MODELING DROPLET HEAT AND MASS TRANSFER IN SPRAYS: PROGRESS AND
CHALLENGES
Shankar Subramaniam
Department of Mechanical Engineering
Iowa State University, Ames, Iowa, USA
The predictive capability of CFD calculations of sprays depends on many factors, principally: (i)
adequacy of the underlying mathematical formulation, (ii) accuracy of the closure relations, especially
those for the physical sub-models representing droplet heat and mass transfer, (iii) numerical accuracy
of the solutions to the model equations, and (iv) the availability of experimental data or analytical
solutions for comparison and validation. Outstanding challenges and recent progress in each of these
areas are discussed. These recent developments motivate an integrated program for predictive spray
model development, which is outlined in this work.
INTRODUCTION
The ultimate goal of spray modeling is to capture those essential, and hopefully universal, principles of
spray dynamics which can successfully represent phenomena that are observed in the wide variety of
spray applications, including: automobile fuel sprays, agricultural sprays, and pharmaceutical sprays
for tablet coating and inhalation drug delivery. The underlying theme of this paper is that progress in
four principal areas, namely
(i)
adequacy of the underlying mathematical formulation,
(ii)
accuracy of the closure relations, especially those for the physical sub-models
representing droplet heat and mass transfer,
(iii)
numerical accuracy of the solutions to the model equations, and
(iv)
validation of models in canonical problems with analytical solutions or experimental
data,
is critical for the development of predictive spray models.
Of all the modeling and simulation approaches that are currently in use, the statistical approach used in
CFD is the most useful for complete spray calculations. This statement is justified by considering the
feasibility of two other simulation approaches that are recently being applied to the spray problem,
namely: Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES).
DNS is currently not a practical approach for complete spray calculations for two principal reasons.
The first reason is that DNS of even fully dispersed droplets in a turbulent gas involves the physical
processes of droplet coalescence and breakup which impose a dynamically changing resolution
requirement that is considerably more demanding than simply resolving all the continuum scales in
single-phase turbulence. In fact molecular dynamics studies of droplet collision1 show that there are
sub-continuum scale physics can affect the outcome of collisions, thereby indicating that DNS of even
dispersed two-phase flows is not a model-free simulation. However, DNS of certain canonical twophase flows can be useful in developing statistical models. The second reason why DNS of full sprays
is currently impractical is because simulation of practical spray systems involves a primary
atomization process.
A rough estimate1 of the computational cost involved in performing DNS of just the turbulent liquid
pipe flow in the nozzle prior to atomization indicates that such simulations are about 20 years in the
future, with current growth in computing power. The scaling of grid points (N) DNS of turbulent pipe
flow with Reynolds number for L/D=5 given by Friedrich et al2 is N=0.004826 Re21/8, and Wu and
Faeth3 report turbulent pipe flow Reynolds numbers for spray atomization experiments in the range of
8400 to 850,000. Currently DNS of turbulent pipe flow for Reynolds numbers up to 8900 are reported
in the literature. The scaling estimate for the number of grid points required for a DNS reveals that a
hundred-fold increase in Reynolds number would require the number of grid points to increase by a
factor of approximately 175000. Even if Moore’s law continues to hold, and we assume computing
power doubles every year, this leads to the conclusion that DNS of realistic turbulent pipe flow will be
feasible in about 17 years. DNS of primary breakup is more computationally demanding because the
resolution requirement will increase with interface pinch-off and formation of small drops.
The LES approach for simulation of single-phase turbulent flow was developed on the basis of two
important assumptions concerning high Reynolds number turbulent flow. One is the Kolmogorov
hypothesis of local isotropy, which states that at sufficiently high Reynolds number the small-scale
turbulent motions are statistically isotropic4. The second is the fact that most of the energy in a singlephase turbulent flow is contained in the large scales, and a simple model for momentum transfer
between the large and small scales can perhaps describe the evolution of the energy containing (largeeddy) motions. The assumption of local isotropy is borne out by experimental observation of high
Reynolds number single-phase turbulent flow4, 5. Although not all sub-grid scale models are simple, it
is now established that LES is useful in simulating single-phase turbulence. However, neither of these
assumptions is shown to be valid for dispersed two-phase flows. LES formulations are not rigorously
developed for two-phase flows, and it appears that significant enhancements or changes to the standard
single-phase LES approach will be needed before it can be applied to two-phase flows.
The statistical approach on the other hand, is useful in its own right as a basis for CFD calculations of
sprays, and its development can also be useful in interpreting and using DNS results. Of the statistical
formulations used for spray modeling, the Lagrangian-Eulerian approach based on Williams’ spray
equation is the most promising for developing predictive spray models6.
The underlying physics always dictate the mathematical representation that must be chosen to model
any problem. For instance, while second-moment closures (RANS models) are often deemed
adequate models for single-phase turbulent flows, a higher level of representation based on the
velocity-composition probability density function (PDF) is used for turbulent reactive flows7. This
is because a highly nonlinear dependence of the reaction rate on composition (mass fraction of
chemical species and enthalpy) implies that the mean reaction rate is not well approximated by the
reaction rate function evaluated at mean composition. Similarly, we need to examine what are the
essential physical phenomena that determine spray dynamics, and how these may differ from singlephase turbulent flows. The first noteworthy difference is that in two-phase flows such as sprays, the
essential dynamics is governed by non-local effectsa). For example, the acceleration of a spray
droplet’s center-of-mass is determined by the pressure and viscous shear exerted by the ambient gas
on the droplet surface, and the same is true for the droplet’s thermal and vaporization dynamics.
Another unique feature of sprays is the presence of drop-drop interactions, which we expect to
become more apparent at higher volume fractions.
a)
It is true that non-local effects due to pressure also arise in single-phase turbulent flows, but pressure is notoriously
difficult to account for in purely Lagrangian PDF methods.
These “action-at-a-distance” phenomena are difficult to capture in traditional single-point PDF
models, which are appropriate for single-phase turbulent reactive flows2. Therefore, if the singlepoint PDF representation is directly applied to spray problems, it is not going to address the issue of
capturing the essential physics. However, a multiphase PDF formulation can be very useful—as is
shown later in this paper—provided the representation is capable of capturing the essential physics.
