AN INVESTIGATION OF THE MESH DEPENDENCE OF THE
STOCHASTIC DISCRETE DROPLET MODEL APPLIED TO DENSE
LIQUID SPRAYS
By
Simone E. Hieber
A THESIS
Submitted in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE IN MATHEMATICS
MICHIGAN TECHNOLOGICAL UNIVERSITY
2001
This thesis, “An Investigation of the Mesh Dependence of the Stochastic Discrete
Droplet Model Applied to Dense Liquid Sprays”, is hereby approved in partial
fulfillment of the requirements for the Degree of MASTER OF SCIENCE IN
MATHEMATICS.
Department of Mathematical Sciences
Signatures:
Thesis Advisor
(Franz X. Tanner)
Department Chair
(Alphonse H. Baartmans)
Date
i
Acknowledgments
I would like to acknowledge my academic advisor Franz X. Tanner for his advice,
support and guidance throughout the course of this thesis. I appreciated the
opportunity to attend the SAE World Congress, Detroit March 2001, funded by
the Department of Mathematical Sciences. I would like to thank Allan Struthers,
Adrian Sandu and Song-Lin (Jason) Yang for serving on my committee. I would
also like to thank David Schmidt for providing the NTC reference algorithm
and for his support, and Rolf Reitz who gave me the opportunity to visit the
Engine Research Center at the University of Wisconsin, Madison. I appreciated
the use of the facilities in the Center of Experimental Computations at Michigan
Technological University.
ii
Abstract
The Stochastic Discrete Droplet Model is widely used to simulate engine sprays.
However, due to inadequate spacial resolution the spray computations can be
strongly mesh dependent. The liquid Void Fraction Compensation (VFC) method
is introduced which compensates for the lack of spatial resolution by correcting
the droplet density in each cell according to a predetermined average liquid void
fraction. This new method has been implemented in the collision and evaporation
models of the KIVA3 computer code.
Computations have been performed for high-velocity dense sprays injected
into cylinders equipped with a coarse, a medium and a fine polar mesh. The mesh
dependence is analyzed for non-evaporating and evaporating sprays where each
type of spray has been investigated under idealized conditions with other models
switched off, and for a realistic spray subject to experimental comparisons. The
evaluation criteria used to judge the model performances include the Sauter mean
radius of the spray, the penetration and the evaporation rate when applicable. In
addition, the grid independent NTC-algorithm by Schmidt and Rutland served
as a reference for the collision behavior of the VFC-method.
The VFC-method applied to the collision model performed well for the idealized sprays. Surprisingly, the evaporation model proved virtually mesh indeiii
pendent for the evaporation behavior and no VFC correction was required. The
real spray still shows mesh dependence which can be attributed mainly to the
insufficient resolution of the liquid-gas momentum transfer.
iv
Contents
1 Introduction
1
2 Background
4
2.1 The Governing Equations . . . . . . . . . . . . . . . . . . . . . .
4
2.1.1
The Gas Phase . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.2
The Liquid Phase . . . . . . . . . . . . . . . . . . . . . . .
10
3 Methods to Reduce Grid Dependence
17
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2 The Collision Algorithm of Schmidt and Rutland . . . . . . . . .
23
3.2.1
Collision Calculations . . . . . . . . . . . . . . . . . . . . .
24
3.2.2
Creation of a Collision Mesh . . . . . . . . . . . . . . . . .
25
3.3 Collision Model of Niklas Nordin . . . . . . . . . . . . . . . . . .
26
3.4 The CLE-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.5 The VFC-Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4 Results and Discussion
33
4.1 Comments on the Type of Grid . . . . . . . . . . . . . . . . . . .
33
4.2 Computation Cases . . . . . . . . . . . . . . . . . . . . . . . . . .
41
v
4.3 The Effect of the Drop Collision Model . . . . . . . . . . . . . . .
4.4
45
4.3.1
The Validation Case . . . . . . . . . . . . . . . . . . . . .
46
4.3.2
The Real Spray Case . . . . . . . . . . . . . . . . . . . . .
51
The Effect of the Evaporation Model . . . . . . . . . . . . . . . .
55
4.4.1
The Validation Case . . . . . . . . . . . . . . . . . . . . .
55
4.4.2
The Real Spray Case . . . . . . . . . . . . . . . . . . . . .
58
5 Conclusions
61
vi
1
Chapter 1
Introduction
The interaction of flow, spray and combustion processes forms a complex system
of physical phenoma whose time and length scales range over a wide spectrum.
The numerical description of such a system relies on spatial and temporal averaging and discretization procedures of the relevant differential equations. This leads
to a loss of physical information, whose recovery is attempted by means of appropriate models. Such models often have a limited range of validity and may require
a fine spatial and temporal resolution for good performance. On the other hand,
the limited capacity of present-day computers often restricts the spatial resolution which for large geometries leads to coarse computational meshes and, along
with the resulting accuracy and stability problems, brings the employed models
to the limit of their applicability.
The computations presented in this thesis have been conducted with a KIVA3based code [2]. KIVA3 is a computer program which numerically solves the equations for three-dimensional, time-dependent, turbulent, chemically reactive flows
which are interacting with fuel sprays. In particular, the gas phase is described
2
by the Reynolds-Favre averaged conservation equations for mass, species, momentum and energy, together with the equations for the turbulence model. The
effects due to sprays and chemical reactions are considered via appropriate source
terms in the gas phase equations. The turbulence model utilized in these computations is the two equations RNG k-² model, modified to include the effects of
the spray-turbulence interaction.
The evolution of the spray is governed by a probability density function which
describes the volume-specific droplet density in a particular state. This leads
to a first order, quasi-linear partial differential equation whose (non-constant)
coefficients and source terms are determined by submodels describing droplet
collisions, deformations and breakups, droplet evaporation, turbulent gas-droplet
interaction and the influence due to gravity.
The stochastic discrete droplet model is used to solve the spray equation.
This approach is a combination of the discrete parcel method and a random
sampling procedure. The discrete parcel method approximates the continuous
spray distribution function by a step function with each step corresponding to a
parcel, i.e. a collection of droplets of identical states.
It is well known that this approach is highly dependent on spatial discretization because the resolution of the mesh is insufficient to accurately reproduce the
spray phenomena. The objective of this thesis is the investigation of the mesh
dependence of dense liquid spray computations and the development of a method
to reduce this effect for the collision and evaporation model.
