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ILASS-Americas 29th Annual Conference on Liquid Atomization and
Spray Systems, Atlanta, GA, May 2017
Effects of Sub-Grid Scale (SGS) Dispersion Modeling on
Large-Eddy Simulation (LES) of Non-Evaporative and Evaporative
Diesel Sprays
C.-W. Tsang* and C. J. Rutland
Department of Mechanical Engineering University of
Wisconsin-Madison
1500 Engineering Drive, Madison, WI 53706, USA
Abstract Performance of the sub-grid models accounting for the
effects of unresolved motions on Diesel spray dispersion was
tested. The models predict the SGS dispersion velocity used for
calculating the slip velocity in Lagrangian-Eulerian LES spray
models. Discussion was made on the effects of different model
formulations and model constants on the simulations of the
constant-volume Engine Combustion Network non-evaporative and
evaporative “Spray-A”. It was found that the SGS dispersion models
have profound impact on the prediction of the spatial distribution
of liq-uid mass. Neglecting the SGS dispersion model results in the
under-predicted width of the lateral projected liquid mass density
profiles. Also, the prediction of the projected liquid mass density
is sensitive to the two model con-stants determining the SGS
dispersion velocity magnitude and turbulence time scale. On the
other hand, the predic-tions of gas-phase velocity profiles, fuel
vapor mass fraction profiles, vortex structures, and liquid
penetration are insensitive to different SGS dispersion model
setups. The primary reason for this is that the motion of
high-momentum liquid blobs in the near-nozzle region leading to air
entrainment and subsequent gas jet development is little influenced
by the SGS dispersion. The SGS dispersion is more critical in
determining the motion of droplets having small inertia.
*Corresponding author: [email protected]
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Introduction
In internal combustion engines, fuel injection pro-cess
determines the mixing of fuel vapor and air, which greatly impact
engine efficiency and emissions. Turbu-lent dispersion of fuel
droplets is one of many im-portant physical mechanisms influencing
the fuel-air mixing process. The existence of droplets may change
the characteristics of turbulent flows, e.g., change of the
turbulent kinetic energy spectrum. In turn, a range of length and
time scales of turbulent flow may have important effects on the
dynamics of droplets. Funda-mental studies on spray-turbulence
interaction are ex-tensively carried out by direct numerical
simulations (DNS) and experiments in simplified flow
configura-tions. Balachandar and Eaton [1] reviewed some im-portant
aspects such as preferential concentration of droplets and
turbulence modulation from the funda-mental DNS and experimental
studies.
Large-eddy simulation (LES) has become a popu-lar approach in
practical engine spray simulations due to the increase of
computational power. The major ad-vantage of LES over the
conventional Reynolds-averaged Navier-Stokes (RANS) approach is
that it can capture more unsteady flow structures, which allows us
to study new phenomena such as cycle-to-cycle varia-tions [2].
Unlike DNS resolving flow length scales down to the Kolmogorov
scales, LES only resolves filtered flow length scales. This means
that the effect of flow length scales below the filter size on
spray dy-namics needs to be modelled. This model is commonly called
a sub-grid scale (SGS) dispersion model.
A number of SGS dispersion models have been proposed and tested
in non-engine flows. They can be categorized into stochastic and
deterministic models. For the stochastic models, Bini and Jones [3]
modeled the velocity increment of Lagrangian particles as the
Wiener vector process pre-multiplied by a diffusion coefficient
matrix, which is a function of the particle-turbulence interaction
time and the sub-grid kinetic energy. This model was validated in
dilute particle-laden mixing layer [3] and evaporating acetone
sprays [4]. Pozorski and Apte [5] modeled the increment of the SGS
dispersion velocity itself by the Wiener pro-cess. The coefficients
in the stochastic PDE of the SGS dispersion velocity are related to
the sub-grid kinetic energy. Preferential concentration of
Lagrangian parti-cles was quantified by a radial distribution
function (RDF). This model was tested in particle-laden forced
isotropic turbulence. It was found that the RDF pre-dicted by this
model is in good agreement with DNS data in a large Stokes number
case, but it did not work well in a small Stokes number case. Also,
the predic-tion is sensitive to the tunable constant determining
the time scale of residual motion. One of the deterministic models
was proposed by Okong’o and Bellan [6]. The
SGS dispersion velocity has a magnitude of the filtered standard
deviation which is determined from the SGS stress model and the
filtered velocity, and its sign is opposite to the Laplacian of the
filtered velocity. As-sumptions needed in this model were verified
by the DNS data of an evaporating mixing layer laden with
evaporating particles in a priori test. Another determin-istic
model is to determine the SGS dispersion velocity from the
approximate deconvolution of a filtered veloc-ity. The main idea is
an approximation of the unfiltered field by truncated series
expansion of the inverse filter operator [7,8]. This model was
later used by Bharadwaj et al. [9] to calculate the spray source
term in the sub-grid kinetic energy transport equation.
