted by the stochastic KH-RT

can be seen. The experimental data shows that the PMD

at y = 0 decreases with increased axial distance, as shown

in Figure 4, but the standard KH-RT model was not able

to capture this trend. The PMDs at y = 0 predicted by the

standard KH-RT model remain almost constant beyond

z ≈ 6 mm. This issue was also reported in the work done by [34] using the same injection model and the breakup

8

model as Case 1. On the other hand, the stochastic KH-

RT model was able to capture the trend although the

PMDs were still over-predicted downstream.

Comparing Case 1 and Case 3, Case 1 predicted larger

spray angle than that predicted by Case 3, as shown in

the contour plots. Also, the injection model plays a more

important role in the prediction of the PMDs upstream.

Before 6 mm, the synthetic eddy model gives much bet-

ter prediction than the cone angle injection model. In

fact, the PMD prediction is sensitive to the prescribed

cone angle in the cone angle injection model. Smaller

prescribed cone angle predicts higher PMD at y = 0, and

vice versa. It was found that reducing the prescribed cone

angle from 15° (current value) to 10° did improve the PMD prediction in the upstream region. However, the

PMD prediction beyond 𝑧 ≈ 6 𝑚𝑚 was even more over-predicted, and the liquid penetration in the vaporizing

sprays was over-predicted as well. Thus, simply adjust-

ing the prescribed cone angle cannot improve the predic-

tions of multiple quantities simultaneously.

As shown in Figure 6(a), the synthetic eddy model im-

proves the lateral PMD profile upstream (z = 5 mm). In

the downstream region at z = 10 mm, the width of the

lateral profile is under-predicted (Figure 6(b)) by Case 4,

suggesting that the spray is less dispersed. In fact, the

prediction of the width of the lateral PMD profiles in the

downstream region not only depends on the breakup and

the injection model but the SGS dispersion model. More

discussion on the effect of the SGS dispersion model on

the prediction of PMD profiles can be found in [35].

Nevertheless, overall Case 4 significantly improves the

prediction of the spatial distribution of liquid mass. The

synthetic eddy injection model improves the prediction

in the upstream region, and the stochastic KH-RT model

improves the prediction in the downstream region.

Figure 4. Mean liquid projected mass density

(PMD) contour plots predicted by Case 1, Case 2, Case

3, and Case 4, and measured by the ECN X-ray non-va-

porizing spray experiment.

Figure 5. Comparison of the mean PMD profiles at

y = 0 predicted by the four cases against the ECN X-ray

data.

9

(a)

(b)

Figure 6. Mean lateral PMD profiles at (a) z =

5mm and (b) z = 10 mm. The line labels are the same as

those in Figure 5.

The next dataset is the ECN vaporing spray A. Liquid

and vapor penetrations are compared. The performance

of the models in predicting the quasi-steady liquid pene-

trations under different operating conditions of the ECN

vaporizing sprays can be quantified by computing the er-

ror defined as

𝑒𝑟𝑟𝑜𝑟 (%) =�̅�𝑠𝑖𝑚 − �̅�𝑒𝑥𝑝

�̅�𝑒𝑥𝑝× 100% (28)

where the overbar means the time-averaging performed

once the liquid penetration becomes quasi-steady. The

errors are shown in Figure 7. The experimental quasi-

steady liquid penetration values are also shown in the

figure. Compared to the standard vaporizing Spray A

case, the high temperature and low density case shows

longer liquid penetration. This is because the drag force

is smaller due to the lower air density although the air

carries higher energy to vaporize the fuel. The lower

temperature case has the longest penetration among

these five operating conditions due to the slower vapori-

zation rate. The penetration of the lower injection pres-

sure case is close to that of the standard case. This occurs

because both air entrainment rate and liquid fuel flow

rate depend linearly on the injection velocity [36]. As a

result, the lower injection pressure and higher injection

pressure cases require the same liquid length needed to

entrain enough air to vaporize the fuel [37]. The Spray-

H penetration is slightly shorter than that of the standard

Spray-A case since the Spray-H fuel, n-heptane, has

lower heat of vaporization than the Spray-A fuel, n-do-

decane, and the ambient temperature is higher in Spray-

H. Droplet size distribution would affect air entrainment

rate and vaporization rate which are important factors in

determining the liquid length, so a good atomization and

breakup model is needed to predict reasonable liquid

lengths.