This problem has been addressed in many studies [1, 4, 19, 20, 14]. A few of
these methods [4, 19, 14] are discussed in this thesis. In addition, a new method
is introduced: the Void-Fraction-Compensation method or VFC-method. The
3
VFC-method compensates the lack of spatial resolution by correcting the droplet
density in each cell according to a predetermined average liquid void fraction. It
has been implemented in KIVA3 to correct the drop collision calculations and the
evaporation rate and tested for non-evaporating and evaporating sprays using a
coarse, a medium and a fine polar mesh. The mesh dependence has been investigated under idealized conditions with other models switched off and for a realistic
spray subject to experimental comparisons. Mesh independence is achieved when
mesh refinements do not change the computational results. For the cold sprays,
the Sauter mean radius and the penetration serve as the evaluation criteria. The
remaining liquid fuel mass, the liquid penetration and the vapor penetration are
considered for evaporating sprays. The results of collision calculations using the
VFC-Method are compared with the performance of the No-Time-Counter (NTC)
algorithm by Schmidt and Rutland [19]. This algorithm is grid independent because a mesh separated from the gas phase grid is generated for the collision
calculations.
The performance of the VFC-method is expected to be most effective in spray
computation where the mesh resolution is very low. Therefore, the VFC-method
might be useful in simulations of large-bore diesel engines as conducted by Tanner
et al. [23, 24, 25]. An application in fast trend analysis is also conceivable where
computations with coarse meshes are used to predict trends in a short run time,
as obtained, for example, by KIVA3V-LITE [7]. The accuracy of the results could
be improved significantly by the VFC-method.
4
Chapter 2
Background
This chapter contains important background information about how the two
phase flow is simulated. In the first part, the physical models are discussed
in the form of governing equations. More information can be found in many
publications, e.g. [3, 5, 6, 11, 12, 16] and details concerning the numerical implementations are documented in [2].
2.1
The Governing Equations
The two phase flow under consideration is described by the continuous droplet
model, in which the gas phase is described by the Reynold-Favre-averaged conservation equations of continuum mechanics, while the liquid phase is modeled
by means of a continuous droplet distribution. Turbulence is described by a k-²
turbulence model. The spray model includes droplet breakup, collisions and aerodynamic effects. Interactions between the two phases are considered in terms of
source terms for mass, momentum, energy and turbulent kinetic energy.
2.1 The Governing Equations
5
Since the equations include ordinary and partial differential equations, initial
and boundary conditions are needed. For the gas phase, this involves a Lawof-the-wall for the velocity and heat transfer at solid boundaries. For the liquid
phase, the injection configurations need to be specified as well as the drop and
wall interactions.
2.1.1
The Gas Phase
The gas phase consists of several components. This approach is needed in the
case where chemical reactions are involved. This requires the use of equations
for a multi-component mixture, together with conservation equations for mass,
momentum, energy as well as the transport equations for the turbulent kinetic
energy and its dissipation rate.
State Equations
The state relations are described for an ideal gas mixture where the individual
species are denoted by the subscript m.
X ρm
Wm
m
X ρm
I(T ) =
Im (T )
ρ
m
X ρm
cpm (T )
cp (T ) =
ρ
m
p = R0 T
hm (T ) = Im (T ) +
R0 T
Wm
(2.1)
(2.2)
(2.3)
(2.4)
R0 is the universal gas constant and Wm and Im (T ) are the molecular weight
and specific internal energy of species m, respectively. The coefficient cp refers
2.1 The Governing Equations
6
to the specific heat at constant pressure and hm (T ) is the specific enthalpy at
the gas temperature T .
Transport Coefficients
The following coefficients are used in several physical models:
The total viscosity
µ = µair + cµ ρ
k2
²
(2.5)
where the air viscosity is given by
A1 T 3/2
T + A2
(2.6)
K=
µcp
Pr
(2.7)
D=
µ
ρSc
(2.8)
µair =
the heat conduction coefficient
and the diffusion coefficient
The gas viscosity µ from equation (2.5) which is used in the momentum equation
(2.12), is the sum of the molecular viscosity µair and the turbulent viscosity
2
µt = cµ ρ k² , where k and ² are the turbulent kinetic energy and its dissipation
rate, respectively, cµ is a constant and ρ is the gas density. The Sutherland
formula (2.6) determines the air viscosity µair , where A1 and A2 are constants.
2.1 The Governing Equations
7
The heat conduction coefficient K is needed to compute the heat flux in the
energy equation (2.15).
Accordingly, the diffusion coefficient D affects the
diffusion term in the species conservation equations (2.9). The Prandtl number
P r and the Schmidt number Sc are dimensionless coefficients.
Species and Mass Conservation
As a multi-component mixture is considered, a continuity equation is needed for
each species marked with the subscript m:
·
µ ¶¸
ρm
∂ρm
+ ∇ · (ρm u) = ∇ · ρD∇
+ ρ̇cm + ρ̇s δm,1
∂t
ρ
(2.9)
ρm is the mass density of the species m, ρ is the total gaseous mass density, and
u is the velocity. Fick’s law is used to determine the diffusion term with the
mass diffusion coefficient D given by equation (2.8). In this study, no chemical
reactions are considered. Therefore, the source terms due to chemistry ρ̇cm are
neglected. The source term ρ̇s takes the evaporation of the liquid into account.
The Kronecker delta function δm,1 restricts the source term to one liquid species
denoted by the subscript 1. The source term’s evaluation requires a weighted
integral over the drop density function f that is illustrated in section (2.1.2):
¶
4π
3
ρd r dv dr dTd dy dẏ
(2.10)
ρ̇ = −
3
P
The sum of the equations (2.9) over all species, ρ = m ρm , leads to the total
Z
s
mass conservation equation:
d
f
dt
µ
2.1 The Governing Equations
8
∂ρ
+ ∇ · (ρu) = ρ̇s
∂t
(2.11)
Momentum Conservation
The momentum conservation equation is
µ
¶
∂(ρu)
2
+ ∇ · (ρuu) = ∇ p + ρk + ∇ · σ + F s + ρg
∂t
3
(2.12)
where p is the gas pressure and g is the constant of gravity. The viscous stress
tensor σ is defined by:
·
2
σ = µ (∇u + (∇u) ) − ∇ · uId
3
¸
T
(2.13)
The superscript T denotes the transpose and Id is the unit dyadic. F s represents
the rate of momentum gain or loss due to the spray per unit volume:
Z
s
F =−
·
d
f
dt
µ
¶
¸
4π
4π
3
3
ρd r v −
ρd r g dv dr dTd dy dẏ
3
3
(2.14)
2.1 The Governing Equations
9
Energy Conservation
The conservation equation for the specific internal energy I is
∂(ρI)
+ ∇ · (ρuI) = −p∇ · u − ∇ · J + ρ² + Q̇s + Q̇c
∂t
(2.15)
Note that the value of I does not include the chemical energy. The heat flux
vector J is the sum of heat conduction and enthalpy diffusion:
J = −K∇T − ρD
X
µ
hm ∇
m
ρm
ρ
¶
(2.16)
where T is the gas temperature and hm is the specific enthalpy of species m. The
source terms Q̇c and Q̇s are due to chemical heat release and the spray interaction,
respectively. The term Q̇s is specified by
Z
s
Q̇ = −
¸
µ
·
1
2
2
f ρd 4πr ṙ Il (Td + (v − u)
2
¶
h
i
4
+ π cl Ṫd + Fg · vr dv dr dTd dy dẏ
3
(2.17)
Since this study does not consider chemical reactions, the source term Q̇c is set
to be zero.
