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Application of a Hybrid RANS-LES CFD Methodology to Primary Atomization in a Coaxial Injector


Abstract and Figures

Non-zonal hybrid RANS-LES models, i.e. those which do not rely on user-prescribed zones for activating RANS or LES, have shown promise in accurately resolving the energy-containing and highly anisotropic large-scale motions in complex separated flows. In particular, the recently proposed dynamic hybrid RANS-LES (DHRL) approach, a method which relies on the continuity of turbulence production through the RANS-to-LES transition zone, has been validated for several different compressible and incompressible single phase flow problems and has been found to be accurate and relatively insensitive to mesh resolution. Time-averaged source terms are used to augment the momentum balance. An added benefit of the DHRL is the ability to directly couple any combination of RANS and LES models into a hybrid model without any change to numerical treatment of the transition region. In this study, an attempt is made to extend the application of this model to multiphase flows using two open literature coaxial two-stream injectors involving non-Newtonian liquids. For the first time, the new model has been successfully implemented in a multiphase framework, combining the SST RANS model with MILES LES approach. Favre averaging is used to ensure consistency between the momentum equations and the density fluctuations. It was found that the momentum source terms must be density weighted in order to ensure stability of the solution. Primary atomization findings with a stable model are encouraging. The spray character with the new model was somewhere between that of a RANS model and the LES result. Droplet sizes, which are indicative of the shear layer energy, for the RANS model were greater than the hybrid results, which were comparable to the LES result and matched the experimental expectation. Additionally, the new approach showed a liquid core breakup length close to that expected from the literature.
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1 Copyright © 2015 by ASME
Proceedings of the ASME 2015 International Mechanical Engineering Congress & Exposition
November 13-19, 2015, Houston, TX, USA
Wayne Strasser
Eastman Chemical Company
Kingsport, TN, USA
Francine Battaglia
Virginia Polytechnic Institute and
State University
Blacksburg, Virginia, USA
Keith Walters
Mississippi State University
Starkville, MS, USA
Non-zonal hybrid RANS-LES models, i.e. those which do
not rely on user-prescribed zones for activating RANS or LES,
have shown promise in accurately resolving the energy-
containing and highly anisotropic large-scale motions in
complex separated flows. In particular, the recently proposed
dynamic hybrid RANS-LES (DHRL) approach, a method
which relies on the continuity of turbulence production through
the RANS-to-LES transition zone, has been validated for
several different compressible and incompressible single phase
flow problems and has been found to be accurate and relatively
insensitive to mesh resolution. Time-averaged source terms are
used to augment the momentum balance. An added benefit of
the DHRL is the ability to directly couple any combination of
RANS and LES models into a hybrid model without any
change to numerical treatment of the transition region. In this
study, an attempt is made to extend the application of this
model to multiphase flows using two open literature coaxial
two-stream injectors involving non-Newtonian liquids. For the
first time, the new model has been successfully implemented in
a multiphase framework, combining the SST RANS model with
MILES LES approach. Favre averaging is used to ensure
consistency between the momentum equations and the density
fluctuations. It was found that the momentum source terms
must be density weighted in order to ensure stability of the
solution. Primary atomization findings with a stable model are
encouraging. The spray character with the new model was
somewhere between that of a RANS model and the LES result.
Droplet sizes, which are indicative of the shear layer energy, for
the RANS model were greater than the hybrid results, which
were comparable to the LES result and matched the
experimental expectation. Additionally, the new approach
showed a liquid core breakup length close to that expected from
the literature.
The agricultural, chemical, food, fire protection, and
energy-production industries have all advanced as a result of
the study and understanding of the breakup and atomization of
jets since Felix Savart in 1833. Controlling the droplet
characteristics and trajectories can be critical for high yields
and productivity of reactors, process equipment, gas turbines,
and reciprocating engines. A thorough introduction to injectors
and sprays is given in Lefebvre [1]. Correlations for Sauter
mean diameter (SMD) are often given in terms of Weber
number (We) and Ohnesorge (Oh) and attempt to approximate
the effects of primary and secondary atomization. These two
dimensionless parameters can be thought of as measures of the
restraining forces for atomization. Designs which reduce the
mean droplet size also tend to narrow the size distribution of
droplets, including an increase in air velocity and/or a reduction
in liquid feed rate.
A fundamental study of primary atomization was carried
out by Lienemann, Shrimpton, and Fernandes [2]. As We
increases, a Kelvin-Helmholtz (KH) instability (driven by
shear) was shown to disturb the balance between momentum
and surface forces and cause sheet flapping. Typical wave
thicknesses were above the theoretical inviscid thickness value.
Wave propagation thins the sheets further. In addition, sheet
perforations can be caused by entrained air and/or boundary
layer (vorticity) development from the orifice and bulk orifice
turbulence. These perforation events depend on the length and
time scales of the turbulence structures. They studied in-tact
sheet radius and show a LCWe-x effect, where “x” ranges from
0.33 to 0.42. A similar power law dependence on phase density
ratio, , was also displayed. Thinner sheets are obviously
advantageous to fine droplets. Close to the rim of the sheet,
waves changed from a sinusoidal shape (linear instability) to a
zigzag pattern (non-linear) as vortices were shed off wave
2 Copyright © 2015 by ASME
crests and move into wave troughs. This is the onset of break-
up. Their work outlined dominant wave frequencies and sheet
thickness dependence on feed conditions and feed stream
angles. They discussed how the local sheet We value can
produce a shift in the preferred mode of instability, with
symmetric waves (in-phase) produced below 1 and dilatational
(phase opposition) above 1. Symmetric modes grew faster.
Lastly, they addressed azimuthal sheet curvature. When the
inner sheet diameter is small compared to the outer (as in a
two-stream injector), it behaved like a full jet. For large values,
like a three-stream injector, it behaved like a thin, inviscid
sheet, favoring linear stability analysis. Above a certain liquid
viscosity, however, linear analysis could vastly over-predict
droplet size.
The effect of viscosity and surface tension for non-
Newtonian fluids on SMD has been explored by Aliseda et al.
[3]. They found that that instability wavelength was
substantially affected by higher viscosities. As expected, higher
viscosities hindered the growth of instabilities resulting in
larger droplets, but only when viscosity was well above ten
times that of water. In fact, there was an extremely pronounced
coupling effect of surface tension and viscosity. They proposed
that the lower surface tension of the non-Newtonian solutions
(about one third that of water) prevented their SMD values
from being orders of magnitude larger than that of water,
instead of just two to three times its value.
Also in 2008, Dumouchel [4] outlined theories and
findings related to cylindrical liquid jets, flat liquid sheets, air-
assisted cylindrical jets, and air-assisted flat liquid sheets. It
was hypothesized that in order for a spray to form, there has to
be initial disturbance(s) at the liquid-gas interface and a
mechanism for those disturbances to grow. The final spray
droplet sizes are individually set by a balance of local
disturbances and local liquid cohesion force. Although [5]
demonstrated the importance of feed liquid profiles for
situations without gas co-flow, [4] found the details of the
liquid flow to be of lesser importance than those of the
surrounding gas for gas-assisted flows. The important effects
include total liquid momentum, gas vorticity, gas phase
turbulence, and the recirculation zone outside of the nozzle.
Similarly, Fuster et al. [6] investigated the effect of gas
momentum ratio (M) and gas boundary layer thickness on the
dominant response frequency of wave growth from a simple
gas/liquid splitter plate arrangement. They modulated the gas
stream and found that the excited wave response dynamics
were dependent upon the thickness the gas boundary layer.
Turbulence Effects
The importance of the surrounding gas phase turbulence
field has already been shown to play a role in primary
atomization. Lasheras et al. [7] showed how breakup from
turbulent resonance with droplets can supersede shear
atomization. Chen et al. [8] discussed resonance in the
turbulence field playing a role in break-up when the small scale
turbulence frequency matches the natural mode of an entrained
droplet. It was surmised that the size of the smallest droplets is
comparable to the size of the coherent turbulent structures that
are just large enough to overcome the surface energy when We
is less than 10. They assumed turbulence isotropy and the
existence of an inertial subrange in the turbulence spectrum.
Their primary atomization models took the common form in
which the SMD is proportional to LCWe-x with modifications
for a “real” radial turbulence kinetic energy (k) distribution, as
opposed to a uniform value. They show SMD related to k-x,
which indicates that droplet size increases as k decreases. The
implication is that, ideally, primary atomization would occur in
a region of high turbulence energy for a lower equilibrium
droplet size.
