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ILASS Americas, 24th Annual Conference on Liquid Atomization and
Spray Systems, San Antonio, TX, May 2012
* Corresponding Author, (currently at Cummins Inc., Combustion
Research): [email protected] or
[email protected]
Study of Spray Induced Turbulence Using Large Eddy Simulations
Siddhartha Banerjee*, Christopher J. Rutland Engine Research
Center, Department of Mechanical Engineering University of
Wisconsin - Madison Madison, WI 53706-1609 USA
Abstract Spray induced turbulence is investigated on a number of
different Computational Fluid Dynamics (CFD) grids of varying mesh
sizes (from 0.5 to 2 mm mesh) using non-viscosity dynamic structure
Large Eddy Simulation (LES) turbulence model. Turbulent flow is
induced inside a quiescent chamber by liquid fuel spray and then
left to decay after end of injection by virtue of its molecular
viscosity and turbulent dissipation. Coherent structures (CS) of
this turbulent flow are constructed and visualized using λ2
definition. Using CS, analysis is performed on the turbulent flow
around the liquid spray jet. The visualization of CS helps to
explain the mechanism of fuel-air mixing obtained from LES results.
It is found that fine mesh LES results predicts fuel-air mixing by
virtue of breaking down of large eddies to number of smaller
eddies. These LES are then compared against the results from RANS
calculations on the same flow situations. It was found that main
difference between RANS and LES flow structures was in its
prediction of break-down of large flow structures into number of
smaller eddies and the nature of diffusion of fuel rich pockets. A
local CFD mesh criteria is derived based on the observation of
these CS for LES calculations. With finer mesh, more flow
structures were predicted resulting in enriched statistic of flow
prediction. It is found that LES dynamic structure model is
effective to resolve turbulent flow structures around spray jets.
CFD grid convergence is obtained in mesh size of ~0.5 mm or less.
Furthermore this study shows that gas phase turbu-lence is induced
due to spray liquid – gas momentum exchange in the secondary
breakup region. Turbulent struc-tures generated in the maximum
spray drag regions are then carried to downstream location due to
large scale sur-rounding motions. Away from spray in downstream
locations, turbulent structures break down to smaller scales in and
produces intermittencies in flow and fuel-air mixing mechanism.
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Introduction
Over last few decades, due to enormous increase in computer
capabilities, it is now generally agreed that LES models can be
applied for engineering applications for CFD calculations. LES
models offer significant advantage over traditional Reynolds
Averaged Navier Stokes (RANS) models. The formulation of LES
mod-els are based on direct treatment of large-scale dynam-ics and
physical modeling of small scale variations, that are universal in
nature [1].
As LES models gains popularity in engineering
CFD user community, set of guidelines for standard practices of
LES are evolving. One of the key variables of interest for CFD
users is grid resolution. In LES model predictions the grid
resolution plays an important role. Turbulent length scales that
are not resolved by the CFD grids are modeled and therefore results
from LES models are essentially CFD grid dependent. The moti-vation
of this study is to come up with a methodology that can be used to
“measure” the quality of CFD re-sults for LES models in spray
induced turbulence. Based on the quality of CFD result, a set of
criteria is proposed for CFD mesh for LES applications.
Overview of Large Eddy Simulation for inter-nal combustion
engine turbulence motions Numerical setup1 adapted by a CFD problem
largely determines solution obtained in the resolved scales. On the
basis of numerical setup, LES model predictions may be classified
as low fidelity or high fidelity. For most engineering applications
including internal com-bustion engines, for LES model applications,
it is com-mon practice to use low fidelity numerical setup due to
complexity of problem and constrains in computer re-sources. Upon
review of LES models for engineering applications, Jhavar et al [2]
reported that for engineer-ing applications such as internal
combustion engines, turbulent intermittencies and flow structures
may be obtained in low fidelity setup using one equation
non-viscosity based dynamic structure model. Upon review of LES
models Rutland [3] recommended following features for LES models in
low fidelity setup. • Non-viscosity model approach: In low fidelity
setup, due to coarse mesh, the numerical viscosity is higher than
that in fine mesh. This added numerical viscosity play a role to
stabilize numerical calculations and add
1 Accuracy of numerical scheme, CFD mesh resolution and time
step.
to the effective eddy viscosity of high fidelity LES models. •
Energy budget: One equation model keeps the model stable without
eddy viscosity and appropriate kinetic energy between resolved and
sub-grid scale (SGS) mo-tions. • Solvability criteria : By dynamic
modeling of the structure of SGS stress, solvability criteria are
met. • Structure of SGS stress: The tensor directions of SGS stress
is important and may be obtained using scale similarity from the
“test filter” scales. Pope [4], stated that it is impossible to
construct LES model that reproduce filtered DNS velocity field
reali-zation by realization. He argued that since the resolved
velocity field does not provide directly the information on SGS
motions, the problem of SGS turbulence mod-eling is therefore
independent from the problem of de-termining CFD mesh resolution.