k - ² Turbulence Model
The turbulence model implemented in KIVA3 is a k - ² turbulence model. In
this model, two additional transport equations are solved, one for the turbulent
kinetic energy k and one for its dissipation rate ²:
2.1 The Governing Equations
10
∂(ρk)
2
+ ∇ · (ρuk) = − ρk∇ · u + σ : ∇u + ∇ ·
∂t
3
µ
µ
∇k
P rk
¶
−ρ² + Ẇ s
µ
∂(ρ²)
+ ∇ · (ρu²) = −
∂t
(2.18)
¶
¶
µ
2
µ
∇²
c² − c²2 ρ²∇ · u + ∇ ·
3 1
P rk
²
+ (c²1 σ : ∇ − c²2 ρ² + cs Ẇ s )
k
(2.19)
The source term Ẇ s appearing in both transport equations describes the effect
of the turbulence on the droplets.
Z
s
Ẇ = −π
f ρd
4π 3
r (Fg · u0 ) dv dr dTd dy dẏ
3
(2.20)
It can be shown that Ẇ s < 0 and, therefore, it only depletes turbulent kinetic
energy.
2.1.2
The Liquid Phase
In this section, the equations of the liquid phase are presented. The liquid phase
is modeled by means of a probability density function f (t, x, v, r, Td , y, ẏ). This
approach requires the description of phenomena such as atomization, droplet
collisions, droplet break up, evaporation and aerodynamic drag .
The Spray Evolution Equation
The droplet distribution function f (t, x, v, r, Td , y, ẏ) is defined such that
2.1 The Governing Equations
f (t, x, v, r, Td , y, ẏ) dv dr dTd dy dẏ
11
(2.21)
is the probable number of droplets per unit volume at position x and time t with
velocities in the interval (v, v + dv), radii in the interval (r, r + dr), temperatures
in the interval (Td , Td + dTd ), distortion parameter in the interval (y, y + dy) and
deformation rate in the interval (ẏ, ẏ + dẏ).
The spray evolution equation of the probability density function
f (t, x, v, r, Td , y, ẏ) is given by
∂f
∂
∂
∂
+ ∇x · (f v) + ∇v · (f F ) + (f ṙ) +
(f ẏ) +
(f ÿ) = f˙coll + f˙bu (2.22)
∂t
∂r
∂y
∂ ẏ
where the source term f˙coll describes the droplet formation due to collision between droplets. The source term f˙bu accounts for the droplet breakup. Both
phenomena are discussed later in this section.
The droplet acceleration F is determined by aerodynamic drag and gravity.
The droplet evaporation leads to a rate of change in the droplet radius ṙ and the
temperature Ṫd . The derivatives of the distortion parameter ẏ and ÿ are described
by the Taylor drop oscillator given later in the section.
Drop Acceleration
The drop acceleration is given by
F = v̇ =
3ρ |u + u0 − v|
(u + u0 − v) CD (Red ) + g
8ρd
r
(2.23)
2.1 The Governing Equations
12
where u is the gas velocity and u0 is its turbulence fluctuation, v is the droplet
velocity, ρg is the droplet density and CD (Red ) is the drag coefficient that is a
function of the drop Reynolds number Red .
Droplet Evaporation
The liquid droplet receives its thermal energy from the gas. This energy is used
to increase the liquid temperature and overcome the latent heat of evaporation
in order to evaporate the fuel. Unless the gas is saturated with vapor fuel, evaporation always takes place and reduces the droplet radius. If the transferred heat
delivered by the gas is insufficient, the droplet temperature will decrease. The
differential equation for the droplet radius is given by
(ρD)gas (T̂ )
d(r2 )
=
Bd Shd
dt
ρd
(2.24)
where (ρD)gas (T̂ ) denotes the fuel vapor diffusivity in the gas at film temperature
T̂ = 31 (T + 2Td ). Td is the droplet temperature and ρd is the droplet density. The
Sherwood number Shd for the mass transfer is given by
1/2
1/3
Shd = (2.0 + 0.6Red Scd ) ln
1 + Bd
Bd
(2.25)
where the mass transfer number is
Bd =
Y1∗ − Y1
1 − Y1∗
(2.26)
2.1 The Governing Equations
13
and the Schmidt number is given by
Scd =
µgas (T̂ )
(2.27)
(ρD)gas (T̂ )
with the fuel vapor mass fraction Y1 =
ρ1
.
ρ
The fuel vapor mass fraction at the
droplets surface is denoted by Y1∗ .
The rate of the droplet temperature change is determined by the energy balance equation
4πr3 ρd cl
dTd
dr
= 4πr2 Qd + 4πr2
L(Td )
dt
dt
(2.28)
where cl is the liquid specific heat. The rate of heat conduction Qd to the droplet
surface per unit volume is obtained by using the Ranz-Marshall correlation:
Qd = Kgas (T̂ )
T − Td
N ud
2r
(2.29)
where the heat conduction coefficient is given by
Kgas (T̂ ) =
K1 T̂ 3/2
(2.30)
T̂ + K2
The Nusselt number is given by
1/2
1/3
N ud = (2.0 + 0.6Red P rd ) ln
1 + Bd
Bd
(2.31)
and the droplet Prandtl number is
P rd =
µgas (T̂ ) cp gas
(ρD)gas (T̂ )
(2.32)
2.1 The Governing Equations
14
The latent heat of evaporation L is a function of the droplet temperature Td and
is defined by
L(Td ) = hl (Td ) − hl (Td , ρv (Td ))
(2.33)
Atomization and Breakup
Atomization is the break up of liquid into tiny droplets at the nozzle exit. This
occurs due to aerodynamic forces. The atomization and drop break up model
used is the cascade drop break up model described in [21, 22]. The fragmented
liquid core is modeled by injecting large droplets which are subject to a sequence
of breakups. Each drop breakup depends on a breakup regime, stripping or bag
breakup, and the breakup criterion is computed from the Taylor drop oscillator
which models a droplet as a forced damped harmonic oscillator. The aerodynamic
drag acts as the external force, the surface tension as the restoring force and the
liquid viscosity as the damping force. The equation for the acceleration of the
normalized droplet distortion parameter y is
ÿ +
8γ
2ρg |u + u0 − v|2
5µl
ẏ
+
y
=
ρl a2
ρl a3
3ρl a2
(2.34)
where µl (Td ) is the viscosity of the liquid phase and γ is the droplet surface
tension.