Maniero et al. [9] discussed treating a droplet (in a liquid-
liquid co-flowing system) as a harmonic oscillator to predict its
secondary breakup. Here, the proposal is that the droplet does
not respond instantly to its surrounding turbulent field, and
some past examples include an eigenfrequency of 43 Hz for a
low-viscosity fluid droplet. The authors conclude that the
droplets in their study had length scales in the inertial subrange
of the continuous phase turbulence, but the Reynolds number of
the study was so low that clear distinctions of turbulent length
scales could not be determined with confidence. The oscillator
forcing function is equal to a constant times the turbulent
Weber number, for which the typical velocity scale is replaced
by a structure function; an Eulerian calculation is employed to
estimate the velocity differences at scales on the order of the
expected droplet size. Only the maximum over the considered
space dimensions is taken as the structure function. Instead of
breakup occurring at a critical Weber number, it occurs at a
critical deformation of approximately unity, making the
constant equal to ~ 0.06. In addition, sub-grid scale (SGS)
VOF modeling has been addressed by Krause et al. [10] for
cases in which droplet coalescence immediately upon droplet
contact is not a true representation of reality. For completeness,
it should be mentioned that the bi-directional coupling of the
turbulence field with entrained materials, i.e. continuous phase
turbulence attenuation/enhancement by droplets or particles has
been addressed by many (Strasser [11] and Ling et al.[12] to
name two); however, that will not be quantified here.
Hybrid Turbulence Modeling
It is well-known that the RANS approach to computation-
ally quantifying the effects of turbulence leaves much to be
desired, even in many single phase flow scenarios and
especially with those involving separation. Adding a second
phase only complicates the situation. Co-axial injectors have
separated shear layers proceeding from the feed nozzle, as well
as separation and boundary layer development around
individual ligament/droplets. Moreover, large density gradients
are present. In RANS modeling, all of the key length scales of
turbulence are lumped into a single model, often employing
gradient diffusion, and the effects of the fluctuations are
superimposed onto the mean. A substantial improvement
would be to invoke the use of LES, in which only the largest
turbulent motions are resolved and characterized; these contain
most of the energy and anisotropy. As a result, treating the
3 Copyright © 2015 by ASME
smaller, more uniform scales in a sub-grid scale (SGS)
statistical model poses less of a risk. There is an obvious
expense associated with LES, as a much higher grid resolution
(and the commensurate smaller time step) is required to
evaluate those larger scale motions. Steep velocity gradients,
such as those near walls, dramatically increase the expense. In
atomization, however, walls are not the only items that are
much slower than the gas; accelerating droplets and ligaments
could be included in this category as well. The term "slip
velocity" is applied to the difference between the two phases. It
would be expected to be the same, or very close to, the value of
the gas velocity near the feed injector.
A hybrid approach is often sought to mitigate the expense
of LES. Using this method, LES is employed primarily for
unsteady separated flow regions, while RANS is employed in
near-wall regions or regions with insufficient mesh resolution
to resolve fine-grained turbulence structures. In general, there
are two categories of hybrid modeling. "Zonal" relies on a
priori knowledge and/or decision-making regarding the
placement of LES and RANS regions. The grid is demarcated
accordingly at the outset. An obvious disadvantage of this,
especially in regards to atomization, is that the LES region
changes in time. A "non-zonal" approach implies that the
location of the RANS and LES regions are allowed to
dynamically adjust to the flowfield of interest, with no need for
explicit demarcation a priori. The acute challenge with non-
zonal models is the handling of the communication of resolved
fluctuating quantities from the LES section(s) of the domain to
the completely modeled quantities in the RANS section(s).
Another critical aspect of the non-zonal is the switching
criteria, i.e. how to determine when to transition from one
model type to the other. Depending on where the dominant
supply of turbulence production occurs, the outcome could vary
A few of these switching mechanisms and criteria are
explored in Breuer et al. [13]. The most common approach is
to use grid resolution metrics, such that the LES model is
invoked in refined mesh zones via suppression of the turbulent
viscosity, either directly or through an increase in modeled
turbulent dissipation rate. The latter, for example, is the method
adopted in the well-known Detached Eddy Simulation (DES)
model. An alternative to this was proposed and discussed by
Bhushan et al. [14] and Walters et al. [15]. The transition
between the two modes is determined by the continuity of total
turbulence production. The advantages include 1) the ability to
couple any RANS model with any LES model, 2) freedom from
explicit grid dependence, and 3) the return to the baseline
RANS model in steady-state flow. This method will be tested
in this work, and more details are provided below.
Multiphase Complications
In gas-liquid multiphase studies, the two immiscible
phases, liquid and gas, often exist in the same computational
cell. Given the large difference in viscosity and density
between liquid and gas, there are substantial jump conditions at
the phase interface. This interface exists at the subgrid (SGS)
level; its reconstruction is often sought through the particular
multiphase algorithm in ways somewhat analogous to how
shocks are captured in high Mach number flows. Aside from
discretization and flux computational challenges outlined in
Rider and Kothe [16], Gueyffier et al. [17], and Davidson [18]
(in regards to the importance of instantaneous pressure-velocity
coupling in LES structure resolution), turbulence production
and transport at the SGS may be compromised by assuming a
homogenous approach. Tavangar et al. [19] studied non-
Newtonian slurry atomization using a homogenous LES and
found strong agreement with experimental results; however,
formal mathematical consideration of the governing equations
shows that filtering operations produce new SGS terms and
their associated instabilities. Liovic and Lakehal [20], [21, 22],
Vallee et al. [23], and Navarro-Martinez [24] highlight various
approaches taking into account these effects as sources of
turbulence in LES and RANS approaches. The authors of [24]
described how SGS surface tension, for example, is important
in cases of high surface tension, unresolved interface curvature,
and flow relaminarization. Obviously, unresolved curvature is
a risk for all CFD modeling. Navarro-Martinez [24] discovered
the fact that SGS effects are not important until after
approximately 20 jet diameters. At the same time, however,
velocity gradients at the interface can induce modeled
turbulence that is actually not present in the real flow and
requires a modeled damping function (Hansch et al. [25]).
Given the fact that these effects are not always important and
are partially offsetting, they are ignored in the present work.
Aside from SGS instability sources, a first order
consideration is the fact that the gas “sees” the liquid as a wall
with a differential velocity of varying degrees, depending on
the local phase slip velocity. Near the nozzle, the gas is moving
much faster than the liquid (hence the purpose of gas-assisted
atomization). The relatively stationary liquid should enhance
turbulence production in the gas phase while decelerating the
gas. At some distance from the nozzle, where the liquid has
been disintegrated, and the droplets are moving nearly the
speed as the decelerated gas, the slip velocity is much lower.
Here, there is very little production between the two phases.
Lastly, in the far-field, the higher density liquid droplets retain
their momentum. The gas, now slower than the liquid, is
dragged along by the liquid, once again enhancing turbulence.
A large body of literature is available to address this for
Lagrangian methods (see Crowe [26] and Strasser [11] for
example). However, for SGS drag, these effects are often
ignored. Liovic and Lakehal [20] described an alternative to the
homogenous approach such that these effects can be included.
They discussed the need to construct phase-dependent velocity
gradient scaling and the associated “wall distance” (actually,
distance from the gas-liquid interface) to feed a Van Driest-
style damping algorithm. Implementing a phase-dependent
turbulence transport methodology will be foregone herein as a
first step towards progress in hybrid modeling. At the macro
scale, these discussed gas-liquid velocity gradients are going to
support turbulence generation, and this will be explored.
4 Copyright © 2015 by ASME
a Speed of sound
D Liquid orifice diameter
F Surface tension body force
i Summation index
k Turbulence kinetic energy
La Laplace num. = Re2/We = Suratman num.
LC Some characteristic length scale
M Gas/liquid momentum ratio = (U2)G/(U2)L
Oh Ohnesorge number = /Re = 1/
p Pressure
Re Reynolds number = UD/
SMD Sauter mean diameter (“D32”)
u Velocity component
U Velocity magnitude
y+ uty/
We Weber number = GUG2D/
Stress DHRL coefficient, 1 = fully LES
Complimentary DHRL stress coefficient
Turbulence dissipation rate
Surface tension
Specific dissipation rate
Stress tensor
Phase volume fraction
γ Outer gas/liquid annular approach angle
μ Molecular viscosity
Subscripts and Superscripts
L Liquid
G Gas
t Turbulent
It is hoped that the broader three-stream coaxial injector
focus of Strasser [27] and the associated works will be unified
and extended in this document as methods enhancements are
sought. Although the overall project is concentrated on a three-
stream self-sustaining pulsatile injector involving compressible
flow and breakup of a non-Newtonian slurry, the new dynamic
hybrid RANS-LES (DHRL) method will be explored and
validated using the studies of incompressible, non-pulsatile
two-stream coaxial primary atomization of non-Newtonian
liquids by Mansour and Chigier [28] and Zhao et al. [29].