The most commonly used SGS dispersion model for engine spray
simulations is inherited from the RANS dispersion model used in the
KIVA engine sim-ulation code [10]. The SGS dispersion velocity is
sam-pled once every turbulence correlation time from the Gaussian
distribution function with variance propor-tional to the sub-grid
kinetic energy. This type of mod-el is widely used in LES of engine
sprays, e.g. Refs. [11,12]. However, studies on the impacts of the
SGS dispersion model on engine spray simulations are few. One of
few studies was done by Jangi et al. [13] inves-tigating the
effects of the stochastic Gaussian disper-sion model on the LES
prediction of Diesel spray. They found that turning off the
stochastic dispersion model results in over-predicted liquid
penetration and the rate of air entrainment is slower.
The objective of this study is to understand how different
modeling strategies of the SGS dispersion and model parameters
influence the prediction of spray-air mixing under Diesel engine
operating conditions. The first part of the paper starts with
introducing the La-grangian-Eulerian governing equations of liquid
and gas flows, followed by the formulation and discussion of the
SGS dispersion model. The second part is the validation and
evaluation of models performance under non-vaporizing and
vaporizing Diesel spray conditions. The paper closes with a summary
and conclusions.
Two-Way Coupled Lagrangian-Eulerian Governing Equations
A two-way coupled Lagrangian-Eulerian approach is employed to
solve the two-phase flow problems. Gas is treated as a continuous
phase. The spray is character-ized by a number of discrete
computational parcels, with each parcel representing a statistical
sample of droplets having the same properties. Properties of each
parcel such as position, velocity, and temperature are tracked by
solving the Lagrangian governing equations with sub-models
predicting physical processes of aero-dynamic drag, atomization,
vaporization, etc.
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The LES gas-phase momentum equation is written as
𝜕𝜌𝒖𝜕𝑡
+ ∇ ∙ 𝜌𝒖𝒖
= −∇𝑝 + ∇ ∙ 𝜌𝜈 ∇𝒖 + ∇𝒖 , −23∇ ∙ 𝒖𝑰 − ∇ ∙ 𝜌𝝉
+ 𝑺232
(1)
where 𝝉 is the sub-grid stress tensor which is modelled by the
mixed dynamic structure model [14]. This mod-el has been shown to
have a superior performance over conventional constant coefficient
viscosity-based mod-els such as the one-equation model [15]. The
momen-tum exchange between the two phases is through the filtered
spray momentum source term, 𝑺232, given as
𝑺232 =1𝑉6
𝑛8,2𝑚8,2𝒖8,2
;<
2=>
− 𝑛8,2𝑭8,2
;<
2=>
(2)
where the first term on the right-hand side is momen-tum source
due to vaporization, and the second term is due to aerodynamic drag
force. The symbol 𝑉6 is the filter volume which is taken to be
equal to the compu-tational cell volume, 𝑛@ the number of
computational parcels in the filter volume, 𝑛8 the number of drops
in a parcel, 𝑚8 the vaporization rate of a drop, 𝒖8 the drop
velocity, and 𝑭8 the drop drag force.
The equation of motion for a droplet is written as
𝑑𝒖8𝑑𝑡
= 𝜏8C> 𝒖 + 𝒖′ − 𝒖8 (3)
where the droplet response time scale is
𝜏8 =431𝐶8𝜌8𝜌𝑑 𝒖 + 𝒖G − 𝒖8 C> (4)
where 𝐶8 is the drag coefficient, 𝜌8 the droplet density, and 𝑑
the droplet diameter, so the drag force 𝑭8 =𝑚8
𝑑𝒖𝑑𝑑𝑡
where 𝑚8 is the droplet mass. Eq. (3) cannot be solved without a
model to close the dispersion velocity term, 𝒖′. The model for 𝒖′
is described in the next sec-tion. Note that the SGS dispersion
velocity is used not only in the drag calculation, but in
atomization, breakup, and vaporization models in which a number of
non-dimensional numbers such as Weber number and Reynolds number
are based on the slip velocity, 𝒖 + 𝒖′ − 𝒖8. In this section only
the momentum conservation equations of gas and liquid phases are
shown since the focus of this study is to study the effect of the
SGS dispersion on the momentum coupling of liquid and gas. Mass and
energy conservation equations of liquid and
gas are also solved in this study. Their detailed formu-lations
can be found in Ref. [14]. SGS Dispersion Models In this study, we
assume that the dispersion velocity can be decomposed into the
deterministic and the sto-chastic parts,
𝒖G = 𝒖HIH + 𝒖HJ3. (5)
The deterministic part, 𝒖HIH, is modeled by the approx-imate
deconvolution method [7],
𝒖HIH = 2𝒖 − 3𝒖 + 𝒖. (6)
The explicit filtering in Eq. (6) is numerically imple-mented by
cell face area-weighted averaging, i.e., 𝒖 =
𝐴H𝒖HH 𝐴HH and 𝒖 = 𝐴H𝒖HH 𝐴HH where the summation is over all