2.1 The Governing Equations
15
Drop Collisions
The standard drop collision model implemented in KIVA is based on the widely
used collision algorithm of O’Rourke [15]. It is well well known [14, 19, 20] that
this collision model is inherently grid dependent. This is especially the case for
dense sprays as they occur in Diesel engines.
The probability Pn that a drop undergoes n collisions with other droplets
follows a Poisson distribution,
Pn = e−µ
µn
n!
(2.35)
where the mean µ = is the mean expected number of collisions of a drop in a
computational timestep ∆t.
µ = ν ∆t
(2.36)
The collision frequency between droplets with subscript 1 and droplets with subscript 2 is given by
ν=
N1 N 2 σ
|v1 − v2 |
V
(2.37)
where V represents the volume of the domain where the droplets can collide, and
N represents the number of droplets corresponding to the subscript 1 or 2. The
term |v1 − v2 | is the relative velocity between droplets and σ is the collision cross
section of the droplets defined by
σ = π(r1 + r2 )2
(2.38)
2.1 The Governing Equations
16
The outcome of the collision can be either coalescence or grazing collision. If
the collision impact parameter b is less than a critical value bcr , the droplets
coalesce. Else, the droplets maintain their size and temperature but undergo
velocity changes, called grazing collision. The critical value bcr depends on the
radii r1 and r2 as well as on the surface tension of the droplets and its exact value
is given in [2].
The implementations in KIVA restricts the collisions to the computational
cells. Therefore, droplets can only collide when they are in the same cell. Therefore, the volume V in the equation for the collision frequency (2.37) is set to be
the cell volume. A first random number specifies the actual number of collisions
according to the Poisson distribution (2.35). If collisions occur, a modified second
random number acts as the collision impact number b explained above.
This drop collision model is inherently grid dependent since the collision frequency depends on the size of the grid cells. Thus, reducing the cell size has
two effects. First, it increases the probability for collision according to equation
(2.37), and second, it reduces the domain in which the droplets can collide. Since
the droplets in different cells have zero probability of colliding, the probability
can also decrease.
17
Chapter 3
Methods to Reduce Grid
Dependence
In this section, the grid dependence is illustrated and recently developed methods which reduce certain aspects of grid dependence are presented. In order to
remove the shortcomings of the collision model, Schmidt and Rutland [19] and
Nordin [14] implemented new algorithms which differ fundamentally from each
other. The Lagrangian-Eulerian Coupling (CLE) model [4] does not affect the
collision calculations but improves the drop-gas coupling mechanism with respect
to momentum and mass transfer. The liquid void fraction compensation (VFC)
method is introduced in this study in order to correct the mesh dependence of
the collision and the evaporation algorithms.
3.1 Motivation
3.1
18
Motivation
Most problems in fluid dynamics are formulated in terms of differential and algebraic equations in a continuum. Due to the complexity of the problems, solutions
are obtained by means of digital computers. This requires spatial and temporal
discretizations which can lead to inaccurate results and/or severe instability problems. Physical models are often needed in order to overcome the lack of spatial
and/or temporal resolution. Therefore, the accuracy of the solution to a problem
depends on the discretization chosen. On the other hand, a fine discretization
leads to enormous central processing unit (CPU) times. Consequently, computational solutions to continuous problems are a compromise between adequate resolution and reasonable CPU times. In this study, the spatial discretization in the
form of mesh dependence is investigated for spray problems in multidimensional
engine modeling. In particular, the coupling between the gas and liquid phase
is under investigation, where the liquid phase is modeled by a discrete droplet
model. It is well known that this approach can exhibit strong grid dependence
as will be discussed below.
3.1 Motivation
19
Figure 3.1: The spray computations have been performed for three different
meshes using the same physical models. The spatial resolution has a significant
effect on the spray behavior.
The problem of mesh dependence for a dense liquid spray injected into a cylinder is illustrated in Figure 3.1. All computations have been performed under the
same conditions. The only parameter which has been varied was the grid resolution. The use of a larger number of computational cells increases the penetration
3.1 Motivation
20
of the spray. The effect is so big that the spray computed using the finest mesh
impinges on the bottom of the cylinder.
Numerical studies of sprays by means of the Stochastic Discrete Droplet Model
have been conducted by many researchers [1, 4, 19, 14, 20], and it was found that
this model exhibits strong mesh dependence. Beard et al. [4] found that one of the
main reason is the inadequate space resolution of the strong velocity and vapor
concentration gradients. The liquid phase is injected typically with a velocity
of hundreds of meters per second into an almost quiescent environment, thereby
creating strong velocity gradients, especially at the nozzle exit. In their opinion,
increasing the grid resolution would violate the assumption that the dispersion
of the liquid phase is very high and would require far more computer resources,
CPU time in particular.
Resolving these extremely large gradients is, in fact, a numerical challenge.
An adequate fine mesh lies outside the capacity of today’s computers. The solution has been found in the use of sub-grid scale models. They describe spray
phenomena by tracking discrete parcels that represent droplets with similar physical properties while the gas phase is solved in a computational grid by using a
finite difference method. The drop-gas coupling terms make the liquid phase
dependent on the gas phase mesh.
The magnitude of the numerical error can vary from cell to cell depending
on the local properties of the spray. Spray submodels and their sensitivity to
the spatial resolution behave differently in the three different spray regions: the
liquid core close to the nozzle, the development region further downstream and
the tip of the spray.
3.1 Motivation
21
Liquid Core
The liquid enters the cylinder with a high velocity and is subject to aerodynamic
forces which lead to a complete atomization of the spray [21, 22]. In order to
simulate this effect large blobs of fuel are injected which then break up into
droplets of various sizes. The liquid volume fraction, the volume fraction occupied
by the liquid mass in each cell, is very high in this region, and the collision
frequency is very high due to the high drop density.
The main shortcoming lies in the insufficiently resolved momentum exchange
since the momentum transfer from the injected droplets to the gas is particularly
high in the liquid core. Therefore, it is essential to have small computational cells
close to the nozzle exit.
Development Region of the Spray
Due to the atomization process, the main body consists of small droplets. They
are almost uniformly spread over several computation cells. In this region, the
dilute spray assumption, required for many spray submodels, holds, i.e. the drop
diameter is small compared to the drop-drop distance. Therefore, the spray submodels are solved correctly but only in completely filled cells. However, many cells
are partially filled by droplets at the periphery of the spray. In large cells the numerical error is bigger in the sense that the collision frequency is under-predicted
and the evaporation rate is over-predicted. Since the droplets are already slowed
down and have accelerated the gas, the gradients are smaller than in the break
up region, and the momentum exchange is better resolved. But the resolution is
still insufficient in partially filled momentum cells.
3.1 Motivation
22
Tip of the Spray
The droplets at the tip of the spray decelerate as a consequence of aerodynamic
drag. Subsequent droplets benefit from the induced flow and decelerate slower,
thereby colliding with their predecessors. This leads to coalescence of droplets
and a lumping of the fuel at the tip of the spray. Usually the spray tip fills cells
only partially, and therefore, the collision frequency is more under-predicted in a
coarse mesh. On the other hand, larger cells promote the creation of big droplets
since there are more droplets available for coalescence in one cell.