Additionally, Strasser and Battaglia [30] showed that the ratio
of integral length scale to mesh element length scale of the two-
stream injectors, relative to the pulsatile injector, were more
conducive to hybrid modeling. It should be noted that all
coaxial atomization may fundamentally be considered
“pulsatile” to some extent in that, from a fixed reference frame
watching the passing spray, there are temporal oscillations in
liquid volume fraction and spray droplet number density. (Plus,
two fluids merging at different velocities will encourage
fluctuations at some scale.) However, the three-stream injector
in the cited works is tuned such that there is substantial mass
flow variability in the bulk feed boundary conditions.
Table 1 (SI units) shows the important parameters in the
first study. Note that the Weber number is the average, but the
instantaneous values of the Weber number can be very different
from the average and can cause the liquid to span multiple
breakup regimes ([31]) caused by polydisperse gas fluid
structures. Of the available liquids from their study, the 0.3%
(by mass) Xanthan gum solution is the closest to the non-
Newtonian liquid of the pulsating injector. Figure 1 offers a
schematic of the computational representation of their nozzle.
The liquid is fed in a central jet of about 3.72 m/s, while gas is
fed in an annular region at 93.4 m/s. The feed liquid Reynolds
number is about 4000 when defined using the high-shear
viscosity, making it transitional. Consequently, in CFD, a flat
profile was used. Although the Xanthan gum solution has a
strong viscosity dependence on shear, they echoed the
commonly known view that only the highest shear viscosity is
the droplet size controlling factor ([3], [32]). Using an optical
method called phase Doppler particle analyzer (PDPA), they
assessed the local droplet SMD at 0.254 meters (10") from the
liquid outlet. As described in Dumouchel [33], the
disadvantages to the method are 1) only spherical droplets can
be measured, 2) only one droplet in an optical volume can be
measured at a time, and 3) a validation study of the ratio of
correctly measured events to the total events was not given. In
effect, if large ligaments (non-spherical) are present at the
measurement location, the reported SMD will be
underestimated. [31] communicated that most objects are non-
spherical at initial breakup. If the view of smaller droplets is
blocked by larger ones, the reported SMD will be
overestimated. Regardless, [28] showed 134 m at the
measurement location for the conditions in Table 1.
Table 1: Important quantities and boundary conditions for
study # 1 (Mansour and Chigier [28]); All are in SI units.
Liquid density
Liquid viscosity at highest shear
Liquid surface tension
Liquid orifice inner diameter
Liquid orifice outer diameter
Liquid mass flow
Liquid velocity
Gas density
Gas viscosity
Gas orifice inner diameter
Gas mass flow
Gas velocity
Gas to liquid mass ratio
Weber number
Ohnesorge number
5 Copyright © 2015 by ASME
Based on the work of Farago and Chigier [34], liquid feed
Reynolds number, and Weber number, this disintegration
processed would be termed “Fiber” type and would be
characterized by fibers peeling off the main liquid core and then
subsequent fiber breakup by non-axisymmetric Rayleigh
breakup. It is “pulsating” in the sense that pockets/bursts of
varying liquid density fields are passing through the domain;
however, it does not contain the bulk axial mass flow rate
fluctuations of a three-stream injector.
Mesh and Boundary Conditions
The 360° azimuthal mesh contained 8.3 million
hexahederal cells, and an interior slice is shown in Fig. 1. As
mentioned in the figure, the mesh length scale was around 80
m near the center line in all three dimensions. Each run
required approximately a month on 32 Intel Xeon E5-2643 3.3
GHz cores. Due to industrial sponsor funding limitations, an
axial span of only 0.0635 m was modeled. For validation
purposes, the resulting time-averaged droplet size from this
work will be compared to those of Mansour and Chigier [28]
from much farther away from the nozzle. An estimate will,
therefore, be sought for the droplet size change from a distance
of 0.0635 m (computational measurement location) from the
liquid nozzle outlet to a distance of 0.254 m (experimental
measurement location) from the outlet. Dumouchel [33],
utilizing the work of Shraiber, Podvysotsky, and Dubrovsky
[35], provided an estimate for the critical droplet size of 90 m
above which secondary atomization is likely. A dimensionless
period of oscillation is estimated to be ~0.029 and a
deformation time scale of about 84 s. That gives plenty of
time for more breakup and a final diameter ratio of about 3.3:1.
In other words, a coarse estimate of the droplet size at the
computational measurement location is ~440 m. As a first
approximation, all of this ignores the fact that the presence of
solid particles will make breakup easier for a given viscosity as
described in Zhao et al. [29]
Fig.1: Computational Mesh of the Co-axial Injector
Governing Equations
A subset of the Eulerian-Eulerian methodology is the VOF.
The method is described in detail in the authors' prior works,
but an overview will be given here. The continuity equation
governing the mass balance of each phase is:
The phase-averaged Reynolds-averaged linear momentum
balance is:
 
 
reft p
It can be seen that the gas and liquid share a common
momentum field, and properties are mass-averaged among the
phase volume fractions present in a cell. With this method, film
formation, ligament production, and droplet onset, as well as
turbulence, are explicitly accounted for. The air density is
assumed constant. Liquid droplet evaporation due to gas
humidity effects has been ignored. Surface energy effects are
treated via the continuum surface force method of Brackbill et
al. [36].
Two groups of turbulence models are compared here. The
first is the shear stress transport (SST) two-equation linear eddy
viscosity model of Menter [37] and is used for computing the
turbulent contributions to momentum such that gradient
diffusion hypothesis has been used to separate the molecular
and turbulent effects in Eq. 2. A homogenous approach is
considered in that only one turbulence field is computed for
both phases. In the SST model, additional consideration is
given to the transport of the principal turbulent shear stress via
1) an eddy viscosity limiting function and 2) a cross diffusion
term in the transport equation for ω. Also, there is a turbulence
production limiter, as discussed in ANSYS [38], preventing the
artificial build-up of fluctuating velocity in regions of
irrotational strain. "Scalable" wall functions, discussed in
ANSYS [38], are an alternative to standard wall functions.
They have the advantage of being less sensitive to variation in
near-wall grid resolution throughout the domain. The distance
of a given computational cell center from the wall is computed
via a Poisson equation with a uniform source value of -1.
The second model compared here is the DHRL approach,
which was introduced earlier and has been documented
thoroughly in the associated references. Additionally, a brief
outline is provided. First, an incompressible version of
equation 2 without body forces is filtered (additional needs
associated with the multiple phases highlighted later) as
  
 
 (3)
with the turbulent stress (last term, any residual stress) given
  
  (4)
Liquid Inlet Gas Inlet
Cell length scale is
~ 80 m in all
three dimensions.
6 Copyright © 2015 by ASME
Now, a decision has to be made on how the ensemble-averaged
velocity field (Reynolds stress) is bridged with the spatially-
filtered velocity field (subgrid stress). To that aim, the
instantaneous velocity () is decomposed as follows.
 
 (5)
where is the velocity extracted from the simulation in each
cell, is the mean (RANS) velocity,  is the resolved
fluctuating velocity, and  is the unresolved fluctuating
velocity. Only  requires modeling through the stress term.
After substitution, assuming there is no correlation between the
resolved and unresolved velocity fluctuations, and applying
scale similarity, the subfilter stress term becomes that below.
  
   
Taking the coefficients and to be complimentary, the result
is a residual stress that is an -weighted function of the SGS
component and the RANS component. Each individual term on
the RHS of equation 6 is normally a linear function of the SGS
and RANS inputs. A secondary filter operation is applied to
dynamically compute equal to:
 
 
 
In other words, ranges from 0 in pure RANS regions (no
resolved fluctuations) to 1 in pure LES regions. Again, there is
no explicit grid dependence. It should be noted that the RANS
inputs are computed based a running time-averaged flow field,
and this is appropriate for statistically stationary flows. In
effect, what is occurring is that a time-averaged RANS source
term is being added back into the momentum equation for the
non-LES regions. This is quite different from a typical
URANS model which adds in an ever-adjusting source term
responding to a local stress/strain balance.