However filtered ve-locity field (resolved motions) implicitly
contains in-formation of SGS motions. Therefore resolution of CFD
grid and SGS modeling are inherently connected. If filtered motions
fully resolve all the turbulent length scales i.e. Direct Numerical
Simulation (DNS), no SGS modeling is required. Pope [5] also argued
that one of the primary goals of LES turbulence model is to resolve
enough energy containing length scales such that solu-tion becomes
grid independent. In engineering applica-tions such as internal
combustion engines, this criterion becomes difficult to achieve.
Diagnosis of turbulence flow field obtained from CFD solution In
order to study in-cylinder turbulence, general expec-tations from
LES are [6]: • More flow structures: Primary and secondary
insta-bilities due to vortical flow motions in resolved scales •
Intermittency : An estimate of time scales of fuel-air mixture
formation from the resolved scale motions. • Energy budget: Energy
balance between resolved scale and SGS motions. One of the methods
to diagnose turbulent flow obtained from CFD solution is to
identify and visualize tempo-rally evolving vortex cores (coherent
structures). Since turbulence flow may be conceptualized as a
tangle of vortex filaments embedded on a base flow, much of
turbulence physics can be explained using the concepts of vortex
dynamics. In turbulent flows such as shear flow layers and mixing
layers are dominated by spatial-ly coherent and temporally evolving
vortical motions [7]. The basis of identifying coherent structures
is that vortex cores contain a region of locally minimum
pres-sure.
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Vortex dynamics, which govern the evolution and in-teraction of
coherent structures with base flow, is prom-ising in understanding
turbulence phenomenon such as entrainment, mixing and aerodynamic
noise. The ap-proach taken in this paper to identify coherent
struc-tures was suggested by Jeong et al. [7] and the method is
popularly called as λ2 vortex core identification (next section).
In order to study in-cylinder turbulence, general expec-tations
from LES are [6]: • More flow structures: Primary and secondary
insta-bilities due to vertical flow motions in resolved scales •
Intermittency : An estimate of time scales of fuel-air mixture
formation from the resolved scale motions. • Energy budget: Energy
balance between resolved scale and SGS motions.
Coherent structures in turbulent flow for CFD modeling Pope [5]
raised some fundamental questions on the conceptual formulation of
LES that revolves around the dependence of LES results on grid size
or length scales resolved by CFD grids. He mentioned that
statistical resolution of all the length scales of turbulent
motions is the most important criteria for any successful LES.
However at high Re, it is impossible to construct statis-tic of
flow motions such as in practical in-cylinder flows. Therefore in
this work, an attempt is made to correlate the flow statistics from
resolved scales by analyzing coherent structure with a model 2D
vortex motion. It is assumed that for all practical purpose the
planer structures of flows in turbulence is a 2D Lamb-Oseen
vortex2. The mathematical description of Lamb-Oseen vortex may be
given by
( ) ( )2
2, 1 exp2 c
rV r t
r r tθ π
Γ −= −
(1)
where, Vθ is velocity in azimuthal directions of 2D vor-tex, r
is radial distance from the vortex center, Γ is circulation
contained inside vortex and rc is radius of vortex core (beyond
which the velocity Vθ decreases). Vortex core radius (rc) is
dependent on molecular vis-cosity (νmol) and time (t).
( ) 4c molr t tν= (2)
2 A line vortex that decays due to viscosity.
Formulation used to identify coherent struc-tures In this paper
coherent structure is interchangeably de-fined as “vortex cores” in
the following section. Coher-ent structures can be mathematically
defined as con-nected regions of space where the tensor ( )kjikkjik
SS ΩΩ+ 3 has at least two negative eigenvalues. Since the tensor (
)kjikkjik SS ΩΩ+ is real and symmetric, their eigenvalues are all
real. In algebraic expansion the tensor may be given by
12
ji k kik kj ik kj
j ik k
uu uuS S
x x x x
∂∂ ∂∂+ Ω Ω = +∂ ∂ ∂ ∂
(3)
If the eigenvalues of the tensors given in equation (1) are λ1,
λ2, λ3, such that λ1 < λ2 < λ3, then region defined by λ2
< 0 defines the region inside coherent structure. The
characteristics of coherent structures identified by equation (1)
are • It defines a region enveloped by region of high azi-muthal
velocity of vortex flow. • It is Galilean invariant and therefore
can describe the intermittency and unsteadiness of coherent
structures. • The region is local to flow field. Interpretation of
λ2 definition of coherent struc-ture in a 2D axisymmetric flow In
this section, the λ2 definition of coherent structure will be
evaluated in a 2D axisymmetric flow. In cylin-drical co-ordinates,
the velocity field of 2D axisymmet-ric flow can be described by
( )0, 0, ,r zu u u V r tθ θ= = = (4) where, r denotes radial
distance from the vortex center and t indicates time. Therefore the
tensor given by equation (1) becomes
0 0
0 0
0 0 0
V Vr
V VSS
r
θ θ
θ θ
′−
′+ΩΩ = −
(5) where, VV
rθ
θ∂′ ≡∂
. The eigenvalues of the tensor
( )ik kj ik kjS S + Ω Ω are as follows
3 Sij and Ωij are symmetric and anti-symmetric parts of strain
rate tensor respectively.