According to Abraham [1], the jet diameter has to be resolved near the orifice
in the case of a gas jet, and a similar criterion must be satisfied in the case of a
spray. Abraham realized that as the grid size decreases and reaches the size of
the orifice diameter, the liquid volume fraction goes to one and the gas volume
fraction approaches zero. This causes numerical instabilities. The standard collision model as well as other spray models were developed with the assumption
that the spray is dilute. Abraham showed that if the grid size is greater than
the orifice radius, computations of transient sprays and gas jets do not reproduce the structure of the jet with adequate accuracy. Abraham also examined
the grid dependence of the spray computations. Two different spatial resolutions of the computation grids were considered. The comparison of physically
identical cases show that the cases computed with low resolutions feature lower
penetrations. The lower penetrations are associated with greater mixing. The
explanation given by Abraham is the lack of conservation of the axial component
of the injection momentum.
In addition to the shortcomings described by Abraham [1], numerical issues in
3.2 The Collision Algorithm of Schmidt and Rutland
23
droplet collision modeling will be addressed in more detail. This topic has been
investigated by Schmidt and Rutland [20] and Nordin [14], both are improving
on the widely used collision model of O’Rourke [15]. As discussed in section
2.1.2 the model of O’Rourke is inherently grid dependent. Schmidt and Rutland
[19] showed that this approach is first order accurate in time and second order
accurate in space. Another weakness is that the collision model takes the relative
velocity of the parcels into account, but it does not consider if they are moving
towards or away from each other. This leads to “clover leaf” patterned sprays in
computations using a Cartesian mesh. This artifact is discussed further in section
4.1.
3.2
The Collision Algorithm of Schmidt and
Rutland
The collision algorithm of Schmidt and Rutland [19] serves as a reference comparison to the VFC-algorithm developed later in this study and is, therefore,
discussed in more detail.
The NTC-algorithm improves the collision calculation. On one hand, the “NoTime-Counter” algorithm (NTC-algorithm) provides a faster and slightly more
accurate collision calculation compared to O’Rourke’s algorithm. On the other
hand, the generation of an optimized collision mesh ensures mesh independence
because the collision calculations do not depend on the gas phase mesh. In
addition, the expected number of droplet collisions is obtained differently than
in the O’Rourke algorithm, as is explained below.
3.2 The Collision Algorithm of Schmidt and Rutland
3.2.1
24
Collision Calculations
The NTC-algorithm is based on the No Time Counter method used in gas dynamics for Direct Simulation Monte Carlo calculations. The algorithm is first order
accurate in time and second order accurate in space. Compared to O’Rourke’s
algorithm it is slightly more accurate because the method avoids the Poisson
distribution, which is only valid if the sample population is left unchanged, i.e.
when collisions do not undergo coalescence. The biggest advantage of the NTCalgorithm is the low computational cost which is proportional to the number of
parcels. This is achieved mainly by a stochastic sub-sampling technique within
each cell.
The NTC method first sorts the parcels which reside in the same cell into a
group. The expected number of collisions Mcoll in a cell over a time interval ∆t
is derived by summing the probability of all possible collisions (Equation (2.36)).
Thus,
Np
Mcoll
Np
1 X X vreli,j σi,j ∆t
Ni
Nj
=
2 i=1
V
j=1
(3.1)
where Np represents the number of parcels, Ni and Nj are the number of droplets
in parcels i and j, respectively. V is the cell volume. The terms vreli,j and
σi,j consider the relative velocity and the collision cross section introduced in
equations (2.37) and (2.38). The cost to evaluate this summation is directly
proportional to Np2 . By using a representative randomly selected sub-sample
from all possible pairs of parcels in a cell, the evaluation cost is reduced. The
number of pairs Mcand is evaluated by the function
3.2 The Collision Algorithm of Schmidt and Rutland
Mcand =
Np2 (qvσ)max ∆t
2V
25
(3.2)
where (qvσ)max is used for scaling. The summation of equation (3.1) turns into
√
Mcand
Mcoll =
X
i=1
√
Mcand
Ni
X
j=1
Nj
vreli,j σi,j ∆t
(qvσ)max
(3.3)
This equation is used in the NTC method and includes a summation over Mcand
terms. As the number of droplets in a parcel q is proportional to 1/Np , the
evaluation cost becomes proportional to the number of parcels Np .
The NTC-algorithm was tested against analytical solutions and it was found to
converge to the exact answer as the cell size approached zero. Due to the sampling
technique, it is recommended to use many computational parcels, in order to
achieve to a good statistical representation which results in a high accuracy of
the collision calculations.
3.2.2
Creation of a Collision Mesh
To group the parcels, the NTC-algorithm requires a mesh for the collision calculations. However, the gas phase grid is usually too coarse for sufficient spatial
resolution. As droplet collisions have no connection to the gas phase according
to the current model, it is useful to create a separate collision mesh. So far, mesh
generation is only available for a single injection system. The collision mesh is
chosen to be polar because a polar mesh better resolves the azimuthal direction
while avoiding the clover leaf effect described in section (4.1). The size of mesh is
3.3 Collision Model of Niklas Nordin
26
determined only by the location of the parcels. The domain is shaped as a cylinder and the axis of the cylinder matches with the axis of the nozzle. It is equally
spaced in the radial and axial directions. Therefore, it is possible to speedily sort
the parcels according to their cells.
3.3
Collision Model of Niklas Nordin
Niklas Nordin [14] formulated and tested a mesh independent drop collision
model. The main goal was to achieve a more accurate way of calculating the
collisions and avoiding the weaknesses mentioned in section (3.1). In this model,
collisions can only occur if the trajectories of two parcels intersect, and the intersection point is reached at the same time within the integration time step. Thus,
two pre-requirements can be used to reject pairs of parcels that cannot collide.
The first requirement for a collision is that the parcels move towards each other.
The second requirement is that the distance these parcels move towards each
other in one timestep must be larger than the distance between them. If both
conditions are fulfilled, there is a chance of collision. Finally, a random number
is chosen between zero and one. The parcels will collide if the chosen random
number is less than P .
¶
µ
r1 + r2 −C2 |α0 −β0 |
∆t
e
P = min 1, C1
∆12
(3.4)
where r1 and r2 are the radii of the droplets considered, ∆12 represents the minimum distance between trajectories at times α0 and β0 , respectively. α0 and β0 are
in the interval ∆t. C1 and C2 are model constants that have to be determined.
3.4 The CLE-Model
27
This collision model has been shown to be grid independent in cases where
only grazing collisions were considered. The case of coalescence has not been
completely modeled yet and, therefore, the model has not be validated by experimental data.