Here we explore the adequacy of the LE statistical approach to modeling sprays. Although the LE
approach is the most suitable of the existing modeling approaches, in its current form it is incapable of
representing droplet-droplet interactions. Droplet-droplet interactions play an important role in spray
heat and mass transfer, and the essence of spray modeling is the successful description of the collective
behavior of the droplets comprising the spray8. While considerable work has been done on the
modeling of isolated droplet heat and mass transfer in a convective environment9, the issue of
modeling drop-drop interactions has received less attention10. We summarize our recent progress in
developing an extended Lagrangian-Eulerian formalism (ELE), which enables the inclusion of dropdrop interaction effects within the existing LE framework. This extended Lagrangian-Eulerian (ELE)
formalism is based on a concept of assumed point fields (APF). The ELE formalism shows promise of
extending the existing LE mathematical formulation to a level that is adequate to describe important
spray physics.
We then turn to the next principal issue, which is the development of accurate closures for heat and
mass transfer in sprays. Considerable work has already been done on the modeling of isolated droplet
heat and mass transfer in a convective environment9, and a plethora of models of varying levels of
detail are available to the modeler. The choice of the appropriate model is ultimately dictated by the
droplet vaporization regime, and this is determined by the spray application under consideration. Our
understanding of the physics of heat transfer in droplet clouds and collective vaporization of droplets
can benefit greatly from DNS, which can be a very useful research tool for developing better physical
sub-models of droplet heat and mass transfer that account for drop-drop interactions. Direct numerical
simulations of scalar transport based on a discrete-time immersed-boundary method (DTIBM)11 are
presented for particle-laden flow. The fluid velocity and scalar evolution are governed by the extended
Navier-Stokes equations. Exact boundary conditions are imposed at the surface of each spherical
particle. These simulations are useful for developing better physical sub-models for heat transfer in
particle-laden and droplet-laden flows that account for particle-particle or drop-drop interactions.
In order for a spray model to be predictive, it is critical to have a numerically convergent model
implementation that also computes accurate solutions to the spray model equations. The importance of
validation tests—whether these are based on comparison with analytical solutions or experiments—as
necessary first step to gain confidence in spray model performance is underscored.
The issue of model accuracy, and criteria drawn from a closely related field of turbulence modeling are
adapted to assess and appraise spray model development. In summary this paper attempts to provide a
perspective on spray model development that is based on the concept of balancing uncertainties/errors
arising from (i) the mathematical representation, (ii) the closure models, (iii) numerical methods, and
(iv) experimental uncertainty.
ADEQUACY OF THE UNDERLYING MATHEMATICAL REPRESENTATION
The principal spray modeling approaches currently used in computational fluid dynamics (CFD)
calculations of sprays are based on either the Eulerian-Eulerian (EE) formulation for two-phase
flows12-17, or the Lagrangian-Eulerian (LE) formulation18, 19. The point-process statistical
representation20, which forms the basis for the mathematical formulation of the LE spray modeling
approach, provides a more suitable framework than the EE approach to represent the droplet size
distribution19. Also the representation of droplets in the Lagrangian frame of reference in the LE
approach is more suitable for model development. In spite of these advantages, we show here that
the LE approach based on Williams’ spray equation is incapable of representing drop-drop
interactions. (The same is also true of other first-order statistical formulations such as the EE twofluid formulation.) This inadequacy of the LE formulation is apparent on reviewing the relevant
basics of stochastic point process theory. The LE formulation is based on Williams’ droplet
distribution function (ddf) f (x, v, r , t ) , which is the number density of droplets in position-velocityradius space20 . The ddf implies an average number of droplets in physical space n(x, t ) , which is
obtained from the ddf by integrating over velocity-radius space:
n(x, t )   f (x, v, r , t )dvdr.
(1)
It is important to note that the ddf and number density are first-order statistics that contain no
information regarding drop-drop interactions. It is also important to note that the same number density
(or average number of drops in a unit volume) can correspond to different spatial arrangements of
droplets in physical space, as shown in Figure 1.
Figure 1. Two monodisperse droplet fields corresponding to the same average number density, but different
spatial arrangements of the droplets. In the left panel the droplet centers can be as close as one drop
diameter, but on the right panel the droplet centers are not allowed to be closer than 1.72 drop diameters.
Isosurfaces of a passive scalar are shown for the same scalar value of 0.5 in both panels. For the same
average number density the left panel shows considerable drop-drop interaction with merging of the
isosurfaces which envelop neighboring droplets, whereas the right panel shows less drop-drop interaction
with distinct scalar isosurfaces.
The natural question to then ask is: What are the implications of different spatial arrangements of
droplets in physical space with same number density for heat and mass transfer modeling in sprays?
Clearly one way in which different spatial arrangements can affect models is because different spatial
arrangements affect LE models for individual droplet heat and mass transfer. A typical LE model21 for
time evolution of the droplet radius r is
(  D) g  Y fs  Y f 
dr

(2)

 Shd ,
dt
2  d r  1  Y fs 
which states that the rate of droplet regression is determined by the transfer number Bd , defined as
 Y fs  Y f 
Bd  
.
(3)
 1  Y s 
f


The transfer number Bd characterizes the vaporization driving force arising from spatial inhomogeneity
of the vapor fuel mass fraction Y f , and is the nondimensional difference between Y fs , the vapor fuel
mass fraction at the surface of the droplet, and Y f , the vapor fuel mass fraction far away from the
droplet surface in the gas phase. In Eq (1.2) d is the thermodynamic density of the liquid fuel
droplet,   D  g is the mass diffusivity of fuel vapor in the gas phase, and Shd is the Sherwood number
characterizing the rate of mass transfer. The Frossling correlation is often used for the Sherwood
number, and this accounts for convective effects through a droplet Reynolds number dependence of
the form
ln(1  Bd )
Shd   2.0  0.6 Re1/d 2 Sc1/ 3 
.
(4)
Bd
A typical LE model21 for time evolution of the droplet temperature Td is
dT
4
dr
 d  r 3Cl d   d 4 r 2 L  4 r 2 qd ,
(5)
3
dt
dt
which states that the total surface heat transfer rate 4 r 2 qd goes into either heating up the droplet (first
term on the left hand side of above equation), or into supplying energy required for phase change
(second term on the left hand side of above equation). In the above equation Cl is the specific heat of
the liquid droplet and L is the latent heat of vaporization. The surface heat flux is calculated in terms
of the droplet Nusselt number through the expression
k (T  T )
qd  g g d Nud ,
(6)
2r
and the Nusselt number is calculated using the Ranz-Marshall correlation which accounts for
convective effects through the following dependence on the droplet Reynolds number
ln(1  Bd )
Nud   2.0  0.6 Re1/d 2 Pr1/ 3 
,
(7)
Bd
where Pr is the Prandtl number in the gas phase.