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1 2 3, , 0V V V V
r rθ θ θ θλ λ λ′ ′= − = − =
(6)
For negative values of λ2, it is necessary that that the
expression 0V Vθ θ′ > , or
2
0Vrθ∂ >
∂. This would mean that
within coherent structure defined by region of negative λ2, the
magnitude of azimuthal velocity Vθ increases with radial distance r
and Vθ is locally maximum when λ2 = 0. Outside the coherent
structure, λ2 > 0, which implies 2
0Vrθ∂ <
∂ indicates that azimuthal velocity Vθ
decreases with radial distance r. The angular velocity vector in
2D vortex in given by ( )10,0, rV
r rθω
∂=
∂ (7)
The vorticity magnitude may is obtained fromω ωi , and for 2D
vortex it is given by 2
2 2V V V
Vr rθ θ θ
θω ω
′= + +′i (8)
Analysis of 2D line vortex using λ2 definition Using Lamb Oseen
formulation given by equation (1) it is possible to calculate λ2 of
equation (6) and correlate with of equation (8). At r = rc, λ2 = 0.
In a model 2D vortex structure given by equation (1), it is
possible to construct relationship between λ2 and ω ωi . Let us
define a vortex turnover timescale based on measure of vorticity
as
1ετ ω ω≡ i (9)
Several model line vortex are constructed with various levels of
circulation (Γ ) [Table 1]. Figure 1 shows the relation between λ2
and vortex turnover timescale τε in Lamb-Oseen model vortexes. From
Figure 1 it can be established that near the vortex core region (r
rc) regions.
Following arguments is placed in the context of estab-lishing
the stability for 2D line vortex. • In eye region (i.e. r ~ 0),
vortex exhibits near solid body rotation (i.e. Sij = 0) • In core
region (r < rc) inflection point theory4 may be applied
(assuming small curvature effects). • Viscous effects dominates
flow pattern in the outer region (i.e. r > rc) Based on the
above assumptions it can be established that at a particular radial
distance from vortex center, vortex rollup may be initiated due to
shear flow insta-bility. This radial distance (ro) is estimated
using inflec-tion point theory such that
( )2
2, 0o
o o
r r
VV r V
rθ
θ=
∂= =∂
(11) The necessary and sufficient condition for instability in
flow is given by ( )
2
2 0oV
V Vr
θθ
∂− <∂ (12)
In the modeled flow it is possible to show that this in-flection
point (ro) lies within the vortex core region [Figure 2]. It is
seen that there exist a region in Lamb-Oseen vortex where the flow
is unstable. In the particu-lar case of 2D vortex, the regions of
instability lies at r ~ 0.55 rc. With this information now the 2D
line vortex may be rearranged into following sub-regions, namely: •
Vortex eye region (r ~ ro): This region is the source of the flow
energy of vortex and exhibits stable flow mo-tions. • Vortex rollup
region (ro ~ r ~ rc): In this region prima-ry and secondary vortex
rollups may occur. In this re-gion, flow energy is both fed from
the vortex eye region and dissipated outwards to the outer region.
• Outer vortex region (r > rc): In this region, viscosity
dominates the flow and flow energy dissipates. This insight of the
2D vortex explains the mechanism of vortex rollup and supports to
identify a region of turbulent vortex core in a complex 3D
flow.
Numerical models Turbulence sub-models to describe high pressure
fuel sprays require a) nozzle flow sub model, b) spray transport,
c) spray breakup d) droplet collision e) wall
4 Inflection point theory establishes criteria for instabil-ity
of flow using Orr-Somerfield equation.
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impingement and f) droplet evaporation. In the context of RANS
turbulence study, Amsden et al. [8] developed the widely popular
CFD code called KIVA and incor-porated these sub-models primarily
for internal com-bustion engine applications. In this study, ERC5
modi-fied version of KIVA-3V release2 is used to perform numerical
simulations. This section elaborates on nu-merical sub-models and
setup for study of spray in-duced turbulence motions. Large Eddy
Simulation – Dynamic Structure Model A non-viscosity based one
equation dynamic structure model [9, 10] used for LES calculation.
The density weighted LES spatial filtering operation on the
Navier-Stokes equation; results in filtered momentum equation:
i j iji ii
j i j j j
u uu upF
t x x x x x
ρ ρτρ µ ∂ ∂∂ ∂∂ ∂+ = − − + − ∂ ∂ ∂ ∂ ∂ ∂
ɶ ɶɶ ɶɶ (13)
The curly overbar in equation (13) and subsequent equations
indicates a spatial Favre-averaged filtering operation. This
filtering operation is never performed in CFD calculations. However
solution obtained in CFD grids (cell centered or node centered
variables) is as-sumed to be representative of the LES filtered
quanti-ties6. The SGS stress term (τij) is unclosed and requires
modeling. In non-viscosity based dynamic structure model the SGS
stress term is modeled using SGS kinet-ic energy and normalized
Leonard term (Lij) as shown by the equation (14).