In the present work, the DHRL is comprised of SST for the
RANS inputs and MILES for the LES input. The premise of
the MILES (implicit filtering) approach is that dissipative
discretization techniques, such as those available in commercial
codes, filter the flowfield a sufficient amount to represent SGS
turbulent diffusion. Any additional SGS model removes too
much of the higher wave number structures. This concept is
described in Pope [39] and is utilized by Semlitsch [40]. The
risks associated with implicit filtering were discussed by Batten
et al. [41] and Lesieur and Metais [42]. The former group
studied acoustic resonance in a cavity with various LES
approaches. They illustrated that the dissipation in an SGS
model is different from the dissipation introduced purely
through upwinding. Properly formulated upwind introduces
zero shear-stress when the shear is perfectly aligned with the
cell interfaces. They showed that effective viscosity can
suppress pressure oscillations, especially when the separation
characteristics are not defined by the geometry. However, the
use of bounded central differencing (BCD) along with MILES
produces a balanced level of dissipation superior to even the
typical dynamic Smagorinsky approach of Germano [43] as
was shown in Adetokunbo et al. [44]. Additionally, the use of a
hybrid approach, where RANS is used to make up for
unresolved LES turbulence, should help mitigate the
uncertainty associated with implicit filtering.
Since the original, single phase formulation of the DHRL
model assumed incompressible fluids, the model has been
extended to take into account strong density gradients. The
source of these density gradients could be high Mach number
flows of a gas, multiphase flows like those shown here, or
compressible multiphase flows like those in related to the cited
three-stream pulsatile injector work. Initially, the removal of
the aforementioned cross-diffusion term in the SST model was
required. This term is the dot product of dk/dxi and d/dxi. In
shear layers, the alignment of these can provide constructive
feedback and unboundedness in the turbulence model. By
removing this term, the standard k- model is approached. The
RANS portion is invoked mainly in boundary layers, where the
SST switching function would transition to the equation for
the length scale. However, the computational issues which
necessitated this were later resolved.
The second modification was much more entailed and
involved Favre averaging all of the velocity and stress
components so that density changes are correlated with phase
velocity changes. Additionally, the momentum source terms
are time-averaged density weighted. The risks associated with
this can be described as follows. The turbulence is being
generated primarily in the gas phase, but the averaged velocity
field is dominated by the liquid phase since it is a density-
weighted average. There is some evidence from industrial
sponsor testing that this leads to falsely smoothing oscillatory
behavior. Early work before employment of density-weighted
sources with a simple multiphase splitter plate showed no
smoothing, while it will be shown here (and in other
unpublished work) that smoothing may be occurring.
The third change involved how RANS and LES
components are assembled in the individual terms of Eq. 6.
Instead of a linear combination, a harmonic averaging is used,
which favors the lower turbulent viscosity values. The guiding
principle is that the boundary layers are typically going to be
present in the gas (lower viscosity) phase. In other words, what
is important to capture is the momentum diffusivity. If air is on
one side of a face and water on the other, the computed
turbulent (dynamic) viscosity will be much closer to the air.
Lastly, there is some discussion necessary around whether
the multiplier on μt gets applied to cell values or face fluxes
in the momentum source terms. Large cell density gradients
can strongly influence the source terms; if gets applied at
faces, there can be a substantial difference in μt at the different
cell faces, resulting in a very high source term applied even if
the mean velocity gradient is smooth. However, to retain the
conservative nature of the flux balances, face assignment was
7 Copyright © 2015 by ASME
Equations 1-7 were solved in ANSYS Fluent’s segregated
single precision commercial cell-centered solver version 14.5.7.
The explicit “geometric reconstruction scheme” [45] is used as
the time-marching scheme to solve equation 1 for cell face
fluxes and for interface assembly; a piecewise-linear function
(“PLIC”) is assumed for the shape of the interface across each
cell. This method was rigorously defined and tested in the
authors' prior works. See Strasser [27], Strasser and Battaglia
[46-48], for details on the PLIC and the droplet size analysis
method. A very small time step size of 4.5×10-7 seconds was
required, along with 10 sub-loops, in order to keep the VOF
reconstruction scheme well-posed and the Courant number
below unity. In fact, there are various bases for quantifying CN
according to Menter [49], most of which are less conservative
for this gas-liquid interface-driven system. Pressure-velocity
coupling is coordinated via the Pressure Implicit with the
Splitting of Operators (PISO) scheme with skewness and
neighbor corrections. A Green-Gauss node-based gradient
method is used for discretizing derivatives and is more rigorous
than a simple arithmetical grid cell center average. The pressure
field is treated with the PRESTO! (PREssure STaggering
Option) scheme. PRESTO! uses mass balances to obtain face
pressures, instead of a traditional second order upwinding
scheme, which uses geometric interpolation to obtain face
pressures from cell-centered pressures. This could be beneficial
in areas of high pressure of volume fraction gradient. The
QUICK scheme, which is formally higher than second order
accuracy on structured meshes, was used on the advection
terms for the SST approach, while BCD was used for the
DHRL turbulence model evaluation. First order upwinding was
used for SST turbulence quantities, which were dominated by
source terms. A total variation diminishing (TVD) slope limiter
(not to be confused with flux limiters) termed
“multidimensional” of Kim et al. [50] is considered for the
second order advection schemes. More details, and the
associated references, can be found in ANSYS [38]. Due to the
additional computational load of the DHRL subroutine, the
time-step solution time increases approximately 20%.
Three models will be discussed. The first is based on the
SST model, while the second is a MILES (no SGS model)
solution, and the third is the DHRL (MILES + SST). The
MILES solution is from 26 convective times (CT) worth of
time-averaging, while the DHRL (referred to as “13E”) is from
20 CT. Unless otherwise noted, all contours are taken from the
same center cut mesh plane shown in Fig. 1.
First, typical instantaneous density (directly related to
liquid volume fraction) distributions are highlighted in Fig 2 at
random sampling times among the three models. Here, red
designates liquid, while blue designates gas. There is a much
less restrained momentum field in the DHRL and MILES
models away from the liquid entry point, allowing more energy
for droplet breakup; however, the opposite is true up near the
liquid entry point. The onset of a response to the shear field is
delayed. This makes sense given that the gradient diffusion
hypothesis of SST is not upheld in DHRL or MILES. In terms
of atomization, the effect is a much larger breakup length,
which is the distance that the liquid core is intact. Menard,
Tanguy, and Berlemont [51] studied the primary atomization of
a liquid jet using DNS and found the extent of the liquid core
was shown to be about 7 jet diameters. Others will be
discussed later. Below this liquid core, the droplets get smaller,
and the spray distribution widens. All three models exhibited
this behavior, but they were different. The SST model broke up
into larger scales but very early on. The MILES approach
produced smaller droplets and also very early. This approach
showed a substantial amount of lateral movement relative to the
others, and this will be revisited later. The DHRL approach, on
the other hand, allowed much longer of a time before breakup
and then a mix of droplet length scales. Similarly, the time-
averaged density field is shown in Fig. 3. It can be see that, on
average, the liquid core length increases from left to right.
More about liquid phase ligament/droplet length scale
quantification will be discussed later.
Fig. 2: Instantaneous density contours; red designates
Fig. 3: Time averaged density contours; red designates
8 Copyright © 2015 by ASME
The relative comparison of instantaneous liquid length
scales is depicted again in in Figs. 4 and 5. Fig. 4 is a front
view of a typical liquid surface outline, similar to Fig. 2 but at
different time stamps. As before, a longer liquid core and
smaller primary liquid fragmentation by the MILES and DHRL
approaches is seen. Fig. 5 illustrates the same time stamp’s
liquid surfaces, except that the view is from the model outlet,
looking back up into the flow. The DHRL appeared to have the
narrowed spray distribution although that is not quantified here.
Fig. 4: Instantaneous liquid surfaces
Fig. 5: Instantaneous liquid surfaces, bottom view of those
in Fig. 4
Figs. 6 and 7 show instantaneous and time-averaged,
respectively, velocity magnitude contours. In both cases, the
contours scale linearly between the colors of blue = 0 m/s and
red = 93 m/s, which is the feed velocity. As expected, the
MILES and DHRL results exhibited a much more spatially
diverse flowfield. Eddy viscosity models are known to diffuse
velocity gradients, and the results corroborate that here. A
much larger range of scales of motion in the MILES, more
similar to those of DNS and LES jet studies, is evident.
Much of the explanation for the slurry breakup behavior
lies in the understanding Fig. 6. As most evident in the MILES
contour set, this two-stream atomization system involves a
wake flow where the liquid core meets the inner gas layer and a
jet flow where the outer gas layer meets the mostly quiescent
outer field. Turbulence of different character is produced in
each of these regions. In the wake, many small scale structures
are produced, while in the jet, larger bulk nearly sinusoidal
shaped waves are produced. The interaction of the two, and
especially the larger scale jet motion, is responsible for the
lateral motion previously discussed. As shown in Fig. 4, the
lateral motion of the MILES case actually creates an area of
liquid discontinuity very early in the breakup process. As will
be shown in upcoming figures of kinetic energy and breakup
length, this lateral motion is too strong. The very thin laminar
layers of velocity gradient (vorticity concentration) produces
too much turbulence and too early. The SST result, on the other
hand, also shows more lateral motion than the DHRL result but
for different reasons. The RANS approach, by design,
computes larger length scales throughout the domain; the lateral
jet energy comes from the wake as much as the jet flow.