Many modeling assumptions underlie the typical LE models21 for droplet regression rate and heat
transfer that are presented here. The important assumptions are a spherically symmetric droplet
undergoing unsteady heating but internal circulation, turbulence and radiative effects are neglected.
The physical effects of internal circulation, turbulence and radiation can be accounted for by
incorporating more complex model expressions for droplet heat and mass transfer9 within the
existing LE framework. Relaxing the spherical droplet assumption is nontrivial, but for small
droplets this assumption is valid. For automotive and gas turbine fuel sprays the droplet sizes are
small.
However, there is one principal assumption underlying LE models that cannot be relaxed within the
existing LE framework. This assumption is that all LE models are for an isolated droplet in a gas
flow and drop-drop interactions are neglected. As noted earlier, accounting for drop-drop
interactions is the essential part of describing the collective behavior of droplets in a spray. In this
sense, the existing LE modeling approach is not adequate to describe the physics of drop-drop
interactions in sprays. In order to further probe the effects of drop-drop interactions, we attempt to
quantify the importance of drop-drop interactions on individual droplet heat and mass transfer by
performing direct numerical simulations. The influence of drop-drop interactions on physical submodels of droplet heat and mass transfer are described in the following section.
There is another effect of drop-drop interactions that must be considered in two-way coupled
models for sprays. This is the effect of drop-drop interactions on the mass, momentum and energy
source terms that arise in the gas-phase conservation equations due to the presence of the spray
droplets. As an example, the mass source term due to droplet vaporization that appears in the gasphase mass conservation equation in the LE approach is
dr
 s    f (x, v, r , Td , t )  d 4 r 2 dvdrdTd .
(8)
dt
The vaporization source term  s can be calculated if the model for the droplet radius regression rate
dr dt and the ddf f (x, v, r , Td , t ) 3 are known. For droplet radius regression rate models that assume
an isolated droplet vaporizing in a gaseous stream, the form of the vaporization source term implies
that in the case of a homogeneous spray with monodisperse droplets having identical initial velocity
and temperature, the source term is linear in the number density. This follows from the decomposition
of the ddf f (x, v, r , Td , t ) into the product of a number density n(x, t ) , and the conditional joint
probability density function of velocity, radius and temperature f VcRTd ( v, r, Td ; x, t ) , that has been
established by Subramaniam19, 20. For the case of a homogeneous spray with monodisperse droplets
having identical initial velocity v 0 , radius r0 , and temperature Td 0 , the ddf decomposition simplifies
to
(9)
f (x, v, r, Td , t )  n(x, t ) f VcRTd ( v, r, Td ; x, t )  n(t ) ( v  v0 ) (r  r0 ) (Td  Td 0 ) ,
from which we deduce the linear dependence of the vaporization source term on the number density
for isolated droplet vaporization models. The second important question that needs to be addressed is
how different spatial arrangements of droplets corresponding to the same average number density
affect the source terms that appear in the gas-phase conservation equations. Specifically, it is important
to know under what conditions the linear dependence of the source terms on the number density is
justified.
Another limitation of existing LE formulations is that they are restricted to dilute sprays with low
liquid volume fraction (in fact, recent analysis shows that the restriction is even stronger, and it
limits the gradient of the volume fraction even in dilute flows). The extended LE (ELE) approach
that is outlined in this paper describes how the existing LE approach can be applied to sprays that
are not dilute. In this case the gas-phase conservation equations also need to be modified to account
for volume displacement effects in the mass, momentum and energy conservation equations.
In summary, we have shown in this section that although the LE approach is the most suitable of the
statistical models for complete spray simulations, in its current form it is an inadequate
The ddf as defined here includes the droplet temperature property, which is an extension of the form of the ddf defined
earlier.
3
mathematical representation of spray physics because it does not account for different spatial
arrangements of droplets that correspond to the same average number density. The effect of these
different spatial arrangements needs to be accounted for because drop-drop interaction effects can
influence both the models for individual droplet heat and mass transfer, as well as the source terms
in the gas-phase conservation equations that represent the influence of the spray droplets.
The principal challenges in developing an adequate mathematical representation are:
(i)
a statistical characterization of the different spatial arrangements of droplets
(ii)
an understanding of the effect of these different spatial arrangements of droplets on the
physics of heat and mass transfer in sprays
(iii)
a formalism to incorporate these effects in statistical models, if possible by extending
the existing LE framework itself.
Second-Order Statistics: A second-order statistical description of droplets in a homogeneous spray
using the pair-correlation function provides a characterization of different spatial arrangements of
droplets10, 22, 23. The pair correlation function g (r ) represents the relative frequency of occurrence of
all possible values of the inter-droplet spacing r . The simplest possible point process model for the
spray droplets is that the droplet centers are distributed according to a homogeneous Poisson
process in physical space23. The independence property of the Poisson process is reflected in its pair
correlation function being unity for all possible values of the inter-droplet spacing r . Since the
location of every point in a Poisson point process is assigned independent of all the other points (the
presence of any Poisson point does not influence the location of any other point), all inter-point
spacing values are equally likely and g ( r )  1 .
The Poisson process is not an appropriate model for droplets because it allows for overlapping
droplets. (The centers of two droplets can be arbitrarily close to each other, and with finite probability
they can lie closer than the droplet radius.) A simple modification to the Poisson process using a
technique called dependent thinning results in a Matérn process that has no overlapping droplets23. In
fact, based on a parameter h , the hard-core distance, one can define a spherical exclusion volume
centered at each droplet within which no other droplet centers are allowed. For a homogeneous spray
of monodisperse droplets with radius r0 , the minimum value of the hard-core distance h  2r0
guarantees non-overlapping droplets (see Figure 2). The Matérn process has an analytic pair
correlation function that is shown in Figure 2 (the blue line is for a hard-core distance h  2r0 ). The
hard-core distance parameter h can be varied in the Matérn process, thereby generating droplets with
different spatial arrangements and known statistics. For a different hard-core distance value of
h  3.46r0 , the droplet field is shown in Fig 1b, and the corresponding pair-correlation function is
shown in Fig 2 (red line). As the hard-core distance is increased for the same average number density,
the droplets are constrained to be farther from each other. There are obviously limits to the maximum
hard-core distance that can be specified for a given average number density23. However, the hard-core
distance parameter in the Matérn process enables us to investigate the effects of different spatial
arrangements of droplets, independent of the average number density (or in other words, the volume
fraction).