�( )�
2
ˆ ˆ,
ijij i j i j sgs
kk
ij i j i j
Lu u u u k
L
where L u u u u
τ ≡ − =
≡ −
ɶ ɶ
ɶ ɶ ɶ ɶ
(14)
The benefits of this model are • Non-viscosity model: Does not
require to model eddy viscosity and instead allow numerical
viscosity to stabi-lize solution. This also allows performing
engineering calculations of flows with minimal computation cost
[3]. • One equation model with SGS kinetic energy transport
maintains kinetic energy budget between re-solved and SGS scales. •
Dynamic coefficient modeling approach allows backscatter. • The
structure of SGS stress term is obtained from Leonard Stress based
on scale similarity argument. The SGS kinetic energy (ksgs) is
solved from the LES derived SGS kinetic energy transport equation.
5 Engine Research Center, University of Wisconsin - Madison 6 LES
filtered quantities are also interchangeably re-ferred to as LES
resolved solution.
32 11
321 2, , ,
sgs i sgs sgsij ij t s
i i i
sgst sgs cell
k u k kS W
t x x x
kwhere C C k V
ρ ρρτ ρε µ
ε ρ µ ρ
∂ ∂ ∂ ∂+ = − − − + ∂ ∂ ∂ ∂
= = ∆ ∆ =∆
ɶɶ ɺ (15)
The spray source term (
sWɺ ) accounts for the two-way
coupling of momentum exchange in SGS scales be-tween liquid
droplets and surrounding gas phase [11]. Using point parcel
assumption, the spray source term is modeled [11, 12].
�
2 3s i iW F u u u
= − +ɺ ɶ (16)
Numerical setup In this work, the multidimensional engine CFD
simula-tions are performed using KIVA [8], a Fortran based 3-D CFD
code. KIVA uses a time splitting numerical scheme for flow solver
with 1st order Euler in time and 2nd order accurate in space. Some
of the features of the numerical scheme of KIVA flow solver are: •
Coupled implicit differencing of diffusion terms and terms
associated with pressure wave propagation. • Sub-cycled calculation
of convection. • Stochastic spray particle injector. The main focus
of this study is to analysis CFD grid for LES calculations of fuel
spray in a non-reacting envi-ronment. In order to access the spray
induced turbu-lence, problem in hand is sub categorized into a)
Evap-orative spray, followed by b) Turbulence decay of flow motion.
The CFD grids employed in this study are shown in Figure 3. The CFD
grids are oriented in Car-tesian coordinates as shown in Figure 4.
An aspect ratio of less than 2 is maintained for all the grids (A,
B, C and D) taken up for this investigation. The grid sizes and the
dimension of computation domain are presented in Table 2. Diagnosis
of CFD solution One of the key aspects of this investigation is to
address the grid convergence for LES solutions in the context of
spray induced droplet breakup and turbulence. The ba-sis of
establishing CFD grid convergence is based on coherent structures
obtained from LES solution of grid resolved flows. In this
investigation it is argued that coherent structures obtained from
resolved flow must be able to reveal certain flow patterns of
orderly vortex surrounding the spray jet, due to the effect of air
en-trainment. LES grid convergence is established based on the
convergence of these flow patterns with succes-sive grid
refinements. In order to identify whether the flow patterns
estab-lished from the LES resolved flow field is consistent
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with the vortex dynamics, scalar values of is compared with
iiωω ~~ and mapped against one which is obtained from 2D
idealized flow from Figure 1. In order to reduce numerical noise
and avoid regions of unstable vortex rollover regions of flow
(refer to previ-ous section on stability analysis of 2D vortex), an
iso-surface of [ ]262 110 s−≤λ is set to identify coherent
structure in CFD solutions in this paper. Furthermore the coherent
structures obtained from RANS mean flow field is also compared as a
baseline case to establish the differences between LES and RANS
flow field in spray induced turbulence.