3.4
The CLE-Model
As discussed in section 3.1 the coupling between the gas and the liquid phase
is in general not accurately reproduced due to inadequate spatial resolution.
Therefore, Beard et al. [4] developed a new Lagrangian-Eulerian Coupling (CLE)
method. This approach doesn’t consider the collision model, but improves the
drop-gas interaction in two aspects:
• The fuel vapor transport and
• the momentum coupling.
The main idea behind the approach is that vapor and momentum are retrained
along parcel trajectories by introducing a gaseous particle and a sphere of momentum influence. These numerical formations follow their associated liquid particles.
This approach helps in resolving steep gradients in a coarse mesh by gradually
releasing vapor and momentum on the mesh following specified diffusion laws.
Evaporated vapor of a liquid particle is confined in a sphere associated with this
drop, and is called a gaseous particle (cf. Figure 3.2).
3.4 The CLE-Model
28
Figure 3.2: The gaseous particle of the CLE-Model
The sphere of momentum corrects the particle relative velocity which is defined in KIVA as the difference between the particle velocity vp and the gas phase
velocity u at the nearest cell vertex. An interpolation of the relative velocity
vrel is applied at the surface of the sphere to reduce the over-estimated relative
velocity (cf. Figure 3.3).
3.4 The CLE-Model
29
Nearest Vertex
U
Vp
Vrel
(CLE)
Vrel
Figure 3.3: Effect of the sphere of momentum on the relative velocity in the
CLE-Model
3.5 The VFC-Method
3.5
30
The VFC-Method
The Liquid Void Fraction Compensation (VFC) method compensates for the lack
of spatial resolution by correcting the droplet density in each cell according to
a predetermined average liquid void fraction computed at each time step. The
method is based on the observation that the numerical schemes of several spray
models utilize the entire volume of the cell which contains a droplet. So far, the
numerical computations interpret the cell volume as the gas volume surrounding
the droplets. The numerical error of the collision model and the drop-gas interactions are particularly severe in peripheral cells which are not completely filled
by the spray. Usually, there are many peripheral cells as illustrated in Figure 3.4
which shows the tip of a spray in a mesh with a typical resolution.
Figure 3.4: At the tip of the spray and the periphery, many computational cells
are partially filled by droplets.
The VFC-method is based on the compensation of this error by correcting
3.5 The VFC-Method
31
the droplet density used in the spray related models. The following steps give an
overview of this approach.
• Identification of cells that are only partially filled by droplets
• Computation of the correction factor according to the liquid volume fraction
in each cell
• Adjustment of the cell volume used in the spray related models by the
correction factor
In this study, the VFC method is applied to the drop collision and the evaporation
model. The void fraction or liquid volume fraction is defined locally for each
computational cell by
Yvoid =
volume of the liquid mass in a cell
cell volume
(3.5)
In order to identify partially filled cells, a comparison value is needed. A reasonable value would be the average void fraction given by
P
Y
∗
void
=
Yvoid
NliqCell
(3.6)
where NliqCell is the number of cells that contain liquid mass. However, the value
∗
of Y void turned out to be far too low because many cells which are included in
the average are just filled with a few droplets. Therefore, the following average
has been used to compute the comparison value
P
Nspraycells Yvoid
Y void =
Nspraycells
(3.7)
3.5 The VFC-Method
32
∗
where Nspraycells is the number of cells with Yvoid > Y void . Suppose the liquid
mass is uniformly distributed over the spray. Then the void fraction Yvoid of
a cell compared to the averaged value Y void gives exact information about the
degree to which the cell is filled. For example, if the void fraction of a cell is half
of the averaged value, then half the cell is filled. This idea motivates the following
selection strategy.
• If a cell has a void fraction equal or greater than the average void fraction
Y void , then it is considered as completely filled.
• If a cell has a void fraction less than the average, it is considered partially
filled and the cell volume needs to be adjusted by a compensation factor.
In this study, the compensation factor is defined by
cV F C =


 Yvoid
if
Yvoid
Y void
<1

1
if
Yvoid
Y void
≥1
Y void
(3.8)
The volume Vspray occupied by the spray in a cell turns into
Vspray = cV F C · Vcell
(3.9)
In this study, the compensation factor of the VFC-method is only used in the
collision and the evaporation model. Applications to other models such as the
liquid-gas momentum transfer are also conceivable.
33
Chapter 4
Results and Discussion
4.1
Comments on the Type of Grid
The liquid has been injected from the top center of a cylinder of different dimensions (cf. Table 4.1). This cylinder has been discretized into computational
meshes of different types and resolutions. The types under consideration are a
Cartesian and a polar grid, as illustrated in Figure 4.1. In the case of a Cartesian
grid, it is important to distinguish between two type of grids. If the cylinder
diameter is spaced by an odd number of cells, the liquid is injected into the cell
at the center of the cylinder. If the diameter is spaced by an even number of cells,
the injector is situated on a vertex and the liquid is injected into the 4 adjacent
cells. This causes different behavior of the spray and will be discussed later.
4.1 Comments on the Type of Grid
34
Figure 4.1: A Cartesian grid with 20 x 20 x 20 cells and a polar grid with
10 x 8 x 20 cells.
Note that due to stability considerations [2], two separate meshes are used.
One mesh is used in the computations of scalar values of the gas phase and the
other is for the computation of gas velocity and is referred to as the momentum
cell mesh. The momentum subcells are centered about the gas phase grid vertices.
4.1 Comments on the Type of Grid
35
Figure 4.2: Simulation with the collision model switched off shows a “clover leaf”
artifact. (View from the top of spray).
U
U
Vp
Vp
Vp
Vp
Nozzle exit
Vp
Vp
Vp
U
Vp
U
Figure 4.3: The momentum grid causes an “clover leaf” artifact
When the drop collision model is switched off, the the spray exhibits a “clover
leaf” artifact in the Cartesian mesh, as is shown in Figure 4.2 for a mesh with
4.1 Comments on the Type of Grid
36
an odd number of cells. The liquid is injected into a single cell of the gas phase
grid at the top center. This situation is shown with a schematic momentum grid
in Figure 4.3. The liquid is injected into 4 momentum subcells. The gas velocity
in each momentum subcell is an average velocity induced by the droplets in the
cell. The resulting gas velocity is in a direction that is roughly diagonal to the
momentum subcells as is illustrated in Figure 4.3. Since the droplets tend to
move with the gas flow, the observed “clover leaf” effect is obtained.
Figure 4.4: The spray’s radial expansion is inhibited due to the drop collision
model.
4.1 Comments on the Type of Grid
Nozzle exit
37
Nozzle exit
Vp
Vp
Vp
Figure 4.5: Due to droplet coalescence, the droplets lose the radial velocity component in the injection cell.