In general, the errors from the stochastic KH-RT

model are much smaller than those from the standard

KH-RT model. Except for the high temperature and low

density case, the errors from Case 2 and Case 4 are less

than 2%. On the other hand, Case 1 using the standard

KH-RT over predicted the liquid penetrations by more

than 25 % in the low temperature and the low injection

pressure cases. To achieve better predictions of these two

cases using the standard KH-RT model, it is unavoidable

to tune the breakup model constants to enhance breakup

rate and thus reduce the liquid penetrations. Error! Ref-

erence source not found. shows that the stochastic KH-

RT model is more capable of predicting the liquid pene-

trations reasonably well in a range of operating condi-

tions without tuning the model constants.

Figure 7. Errors of the quasi-steady liquid penetra-

tion predictions of the ECN vaporizing sprays under

different operating conditions: (a) Spray A, (b) Spray A

high T and low Den, (c) Spray A low T, (d) Spray A

low inj. P, and (e) Spray H. The horizontal labels repre-

sent the different model setups listed in Table 5. The

experimental values of the quasi-steady liquid penetra-

tions are also shown in the figures.

10

Error! Reference source not found. shows the “Vap

A” mass-weighted and spatially averaged mean and root-

mean-square (RMS) values of the droplet diameter ver-

sus distance from the injector. Three distinct slopes can

be seen in Figure 8. The three distinct slopes imply dif-

ferent breakup mechanisms from upstream to down-

stream of the spray. In the near-nozzle region, the droplet

size decreases gradually with increased distance. The

KH breakup is the primary mechanism in this region. In

the breakup model, parent parcels become smaller, and

child droplets are formed with the diameter proportional

to the KH wavelength. The RMS values of the droplet

diameter are larger predicted by the stochastic model

since the KH breakup time used in the rate equation (Eq.

(11)) is stochastically sampled from the log-normal dis-

tribution. The mean values of the droplet diameter start

to decrease more dramatically at certain axial distances

which are identified as the breakup length. Beyond the

breakup length, the RT breakup, which tends to produce

small drops, is activated. As shown in the figure, the

breakup length of the “Vap A” case is approximately

5𝑚𝑚. In this secondary breakup region, the decrease in the mean droplet size with increasing distance predicted

by the stochastic model is more gradual than that pre-

dicted by the standard model, and the RMS values are

larger. The more gradual decrease is due to the stochastic

breakup length (Eq. (12)), which is determined not only

by the fluid density ratio but the PDF with the shape pa-

rameters dependent on local flow conditions. The larger

RMS values of the droplet size are due to the stochastic

RT wavelength and the stochastic RT breakup time. Near

the end of the liquid sprays, the droplet size becomes

more stable since in this region the liquid slip velocity is

not large enough to further break up drops through either

of the mechanisms. In general, the stochastic KH-RT

model, attempting to account for breakup mechanisms

other than the primary wave instabilities such as turbu-

lence, predicts a wider range of the droplet size distribu-

tion due to the stochastic model parameters. This may be

one reason why the stochastic model gives better predic-

tions of the liquid penetrations in a range of operating

conditions (see Figure 7).

Figure 8. “Vap A” mass-weighted and spatially

averaged mean and RMS values of the droplet diameter

versus distance at 1.5 ms.