Fig. 6: Instantaneous velocity magnitude
(red = feed velocity)
Fig. 7: Time averaged velocity magnitude (red = feed
9 Copyright © 2015 by ASME
Fig. 8: Time-averaged turbulent kinetic energy
(red 500 m2/s2)
Fig. 9: Time-averaged turbulent specific dissipation rate
contours (red 1×105 s-1)
In considering DHRL, the jet flow is shown to be much
diffused. There are nearly no structures seen in the outer gas
periphery. This is one reason for the longer liquid core and
brings into question the source of the jet smoothing. Certainly
the incorporation of a hybrid approach is partially responsible.
However, the breakup length is too much retarded. Smoothing
results from either density weighting or from the first order
transient stencil. Based on unpublished work from the authors,
it is much more likely that the transient stencil is responsible
for the jet smoothing. The mandate for using first order in time
is set by an ANSYS solver limitation. More work would be
required to develop and implement a custom higher order
transient scheme.
The time-averaged SST velocity does not look very
different from the instantaneous liquid velocity, implying that
the MILES and DHRL motion scales vary not only more in
space but also in time. This makes sense given that the RANS
approach is already time-averaging, to a large extent, the
various scales of motion. The MILES field is widened after the
initial breakup of the liquid central core, which is consistent
with the increased MILES radial spreading of the spray shown
in Fig. 2. The DHRL solution showed results somewhere in
between the two extremes with a much higher axial penetration
of the mean velocity. An interesting feature of the MILES
result, however, is unexpected striations (axial gradients) in the
time-averaged velocity field of Fig. 7. These striations result
from Favre averaging the velocity field and the “scarring”
caused by passing droplets. The droplets raise the local density,
and many CT are required to wash these gradients out. The
more energetic and dispersed (sparsely passing a single point)
the droplets, the more CT is required for a smooth field. Notice
how 13E has a smooth field after 20CT due to the less energetic
There is dramatically larger time-averaged turbulent
kinetic energy in Fig. 8 for the MILES approach. Here, blue =
0, while red 500 m2/s2 resulting from too steep of a velocity
gradient without the stabilizing effect of turbulent viscosity.
Fig. 9 illustrates time-averaged contours of specific dissipation
rate over the range blue = 0 to red 1x105 s-1. From this figure,
the main distinguishing feature is the widening of the flowfield
at the end of the MILES liquid core as has been already
Further elucidation of the relative comparison between the
two results will be provided in axial plots Figs. 10-15. Each of
these reflects time-averaged and mass flow-weighted area-
averages of a given quantity on 10 evenly spaced planar cross-
cuts. All x-axes are distances normalized by liquid orifice ID.
“13E” utilizes second order upwinding momentum
discretization, while “13D” employs bounded central
differencing. Both incorporate 20 CT of time-averaging. The
lack of differentiation between these two cases illustrates that
the space stencil is not a limiting issue.
The first plot, Fig. 10, quantifies k. As with the contour
plots, a strikingly different k field is present in the MILES
model, so much so that the MILES results are plotted on the
secondary (right) axis. The units are the same. All models
show a generally increasing trend as the shear layers take
energy from the mean flow and add it to the fluctuations. The
DHRL results are like one another, similar to SST, but increase
approximately linearly. Higher turbulence intensities indicate a
wider range of turbulent gas phase length scales and, through
fluctuating Rayleigh-Taylor instabilities, can produce a wider
range of droplet sizes [31].
Axial omega plots are shown in Fig. 11. All models
produce similar trends. Omega peaks at 4 liquid orifice
diameters from the orifice and then generally decreases for SST
and MILES models, but they were fairly smooth for the two
DHRL versions. An estimate for the integral time scale (ITS)
of motion is offered by Fig. 12 with results nearly overlapping
between the SST and MILES models; however, the two DHRL
models produce ITS values about double that of the former.
Liquid breakup occurs on timescales the local ITS, because it
takes time for the instabilities to be communicated throughout
the liquid surface [31]; at higher turbulence intensities
(fluctuating We), ITS is the controlling factor.
10 Copyright © 2015 by ASME
The next two plots are concentrated on the various length
scales present. Fig. 13 shows the estimate for the raw integral
length scale (ILS), while Fig. 14 compares these values to other
scales. Any reference to the ILS implies a local computation of
0.5k1.5/. This is a RANS-based estimate of the a length scale
such that a range of 1/6 ILS to 6 ILS contains 80% of the
energy of motion for an ideal spectrum (Pope [39]). Its
invocation herein is only meant to imply an order of magnitude
estimate for the largest scales of energy (and isotropy)-
containing motion. It is apparent from Fig. 13 that the MILES
solution exhibited much larger length scales. Both the SST and
MILES showed similar increasing trends, while the two DHRL
results are consistently lower. The ratio of this scale to the
mesh length scale is shown in Fig. 14. Davidson [18] discussed
measures “adequate” LES resolution (and making SGS length
scale be a function of magnitude of the resolved strain) says
this ratio should be 10. The MILES solution is in that range
for some of the domain, while the DHRL models showed
values around unity. This implies that, from a laterally
averaged standpoint, mesh resolution may not be ideal;
however, the consistency between the results of 13D and 13E
reminds that the spatial stencil is not a limiting issue.
Additionally, the data down the centerline of the model (not
shown) shows higher ratios for the DHRL models. Moreover,
the ILS is only an order-of-magnitude estimate.
Fig. 10: Time-averaged mass flow-weighted area-averaged
turbulent kinetic energy on 10 evenly spaced cross-cuts.
Fig. 11: Time-averaged mass flow-weighted area-averaged
specific turbulent dissipation rate on 10 evenly spaced
Fig. 12: Time-averaged mass flow-weighted area-averaged
integral time scale on 10 evenly spaced cross-cuts
Fig. 13: Time-averaged mass flow-weighted area-averaged
integral length scale on 10 evenly spaced cross-cuts
11 Copyright © 2015 by ASME
Fig. 14: Time-averaged mass flow-weighted area-averaged
ratio of integral length scale to the average mesh length
scale on 10 evenly spaced cross-cuts
Fig. 15: Time-averaged mass flow-weighted area-averaged
ratio of droplet length scale to mesh length scale
Fig. 16: Time-averaged mass flow-weighted area-averaged
ratio of integral length scale to average droplet length scale
Now, attention will be shifted to quantifying the scales of
liquid fragments and droplets, and the results are embodied in
Fig. 15. Instead of the raw data, however, the ratio of droplet
length scale to mesh length scale is given. It is important to
always test and make sure that there are enough cells present to
reconstruct the interface of a droplet or liquid contiguous body.
It has been shown by Herrmann [52] that values ranging from
at least 2 to 6 are sufficient, so both models pass. This depends
on the droplet shapes/curvature as discussed in Strasser and
Battaglia [30]. The raw values at the end of the modeled
domain are given on the plot. Note specifically that the MILES
value of 450 m is close to the adjusted experimental coarse
estimate of 440 m, while the SST value is more than double
that. The use of a differential Reynolds stress turbulence (linear
pressure strain) model resulted in an ending droplet size (not
shown) of around 2/3 of the SST value. It is clear that DHRL
ended up at nearly the same slurry length scale, and the scale
diffusivity (rate of decline, [53]) was much greater after a
longer liquid core delay. Though the ending value was higher
than target, the decay rate implies that the slurry lengths scale
would hit the target only slightly beyond the modeled domain.
In Fig. 16 the ratio of the integral scale of turbulence is
compared to the droplet length scale. For the SST and DHRL
outcomes, the droplet scale was about ten times the integral
scale, but for the MILES result, the two scales are about the
same order. This implies that the droplets would efficiently
directly interact with the turbulent structures for only the
MILES approach.
Early figures displayed the time-averaged volume fraction
contours, giving indication of a continuous liquid core length.
Table 2 shows the expected breakup length values (normalized
by the liquid orifice diameter) from various authors when their
correlations are framed for this geometry and flow conditions.