The “clover leaf” artifact disappears when the collision model is switched
on. However, for the mesh under consideration, another artifact occurs which is
illustrated in a side view in Figure 4.4. The figure shows that the droplets hardly
move in radial direction. This phenomena is explained as follows: Figure 4.5
shows the situation of droplets with some radial velocity in the same cell. The
collision algorithm checks if a pair of droplet parcels are in the same cell, but
it does not check if they are moving towards each other. Since they are in the
same cell and the relative velocity between these parcels is relatively high, they
are likely to collide and exchange momentum. In this process, the radial velocity
components cancel each other out, and after coalescence, the droplets move only
in the axial direction.
The second artifact can be eliminated by using a Cartesian grid with an even
number of cells over the diameter of the cylinder. But the drop collision model
4.1 Comments on the Type of Grid
38
still causes a “clover leaf” in the radial direction, as is shown in Figure 4.6. This
effect was also observed by Schmidt and Rutland [19]. The reason is basically
the same as discussed above. Droplets which move radially apart by almost 900
can be located in the same cell of a Cartesian mesh near the spray origin and,
therefore, are likely to collide.
Figure 4.6: “Clover leaf” artifact due to the drop collision model.
In conclusion, the mesh resolution used for the Cartesian grid around the
nozzle exit is insufficient. In order to increase the number of cells in this region
without unnecessarily increasing the overall number of cells, a polar grid was used
in this study. Polar meshes provide a better resolution in an azimuthal direction
and hence avoid the above discussed artifacts. In addition to the artifacts, the
Cartesian meshes can also cause instabilities due to large spray source terms and
insufficient spatial resolution. It is possible to stabilize the simulation of the non-
4.1 Comments on the Type of Grid
39
evaporating case by using a small timestep (∆tn =1µs), but in the evaporating
case even this measure fails. Apparently, the cells around the spray are too large
to resolve the high velocity gradients. Therefore, the computations discussed in
the following section have been performed with polar grids. A polar grid does not
need as many overall computational cells as a Cartesian grid to achieve similar
accuracy. The reason for the smaller number of cells is the structure of the polar
grid. First the polar grid matches better with the shape of a spray which usually
features a radial and an axial expansion. Moreover, the highest resolution is
in the center of the cylinder which, in this study, coincides with the injection
axis. To enhance further the resolution, the nodes are not equally spaced on the
axes. They are moved closer to the nozzle exit, and therefore, more cells are
concentrated around the injection axis. More details can be found in Table 4.1.
The cylinder dimensions may vary from case to case to optimize the number of
cells and hence the computation time.
4.1 Comments on the Type of Grid
40
Table 4.1: Overview of mesh data: Polar grids.
Coarse
Medium
Fine
30 x 50
30 x 50
30 x 50
4 x 8 x 10
6 x 12 x 15
9 x 18 x 22
320
1080
3564
2.6, 2.8, 45o
1.6, 1.7, 30o
1.0, 1.0, 20o
30 x 100
30 x 100
30 x 100
4 x 8 x 20
6 x 12 x 30
9 x 18 x 45
640
2160
7290
2.6, 2.4, 45o
1.6, 1.4, 30o
1.0, 0.9, 20o
110 x 100
30 x 110
110 x 100
10 x 8 x 20
15 x 12 x 30
23 x 18 x 45
1600
5400
18630
3.1, 2.4, 45o
1.9, 1.4, 30o
1.1, 0.9, 20o
Validation case (Collision)
Cylinder:
Diameter [mm] x height [mm]
Number of cells:
Radial, azimuthal, axial
Total
Smallest cell:
Radius [mm], length [mm], angle
Validation case (Evaporation)
Cylinder:
Diameter [mm] x height [mm]
Number of cells:
Radial, azimuthal, axial
Total
Smallest cell:
Radius [mm], length [mm], angle
Experimental cases
Cylinder:
Diameter [mm] x height [mm]
Number of cells:
Radial, azimuthal, axial
Total
Smallest cell:
Radius [mm], length [mm], angle
4.2 Computation Cases
4.2
41
Computation Cases
The computations have been performed for non-reacting, solid-cone diesel fuel
sprays injected into nitrogen with zero mean flow and zero initial turbulence. The
geometry for the experimental cases models a cylinder of ø 110 mm x 100 mm
which is equipped with a one-hole injector at the top center, directed in the axial
direction. Model diesel fuel df2 is injected from the top center along the cylinder
axis at a constant injection rate. The evaporating case features a high initial gas
temperature. More details can be found in Table 4.2.
Table 4.2: Overview of computation data.
Non-evaporating case
Evaporationg case
ø 110 mm x 100 mm
ø 110 mm x 100 mm
0.3 mm
0.2 mm
Initial gas temperature
300 K
800 K
Gas pressure
11 bar
50 bar
0.01528 g
0.006 g
Injection duration
2.5 ms
1.3 ms
Experimental Data
Hiroyasu et al.[8]
Koß [10]
Cylinder dimensions
Nozzle diameter
Total injected mass
Four sets of computations have been performed, two for the non-evaporating
case and two for the evaporating case. The purpose of the computations is the
examination of the mesh dependence discussed in chapter 3. The collision algorithms are tested for non-evaporating cases, one validation case and one real
spray case. The mesh dependence of the evaporating spray is investigated for a
validation case and in a real spray case. The validation cases are unrealistic cases
4.2 Computation Cases
42
where the collision model and the evaporation model are considered isolated by
switching off other influences. Meshes with three different resolutions are used to
investigate the mesh dependence. They are described in Table 4.1. The realistic
cases are compared with experimental data.
Table 4.3: Overview of validation case models.
Collision
Evaporation
set to zero
set to zero
Aerodynamic drag
on
off
Collision model
on
off
Liquid evaporation
off
on
Breakup model
off
off
Turbulence model
off
off
Gas velocity
Validation Case for the Collision Model
In this case the cylinder dimensions have been reduced to ø 30 mm x 50 mm in
order to reduce the CPU time. All other data and parameters coincide with the
realistic case and are presented in Table 4.2. The main idea is to investigate the
effect of the collision model while isolated from other influences. The evaporation,
the breakup and turbulence models are switched off. In addition, the liquid-gas
momentum transfer has been switched off, i.e. the gas velocity is set to zero,
but the droplets still face the aerodynamic drag (cf. Table 4.3). Since the gasdrop interaction is reduced to aerodynamic drag at zero gas flow, it is possible to
reduce the physical domain as required by the spray. This allows the reduction
of the cylinder diameter to 3 cm and height to 5 cm. The injected droplets have
4.2 Computation Cases
43
a constant radius of 10 µm. In order to achieve a dilute spray, the spray angle is
set to 200 .