Results and discussion For validating the numerical framework,
experimental data from Sandia National Laboratory’s “Spray A”
con-figuration is chosen [13]. In the Spray A setup, diesel
surrogate fuels are injected in a high pressure quiescent chamber
filled with products of complete combustion. Desired temperature
and pressure in the chamber is reached by igniting premixed
combustible mixture prior to the start of spray injection. The
conditions for these experiments are tabulated in Table 3. For
comparing the numerical results with the experi-mental
measurements, time resolved liquid droplet and fuel vapor
penetrations are considered. In this study, liquid penetration is
defined as the axial distance from the injector nozzle encompassing
97 % of the injected liquid mass and fuel vapor penetration is
determined using farthest downstream location of 0.05 fuel mass
fractions. For benchmarking the results obtained from the
numer-ical simulations, fuel liquid and vapor penetration data are
compared against the experimental observations (Figure 5, Figure
6). It is observed that, the steady liq-uid spray penetrations
agree well with the experimental measurements. However time
required for attaining steady penetration is not predicted
accurately by simu-lations (which in LES predictions are ~0.3 ms
ASOI). There is considerable uncertainty in prediction of vapor
penetration and subsequent fuel-air mixing due to suc-cessive grid
refinements (Figure 6). The vapor penetra-tion predictions
presented in Figure 6 obtained numeri-cal simulation, is from one
single realization of numeri-cal simulation and compared with
ensemble averaged vapor penetration result of experimental
observation. Due to influence of turbulence on the formation
fuel-rich pockets, transient vapor penetration predicted by
numerical simulations are intermittent in nature. Shape of
turbulence structures strongly influences the predic-
tions in vapor penetration. This is prominent in Figure 7 and
Figure 8 where the differences in the evolution of fuel-air mixture
predicted from different numerical grid refinement levels are
evident. To evaluate the accuracy of the results obtained from CFD
numerical setup, further investigation is required on turbulence
produced due to spray induced motions surrounding spray jet. To
validate the numerical mod-els, Figure 9 and Figure 10 shows the
comparison be-tween numerical predictions of liquid and vapor
sprays with the experimental images obtained from optical diagnosis
at various time after start of injection (ASOI). It can be seen
that general contours of fuel vapor mix-ture agrees well with the
experimental images [13]. Evaluation of CFD grid for LES model and
comparison with RANS It is observed that CFD grid has a strong
influence on prediction of fuel vapor mixture formation in DI
sprays. Visualizing coherent structures obtained from resolved flow
field, (Figure 11) shows that • Evolution of varicose instability
modes surrounding the spray liquid jet is represented by spiral
coherent structures. • Detailed flow structures emerge with
successive mesh refinements. • Prediction of spray downstream
fuel-air mixture and vapor penetration is largely correlated with
the evolu-tion of flow structures. Recent study conducted by the
authors [14] demon-strated using analysis of coherent structures of
turbulent motions that LES models exhibit grid independence when
CFD cell size is ~0.5 mm or less. A set of criteria were set to
evaluate grid quality for LES solution. Be-sides evaluating flow
statistic in
− 22 1,ετ
λ space for
upper linear bound, the visual representations of coher-ent
structures of grid resolved solutions in LES were carefully studied
(Figure 12). In the figure, grids E and F are successive
refinements of CFD cell sizes of (0.25 mm and 0.125 mm)
respectively. It was observed that while grids A failed to
reproduce varicose instability modes in coherent structures, grids
B and C provide some degree of resolution in flow structures.
However the distance between two successive structures in the
direction of spray does vary between solutions obtained from grids
B and C (marked in Figure 12). Varicose instability modes obtained
by coherent structures offer a better insight to evaluate LES
resolution of turbu-lence. It is found that distance between two
successive structures in the direction spray does not vary in the
solutions obtained from grids D, E and F. This led to the
conclusion that from LES dynamic structure model,
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in spray induced turbulence grid convergence is ob-tained when
coherent structures no longer provides more structures in between
two successive varicose instability modes in coherent structures
[6, 14]. Author extended the study to explore the coherent
structures obtained from various LES models. It was found that LES
modeling methodology also contributes largely to production of flow
structures in grid resolved scales (Figure 13). RANS exhibits grid
independent turbulent flow sur-rounding the spray jet with CFD cell
size about ~2 mm. Spatial evolution of spray liquid and fuel vapor
is strongly correlated with the nature of coherent struc-tures
around sprays. In the Sandia Spray A case this is also demonstrated
in the Figure 14 for spray and vapor penetrations and Fig-ure 15
for evolution of coherent structures. It is clear from these
figures that • RANS model exhibits fuel spray and vapor
penetra-tions predictions independent of CFD grid sizes in grids B,
C and D. • Coherent structures obtained from RANS mean flow does
not show evidence of varicose instabilities or in-termittencies
when compared to LES resolved flow structures It is found that LES
predicts fuel-air mixing by the dual mechanism of breaking down of
large eddies into smaller eddies as well as diffusion. For RANS
predic-tion, fuel-air mixing is driven only by diffusion
mecha-nism. Furthermore from Figure 16 it is observed that for same
CFD grid, vapor penetration prediction from RANS is underestimated
while LES shows intermitten-cies in one single realization of fuel
injection. Evolution of spray and gas phase turbulence In this
sub-section, the evolution of spray and gas phase turbulence is
analyzed obtained from the LES Grid D results. Figure 17 shows that
after attaining steady state penetration, at 0.3 ms ASOI, spray
droplets may be distinguished into two distinct regions of breakup
by measuring the droplet drag forces. In the figure, liquid
particles are colored by droplet drag force, and it is ob-served