Lasheras et al. [7] explained that the liquid core length
definition could vary among authors and could be at volume
fraction equal to 0.5. Additionally, it is not clear if authors of
the experimental studies, using frame-by-frame video analysis,
considered liquid cores as being continuous when the liquid jet
was laterally oscillatory and off-centerline. Another issue that
is not discussed in these sources is the fact that the turbulence
intensity (through We fluctuations) can have a strong effect on
breakup length; higher gas intensity values reduce the length
[31]. Even though the primary instability wavelength dictates
ligament sizes, the gas phase turbulence can determine the
ligament forming distance. As a result, there is some
uncertainty in how to directly compare CFD with the
experimental counterpart. Core length will be reported at two
liquid volume fractions, 99.5% and 50%, respectively:
2.70/4.16 (SST), 3.88/5.44 (MILES), and 6.93/14.4 (DHRL). It
is evident that only the DHRL solution comes close to the
expected values. A laminar liquid jet relies on the gas phase
structures and gas boundary layer thickness to deform the liquid
interface [54]. That buffeting then competes with surface
tension for the primary instability mechanism. The liquid feed
Reynolds number is about 4000 (based on the high-shear
12 Copyright © 2015 by ASME
viscosity), making it transitional and the condition of the
boundary layer uncertain. The SST model assumes boundary
layers are everywhere turbulent. Since that is not the case in
the liquid feed, the use of SST likely causes breakup too early.
The DHRL and MILES models produced a more realistic liquid
feed, but the MILES approach had too much kinetic energy.
Only the DHRL model addresses both of these issues, although,
as previously discussed, the lateral motion of the DHRL result
is artificially smoothed.
Table 2: Normalized breakup length estimates from other
Zhao et al. [29]
Eroglu et al. [55]
Engelbert et al. [56]
Lasheras et al. [7]
Porcheron et al. [57]
Leroux et al. [58]
Matlab routines were written to quantify velocity spectra
using the Multi-taper method”. A time-bandwidth product of
4, FFT length of the nearest 2n data set size, and Thomson’s
adaptive nonlinear combination were utilized. Experimentation
showed these settings to be the most resolute. Fig. 17 shows
energy spectra from a point 12.5 orifice diameters downstream
of the liquid outlet and about 1.74 orifice diameters off the
centerline for all models (except MILES, which is not
available) at a sampling frequency of 1 million Hz and 214 data
points. The point was placed in the shear zone towards the
outer liquid layer. For reference, the grid wavenumber (/grid
length scale since two cells per period are needed to satisfy the
Nyquist criterion) was on the order of 104. The results imply
that not much changes at this particular point with changes in
turbulence model.
Fig. 17: Energy spectra
The time-averaged fraction resolved by LES is depicted in
Fig. 18 for the DHRL model. The blue regions represent areas
using RANS only, while the red regions represent areas using
LES only. Values in between indicate that the solution uses a
blending between RANS and LES. There is a fair amount of
RANS support needed in the interfacial shear layers, indicating
diffusivity there. This diffusivity can be related to the temporal
stencil, time step size, spatial stencil, grid size, and/or or SGS
model (zero in this case). From a cross-sectionally averaging
standpoint, the mesh resolution was likely sufficient. However,
locally, there are areas of sufficient and insufficient resolution.
Additionally, the temporal stencil is based on first-order
upwinding, and that is a current limitation with Fluent. It is
apparent, based on the droplet size analysis that the DHRL
RANS compensation balances these issues.
Fig. 18: Time-averaged (fraction resolved by LES)
contours from DHRL model
Second Case Study
In a similar way as the work of Mansour and Chigier was
mimicked in CFD, the coaxial system of Zhao et al. [29] was
investigated computationally using a similar transient PLIC
method and 36mesh layout. The mesh element length scale
for the latter work was closer to 120 microns down the
centerline. Two models will be discussed, an SST result and a
MILES result; time did not permit running this with a DHRL
model. Comparisons of mesh, slurry, and integral length scales
13 Copyright © 2015 by ASME
for the SST result can be found in Strasser and Battaglia [30].
Table 3 (SI units) shows important setup information for their
experimental case designated “A1”. The liquid feed to the gas
was laminar-parabolic in the experiment and in CFD. Based on
the work of Farago and Chigier [34], the liquid Reynolds
number (~1, not shown) and Weber number, this disintegration
processed would be termed “Superpulsating” type and would
be characterized to be similar to “Fiber” type, except with
larger waves between major bursts of droplets. If the liquid
flow were larger, it would be more like the Fiber type.
“Superpulsating” should not be confused with the bulk
boundary mass flow variations of the three-stream injector of
the cited works.
A sample energy spectrum for the SST result at a sampling
frequency of 1 million Hz at a point located 2 orifice diameters
below the phase meeting point and half of an orifice diameter
off centerline. The MILES data are not available. For
reference, the grid wavenumber (/grid length scale since two
cells per period are needed to satisfy the Nyquist criterion) was
on the order of 3×104. There appears to be no isolated inertial
range, as most of the lower wavenumbers follow the -5/3 slope.
Table 3: Important quantities and boundary conditions for
study number 2 (Zhao et al. [29] case designated “A1”); All
are in SI units.
Liquid density
Liquid viscosity at reference shear
Liquid surface tension
Liquid orifice inner diameter
Liquid orifice outer diameter
Liquid mass flow
Liquid velocity
Gas density
Gas viscosity
Gas orifice inner diameter
Gas mass flow
Gas velocity
Gas to liquid mass ratio
Weber number
Ohnesorge number
Fig. 19: Energy spectrum from SST model with -5/3 slope
Fig. 20: Instantaneous slurry spray surface
Fig. 20 illustrates instantaneous surfaces of slurry from the
SST and MILES simulations at random uncorrelated times. No
obvious differences arise given there is substantial variability in
time. It appears as though there may be slightly more lateral
motion in the spray for the MILES approach as was seen in the
prior study.
Fig. 21 illustrates instantaneous velocity contours ranged
from blue = 0.0 to red = 100 m/s (gas feed velocity). The rich,
energetic character of developing shear layers is evident. The
fraction of flow field resolved by LES is shown in Fig. 22.
Even though there is no hybrid solution being sought, this
represents a preliminary look at how much RANS support
(shades of blue and green) is necessary. It is evident that most
of the domain is treated with RANS.
14 Copyright © 2015 by ASME
Fig. 21: Instantaneous velocity contours; Red = feed
Fig. 22: Time-averaged (fraction resolved by LES)
contours from MILES model
Fig. 23: Time-averaged mass flow-weighted area-averaged
ratio of
Laterally averaged slurry length scales versus axial
distance (scaled by mesh length scale) are shown in Fig. 22.
The SST and MILES solution shows similar trends, which
corroborates the prior figure. Clearly, more fluctuating velocity
components does not imply a correct turbulence model as is
noted in Pope [39].
Investigations into the ratio of the integral length scale to
the mesh length scale revealed ratios in the range of 2 (near the
injector) to 3 (far-field), which is similar to the SST values
(prior work). Because this ratio is below 10, adequate LES
resolution is not expected.
In the original paper, the authors discussed the breakup
frequency. For the conditions of the test used herein, their
relations dictated a frequency of 105 Hz. Data were sampled at
various locations throughout the domain, and passing liquid
volume fraction was logged. It was found that the results were
extremely unpredictable, in that massive amounts of run time
would have been required to sequester enough data to make
meaningful liquid passing conclusions at any particular point.
Frequency results ranged from around 10 Hz at one point to
around 400 Hz at another. When comparing the SST
simulation data to the MILES data, there were no consistent
correlations between the two runs. More work would be
needed to asses this. However, it was found that more stable
frequency information could be extracted from droplet size
transient information in the 10 axial sampling volumes. For the
SST run, values were typically about 80 Hz and did not
strongly depend on sampling volume location. For the MILES
simulation, the results were closer to 60 Hz and also did not
strongly depend on sampling volume. This is consistent with
private communications with the authors in which it was
discussed that the frequency was not dependent upon
measurement location except very close to the nozzle. In other
words, the more energetic velocity field of the MILES run
produced less bulk cyclicality in the droplet size burst results.
The smaller length scales in the MILES approach apparently
damped the rate of droplet size temporal variability.
As with the earlier case study breakup length was
investigated. The 50% and 99.5% liquid volume fraction core
length values were 0.87/1.9 (SST) and 1.2/2.0 (MILES). The
experiment reported values of 5.7. Just like the earlier study,
the SST and MILES solutions produced far too low of a
breakup length. It is expected that, similar to above, more
accurate prediction of breakup length may be possible with a
hybrid RANS-LES approach, to be investigated in future work.
A computational evaluation of a dynamic hybrid RANS-
LES (DHRL) modeling approach was carried out in the context
of the turbulent atomization of a non-Newtonian fluid. Inherent
to this type of flow field are separated shear layers and strong
density gradients. The DHRL offers the advantages of 1) no
explicit mesh dependence, 2) ability to couple any RANS or
LES approach, and 3) return to the baseline RANS model for a
numerically steady simulation. The framework utilizes the
15 Copyright © 2015 by ASME
continuity of turbulence production to communicate
information between areas treated with LES and those treated
with RANS and has been validated for single phase flows in
prior works. The purpose of this work was to enhance the
model for use with multiphase flows and then compare its
performance to the well-known shear-stress transport (SST).