Figure 2. Two monodisperse droplet fields generated using the Matérn process that removes points from a
Poisson process using the technique of dependent thinning. Both droplet fields have the same average
number density (and volume fraction), but the hard-core distance parameter for the field shown above is
chosen to be twice the droplet radius, while it is 3.46 times the droplet radius for the field shown below. The
corresponding pair-correlation functions (which are analytically known for the Matérn process) are also
shown. Note that the pair-correlation function goes to zero for droplet center separation distances below the
hard-core distance, and falls off to the Poisson value of one for separation larger than twice the hard-core
distance.
Other researchers have attempted to characterize drop-drop interaction effects starting with the early
work of Tishkoff24 where a drop-in-bubble model was proposed. More recently Chiu10 has shown that
second-order statistics can be used to describe drop-drop interactions. Chiu’s proposal for the paircorrelation function10 is more typical of the pair-correlation of molecules in liquid phase, and more
than one parameter is needed to specify this pair-correlation function. Furthermore, it is non-trivial to
generate a point process that corresponds to a specified pair-correlation function, even if the paircorrelation function has an analytic form. The principal advantage of the Matérn point process
model22, 23 is that it is easy to simulate a realization of this point process, and this process has a known
analytic pair-correlation function. The main reason why the Matérn point process model in this work is
that if the point process that corresponds to a specified pair-correlation function can be generated, then
this point process can be used to initialize a droplet field with known statistics for DNS.
With this statistical description of different spatial arrangements to address the first challenge, we
now investigate the effect of these different spatial arrangements of droplets on the physics of heat and
mass transfer in sprays. In the following section we quantify the influence of different spatial
arrangements on individual droplet heat and mass transfer by performing DNS of passive scalar
transport over spherical particles. We also show how DNS can also be used to quantify the influence
of drop-drop interactions on the source terms that arise in the gas-phase conservation equations.
PHYSICAL SUB-MODELS: DROPLET HEAT AND MASS TRANSFER
Accurate modeling of heat and mass transfer between the droplet liquid phase and the vapor phase
is critical for predictive spray computations. Even for simple thermodynamically sub-critical
conditions, this is a serious challenge. While considerable work has been done on the modeling of
isolated droplet heat and mass transfer in a convective environment, the issue of modeling drop-drop
interactions has received less attention. Although more sophisticated LE models than the typical LE
model presented in the previous section can be used, they all assume an isolated droplet vaporizing
in a gaseous stream.
The principal challenge for statistical models is describing the collective behavior of spray droplets
from our knowledge of heat and mass transfer in single droplets, droplet arrays, and droplet groups.
In this work we show that direct numerical simulations based on solution of the Navier-Stokes
equations with exact boundary conditions on the surface of spherical particles are a promising
approach to understanding the physics of drop-drop interactions on heat and mass transfer in sprays.
Direct Numerical Simulation Approach: The coupling of the fluid and particles is accomplished
through the DTIBM algorithm of Yusof11, which imposes the exact no-slip and no-penetration
boundary conditions at the particle-fluid interface. This new capability allows us to perform
simulations of large-scale polydisperse fluid-particle systems over the entire range of volume
fraction—from dense granular assemblies and dilute fluid-particle flow—thus giving us reliable, direct
access to details of the complex physics underlying these systems.
The spectral NS/DTIBM is a numerical method for solving the incompressible, forced Navier-Stokes
(NS) equations of the form:
u
  u   u  P   2u  g (x, t ),
(10)
t
 u  0
where g(x,t) is a body force term which may be a function of space and time. In the DTIBM approach
g(x,t) is the force on the fluid due to the presence of solid particles. These equations are solved using a
pseudo-spectral method that employs Fourier transforms in the y- and z- directions, and space-centered
finite differences in the x- direction. Time stepping is performed using a Crank-Nicholson scheme for
the viscous terms and a second-order Adams-Bashforth scheme for all other terms. A second-order
velocity-pressure predictor scheme is used, followed by a pressure correction step that ensures that the
flow is incompressible (i.e., divergence free). The Discrete Time Immersed Boundary Method couples
the solid and fluid flow through the spatially and temporally varying force on the fluid, g(x,t). For a
no-slip boundary condition between the fluid and a given particle the fluid velocity, u, must equal the
velocity of the surface of a particle, v. Using the discrete-time Navier-Stokes equations and noting
that at time n+1
u n1  v
On the particle surface this gives
1
g(, t )   u   u  P  2u  ( v  u n )
(11)
t
for the force, g on the fluid due to the particle surface Ω. Finally, for the no-slip condition, the
smoothest local tangential velocity is achieved by simply reversing the tangential velocities between
flow reversal pairs on either side of the solid boundary, and scaling the internal tangential velocity to
get no-slip at the desired location. This method allows for many complex moving boundaries to be
included within the solution of the NS equations because the computational cost scales very favorably
with the number of boundaries (computational cost is almost independent of number of boundaries).
Therefore DTIBM is well-suited for simulation of a large number of particles in flows at Reynolds
numbers O(100) .
We have extended the NS/DTIBM approach to incorporate the transport of a passive scalar field  ,
which obeys a transport equation

  u      2  h(x, t ) ,
(12)
t
where  is the scalar diffusivity, and h(x,t) is a scalar forcing term which may be a function of space
and time. For the scalar DTIBM, h(x,t) is the scalar forcing on the fluid due to the presence of solid
particles. Currently the code is capable of imposing the Dirichlet boundary condition on all the particle
surfaces. It is possible to impose Neumann and mixed boundary conditions on the particle surfaces,
and we plan to implement these in the near future. Using the discrete-time equations and noting that at
time n+1
 n 1  s
On the particle surface this gives
1
h(, t )   u      2  (s   n )
(13)
t
for the scalar forcing h(x,t) on the fluid due to the particle surface Ω.