that in the primary breakup region (near to spray nozzle) the drag
forces experienced by droplets are lower that the drag force
experienced by droplets away from the nozzle. It is also observed
that the region of generation of coherent structures surrounding
the sprays is strongly correlated with the droplet drag. In the
sec-ondary breakup region, spray induced turbulence is produced due
to two way interactions between droplet-gas phase. These coherent
structures produced in the secondary breakup region are carried
downstream by
the motion of large scale gas jet. It is also observed that in
the spray breakup region the length scales of coher-ent structures
is in the same order of magnitude of the spray jet plume. In the
Figure 18 and Figure 19, evolution of coherent structures is
investigated at different times from the start of injection. It is
observed that • Coherent structures are generated first in the
regions of secondary breakup • These coherent structures are
carried by gas motions downstream of spray jet • Instabilities in
coherent structures causes breakup of the coherent structures in
the downstream locations • In the direction of spray, the coherent
structures in-creasingly become intermittent in nature, and
transition from orderly to less organized structures. To
investigate the transfer of turbulent energy from upstream to
downstream, time spectra of turbulent flows at various distances
downstream of nozzle are presented for both LES and RANS
predictions in Figure 20 and Figure 21 respectively. It is noted
that in LES prediction, the time spectrum shows increasing order of
turbulent energy from the upstream to downstream lo-cations. In LES
resolved scales, it also predicts that turbulent energy cascades
from flow motions of large time scales to smaller time scales. This
is consistent with the traditional view of equilibrium turbulence
en-ergy cascade [4]. However when the time spectra are compared
between LES and RANS flow field, the pri-mary difference is in the
transfer of flow energy in the direction of spray. In RANS
calculation turbulent flow energy peaks at 35 mm downstream of
nozzle and dis-sipates quickly afterwards in the spray downstream
locations. Coherent structure and vortex turnover time-scale In
previous section (analysis of 2D line vortex) it is shown that
coherent structure may be analyzed using relation between λ2 and
τε. In successive grid refine-ments relation between λ2 and τε
obtained from resolved velocity field in LES and mean flow field in
RANS calculations are compared in Figure 22 and Figure 23
respectively. From these figures, it is possible to draw the
following conclusion: • LES resolved flow results shows a distinct
upper line-
ar bound between -λ2 and 21ετ
at 0.9 ms ASOI, after
steady state is achieved for spray liquid penetrations.
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• The scatter plots between -λ2 and 21ετ
obtained from
LES resolved fields are self-similar between various grid
refinement levels. • RANS mean flow results fails to show the
distinct upper linear bound between -λ2 and 21
ετ at 0.9 ms
ASOI, i.e. after steady state is reached for spray liquid
penetrations. Spray induced turbulence may be perceived as
superpo-sition of vortical motions of all length scales. Results
that reveal more statistics in coherent structure (rela-tions
between –λ2 and τε) are able to resolve more scales of turbulence
motion in the CFD grid itself. Flow structures that exhibits “sold
body rotation” like mo-tions in vortex eye regions are stable and
more likely to dissipate energy to vortex roll-up and outer vortex
re-gions due to viscosity. This is one of the mechanisms of
turbulent dissipation in the inertial sub-ranges of turbu-lent
flow. Therefore in order to resolve energy contain-ing scales, it
is important to resolve flow statistics in the vortex eye region
which is represented by upper linear bound in
− 22 1,ετ
λ space.
In RANS prediction, due to the nature of turbulence modeling
with eddy viscosity the distinct upper linear bound in
− 22 1,ετ
λ space is not captured by the mean
flow motions.
Summary In this paper, spray induced turbulence is studied using
LES as well as RANS calculations. The spray liquid and vapor
penetrations from the simulations are com-pared against the
experimental measurements in Sandia National Laboratory’s Spray A
configuration. Addi-tionally CFD grid sensitivity study is
presented to es-tablish criteria for CFD mesh size for accurate
predic-tions of fuel-air mixing based on coherent structures from
resolved flow motions. An analysis based on re-solved flow
statistics in
− 22 1,ετ
λ space is proposed.
From this study following remarks can be made: • With RANS
turbulence predictions the mechanism of fuel-air mixing is governed
by diffusion in the direction of maximum gradient. In LES
predictions, in addition to diffusion, fuel-air mixture is also
predicted due to the contribution of breakup of large fuel rich
pockets into smaller structures. • With sufficient CFD grid
resolutions (~0.5 mm), LES scalar mixing exhibits intermittencies
and instabilities of turbulent motions.
• In LES one equation, non-viscosity, dynamic structure model
CFD mesh of 0.5 mm average grid size can pro-vide sufficient
resolution of turbulence flow field sur-rounding the spray jet to
achieve grid convergence and provide better predictions of fuel-air
mixing. • In spray induced flows, turbulent structures are
gener-ated in the region of secondary breakup due to drag induced
motions. These structures are then carried by large scale motions
in the downstream locations of spray. Turbulent instabilities are
predominant in the location away from nozzles (> 50 mm) •
Turbulent time spectra shows energy cascades from flow motions of
larger time scales to smaller time scales. It is also observed that
LES predicts transfer of energy to further downstream locations
(~65 mm from nozzle) than RANS (where turbulent flow energy peaks
at ~35 mm from nozzle). In conclusion this study provides a
comprehensive study of LES and RANS predictions of Sandia Spray A
and provides analysis of fuel-air mixture predictions based on
evolution of spray induced turbulence. It is recommended that this
work may be extended to other spray cases of varying injection
pressures to provide valuable insight to correlate the regions of
sprays and fuel-air mixture. Authors would like to acknowledge the
funding support from Department of Energy Award DE-EE0000202 and
computational resource at Engine Research Center, University of
Wisconsin – Madison.