DHRL shows a much smaller droplet size distribution with
values in close agreement with the referenced experiment.
Additionally, the jet breakup length is closer to what is
expected from the open literature. For differing reasons, the
SST and implicit LES approaches, separately, produced too
short of a liquid core. There was some evidence that turbulence
via local instabilities was being suppressed by Favre averaging
and density-weighted sources. It is unknown, however,
whether this was a modeling flaw or diffusion due to a first
order transient stencil. To be certain of the applicability and
accuracy of the proposed modeling paradigm, more data would
have to be available for comparison. An interesting
consideration is the unknown effect that the hybrid modeling
will have on bulk pulsations of three-stream unknown, i.e. bulk
boundary feed pulses versus local instabilities.
The support of a multitude of Eastman Chemical Company
personnel is greatly appreciated. Specifically, George
Chamoun and Jason Goepel deserve special recognition for
processing transient signal data sets using various methods.
Additionally, discussions with Mihai Mihaescu from Royal
Institute of Technology (KTH), Marcus Herrmann from
Arizona State University, David Schmidt from the University of
Massachusetts, Mario Trujillo from University of Wisconsin
Madison, Daniel Fuster of Institut Jean Le Rond D'Alembert
UPMC, and Christophe Dumouchel of Université et INSA de
Rouen were extremely beneficial.
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... Designs typically involve an "assisting" gas on the outside with the liquid fuel on the inside, including the efficient "flow blurring" design implemented in [15] for disintegrating algae oil. Some designs entail co-flowing gas streams on both the inside and the outside of the liquid sheet, called "airblast", and those also are known to exhibit pulsations [16][17][18][19][20][21][22][23][24][25][26]. ...
... The mesh size gradient is a consequence of early studies giving very preliminary estimates of droplet size, as well as learnings from [16][17][18][19][20][21][22][23][24]. The azimuthal boundaries of the 1/32 nd wedge are treated with periodic planes, because atomization is not axisymmetric. ...
... The azimuthal boundaries of the 1/32 nd wedge are treated with periodic planes, because atomization is not axisymmetric. The risks of using a reduced order model are discussed in [16][17][18][19][20][21][22][23][24], where it was found that the Sauter mean diameter (SMD) was affected only about 10% by the incorporation of 11.25° wedge boundaries. That risk is acceptable in this scoping effort. ...
... Dhakal et al. compared several LEVs with DRSM and found that DRSM had better correlation with experimental data than SST [32]. Departing from RANS and resolving the largest, anisotropic, energy-containing turbulent structures, one can utilize LES [33] or hybrid RANS-LES [34]. However, we seek to show herein that both SST and RSM provide necessary rigor for our objectives. ...
... The benefit of resolving scales is that the model can incorporate true turbulence and not simply the superimposed effects of turbulence onto the mean flow. While there are many measures for LES mesh sufficiency [34], an important consideration when employing LES is the ratio of the integral length scale (ILS) to the mesh length scale (MLS). The ILS is approximately the logarithmic middle of the energy containing range for a Kolmogorov spectrum, while the MLS is often the filter size used to differentiate between a modelled and resolved quantity. ...
Full-text available
CFD was employed to develop a rigorous model of an LDPE autoclave reactor. Different numerical settings within the solver are evaluated to eliminate false diffusion and to reflect the sensitive heat generation taking place during free radical polymerization. An accurate model can allow geometry and process adaptations to be evaluated for much lower costs than physical experiments. Improving the reactor design allows for longer run times and a higher degree of catalyst conversion. The rigorous CFD model employed reaction kinetics, PID-automated thermal management, and a rotating stirrer shaft. Validation was carried out to determine the sensitivity to time-step size, turbulence model, and grid resolution. Data were compared to an industrial scale plant autoclave to guide the development of CFD. In a comparison of turbulence models, the shear stress transport (SST) model was found to predict higher concentrations of turbulent kinetic energy (TKE) resulting in a lower temperature distribution throughout the reactor than the differential Reynolds stress model (DRSM). The less diffusive DRSM was recommended for future studies. A mesh refinement study revealed slight variation in the results between the base mesh of 6 million computational elements and the refined mesh consisting of 40 million. Ultimately, the variation between different grid resolutions was not significant enough to justify slowing down the solver speed by 14X by using the refined mesh. Increased rigor improved the model’s ability to match plant data, and CFD thermocouples were within 2.5% of temperatures from plant data.
... Volume of fluid (VOF) model based on a surface-tracking technique is widely used to predict a jet breakup of two immiscible fluids [14]. The jet breakup of a primary atomisation can be simulated by VOF approach in the hybrid LES-RANS simulation, and the liquid core length can be measured [15]. ...
... Thus, the flamelet profiles were convoluted with the assumed β-PDF and the mean mixture scalars are determined from the look-up table as in Equation (12). The non-adiabatic β-PDF tables have the dimensions of Equations (13)(14)(15) to obtain the cell temperature, density and species mass fraction. ...
The spray combustion of LOX/GCH4 in the pintle injector is simulated to study the combustion characteristics at a subcritical condition by considering the effects of pintle opening distance and pintle tip angle. To perform the simulation, first, the VOF simulation was used to estimate the LOX sheet thickness and breakup length. Second, the breakup constants for the secondary breakup model were calculated using the results of VOF and the Lagrangian approach was used to estimate the Sauter mean diameter around the injector exit. Finally, the spray combustion of LOX/GCH4 simulation was performed using the estimated SMDs data. Subsequently, the spray combustion characteristics were discussed by considering the individual effects of pintle opening distance at fixed pintle tip angle, and pintle tip angle at fixed pintle opening distance using various contours such as temperature, mass fraction of OH, and discrete phase model evaporation of LOX droplets. Following this, the characteristic velocity and combustion efficiency were shown for various cases. The optimised design of the pintle injector based on the simulated results was summarised for future liquid rocket pintle injectors.
... The methodology is implemented via time-averaged source terms directly in the three momentum equations. The strong performance of this approach has been demonstrated in stationary systems involving both incompressible and compressible flows [11], as well as non-stationary multiphase systems [12] in a more tentative manner. ...
Conference Paper
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Coolant flow in advanced reactors can be characterized by oscillatory mixing of non-isothermal liquid coolant streams, and is a potential cause of thermal fatigue damage in fuel and reactor components. The NEAMS IRP Challenge Problem 2 (CP2) seeks to develop an accurate, yet computationally affordable, turbulence modeling option for thermal-striping predictions. The designs of upper internal structures, as well as lower plena and heat exchanger configurations, are strongly affected by the ability to efficiently mix the coolant streams and further affects the material selection as well as operation and maintenance of the systems. This challenge problem generalizes specific needs related to the TerraPower and General Atomics designs by developing a set of benchmarks in order to advance, demonstrate and quantify the accuracy of the thermal striping modeling approach. The focus of the activities is to advance and demonstrate a modeling practice capable of accurately representing the performance of the structural reactor components under the influence of thermal striping. Assessment against adiabatic quasi-2D jets striping has demonstrated great promise for the proposed turbulence approaches, with 2 orders of magnitude acceleration from the reference Large Eddy Simulation (LES) solution. More recent efforts have extended the quasi-2D validation to diabatic conditions, leveraging the demonstrated accuracy of the highly resolved LES methods, and with a new set of experimental data for heated parallel round jets. Further upscaling through the use of reduced order models is being evaluated, and a two-step machine learning approach has been demonstrated on thermal striping for 3 parallel jets.
This study assesses different turbulence modeling approaches for simulation of two-phase coaxial annular swirling jet flows. The problem selected from literature for comparison involves an analytical inlet profile for an annular liquid sheet sandwiched between two coaxial annular gaseous jets. The liquid-gas interface is resolved using the volume-of-fluid model with continuum surface force approximation. Reynolds-averaged Navier-Stokes simulations and detached eddy simulations (DES) are conducted to obtain transient multiphase numerical solutions. Different turbulence models explored include the k-ε renormalization group (RNG) with swirl modification, the Reynolds stress model (RSM), RSM with scale-adaptive simulation (RSM-SAS), and DES. Comparisons with the direct numerical results from literature suggest that the k-ε RNG and RSM approaches simulate only the streamwise shear of the liquid jet and are inadequate in capturing the swirling aspect of the jet flow and expected instabilities. DES can predict several expected features such as radial asymmetry, surrounding gas vortices causing jet instabilities, and eventual jet breakup with reasonable accuracy. While RSM-SAS predicts radial asymmetry, some jet instability, and is much more accurate than k-ε RNG and RSM, it fails to predict instabilities as good as DES and does not predict a complete jet breakup. RSM-based methods are found to be computationally very expensive compared to the k-ε RNG model, suggesting DES as the better alternative than RSM methods for such applications if resources are available.