faces (area=𝐴H) of a computa-tional cell. Theoretically, the
approximate deconvolu-tion method can only recover the sub-grid
scales on the order of the cutoff wave number. Information from
smaller scales cannot be recovered, as discussed in Refs [5,6]. In
this study, the small-scale motion is rep-resented by the
stochastic part of the dispersion veloci-ty. It is assumed to be
isotropic and Gaussian-distributed with zero mean:
𝑓 𝑢HJ3,O =12𝜎Q𝜋
𝑒𝑥𝑝 −𝑢HJ3,OQ
2𝜎Q, (7)
where the subscript i is denoted as the ith component of a
vector, and the variance 𝜎Q is proportional to the sub-grid kinetic
energy,
𝜎Q = 𝐶HOI23𝑘HIH (8)
where 𝐶HOI is a model constant and 𝑘HIH is solved by a transport
equation 𝜕𝜌𝑘HIH𝜕𝑡
+ ∇ ∙ 𝜌𝒖𝑘HIH
= −𝜌𝝉: 𝑺𝒊𝒋 − 𝐶Z𝜌𝑘HIH[ Q
Δ+ ∇ ∙ 𝜌𝐶]∆𝑘HIH
>Q ∇𝑘HIH + 𝑊HIH
(9)
where 𝑺𝒊𝒋 is the filtered strain rate tensor, 𝐶Z and 𝐶] are
model constants, ∆ is the cell size, and the sub-grid kinetic
energy spray source term is
𝑊HIH =1𝑉6
𝑛8,2𝑭8,2 ∙ 𝒖2G;<
2=>
. (10)
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From Eqs. (2), (3), and (10) we can see that the disper-sion
velocity 𝒖′ appears both in the gas- and liquid-phase equations, so
the effect of the SGS dispersion is two-way coupled.
The SGS dispersion velocity of a parcel is sampled once every
turbulence correlation time, 𝑡J`ab . This concept was also used in
the RANS dispersion model [10]. In RANS, the turbulence correlation
time is the minimum of the eddy breakup time and the time for a
droplet to traverse a turbulent eddy. The length and the time
scales in the RANS model are determined from the turbulent kinetic
energy and its dissipation rate. We can extend this model to be
used in the LES framework by simply replacing the turbulent kinetic
energy by the sub-grid kinetic energy, and the turbulent kinetic
ener-gy dissipation rate by the 𝑘HIH dissipation rate, 𝜀HIH , which
is modelled as the second term on the right-hand side of Eq. (9)
divided by the filtered density. Thus, we can write 𝑡J`ab as
𝑡J`ab = 𝑚𝑖𝑛𝑘HIH𝜀HIH
, 𝐶@H𝑘HIH[/Q
𝜀HIH1
𝒖 + 𝒖′ − 𝒖8 (11)
where 𝐶@H = 0.16432 is an empirical constant derived from the
RANS 𝑘 − 𝜀 model. Another possible formulation of 𝑡J`ab can be
related to the deterministic part of the SGS dispersion
velocity,
𝑡J`ab = 𝐶HIH2∆
𝒖HIH − 𝒖8 (12)
where 𝐶HIH is a model constant, 2∆ respresents the smallest eddy
LES can resolve. The physical meaning of Eq. (12) is the time for a
droplet traveling through the largest unresolved eddy moving with
the velocity of 𝒖HIH. The underlying assumption of this formulation
is that droplets are smaller than the largest unresolved eddy.
Typically, the grid size is on the order of (usually larger than)
the nozzle hole diameter in LES Lagrangi-an-Eulerian simulations,
so the assumption is reasona-ble. It is of interest to compare
individual droplet trajec-tories predicted by Eqs. (11) and (12).
Figure 1 shows the trajectories of one of liquid blobs and its
child drop-lets due to the Kelvin-Helmholtz breakup under the
Engine Combustion Network (ECN) “Spray-A” condi-tions [16]. The
trajectories predicted by Eq. (12) are apparently much more
reasonable than those predicted by Eq. (11). We can see that in
Figure 1(a) all the child droplet trajectories are straight lines.
By examining the magnitude of 𝑡J`ab of child droplets predicted by
Eq. (11), we found that it is on the order of 10C[ seconds, which
is the same order of magnitude of the injection duration. Thus,
after a child droplet is born due to the
Kelvin-Helmholtz breakup with a randomly chosen velocity vector
orthogonal to the slip velocity vector of its parent blob, its
dispersion velocity almost never gets updated or only gets updated
once, resulting in the straight trajectories as shown in Figure
1(a). On the other hand, the turbulence correlation time predicted
by Eq. (12) is on the order of 10Ch seconds, so the disper-sion
velocity of child droplets does get updated by Eq. (7) multiple
times, resulting in the droplet trajectories more similar to the
expected stochastic diffusion pro-cess. This comparison suggests
that the simple modifica-tion from RANS-type spray models to LES
may not work well without consideration of different modeling
methodologies between RANS and LES, e.g., turbulent kinetic energy
versus sub-grid kinetic energy. In the subsequent study, Eq. (12)
is used to calculate the tur-bulence correlation time.
(a)
(b)
Figure 1. Trajectories of one of liquid blobs and its child
droplets due to the Kelvin-Helmholtz breakup predicted by (a) Eq.