Nomenclature Acronyms ASOI After Start of Injection CFD
Computational Fluid Dynamics DNS Direct Numerical Simulation LES
Large Eddy Simulation RANS Reynolds Averaged Navier Stokes SGS
Sub-Grid Scale Non-dimensional numbers Re Reynolds Number Roman
Symbols ksgs Sub-grid scale kinetic energy of flow motion
[m2/s2] p Pressure [N/m2] r Radius [m] rc Radius of vortex
corresponding to maximum
azimuthal velocity [m] t Time [s] Vcell Volume of CFD grid cell
[m
3] Vθ Azimuthal velocity of vortex [m/s] Vθ,max Maximum
azimuthal velocity of vortex [m/s]
-
sWɺ Source term from droplet – gas phase kinetic ener-
gy exchange in sub-grid scale [kg/m-s3] Greek Symbols ∆ LES grid
filtered length scale [m] Γ Circulation of vortex [m2/s] λ1, λ2, λ3
Eigenvalues of symmetric tensor to define
coherent structure [1/s2] µ Dynamic viscosity of fluid
[kg/m-s]
tµ Turbulent dynamic viscosity [kg/m-s] νmol Molecular kinematic
viscosity [m2/s] ρ Density of fluid [kg/m3] τε Vortex turnover
timescale [1/s] Vector Symbols
iF Momentum interaction (drag force per unit volume) from liquid
spray to gas phase flow [kg/m2-s2]
ui, Velocity vector of gas phase [m/s]
ωω ,i Angular velocity vector of gas phase [1/s] xxi , Position
vector in Cartesian coordinate [m]
Tensor Symbols L ij Leonard term [m
2/s2] Sij, Symmetric strain rate tensor [1/s] τij Sub-Grid Scale
stress term [m2/s2] Ωij,Ω Anti-symmetric strain rate tensor [1/s]
Symbols, Embellishments, Superscripts (Φ represents a generic
variable)
φ To denote LES Grid-filtered quantity φ~ To denote Favre
averaged quantity
φ̂~ To denote LES test filtered quantity References
1. Manley D. K., McIlroy A., Taatjes C. A., (2008), “Research
needs for future internal combustion en-gines,” Physics Today,
(November), pp. 47-52. 2. Jhavar R., Rutland C. J., (2006), “Using
Large Eddy Simulations to Study Mixing Effects in Early Injection
Diesel Engine Combustion,” SAE Technical Paper 2006-01-0871. 3.
Rutland C. J., (2011), “Large Eddy Simulations for Internal
Combustion Engines – A Review,” Internation-al Journal of Engine
Research, 12(5), pp 421-451.
4. Pope S. B., (2000), Turbulent Flows, Cambridge, University
Press. 5. Pope S. B., (2004), “Ten questions concerning the
large-eddy simulation of turbulent flows,” New Journal of Physics,
6, pp. 35-35. 6. Banerjee S., (2011), “Study of Low Temperature
Combustion Using Large Eddy Simulations,” PhD dis-sertation,
University of Wisconsin – Madison. 7. Jeong J., Hussain F., (1995),
“On identification of vortex,” Journal of Fluid Mechanics, 285(1),
pp. 69-94. 8. Amsden A. A., O’Rourke P. J., Butler T. D., (1989),
“KIVA II: A Computer Program for Chemically Reac-tive Flows with
Sprays,” Los Alamos National Labora-tory, Los Alamos. 9. Pomraning
E., (2000), “Development of Large Eddy Simulation Turbulence
Models,” PhD Dissertation, University of Wisconsin – Madison. 10.
Jhavar R., Rutland C. J., (2006), “Using Large Eddy Simulations to
Study Mixing Effects in Early Injection Diesel Engine Combustion,”
SAE Technical Paper 2006-01-0871. 11. Bharadwaj N., Rutland C. J.,
Chang S., (2009), “Large eddy simulation modeling of spray-induced
tur-bulence effects,” International Journal of Engine Re-search,
10(2), pp 97-119. 12. Banerjee S., Bharadwaj N., Rutland C. J.,
(2009), “ Investigation of In-Cylinder Mixing Using Large Eddy
Simulation Models for LTC Diesel Applications,” ASME 2009 Internal
Combustion Engine Division Spring Technical Conference, ASME,
Milwaukee, WI, pp. 521-527. 13. Pickett L., Genzale, C., Bruneaux,
G., Malbec, L., Hermant L., Christiansen C., Schramm J., (2010),
"Comparison of Diesel Spray Combustion in Different
High-Temperature, High-Pressure Facilities," SAE Int. J. Engines
3(2):156-181, 2010. 14. Banerjee, S., Rutland, C., (2012) "On LES
Grid Criteria for Spray Induced Turbulence," SAE Technical Paper
2012-01-0141.