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We reveal mechanisms driving pre-filming wave formation of non-Newtonian banana puree inside a twin-fluid atomizer at a steam-puree mass ratio of 2.7%. Waves with a high blockage ratio form periodically at a frequency of 1000 Hz, where the collapse of one wave corresponds to the formation of another (i.e., no wave train). Wave formation and collapse occur at very regular intervals, while instabilities result in distinctly unique waves each cycle. The average wave angle and wavelength are 50{degree sign} and 0.7 nozzle diameters, respectively. Kelvin-Helmholtz instability (KHI) dominates during wave formation, while pressure effects dominate during wave collapse. Annular injection of the puree into the steam channel provides a wave pool, allowing KHI to deform the surface; then, steam shear and acceleration from decreased flow area lift the newly formed wave. The onset of flow separation appears to occur as the waves' rounded geometry transitions to a more pointed shape. Steam compression caused by wave sheltering increases pressure and temperature on the windward side of the wave, forcing both pressure and temperature to cycle with wave frequency. Wave growth peaks at the nozzle exit, at which point the pressure build-up overcomes inertia and surface tension to collapse and disintegrate the wave. Truncation of wave life by pressure build-up and shear-induced puree viscosity reduction is a prominent feature of the system, and steam turbulence does not contribute significantly to wave formation. The wave birth-death process creates bulk system pulsation, which in turn affects wave formation.
Full-text available
Characterization of viscous, non-Newtonian atomization by means of internal waves is presented for a twin-fluid injector. Atomization of such fluids is challenging, especially at low gas-liquid mass ratios (GLRs). This paper details mechanisms that enhance their disintegration in a "wave-augmented atomization" process. The working fluid, banana puree, is shear-thinning and described by the Herschel-Bulkley model. Unlike a conventional airblast injector, an annular flow of banana puree is injected into a core steam flow, encouraging regular puree waves to form inside the nozzle. A pulsing flow develops with three distinct stages: stretch, bulge, and burst leading to an annular puree sheet stretching down from the nozzle exit. Rayleigh-Taylor instabilities and viscosity gradients destabilize the surface. During wave collapse, the puree sheet bulges radially outward and ruptures violently in a radial burst. Near-nozzle dynamics propagate axially as periodic fluctuations in Sauter mean diameter occur in a wave pattern. Numerical simulations reveal three atomization mechanisms that are a direct result of wave formation: 1) wave impact momentum, 2) pressure buildup, and 3) droplet breakaway. The first two are the forces that exploit puree sheet irregularities to drive rupture. The third occurs as rising waves penetrate the central steam flow; steam shear strips droplets off, and more droplets break away as the wave collapses and partially disintegrates. Waves collapse into the puree sheet with a radial momentum flux of 1.7 × 10 ⁵ kg/m-s ² , and wave-induced pressure buildup creates a large pressure gradient across the puree sheet prior to bursting.
Full-text available
Although coaxial airblast primary atomization has been studied for decades, relatively little attention has been given to three-stream designs; this is especially true for transonic self-pulsating injectors. Herein, the effects of nozzle geometry, grid resolution, modulation, and gas flow rate on the acoustics and spray character within an industrial scale system were investigated computationally using axisymmetric (AS) and three-dimensional (3D) models. Metrics included stream pressure pulsations, spray lift-off, spray angle, and primary droplet length scale, along with the spectral alignment among these parameters. Strong interactions existed between geometry and inner gas (IG) feed rate. Additionally, inner nozzle retraction and outer stream meeting angle were intimately coupled. Particular attention was given to develop correlations for various metrics versus retraction; one such example is that injector flow capacity was found to be linearly proportional to retraction. Higher IG flows were found to widen sprays, bringing the spray in closer to the nozzle face, and reducing droplet length scales. Substantial forced modulation of the IG at its dominant tone did not strongly affect many metrics. Incompressible 3D results were similar to some of the AS results, which affirmed the predictive power by running AS simulations as surrogates. Lastly, normalized droplet size versus normalized distance from the injector followed a strikingly similar trend as that found from prior two-fluid air-slurry calibration work.
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Acoustics and ligament formation within a self-generating and self-sustaining pulsating three-stream injector are analyzed and discussed due to the importance of breakup and atomization of jets for agricultural, chemical, and energy-production industries. An extensive parametric study was carried out to evaluate the effects of simulation numerics and boundary conditions using various comparative metrics. Numerical considerations and boundary conditions made quite significant differences in some parameters, which stress the importance of using documented and consistent numerical discretization recipes when comparing various flow conditions and geometries. Validation exercises confirmed that correct droplet sizes could be produced computationally, the Sauter mean diameter (SMD) of droplets/ligaments could be quantified, and the trajectory of a droplet intersecting a shock wave could be accurately tracked. Swirl had a minor impact by slightly moving the ligaments away from the nozzle outlet and changing the spray to a hollow cone shape. Often, metrics were synchronized for a given simulation, indicating that a common driving mechanism was responsible for all the global instabilities, namely, liquid bridging and fountain production with shockletlike structures. Interestingly, both computational fluid dynamics (CFD) and the experimental non-Newtonian primary droplet size results, when normalized by distance from the injector, showed an inversely proportional relationship with injector distance. Another important outcome was the ability to apply the models developed to other nozzle geometries, liquid properties, and flow conditions or to other industrial applications.
The second edition of this long-time bestseller provides a framework for designing and understanding sprays for a wide array of engineering applications. The text contains correlations and design tools that can be easily understood and used in relating the design of atomizers to the resulting spray behavior. Written to be accessible to readers with a modest technical background, the emphasis is on application rather than in-depth theory. Numerous examples are provided to serve as starting points for using the information in the book. Overall, this is a thoroughly updated edition that still retains the practical focus and readability of the original work by Arthur Lefebvre.
An analysis of experimental data on the critical conditions of drop breakup by aerodynamic forces is given. Possible causes of divergence among data on the critical Weber number, We c, obtained by different authors is discussed. It is shown that the most important of these causes is incorrect technique for determining We c (at the point of breakup), and disregarding the influence of the rate of change of the flow action upon the drop. Thorough experimental investigation of peculiarities of drop fragmentation under critical and overcritical conditions was carried out. Temporal characteristics of the breakup process and the mean size of fragments being formed is measured. Generalizing formulas for We c, the induction time, and the period of drop natural oscillations are obtained.
Large eddy simulation (LES) is known to suffer from two primary error sources - the subgrid stress (SGS) model and the numerical discretization scheme. These cannot be separately quantified for finite-volume numerical simulations, but an appropriate combination can yield ‘engineering accurate’ prediction of turbulence dynamics. This paper seeks to evaluate combinations of commonly used second-order numerical schemes and Smagorinsky-type SGS models for practical LES. Error assessments are performed for isotropic decaying turbulence using both pseudospectral and finite-volume solvers, followed by validation for a complex turbulent flow of engineering interest. Error assessment using pseudospectral techniques is performed to isolate finite-differencing and modeling errors by explicitly adding numerical derivative error terms to the simulations. Error assessment using the finite-volume method focuses on identification of optimal model-discretization combinations for the best LES predictions using application solvers. The pseudospectral and finite-volume approaches show consistent predicted behavior of the interplay between numerical and modeling errors for different model-discretization combinations. Of those studied here, the two most optimal combinations are identified as: (1) standard or dynamic Smagorinsky SGS model with a bounded central difference scheme; and (2) Monotonic Integrated LES (MILES) with either a second-order upwind or QUICK scheme. These combinations were applied for an axisymmetric jet flow at Re ∼ 105, using an ‘engineering quality’ mesh, where the MILES model with either an upwind or QUICK scheme showed the best predictive capability.
The main objective of this study is to gain insight into dynamic behavior of coal–water slurry (CWS) drops during secondary breakup. In the case of Bingham liquids (CWS) considered here, this endeavor requires a thorough understanding of the influence of the shear-thinning and viscoplastic fluid rheology on the flow inside the deforming drops. In numerical analysis, a coupled technique including Large Eddy Simulation (LES) and smoothed volume of fluid (VOF) was used with a dynamic adaptive mesh. The effects of variations in secondary breakup and viscoplastic liquid rheological parameters on Sauter mean diameter (SMD), droplet aspect ratio, and dynamic average viscosity were studied in detail. The results show good predictions for the breakup regimes. Moreover, an effective constant viscosity leading to a similar Newtonian breakup regime was detected in all cases. According to the analogy between secondary and primary breakups, the results can be used to better understand the CWS jet breakup.