Droplet Field Initialization: The effects of different spatial arrangements of droplets on heat and mass
transfer are quantified by performing DNS of passive scalar transport over monodisperse spherical
particles that are generated by simulations of the Matérn process. In one set of simulations the average
number density of the particles is kept constant (volume fraction is constant) and the hard-core
distance is systematically varied to investigate the effect of drop-drop interactions. Several such sets of
calculations over a range of number density (volume fraction) values were performed to investigate the
effect of number density (volume fraction).
The droplet field and initial conditions are chosen to be representative of typical gasoline direct
injection spray conditions with 60 m diameter droplets vaporizing in air at a temperature of 1200K.
A droplet’s region of influence is first determined by performing DNS of steady flow with passive
scalar transport over a single sphere with the particle Reynolds number Rep=20, and Pr=0.7 (in this
case the passive scalar is taken to be temperature with the thermal diffusivity equal to the scalar
diffusivity,    ). The ambient gas temperature is initially uniform with value  , and the sphere is
maintained at a temperature s for all time. The nondimensional temperature field is defined as
  s
T
,
  s
so that it is equal to one in the ambient gas and zero on the sphere. The drop region of influence is
defined by the length of the T  0.5 contour, which is denoted d in the longitudinal (streamwise)
direction, and d  in the transverse (cross-stream) direction. Figure 3 shows two cut-planes through the
flow which reveal d  3.37 r0 and d  2.77r0 for Rep=20, and Pr=0.7, where r0 is the radius of the
spherical droplet. Estimates of the drop’s region of influence are used to determine the range over
which the hard-core distance parameter h is varied.
Figure 3. Contours of a passive scalar (denoted T) obtained from direct numerical simulation of steady flow
at Reynolds number of 20 (based on sphere diameter) past a sphere with Pr=0.7. Two cut-planes are shown
to characterize the drop region of influence in streamwise and cross-stream directions based on the
dimensions of the T=0.5 contour.
Computational requirements for accurate resolution of the flow around each sphere require a minimum
of 8 grid points on the diameter of each droplet. The total number of grid points ( 256 128 128 ) is
limited by processor memory, which in turn determines the size of the computational box. The average
number density value is chosen to simulate around 100 droplets in the computational domain, resulting
in a number density of 175739 drops/cc. (It should be noted that the computational cost increases very
little with increase in number of droplets, therefore simulating higher number densities is not a
limitation of the DTIBM method.) For droplet radii of 31.5 microns this results in a volume fraction of
0.023, which corresponds to a fairly dilute spray regime.
Initial and Boundary Conditions: The steady flow past the droplet field is computed at a nominal
Rep=20 (based on an isolated sphere). Then the scalar field is introduced with T  1 in the ambient gas
and T  0 on the sphere surface. The inflow condition is at x  0 and outflow at the end of the
computational domain into a buffer region to prevent artificial pressure wave reflection.
DNS Results: The T  0.5 isosurfaces (orange), as well as the T  0 isosurface (in violet denoting the
sphere surfaces) are shown for a volume fraction of 0.02 and various hard-core distance values in
Figure 4. For the lowest hard-core distance of h  2r0 there is considerable overlap of the T  0.5
isosurfaces, which in some cases enclose several neighboring spheres indicating considerable dropdrop interaction. On the other hand, at the same volume fraction but with higher hard-core distances
up to h  3.46r0 there is decreasing overlap of the T  0.5 isosurfaces indicating minimal drop-drop
interaction at the highest hard-core distance.
Figure 4. Isosurfaces of T=0.5 from DNS of steady flow past different spatial arrangements of spheres
initialized using a Matérn point process. All cases are for an average volume fraction of 0.02. As the hardcore distance is increased, the drop-drop interaction effects are less pronounced.
The surface scalar flux was calculated for each sphere, and an average surface scalar flux was
determined for the entire droplet field. The variation of this average surface scalar flux (scaled by the
isolated sphere’s surface scalar flux) with increasing hard-core distance is examined at a fixed volume
fraction of 0.02. See Figure 5. For the lowest hard-core distance where maximum drop-drop
interactions are expected, it is found that the average surface scalar flux can be about 18% less than the
isolated sphere’s surface flux. As the hard-core distance is increased, the effects of drop-drop
interactions are weaker, and the ratio tends towards one. These preliminary results are obtained for a
single realization of the droplet field, and need to be repeated over multiple independent initializations
to obtain reliable averages and confidence intervals. Average values formed from multiple
independent simulations may also revise the non-monotonic trend seen in this figure.
Figure 5. The variation of the ratio of average surface scalar flux (obtained by averaging over all the spheres
in the domain) to the surface scalar flux for an isolated sphere, with increasing hard-core distance. The
volume fraction is 0.02 for all the cases.
The interphase scalar source term from the droplet cluster that would appear in the gas-phase scalar
conservation equations is also calculated, and it is scaled by the interphase scalar source term that
would appear for a single isolated droplet. If the source terms are indeed linear in the number density
for this homogeneous problem (which is an assumption inherent in the LE approach), then the ratio of
these interphase scalar source term for the droplet cluster should just be the number of droplets (102,
in this case) times the interphase scalar source term for a single isolated droplet. However, Figure 6
shows that the effect of drop-drop interactions even at this low volume fraction (2%) results in a
scaling of the interphase source terms that is not linear in the number density.
Figure 6. Ratio of the scalar interphase source term calculated from the droplet cluster composed of 102
droplets, to the source term arising from an isolated droplet, as a function of hard-core distance. The source
terms computed in the current LE approach assume a linear scaling with the droplet number density for
homogeneous droplet clusters, which corresponds this ratio being a constant value of 102.
Finally the effects of varying the volume fraction on the average surface flux for a single sphere are
shown in Figure 7. The effects of hard-core distance become more apparent as the volume fraction
increases, and they are non-negligible even at a volume fraction of 0.02 for a hard-core distance of
h  2r0 .
Figure 7. The ratio of average surface scalar flux (obtained by averaging over all the spheres in the domain)
to the surface scalar flux for an isolated sphere, with increasing hard-core distance for different values of the
volume fraction.