-
Table 1: Model Lamb-Oseen vortex parameters
LOV-1 LOV-2 LOV-3 LOV-4
Γ [m2/s] 62.83 125.66 314.16 628.32 rc [m] 0.01 0.01 0.01
0.01
Table 2: CFD grid specifications
Grid Name Number of grids Nx, Ny,
Nz
Δx ≈ Δy (mm) Δz (mm)
Grid A 15, 15, 25 2.0 4.0
Grid B 30, 30, 50 1.0 2.0
Grid C 60, 60, 100 0.5 1.0
Grid D 60, 60, 200 0.5 0.5
Table 3: Specification of Sandia National Lab "Spray A"
experiments [13]
Ambient gas temperature 900 K
Ambient gas pressure ~60 bar
Ambient gas density 22.8 kg/m3
Ambient gas velocity Near quiescent
Ambient gas oxygen ~0 % (by mass)
Fuel injection pressure 1500 bar
Fuel n-dodecane
Fuel temperature at nozzle 363 K
Injection duration 1.5 ms
Fuel injection quantity ~3.6 mg
-
Figure 1: Relation between λλλλ2 and vortex turnover timescale
for number of model Lamb-Oseen vortexes
-
Figure 2: Normalized velocity profile (top) and Instability
analysis (bottom) of Lamb-Oseen Vortex
-
Figure 3: CFD grids for spray induced turbulence study
Figure 4: Top view (left) and mid-section view (right) of CFD
grid C
-
Figure 5: Spray liquid penetrations for Spray A
-
Figure 6: Spray vapor penetration for Spray A
Figure 7: Spray liquid and vapor penetrations from LES at 1.4 ms
ASOI
-
Figure 8: Spray liquid and vapor penetrations from LES at 2.5 ms
ASOI
Figure 9: Spray liquid and vapor penetrations of spray A at 0.28
ms (left) and 0.53 ms ASOI with experi-mental images in top and LES
grid C results in bottom
Figure 10: Spray liquid and vapor penetrations of spray A at 0.8
ms (left) and 1.55 ms (right) ASOI with ex-perimental images in top
and LES grid C results in bottom
-
Figure 11: Visualization of coherent structures colored by
velocity and spray droplets (black) from the LES calculations of
spray A at 1.4 ms ASOI
Figure 12: Visual representation of coherent structures from LES
resolved solutions of spray induced turbu-lence with successive
grid refinements [6]
-
Figure 13: Turbulent coherent structures from LES resolved flow
solutions at the end of injection of spray induced turbulence.
Left: One-equation eddy viscosity model, Middle: Zero Equation
Smagorinsky, Right:
Non-viscosity dynamic structure model [6]
Figure 14: Spray liquid and vapor penetrations from RANS at 1.4
ms ASOI
-
Figure 15: Visualization of coherent structures colored by
velocity and spray droplets (black) from RANS calculations of spray
A at 1.4 ms ASOI
Figure 16: Spray vapor penetration predictions from LES and RANS
calculations
0
10
20
30
40
50
60
70
80
90
100
0.0 0.5 1.0 1.5 2.0 2.5
Va
po
r p
en
etr
ati
on
[m
m]
Time ASOI [ms]
LES - Grid D
RANS - Grid D
Experiment
-
Figure 17: Spray droplets colored by droplet drag from LES Grid
D prediction at 0.3 ms ASOI (left: droplets only, right: droplets
and coherent structures around droplets)
Figure 18: Evolution of coherent structures (grey surfaces) due
to spray induced turbulence in the down-stream of spray jet
Primary breakup region
Secondary breakup region
Generation of coher-ent structures
Breakup and intermittence in coherent struc-tures
-
Figure 19: Evolution of coherent structures (gray surfaces) due
to spray induced turbulence in the down-stream of spray jet
Figure 20: Turbulence time spectrum from LES calculations (Grid
D) at various locations downstream of nozzle
10-2
10-1
100
10-4
10-2
100
102
Frequency [1/s]
Ene
rgy
Spe
ctru
m [m
2 /s]
Sandia Spray A Time Spectrum - LES Grid D
15 mm from nozzle35 mm from nozzle55 mm from nozzle65 mm from
nozzle
-5/3 slope
-
Figure 21: Turbulence time spectrum from RANS calculations (Grid
D) at various locations downstream of nozzle
10-2
10-1
100
10-4
10-2
100
102
Frequency [1/s]
Ene
rgy
Spe
ctru
m [m
2 /s]
Sandia Spray A Time Spectrum RANS Grid D
15 mm from nozzle35 mm from nozzle55 mm from nozzle65 mm from
nozzle
-5/3 slope
-
Figure 22: Scatter plot of -λ2 vs. vortex turnover time scale
(ττττε) at 0.9 ms ASOI for LES results from various CFD grid
configurations
-
Figure 23: Scatter plot of -λ2 vs. vortex turnover time scale
(ττττε) at 0.9 ms ASOI for RANS results from various CFD grid
configurations