Figure 9 compares the predicted vapor penetrations

against the ECN experimental data under the different

operating conditions detailed in Table 3. The vapor pen-

etration in simulations is defined as the maximum dis-

tance from an injector where the fuel vapor mass fraction

is 0.1 %. As shown in the figures, the injection model has

bigger effect than the breakup model on predicting the

vapor penetrations, particularly at early times. Within

0.25 ms, the tip penetration speeds (slopes of the vapor

penetrations) are over predicted by the cone angle injec-

tion model. The issue of the over-predicted vapor pene-

trations at early times (within 1 ms) in LES of engine

sprays has been reported in many studies [4], [20], [38]–

[42]. The primary cause is that the turbulent diffusion

due to interactions of different instability modes is not

well-captured. One solution is to increase grid resolution

with higher-order numerical schemes so that flow struc-

tures in the near-nozzle region can be better-resolved,

leading to an earlier development of the turbulent diffu-

sion and shorter vapor penetrations. However, reducing

grid size by 1/2 in each side of cells may increase com-

putational time by approximately 16 times (number of

cells increases by 8 times and the time step size needs to

reduce by 1/2 to keep the same Courant number). To

keep a reasonable computational cost while having rea-

sonable predictions of the vapor penetration, the syn-

thetic eddy injection model is an alternative. As shown

in the figures, the synthetic eddy injection model im-

proves the predictions of the tip penetration speeds at

early times of fuel injection. The model helps develop

flow structures faster by artificially providing time- and

space-correlated injection velocity data.

11

(a)

(b)

(c)

(d)

Figure 9. Comparison of the (a) “Vap A”, (b) “Vap

A high T and low den”, (c) “Vap A low inj P”, and (d)

“Vap H” vapor penetrations predicted by the different

spray models against the ECN experimental data. The

shaded area is measured mean uncertainty.

Figure 10 compares the vapor penetrations predicted

by Case 1 and Case 4 using the different mesh resolu-

tions. The vertical dashed lines in these figures associate

with the times at which the tip penetration speeds start to

decrease due to the formation of small-scale flow struc-

tures downstream of the jet. The vertical lines move to

the left as the mesh is refined, which is expected since

flow structures are developed faster with the finer

meshes. Also, with the 0.25 mm and the 0.5 mm meshes,

Case 4 predicted earlier developments of flow structures

and improved the predictions of the vapor penetrations.

As shown in Figure 11 of fuel vapor contours, more fine

structures can be seen in the 0.125 mm mesh, as ex-

pected. For the coarser meshes, 0.5 mm and 0.25 mm,

Case 4 predicts larger jet angle, suggesting that turbulent

diffusion is better represented. Overall, the vapor pene-

trations predicted by Case 4 are less grid-sensitive.

12

(a)

(b)

Figure 10. Vapor penetrations of the “Vap A” case

predicted by (a) Case 1 and (b) Case 4 using different

grid resolutions. The penetration speeds reduce at cer-

tain times indicated by the blue dashed lines, due to the

development of flow structures downstream of the jet.

Figure 11. Fuel vapor contour plots of the “Vap

A” case at 1.0 ms predicted by Case 1 and Case 4 using

the three different grid resolutions, 0.5 mm, 0.25 mm,

and 0.125 mm.

Figure 12 compares side views of the fuel vapor contours

of one of the six plumes in the ERC optical engine. The

fuel vapor mass distributions predicted by the standard

KH-RT model (Case 1 and Case 3) and the stochastic

KH-RT model (Case 2 and Case 4) are remarkably dif-

ferent. At 139 𝜇𝑠, the standard KH-RT model predicts two vapor regions. There are gaps with very low fuel va-

por mass fractions between the two regions with higher

mass fractions. The location of the vapor region down-

stream corresponds to the deterministic breakup length

(~ 10 mm). Beyond the breakup length, vaporization rate

is faster due to the RT breakup. In the stochastic KH-RT

model transition from the primary breakup to the second-

ary breakup does not occur at a fixed breakup length, re-

sulting in the smoother centerline mass fraction profiles.

Also, the stochastic KH-RT model predicts higher mass

fractions in the near-nozzle region.

13

Figure 12. Fuel vapor mass fraction contours of the injection pressure 52 MPa and ambient density 15 kg/m3 case

predicted by Cases 1, 2, 3, and 4 at three different times after start of injection. The locations of the small vapor regions

predicted by the standard KH-RT model at 𝟏𝟑𝟗 𝝁𝒔 correspond to the location where the RT breakup is activated.