Conclusions from DNS study: The principal conclusions from the DNS study are that drop-drop
interactions can be important even at relatively low volume fraction, and they can affect the LE model
for individual droplet heat transfer that assumes an isolated droplet. Also the interphase source terms
are influenced by drop-drop interaction effects, and depending on the spatial arrangement of the
droplets they may not scale linearly with number density even for a homogeneous spray. The form of
the interphase source terms in an LE approach assumes a linear dependence on the number density for
a homogeneous spray. Therefore, even these preliminary DNS results point to the importance of
accounting for drop-drop interactions in spray models of heat and mass transfer.
Extensions to current DNS study: The DNS results presented here are very preliminary and there are
several restrictions and assumptions inherent in the simulations. Many of these restrictions can be
easily removed by code development and extension, while others are fundamental to the formulation
and are considerably harder to overcome. The extensions that are relevant to this work are itemized
below:
1. Multiple Independent Simulations (MIS): The DNS need to be repeated for different
initializations of the droplet field for each hard-core distance specification in every set of
calculations performed at a fixed volume fraction. Such MIS are trivial to perform and will
enable the calculation of converged statistics with confidence intervals.
2. Thermal Boundary Condition and Coupling: The thermal boundary condition can be extended
from pure Dirichlet to specified heat flux (including the zero heat flux insulated condition).
The thermal coupling to the droplet can be improved by first considering unsteady heating of
the droplet assuming a lumped capacitance approach that is valid for small Biot number.
3.
4.
5.
6.
7.
8.
Solution of the conduction equation for the temperature field within the sphere can also be
incorporated based on semi-analytical procedures.
Effects of upstream turbulence: This is easily incorporated in the pseudo-spectral DTIBM code
that was designed to study particle-turbulence interactions. Testing and validation of the
turbulence calculation in underway and the upstream turbulence effects will be incorporated in
the code in the near future.
Droplet motion: This is already incorporated in the code. Combining droplet motion with
turbulence will enable us to study the effect of dispersion of the droplets in a more natural way,
than forcing them to occupy fixed locations based on a point-field model. The evolution of the
pair-correlation function due to droplet-turbulence interactions can be studied using these
enhancements to the current DNS capability.
Collisions: An important issue is how to treat collisions. If the spheres are treated as solid
particles undergoing elastic collisions, then this can be easily incorporated.
Droplet coalescence: If the DNS is to truly represent droplets, then the phenomenon of droplet
coalescence or bounce-back after collisions needs to be accounted for. As noted previously,
MD calculations indicate that the physical outcome of coalescence or bounce-back after
droplet collision may depend on sub-continuum phenomena. These outcomes can be
incorporated in a somewhat ad hoc fashion, but this will undermine the model-free
characteristic of the true DNS calculations presented here.
Droplet internal flow and droplet deformation: The DTIBM technique generates a fictitious
flow within the sphere. Extending the hydrodynamic boundary condition at the droplet surface
from no-slip to a stress boundary condition is a non-trivial extension, and possibly one of the
principal challenges in extending the DTIBM technique to realistic droplet calculations.
Droplet vaporization: Currently the DTIBM technique implements a no-penetration
hydrodynamic boundary condition at the surface of the sphere, corresponding to a solid
surface. In order to represent vaporization, this boundary condition needs to be extended to
include non-zero normal velocity corresponding to droplet surface regression due to mass
transfer. Also the gas-phase mass conservation equations need to be extended to include
variable-density effects.
Having established through preliminary DNS calculations that drop-drop interactions can be
important even at relatively low volume fraction, we show in the next section how the LE approach
may be extended using the assumed point field (APF) formalism to incorporate drop-drop
interaction effects.
THE ASSUMED POINT FIELD FORMALISM
We have already noted that other works by Chiu10 have already demonstrated the use of second-order
statistics such as the pair-correlation function to characterize drop-drop interactions. Here we extend
this characterization to augment existing LE models using the concept of an assumed point field. Just
as the mean and variance of a random variable are only incomplete statistical characterizations
whereas the probability density function (PDF) is needed to completely characterize the random
variable, similarly the average number density is an incomplete statistical characterization of a
stochastic point process whereas the complete characterization of a point process20 requires
specification of additional information. It is common in assumed PDF models7 to assume a Gaussian
or normal PDF because the mean and variance completely specify the Gaussian PDF. By analogy, we
may ask what is the simplest point process that is completely specified by the average number density.
This is the Poisson point process, but it is unsuitable for describing the location of droplet centers
because the complete independence property of the Poisson process allows for overlapping droplets,
which results in an unphysical model. One can remove overlapping drops in the Poisson model by
using a thinning procedure on a Poisson point process22, 23, which is the procedure to simulate a
realization of a Matérn point process. The Matérn point process model has an analytic expression for
the pair correlation that is completely specified by the hard-core distance parameter. The advantage of
using a Matérn point process model is that it is easy to simulate this point process and it has an
analytic pair-correlation function that is specified in terms of the hard-core distance and number
density of the base Poisson process. It is not obvious how to simulate point processes for arbitrary pair
correlation functions. However, it is important to also note that there are limits on the volume fraction
that can be simulated using the Matérn point process.
In the next section we describe the extended LE (ELE) approach that consists of locally representing
the spray as a realization of a Matérn process of non-overlapping droplets, that corresponds to the
average number density obtained from the LE approach. This allows for a more realistic calculation
of individual droplet heat and mass transfer as well as the interphase heat and mass transfer source
terms, by accounting for drop-drop interaction effects through the pair-correlation function of the
Matérn process.
THE EXTENDED LE CONCEPT
A full second-order closure would require a solution of the evolution equation of the pair-correlation
function at each cell in an LE calculation. This approach is deemed to not be practical for
computational and modeling reasons at this stage. The APF provides an alternative route to
incorporating second-order information in the existing LE approach by assuming the underlying point
field and pair-correlation in each cell in the LE calculation. The schematic in Figure 8 describes the
ELE concept. The Matérn number density is required to match the local average number density
calculated by the LE model. All that remains is the specification of the hard-core distance parameter.
Figure 8. The extended LE concept. The quantities (g(r)) represent parameters or moments of the paircorrelation function.