(11) and (b) Eq. (12) with 𝑪𝒔𝒈𝒔 =𝟎. 𝟓. Parent droplets/blobs are
shown in solid trajecto-ries, and child droplets are shown in
dashed lines.
Effect of the variance of the Gaussian distribution and the
turbulence correlation time on particle trajecto-ries is studied in
a simplified case before running full
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simulations. Consider a particle is located at 𝑧, 𝑦 =0,0
initially, moving with a constant mean speed 𝑈 in
the z-direction, and the dispersion velocities in z- and
y-directions are 𝜁 and 𝜉 respectively. The dispersion velocities
are sampled once every constant time inter-val ℎ by the normal
distribution with zero mean and a constant variance of 𝜂Q. Assume
there is no drag, no collision, and no breakup, then after time T
the z- and y-positions of the particle are simply
𝑧 𝑇 = 𝑈𝑇 + 𝜁Oℎ,/v
O=>
(13)
and
𝑦 𝑇 = 𝜉Oℎ,/v
O=>
. (14)
The y-position, 𝑦 𝑇 , is also Gaussian distributed with mean
zero and variance of
𝑣𝑎𝑟 𝑦 𝑇 =𝑇ℎℎQ𝜂Q = 𝑇ℎ𝜂Q. (15)
Hence, larger variance, 𝜂Q, or longer duration between two
sampling events, ℎ, lead to larger variance of y-positions and thus
larger spreading angle. This is veri-fied by plotting sample paths
of particles whose posi-tions are determined by Eqs. (𝟏𝟑) and (14),
as shown in Figure 2.
Figure 2. Particle sample paths calculated from Eqs. (13) and
(14) with different variances and time inter-vals between two
sampling events.
Numerical Setups of the ECN Spray-A
“Spray-A” is a series of spray experiments carried out by Sandia
National Laboratories and their collaborators, whose measured data
is available from the online database, “Engine Combustion Network
(ECN)” [16]. Spray is injected from a single-hole injec-
tor with a speed of ~500 m/s into a constant-volume pressurized
chamber. Its temperature and oxygen con-centration can be
controlled. Experimental conditions are listed in Table 1. Note
that for the room tempera-ture case, vaporization of spray is
negligible, so it is referred to as a non-vaporizing spray. In this
study, we are interested in spray behaviors under non-reacting
conditions, so the oxygen concentration is zero. The top-hat
injection profiles of the non-vaporizing and vaporizing sprays are
shown in Figure 3.
Fuel 100 % n-dodecane Oxygen concentration 0 %
Ambient gas temperature [K] 900 (vaporizing) and 300
(non-vaporizing)
Ambient gas density [kg/m3] 22.8 Nozzle hole diameter [mm]
(single hole) 0.09
Nozzle discharge coefficient 0.89 Fuel injection pressure [MPa]
150
Injection duration [ms] 6 (vaporizing) and 1.5
(non-vaporizing)
Fuel temperature [K] 373
Table 1. ECN Spray-A operating conditions.
Figure 3. Top-hat rate of injection profiles.
The CFD toolbox, OpenFOAM version 2.1.1
[17], is used to numerically solve the governing equa-tions of
liquid and gas with a number of spray sub-models. Detailed
numerical algorithms for solving the partial differential equations
can be found in Jasak’s PhD thesis [18]. The authors’ previous work
[19] also evaluated the performance of different numerical schemes
for solving the gas-phase momentum equation, and found that the
central cubic scheme for the convec-tion term with the implicit
Euler time integration scheme gave the best prediction.
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A two-dimensional cut-plane of the three-dimensional domain and
grid sizes are shown in Figure 4. Note that all cells are
isotropic, namely ∆𝑥 = ∆𝑦 =∆𝑧. The grid size is refined to 0.25 mm
in the main spray development region. The computational time step
size is fixed at 2.5e-07 seconds, resulting in maximum Courant
number of ~0.3. The Euler implicit method is used to numerically
integrate the Lagrangian equation of motion, Eq. (3). The time step
size for solving this equation is restricted to one-tenth of the
droplet re-sponse time scale, 𝜏8 . This constraint improves the
numerical stability. It was found that when the variance of the
Gaussian distribution, Eq. (8), is large or the turbulence
correlation time, Eq. (12), is short the solu-tions are likely to
be unstable without this constraint.
Spray Models
Liquid blobs are introduced into a computa-tional domain at a
given injection position. These blobs experience aerodynamic drag,
atomization, and breakup, and smaller droplets are formed. The drag
model by Liu et al. [20] takes account for the defor-mation of
spherical droplets by correcting the standard drag coefficient. The
stochastic Kelvin-Helmholtz/Rayleigh-Taylor (KH-RT) breakup model
[19,21] is employed to simulate the atomization and breakup
process. This is a modified version of the clas-sical KH-RT model.
The model constant determining the KH breakup time scale is
determined stochastically and dynamically. For vaporizing sprays,
Frossling cor-relation [22] is employed to calculate the
vaporization rate of a spherical droplet. Collision and coalescence
of droplets are not considered in this study.