Figure 13 compares the predicted liquid and vapor

penetrations against the experimental data. The meas-

ured liquid penetration becomes quasi-steady at approx-

imately 250 𝜇𝑠. This time was accurately predicted by Case 2 and Case 4 using the stochastic KH-RT model.

This good prediction also means that the air entrainment

due to the fuel injection was captured reasonably well.

On the other hand, the standard KH-RT model predicted

overshoots of the liquid penetrations. In addition, the liq-

uid penetrations predicted by Case1 show larger standard

deviations, indicating that the predictions are more grid-

sensitive. Overall, Case 4 gives the best prediction of the

liquid penetration. The vapor penetrations predicted by

Case 2 and Case 4 before 150 𝜇𝑠 match well with the data. The sharp increases in the vapor penetrations pre-

dicted by the standard KH-RT model from ~5 mm at 100

𝜇𝑠 to ~10 mm at 139 𝜇𝑠 correspond to the small vapor regions that formed downstream as shown in the contour

plots in Figure 12.

(a)

(b)

Figure 13. (a) Liquid and (b) vapor penetrations of

the injection pressure 52 MPa and ambient density 15

kg/m3case. The symbols and the error bars represent

means and standard deviations of the liquid and vapor

penetrations from the six plumes. The shaded area is the

experimental 95 % confidence interval band.

Summary and Conclusions

Two new LES specific Diesel spray models were

developed. They are the synthetic eddy injection model

and the stochastic KH-RT breakup model. Key ideas and

features of the two models are summarized as follows.

The synthetic eddy injection model provides time-

and space-correlated initial velocities for injected La-

grangian parcels. The model simulates turbulence at the

nozzle outlet and helps LES develop instability modes

not only through the basic equations solved on the com-

putational mesh but also by artificially, but coherently,

perturbing the injection velocity. The injection velocity

is modeled by the superposition of coherent structures

characterized by shape functions dependent on the struc-

ture positions and a prescribed length scale. The two

model parameters, the Reynolds stresses and the length

scale, were derived from the high-fidelity VoF spray

simulations.

The stochastic KH-RT model predicts the atomiza-

tion and breakup processes of Diesel sprays. The model

attempts to reduce the sensitivity of the model constants

in the original KH-RT model. The sensitivity partly re-

sults from the limitation of predicting breakup mecha-

nisms only from wave instabilities. Compared to the

original KH-RT model, the stochastic KH-RT model de-

termines time and length scales of the growth of instabil-

ity waves stochastically and dynamically. All these

scales are functions of the stochastic droplet size, which

is sampled from a log-normal distributions with shape

parameters determined from local flow conditions.

The models were systematically tested using the two

experimental datasets, the ECN constant-volume sprays

database and the ERC optical engine sprays database. It

was found that the predictions of near-nozzle flow fields

are sensitive to the spray injection model. In the non-va-

porizing Spray-A case, the prediction of the near-nozzle

liquid PMD profiles was greatly improved by the syn-

thetic eddy injection model. The cone-angle injection

model significantly under-predicted the PMDs at y = 0

in the upstream region. In the vaporizing ECN spray

cases, the synthetic eddy injection model improved the

vapor penetration predictions at early times of spray in-

jection under a range of operating conditions. Also, it

14

was found that the gas-phase solutions predicted by the

synthetic eddy injection model are less sensitive to grid

resolutions.

The major advantage of the stochastic KH-RT

model over the standard KH-RT model is that it is able

to give reasonable predictions of the liquid penetrations

in a wide range of operating conditions without tuning

the model constants. The stochastic KH-RT model can

correctly capture the liquid penetration trends varying

with injection pressures, ambient densities, and ambient

temperatures. Results showed that the stochastic KH-RT

model predicted a wider range of the droplet sizes and a

smoother transition from the primary KH breakup to the

secondary RT breakup.

Overall, compared to the commonly used cone-an-

gle injection and the standard KH-RT breakup models,

the synthetic eddy model and the stochastic KH-RT

breakup model together improve the LES predictability

of Diesel sprays. Quantities such as liquid penetration,

vapor penetration, spatial distribution of liquid fuel and

fuel vapor were well-captured.

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