Here we propose a simple specification of the hard-core distance parameter h . This quantity has a
lower bound of 2r0 arising from the requirement of non-overlapping droplets. We also note that the
Matérn pair-correlation function takes the Poisson value of one for values of r  2h . See Figure 2.This
implies that in the Matérn model, for droplet separations beyond this correlation distance the droplets
behave practically independently of each other. We argue that a pair of droplets that are initially very
close to each other will remain correlated over a length scale l  d characteristic of the turbulent eddy
whose timescale  is the same as  d , the droplet’s momentum response time. Over this length scale
l  d we assume the pair of droplets is “trapped” in the same eddy, and that they evolve independently
once their separation distance exceeds this length scale. This independence is reflected in their paircorrelation going to unity beyond this length scale. Assuming inertial range scaling of turbulent eddies
in high Reynolds number turbulence4
u3
u 2
,
l
, (l ) 


we estimate the length scale of the eddy whose time scale is the same as the droplet’s momentum
response time to be
2
1/ 2 2 ro  d

l  d  u  d  k
(14)
9 g  f
The hard-core distance is then simply specified to be
l 

h  max  2r0 , d
2


.

(15)
The above specification is adequate for monodisperse droplets. For polydisperse sprays some
modifications are needed. The dependent thinning procedure for generating a Matérn point process
from an underlying Poisson point process can be extended to the polydisperse case quite easily.
However, to our knowledge there is no analytic form for the pair-correlation function in the
polydisperse case. Nevertheless, the pair-correlation function can be estimated by numerical
simulation.
EFFECTS OF TURBULENCE ON DISPERSION
It is instructive to examine what affects the evolution of the pair-correlation function. Clearly
turbulence in the ambient gas plays a central role in pair- or relative- dispersion because it affects the
relative velocity between droplet pairs. It is important to note that this is an effect that manifests itself
at the level of second-order statistics, and has been denoted preferential concentration by Eaton and
co-workers25-28.
However, turbulence in two-phase flows, and its effect on dispersion of droplets pose significant
modeling challenges even at the level of first-order statistics. The effects of turbulence on first-order
statistics of droplet dispersion are non-trivial to reproduce since it is coupled to the dynamics of
interphase turbulent kinetic energy transfer. The problem is particularly challenging because it is
difficult to simultaneously reproduce the disparate trends of turbulent kinetic energy decay and droplet
dispersion with variations in droplet Stokes number. A recently developed dual-timescale Langevin
model29 has been shown to be capable of reproducing these effects simultaneously. The question of
droplet relative dispersion is an important one, and it will need careful DNS calculations to resolve the
question of how the pair-correlation function evolves under the effects of turbulence.
NUMERICAL CONVERGENCE AND ACCURACY
Devising a numerically convergent scheme to solve the system of equations in the LE approach, and
validating the accuracy of the numerical solution is a necessary step for predictive spray modeling.
Many studies have demonstrated the non-convergence of popular numerical implementations of the
LE approach to spray modeling30-32. A systematic investigation of the numerical convergence
characteristics of the KIVA implementation of the LE approach led to the conclusion that even
global spray characteristics such as the spray penetration length did not show any trend towards
convergence to an asymptotic value as the numerical parameters such as grid size, time step and
computational particles were varied toward their limiting values. Similar concerns regarding the
convergence of numerical implementations of the LE approach have also been reached by works of
Abraham31, and Schmidt33. Numerical implementations that do not converge, and that are not
accurate, can seriously undermine the predictive capability of a spray model.
In the numerical convergence study a systematic approach to establishing convergence was also
outlined30. Considerable success has been attained in establishing numerical convergence for the
vaporization sub-model of the LE implementation34. In recent work34 we show how the numerical
implementation of vaporization in the LE approach can be improved. A simple vaporization test
problem is proposed, that admits analytic solutions to the spray equation, and is useful for testing the
accuracy of numerical solutions. This study shows that a simple particle method solution using
uniform sampling of the ddf yields an accurate solution to the simple vaporization test problem.
However, many spray codes such as KIVA use importance sampling of the mass-weighted ddf on the
grounds that this is more computationally efficient. The implementation of importance sampling in
KIVA results in an inaccurate numerical solution of the spray equation that does not converge to the
analytic solution for the simple vaporization test, even for a very large number of computational
particles. We show that importance sampling can be accurate and computationally efficient if
statistical weights are correctly assigned to match the initial radius distribution. Simulations also
reveal that the discontinuous evolution of statistical weights corresponding to vaporization in existing
particle methods results in numerical estimates of spray statistics that do not unconditionally converge
to a continuous asymptotic limit as the time step is decreased. A continuously-evolving-weights
algorithm is developed that yields numerically convergent results that also match the analytic solution
very well. These improvements to the particle method solution of the spray equation, which result
in an excellent match of numerical predictions with the analytical solution in the test problem, can be
expected to dramatically improve the accuracy of complex spray calculations at minimum
computational expense. This success indicates that if proper attention is paid to testing and validation
of the numerical implementation following the systematic approach outlined in Ref.30, it is possible to
obtain accurate, numerically convergent solutions using particle methods.
The next step in the approach to establishing numerical convergence of LE implementations is
accurate coupling of the particle method used to solve the spray equation, to the finite-volume solution
to the Eulerian gas-phase conservation equations. This coupling requires numerically convergent and
accurate estimation of the number density and spray source terms. In earlier work where particle
methods were combined with finite-volume calculations, it is shown that kernel estimation methods35
offer a grid-free estimation approach for the number density and spray source terms. Some ideas for
developing test problems that can be used to validate the numerical convergence and accuracy
of spray computations using such estimation methods are now discussed.
In our efforts to demonstrate convergence of interphase source terms obtained from particle data, we
are currently investigating three different approaches for estimation of mean statistical quantities on an
Eulerian grid from Lagrangian particle data. These three algorithms are:
(i)
Lagrangian Polynomial Interpolation (LPI),
(ii)
Cubic Spline Interpolation (CSI), and
(iii)
a Two-Stage Kernel Estimation (TSE) algorithm based on local least-squares proposed by
Dreeben & Pope (details of which are given in Ref.35)
The three algorithms are tested by specifying an analytic number density function from which
Lagrangian particle data is generated. See Figure 9. The algorithms then estimate the number density
from the Lagrangian particle data, and the error with respect to the analytic number density function is
calculated on a finer test grid. For the number density estimation our first finding is that a nonnegativity constraint for number density has to be imposed on the TSE algorithm that uses local leastsquares. For number density estimation the CSI and TSE algorithms perform better than the LPI
algorithm, as seen in Figure 10. If estimates of the number density gradient are considered (not shown
here), then the TSE algorithm performs better than the other two algorithms.