Figure 4. 2-D cut-plane of the computational mesh.
Test Cases To investigate how different modeling methodologies
of SGS dispersion impact the prediction of spray dy-namics,
simulations with different model setups as listed in Table 2 were
run. Case #1 does not use any dispersion model. Namely, the
dispersion velocity 𝒖′ is zero. Case #2 only uses the deterministic
model in which 𝒖G is equal to the sub-grid velocity computed from
Eq. (6). Cases #3 through #5 are used to study the effect of the
magnitude of the variance of the Gaussian distribution, and Cases
#3, #6, and #7 are used to study the effect of the turbulence
correlation time.
Case No. Model
𝑪𝒔𝒊𝒈 in Eq. (8)
𝑪𝒔𝒈𝒔 in Eq. (12)
#1 No model 𝒖G = 0
N/A N/A
#2 Deterministic 𝒖G = 𝒖𝑠𝑔𝑠
N/A 0.5
#3 Deterministic + stochastic 𝒖G = 𝒖HIH + 𝒖HJ3
0.5 0.5 #4 0.75 0.5 #5 0.25 0.5 #6 0.5 1.0 #7 0.5 0.25
Table 2. Different model setups to be tested.
Non-Vaporizing Spray-A Results: Spatial Distribu-tion of Liquid
Mass
The impact of the SGS dispersion model on the spatial
distribution of liquid mass is shown in Fig-ure 5, which compares
the predicted liquid projected mass density (PMD) profiles against
the experimental results from the ECN database. The PMD data were
obtained from X-ray radiography path-integrated measurements at
Argonne National Laboratory. The data provide a two-dimensional
projection of liquid fuel mass onto a plane whose normal vector is
orthog-onal to the spray axis. Detailed technique of the
meas-urement can be found in Ref. [23]. The data were
en-semble-averaged at quasi-steady state at the region of interest.
In the simulations, we performed time-averaging (0.4 ms – 1.2 ms)
and spatial-averaging (10 different viewing angles) to mimic the
ensemble-averaging in the experiments. As shown in Figure 5,
without using the stochastic model (Case #1 and #2) the PMDs at y =
0 are significantly over-predicted, and the spray width is
under-predicted at downstream re-gion. The stochastic model
predicts wider spray and improves the prediction at y = 0. These
differences can be explained by Figure 6, showing the mass-weighted
and spatially averaged dispersion velocity 𝑢2Z;G and
-
𝑢a2HG at different axial locations at 0.6 ms. The calcula-tions
of 𝑢2Z;G and 𝑢a2HG are
𝑢2Z;G (𝑍) =𝑚8,2𝑛8,2𝑢2G2=> 𝜒 𝑧8,2, 𝑍
𝑚8,2𝑛8,22=> (16)
and
𝑢a2HG (𝑍) =𝑚8,2𝑛8,2 𝑢2G − 𝑢2Z;G (𝑍)2=>
Q 𝜒 𝑧8,2, 𝑍𝑚8,2𝑛8,22=>
(17)
where
𝜒 𝑧8, 𝑍 =1, 𝑍 − ∆𝑍 ≤ 𝑧8 ≤ 𝑍 + ∆𝑍0,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (18)
𝑁 the total number of parcels in the computational do-main, 𝑧8
the z-coordinate of a droplet position, and ∆𝑍 is 0.5 mm. The RMS
of the dispersion velocity, shown by the error bars in Figure 6, in
the two lateral direc-tions (x and y) predicted by the
deterministic model is much smaller than those predicted by the
stochastic model. Hence, it is less probable for a droplet moving
further away from the centerline, resulting in narrower sprays.
Another observation in Figure 6 is that the predicted dispersion
velocity is not isotropic upstream of the spray. The mean values in
the two lateral direc-tions are approximately zero, and the RMS
values in the two lateral directions are close to each other. On
the other hand, the mean values in the streamwise direction is
positive upstream, and the RMS values is larger than those in the
two lateral directions. Since the stochastic part of the model is
isotropic, the anisotropy must re-sult from the approximate
deconvolution method. The stochastic part of the dispersion model
enhances the diffusion of droplets, resulting in better prediction
of the PMD profiles. Next, the effect of the model constants, 𝐶HOI
and 𝐶HIH, is investigated.
(a)
(b)
Figure 5. Liquid projected mass density profiles at (a) z = 5 mm
and (b) z = 10 mm predicted by the different SGS dispersion
models.
(a)
(b)
Figure 6. Mass-weighted and spatially averaged dis-persion
velocity predicted by (a) Case #2 and (b) Case #3. Symbols are mean
values, and error bars are RMS values.
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Figure 7 shows the qualitative comparison of instantaneous PMD
contours predicted by different values of 𝐶HOI and 𝐶HIH. The
display range of PMD is from 0 to 1 𝜇𝑔/𝑚𝑚Q. The value of 1 is
chosen in that Pickett et al. [24] suggested that spray edge
observed from optical diffused backlighting measurements
cor-respond to the PMD of 1 𝜇𝑔/𝑚𝑚Q measured from the x-ray
measurements. Also, the PMD contours are shown up to z = 20 mm to
observe the predictions in the near-nozzle region more closely,
although the pene-tration is ~35 mm at 0.55 ms. As shown in Figure
7, larger values of 𝐶HOI or 𝐶HIH predict wider spray. This is
consistent with Figure 2 in which the simplified case is
considered. Figure 8 and Figure 9 show the quantitative comparison
of PMD profiles predicted by different values of 𝐶HOI or 𝐶HIH,
respectively. Different values of the two constants predict similar
PMD profiles at y = 0.
Comparing the lateral profiles at two different axial distances,
the prediction at z = 10 mm is more sensitive to the two constants
than at z = 5 mm. Larger values of 𝐶HOI or 𝐶HIH predict more
diffusion of droplets and thus wider profiles. This suggests that
the SGS dispersion model has larger impact on the spatial
distribution of liquid droplets having small inertia at downstream
of the spray.
Figure 7. Instantaneous PMD contours at 0.55 ms predicted by
different values of 𝑪𝒔𝒊𝒈 and 𝑪𝒔𝒈𝒔 in the SGS disper-sion model,
Eqs. (8) and (12).
(a) (b) (c)
Figure 8. PMD profiles at (a) y = 0 mm, (b) z = 5 mm, (c) z = 10
mm predicted by different values of 𝑪𝒔𝒊𝒈.
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(a) (b) (c)
Figure 9. PMD profiles at (a) y = 0 mm, (b) z = 5 mm, (c) z = 10
mm predicted by different values of 𝑪𝒔𝒈𝒔.
Non-Vaporizing Spray-A Results: Gas-Phase Flow Structures and
Statistics
In the previous section, it is found that the dispersion
velocity has profound effect on the spatial distribution of liquid
mass. In this section, the effect of the SGS dispersion model on
gas-phase solutions are examined. Qualitative comparison is made
first by vis-ualizing vortex structures by plotting iso-surfaces of
the Q-criterion [25]. The Q-criterion is defined as
𝑄 =12𝛀Q − 𝑺Q (19)
where 𝛀 is the rotational rate tensor and S the strain rate
tensor. Figure 10 shows the iso-surfaces of the Q-criterion
predicted by the different model setups listed
in Table 2. Unlike the notable difference of the liquid spray
angles shown in Figure 7, spreading angles of the gas jet predicted
by the different dispersion models are similar to each other.
Length scales of the vortices ap-pear to be similar as well. Mean
and root-mean-square (RMS) gas veloc-ity radial profiles at three
different axial locations are plotted in Figure 11 and Figure 12.
Time-averaging from 0.4 ms to 1.2 ms and spatially averaging in the
azimuthal direction were performed to obtain mean and RMS values.
Although there are some differences of the mean and RMS values at
centerline at z = 5 mm and 15 mm, the overall shapes of the
profiles predicted by the different model setups are close to each
other. These results indicate that the SGS dispersion model plays a
less crucial role in predicting the air entrain-ment using the
current LES L-E approach under Diesel spray conditions.
Figure 10. Q-criterion iso-surfaces at 0.4 ms, 𝑸 = 𝟏. 𝟓×𝟏𝟎𝟖𝟏/𝒔𝟐,
colored by vorticity magnitude predicted by different model setups.
#1: 𝒖′ = 𝟎; #2: 𝒖′ = 𝒖𝒔𝒈𝒔; #5: 𝒖′ = 𝒖𝒔𝒈𝒔 + 𝒖𝒔𝒕𝒐, 𝑪𝒔𝒊𝒈 = 𝟎. 𝟐𝟓;
#3:𝒖′ = 𝒖𝒔𝒈𝒔 + 𝒖𝒔𝒕𝒐, 𝑪𝒔𝒊𝒈 =𝟎. 𝟓; #4: 𝒖′ = 𝒖𝒔𝒈𝒔 + 𝒖𝒔𝒕𝒐, 𝑪𝒔𝒊𝒈 = 𝟎.
𝟕𝟓.
-
(a) (b) (c)
Figure 11. Mean gas velocity profiles predicted by different
model setups at three axial locations, (a) 5 mm, (b) 15 mm, and (c)
25 mm. #1: 𝒖′ = 𝟎; #2: 𝒖′ = 𝒖𝒔𝒈𝒔; #5: 𝒖′ = 𝒖𝒔𝒈𝒔 + 𝒖𝒔𝒕𝒐, 𝑪𝒔𝒊𝒈 = 𝟎.
𝟐𝟓; #3:𝒖′ = 𝒖𝒔𝒈𝒔 + 𝒖𝒔𝒕𝒐, 𝑪𝒔𝒊𝒈 =𝟎. 𝟓; #4: 𝒖′ = 𝒖𝒔𝒈𝒔 + 𝒖𝒔𝒕𝒐, 𝑪𝒔𝒊𝒈 =
𝟎. 𝟕𝟓.