Figure 9. A test for number density estimation where particles are distributed according to a specified
analytic number density function that is representative of sprays. Errors incurred by different numerical
algorithms in estimating the number density from particle data can be compared.
Figure 10. Estimates of the number density from particle data using three different algorithms are compared
with the analytic specification for the number density (top left panel). The two-stage estimation (TSE)
algorithm based on local least-squares with an imposed non-negativity constraint on the number density (top
right panel), and the Cubic Spline Interpolation (CSI) yield good results. The Lagrangian Polynomial
Interpolation (LPI) is subject to large fluctuations in the number density estimate, especially near the edge of
the spray.
DISCUSSION: A PERSPECTIVE ON SPRAY MODEL DEVELOPMENT
If the objective of spray modeling is to develop robust and predictive models of sprays for use in
engineering problems, it is useful to speculate on what determines model accuracy. When a model
prediction is compared with experimental data, there are many reasons why differences can arise
between the two. Even the perfect spray model will not give exact agreement with experimental data
due to experimental uncertainty. In general, differences between model prediction and experimental
data can arise due to
(i)
inadequacies in the mathematical representation,
(ii)
inaccurate closure models for physical phenomena such as droplet heat and mass transfer,
(iii)
numerical inaccuracy due to inadequate grid resolution, finite number of computational
particles, time step not being small enough to resolve relevant time scales, and finally,
(iv)
experimental uncertainty.
From an engineering perspective, the optimal model should balance the uncertainties and errors arising
from each of these sources, so that modeling and computational efforts are efficiently focused on those
sources that contribute the most to the predictive capability.
Another perspective on what constitutes a good spray model can be gained by applying Pope’s criteria4
for assessing turbulence models to the spray problem. The principal criteria advanced in that analysis,
and their relevant interpretation in the spray context are:
(a) Level of description: in the spray context this means does the model represent the SMD, or
the entire distribution of droplet sizes. A higher level of representation usually requires more
complex models and more computation time
(b) Completeness: in the spray context this means the number of additional inputs required by
the model, e.g., is the spray angle a required input to the model, or is that quantity predicted
by the model itself
(c) Cost and ease-of-use: Is the model used for repetitive calculations in industry, or is it used in
large-scale one-of-a-kind research computations
(d) Range of applicability: Can the model be applied equally to internal combustion engine
sprays as it is to pharmaceutical inhalers?
(e) Accuracy: Model accuracy, as discussed earlier in this section.
The use of such criteria can be very useful in performing legitimate comparisons of models. It is of
particular importance to not expect models with different levels of representation to perform equally
well in predicting sprays to the same level of detail.
With these perspectives, the extended LE modeling approach proposed in this work may be viewed as
striking a balance between incorporating adequate mathematical representation to model important
spray physics, while retaining the cost and ease-of-use for repetitive spray calculations in industrial
applications. The use of DNS to improve physical sub-models in the LE approach is also an attempt in
this direction.
SUMMARY AND CONCLUSIONS
Significant progress has been made in areas that are critical to improving the predictive capability of
spray models, such as (a) the mathematical formulation of spray theory10, 18-20, 36, (b) improving
accuracy of the closure relations for the physical sub-models representing droplet heat and
mass transfer9, 37, and (c) improving the accuracy of numerical solution to the model equations31, 33, 34.
Specifically, in this work we describe recent advances in developing an extended LE modeling
approach, the use of DNS to improve closure relations for physical sub-models representing heat and
mass transfer, and the use of importance-sampling and kernel estimation methods to improve the
accuracy of numerical solutions to the model equations. The extended LE approach not only provides
a mathematical formalism for incorporating drop-drop interactions, but it also permits the extension of
existing LE formulations which are restricted to dilute sprays with low liquid volume fraction to
dense sprays. In fact, even collision modeling in sprays can be improved using the extended ELE
approach, although those details are not presented here.
In spite of this progress, the ultimate goal of developing a truly predictive and robust CFD spray
simulation remains a challenging, unsolved problem. While most research efforts have focused on
developing accurate closure relations for droplet heat and mass transfer9, the importance of other
factors, such as numerical accuracy, and adequacy of the underlying mathematical formulation, is also
now being recognized. An important conclusion of this work is that integration of knowledge in the
specialty sub-areas is critical if we are to make progress in solving the spray problem, even for
the simple test case of a single-component sub-critical vaporizing spray. An integrated approach to a
CFD spray simulation requires assessing the impact of (i) the adequacy of mathematical formulation,
(ii) the accuracy of closure relations for droplet heat and mass transfer, and (iii) the numerical accuracy
of the computational scheme, on prediction of spray characteristics that are relevant to design
engineers.
Such an assessment would help identify where spray modelers should focus their efforts. Systematic
studies of the sensitivity of predicted of global spray characteristics to numerical parameters, model
constants, and underlying mathematical formulation can be very useful in guiding a program of spray
modeling development. Preliminary studies indicate that spray predictions using Lagrangian-Eulerian
codes exhibit strong sensitivity to numerical parameters31. It has been conclusively demonstrated that
these problems can be addressed through using better numerical schemes and algorithms34, 35. A
similar systematic assessment of sensitivity to the model constants has not yet been reported in the
spray literature. It is considerably more difficult to assess the sensitivity of model predictions to
the underlying mathematical formulation.
It is anticipated that the issue of obtaining accurate numerical solutions to the spray problem will be
resolved in time, given the increasing availability of inexpensive computational power, and
the development of better algorithms for solving the model equations. The same is true for the
development of more accurate closures for droplet heat and mass transfer using direct numerical
simulations and experimental data. What is likely to remain a more long-term challenge is the
development of better mathematical formulations to address the complex multiscale, nonlocal and
nonlinear interactions in sprays. A rigorous mathematical formulation that bridges the internal flow in
the nozzle through breakup and atomization into dispersed spray droplets of a given size distribution
and initial spray angle, would be a significant breakthrough in spray modeling and simulation. In
summary, the predictive capability of CFD calculations of sprays will continue to improve with robust
numerical implementations, accurate sub-models, and significant enhancements to the underlying
mathematical formulation, examples of which have been described in this paper.
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