(a) (b) (c)
Figure 12. Room-mean-square (RMS) gas velocity profiles
predicted by different model setups at three different axial
locations, (a) 5 mm, (b) 15 mm, and (c) 25 mm. #1: 𝒖′ = 𝟎; #2: 𝒖′ =
𝒖𝒔𝒈𝒔; #5: 𝒖′ = 𝒖𝒔𝒈𝒔 + 𝒖𝒔𝒕𝒐, 𝑪𝒔𝒊𝒈 =𝟎. 𝟐𝟓 ; #3:𝒖′ = 𝒖𝒔𝒈𝒔 + 𝒖𝒔𝒕𝒐, 𝑪𝒔𝒊𝒈
= 𝟎. 𝟓; #4: 𝒖′ = 𝒖𝒔𝒈𝒔 + 𝒖𝒔𝒕𝒐, 𝑪𝒔𝒊𝒈 = 𝟎. 𝟕𝟓.
This fact is further examined. Figure 13 shows
time series of the convection, sub-grid stress, pressure
gradient, and spray momentum source terms in the gas-phase
z-momentum equation, Eq. (2), predicted by Case #5 at different
positions. At the upstream region, 5 mm from the injector, the
magnitude of the spray momentum source term is comparable to the
convec-tion term, meaning a significant amount of momentum transfer
from liquid to gas. Figure 14 also shows a sharp decrease of the
drag force within 5 mm. Moving further downstream to 10 mm and 15
mm from the injector, the magnitude of the spray momentum source
term is much smaller than the convection, sub-grid stress, and
pressure gradient terms, and it is negligible at 15 mm. This
corresponds to small drag force down-stream as shown in Figure 14.
Thus, air entrainment and subsequent gas jet development due to
spray injec-tion is mainly initiated in the near-nozzle region
(with-in ~5mm). In this region, liquid blobs injected from the
nozzle carry significant amount of momentum and ex-perience
little deceleration as shown in Figure 16. Fig-ure 15 shows that
within 5 mm the relative dispersion velocity, 𝒖′ 𝒖 − 𝒖8 , is
smaller than that at the downstream region. This suggests that the
motion of the liquid blobs leading to air entrainment and
subse-quent gas jet development is little influenced by the SGS
dispersion. This may be one of the reasons for similar gas velocity
profiles predicted by the different dispersion model setups.
At the downstream region (z > 10 mm), the dispersion velocity
magnitude is comparable to the resolved slip velocity as shown in
Figure 15, indicating that the dispersion model is important in
determining droplets motion. This is further justified by Figure
16, showing that the larger the dispersion velocity magni-tude the
larger the acceleration magnitude. This is also consistent with
Figure 8(c), indicating that larger dis-persion velocity results in
more diffusion of droplets. Another note is that at the downstream
region of the
-
spray, it can be approximated as one-way coupled since the
momentum coupling term in the gas momentum equation can be
negligible as shown in Figure 13(c) and (d), but droplets still
experience large accelera-tion/deceleration due to the stochastic
SGS dispersion (Figure 16).
The main point from Figure 13 through Figure 16 is that the SGS
dispersion model has small effect on the air entrainment process,
but has significant impact on the droplets motion downstream of the
spray where momentum transfer from liquid to gas is negligible. In
fact, we can clearly see the correlation between the relative
dispersion velocity and the drag force in Figure 17: the more
important the SGS dispersion the smaller the drag force.
(a)
(b)
(c)
(d)
Figure 13 Time series of z-momentum budgets at (a) 𝒙, 𝒚, 𝒛 = 𝟎,
𝟎, 𝟓 , (b) 𝒙, 𝒚, 𝒛 = 𝟎, 𝟎, 𝟏𝟎 , (c) 𝒙, 𝒚, 𝒛 = 𝟎, 𝟎, 𝟏𝟓 , and (d) 𝒙,
𝒚, 𝒛 = 𝟎, 𝟐, 𝟏𝟓 .
The injection position is 𝒙, 𝒚, 𝒛 = 𝟎, 𝟎, 𝟎 .
Figure 14. Mass-weighted and spatially averaged drag force
magnitude versus distance from injector predicted by the different
model setups.
Figure 15. Mass-weighted and spatially averaged ratio of
dispersion velocity magnitude to resolved slip veloc-ity magnitude
versus distance from injector predicted by two different values of
𝑪𝒔𝒊𝒈.
-
Figure 16. Mass-weighted and spatially averaged drop-lets
acceleration versus distance from injector predicted by the
different model setups.
Figure 17. Drag-relative dispersion velocity scatter plot.
Vaporizing Spray-A Results
In this section, simulation results of the vapor-izing Spray-A
are compared against the available ex-perimental data. As shown in
Figure 18, in average Case #4 (largest 𝐶HOI) predicts the shortest
liquid pene-tration, and Case #1 (no model) predicts the longest
liquid penetration. This is consistent with the PMD results showing
the larger the constant 𝐶HOI the larger the predicted liquid spray
angle. Nevertheless, the dif-ference of liquid penetrations
predicted by the different model setups is within 1mm, which is
small compared to the liquid penetration, ~10mm. This finding is
con-sistent with the finding by [26]. Some differences of vapor
penetrations predicted by the different model setups exist as shown
in Figure 19. Also, vapor pene-tration does not increase or
decrease monotonically with the value of 𝐶HOI. The cause of these
differences
might be explained by the different centerline gas ve-locity
predictions as shown in Figure 11and Figure 12. In Figure 20, mean
fuel vapor mass fraction profiles are compared. To compare the
simulation results to the ensemble-averaged experimental data,
time-averaging from 1.5 ms to 2.5 ms and spatial averaging in the
azi-muthal direction were performed. Except for the cen-terline
fuel vapor mass fraction within z = 15 mm, the spatial
distributions of fuel vapor mass predicted by the different models
and model constants are close to each other, which is consistent
with the similar velocity pro-files as shown in Figure 11. From
these vaporizing spray results, again we see that the effect of the
SGS dispersion modeling on gas entrainment is small.
Figure 18. Liquid penetration predicted by the differ-ent model
setups.
Figure 19. Vapor penetration predicted by different model
setups.
-
(a)
(b)
(c)
Figure 20. Mean fuel vapor mass fraction profiles at (a)
centerline, (b) 25 mm, and (c) 40 mm.
Summary and Conclusions
This work is focused on the development and evaluation of the
SGS dispersion model used in large-eddy simulation of engine
sprays. The model computes the dispersion velocity term needed in
the calculation of the slip velocity used in a number of spray
models. It is assumed that the dispersion velocity is decomposed
into the deterministic and the stochastic part. The de-terministic
part is calculated by the approximate de-convolution method, which
recovers the largest unre-solved scales from the solution of the
filtered velocity. Small-scale sub-grid motions are taken account
by the stochastic part of the model. The stochastic part of the
dispersion velocity is assumed to be isotropic and
Gaussian distributed. The dispersion velocity is sam-pled once
every turbulence correlation time, the time needed for a droplet to
traverse an eddy in sub-grid scales. We found that the RANS-type
turbulence corre-lation time model predicts unphysical droplet
trajecto-ries since the predicted correlation time is on the same
order of the injection duration, meaning that the disper-sion
velocity almost never gets updated. Thus, a new form of the
turbulence correlation time is proposed. It is related to the
deterministic part of the model and the filter/computational cell
size, Eq. (12).
The importance of the stochastic part of the model was
evaluated. It was found that without the stochastic part, the
liquid spray angle is under-predicted. Using the stochastic model
enhances the radial diffusion of droplets, resulting in improved
predictions of the PMD profiles.
The effect of the two model constants, 𝐶HOI and 𝐶HIH , was
investigated. Larger values of the constant 𝐶HOI and 𝐶HIH predict
larger variance of the Gaussian distribution and longer turbulence
correlation time, respectively. It was found that larger variance
or longer turbulence correlation leads to larger liquid spray
angle, which is consistent with the analytical analysis in a
simplified case (two-dimensional, no drag, no breakup, and no
collision).
The influence of the SGS dispersion modeling on gas-phase flow
structures and statistics was studied. Different dispersion model
setups listed in Table 2 pre-dict similar gas jet spreading angle,
similar length scales of vortices, and similar mean and RMS values
of gas velocity. The reason for this was further investigat-ed. It
was found that the SGS dispersion model is more important in
determining the motion of droplets having small inertia and thus
contributing less to momentum transfer from liquid to gas. On the
other hand, for the liquid blobs having large inertia and
initiating gas en-trainment and subsequent gas jet development, the
dis-persion velocity magnitude is small compared to the resolved
slip velocity, so it does not affect the drag force prediction
significantly.
It is suggested that one cannot neglect the SGS dispersion model
in LES of engine sprays since it pro-vides more reasonable
prediction of the spatial distribu-tion of liquid mass, although
the model plays a less important role in predicting gas entrainment
due to high-speed spray injection.
Nomenclature DNS direct numerical simulation ECN Engine
Combustion Network KH-RT Kelvin-Helmholtz/Rayleigh-Taylor LES
large-eddy simulation PMD projected mass density
-
RANS Reynolds-averaged Navier-Stokes RMS root-mean-square SGS
sub-grid scale 𝐶8 drag coefficient 𝐶HOI dispersion model constant,
Eq. (8) 𝐶HIH dispersion model constant, Eq. (12) 𝑘HIH sub-grid
kinetic energy 𝑚8 droplet mass 𝑛8 number of drops in a parcel 𝑝
filtered pressure 𝑺232 filtered spray momentum source term 𝑡J`ab
turbulence correlation time 𝒖 filtered gas velocity 𝒖′ sub-grid
dispersion velocity 𝒖8 droplet velocity 𝑉6 filter/computational
cell volume 𝑊HIH sub-grid kinetic energy spray source term ∆
computational cell size 𝜀HIH sub-grid kinetic energy dissipation
rate 𝜌 filtered density 𝜌8 droplet density 𝜏 sub-grid stress tensor
𝜏8 droplet response time scale 𝜈 molecular kinematic viscosity
Acknowledgments This work was supported by Caterpillar Inc. The
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