The results of the VFC-method are compared with results of the original
KIVA collision algorithm and the mesh independent NTC-algorithm. Since the
NTC-algorithm uses a sampling technique, it is recommended to use many computational parcels, at least 8000 parcels, to achieve high accuracy for the collision
calculations. In this study, 20000 parcels are used after consultation with David
Schmidt, a coauthor of the NTC algorithm, because the computations involve
a dense spray with a small spray angle. All other computations have been performed with 5000 parcels, because the validation case is insensitive to the number
of parcels. In fact, the difference in the SMR is about 0.5% between computations
using 5000 parcels or 20000 parcels with the fine mesh. Because of numerical instabilities, the NTC algorithm needed the introduction of a minimum cell size.
This was especially needed at the beginning of the injection, where the original collision mesh contained cells smaller than the drop size. This modification
conforms to the requirements given in [1].
Non-Evaporating Real Spray Case
In comparison with the validation case, all models are activated in order to measure the performance of the methods with a realistic problem. It allows a comparison with experimental data obtained by Hiroyasu & Kadota [8]. As motivated
in [22], the model diesel fuel df2 is injected according to the distribution given by
f (r) =
n+1 r n
( )
r0 r0
where r and r0 are the drop and nozzle radii, respectively, and n = 0.5.
(4.1)
4.2 Computation Cases
44
Validation Case for Evaporating Spray
In analogy to the validation case for the collision model, this case is used to isolate
the evaporation model from the other models (cf. Table 4.3). The physical
situation is the same as in the real spray case (cf. Table 4.2). However, the
collision model, the breakup model and the turbulence model are switched off.
The injected droplets have a constant radius of 10 µm and form a spray with an
angle of 200 . The main difference from the collision validation case is that the
droplets are not subject to aerodynamic drag. As a consequence, the droplets
pass through the ambient gas at a constant velocity. Otherwise, the tiny droplets
with radius of 10 µm would stop moving shortly after the injection and hardly
evaporate because the relative drop-gas velocity promotes the evaporation rate
significantly. In order to prevent the droplets from impinging on the cylinder
bottom, the cylinder height is extended to 10 cm while the diameter can maintain
the value of 3 cm.
Real Spray Case for Evaporating Spray
The computational results are compared with measurements of Koß et al. [10].
The liquid phase is injected in the same way as in the non-evaporating case in
form of droplets with a mean radius that is equal to the nozzle radius. However,
the injection time is shorter, and less fuel mass is injected (cf. Table 4.2). The
temperature of the gas phase is high enough to evaporate the total injected mass
within 2 ms.
4.3 The Effect of the Drop Collision Model
45
Properties Used for Validation
Three spray properties are used to quantify the mesh dependence. For the collision algorithm, the mesh dependence is measured by the behavior of the Sauter
mean radius and the liquid penetration. For the evaporation algorithm, the liquid
penetration and the vapor penetration are compared. Their definitions are given
by
• Sauter mean radius (SMR):
The Sauter mean radius is averaged over all droplets N in a spray by means
of the equation
P 3
r
SM R = PN 2
N r
(4.2)
• Liquid penetration:
The liquid or spray penetration is the distance between the nozzle and the
tip of the spray. In this study, the tip is defined as the center of first 20
drop parcel of the spay.
• Vapor penetration:
The vapor penetration is the distance between the nozzle and the tip of the
fuel vapor cloud. In order to determine the border of the cloud the lower
bound of the fuel mass fraction is set to be 0.01 in this study.
4.3
The Effect of the Drop Collision Model
In order to investigate the effect of the drop collision model, only non-evaporating
cases are considered. The VFC-method is compared with the standard algorithm
4.3 The Effect of the Drop Collision Model
46
and the grid independent NTC-algorithm. This is first done in an imaginary case
called a validation case, where all other relevant models are switched off. Finally,
the computations of a realistic case reveal the relevance of the collision model in
simulating dense liquid sprays.
4.3.1
The Validation Case
The drop collisions have a big effect on the droplet radius because collisions
promote the coalescence of droplets. This effect can be measured by the Sauter
mean radius defined in section 4.2. Since the collision model is inherently grid
dependent, the value of the Sauter mean radius is also expected to be grid
dependent.
60.0
50.0
40.0
SMR [µm]
SMR [µm]
50.0
60.0
fine
medium
coarse
30.0
20.0
10.0
0.0
0.0
fine
medium
coarse
40.0
30.0
20.0
10.0
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.7: Sauter mean radius for
three spatial resolutions (O’Rourke collision model).
0.0
0.0
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.8:
Sauter mean radius
for three spatial resolutions (NTCalgorithm). The result is virtually
mesh independent.
This is validated in Figure 4.7 which shows the Sauter mean radius aver-
4.3 The Effect of the Drop Collision Model
47
aged over all droplets versus time. The Sauter mean radius is smaller in the
coarser mesh because of an underestimation of the collision probability (cf.
Equation (2.35)). In particular, in a coarse mesh, the numerical error of the
collision calculations is large in partially filled cells. In contrast, the results of
the NTC-algorithm shown in Figure 4.8 are grid independent because it utilizes
a collision mesh that is isolated from the gas phase mesh. In every timestep,
the algorithm adjusts the mesh size according to the spray dimensions. The cell
volume of the collision mesh is on an average smaller that the cell volume of the
fine mesh. Therefore, the collisions are better resolved thus leading to a higher
Sauter mean radius.
5.0
4.0
Penetration [cm]
Penetration [cm]
4.0
5.0
fine
medium
coarse
3.0
2.0
1.0
0.0
0.0
fine
medium
coarse
3.0
2.0
1.0
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.9: Spray penetration for three
spatial resolutions (O’Rourke collision
model).
0.0
0.0
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.10: Spray penetration for
three spatial resolutions (NTCalgorithm).
The penetration behaves in a similar manner (cf. Figures 4.9 and 4.10 ).
The finer mesh contains larger droplets which penetrate further. Therefore,
the Sauter mean radius correlates directly with the spray penetration in the
4.3 The Effect of the Drop Collision Model
48
validation case, i.e. the penetration is grid independent when the NTC-algorithm
is used for the collision calculation. The standard collision model used in a finer
mesh causes a deeper penetration.
0.10
fine
medium
coarse
Average void fraction
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0.000
0.001
Time [s]
0.002
Figure 4.11: Average void fraction for different meshes.
Before the performance of the VFC-method is presented for this case, it
is necessary to take a look at the value of the average void fraction Y void (cf.
Figure 4.11). In the finer mesh, Y void is larger and might be more accurate
because more completely filled cells are considered when the average is evaluated.
It is also worth noting that the average void fraction is not a constant as it would
be in an ideal case. This is especially true in the beginning of the injection when
none of the cells are completely filled and the averaging method fails since the
average value is too low. When enough cells are filled, the average void fraction
seems to settle down at a specific value. According to Figure 4.11, it seems to be
reasonable to take 0.06 as an estimate for the average void fraction Y void .
4.3 The Effect of the Drop Collision Model
5.0
60.0
fine
medium
coarse
4.0
Penetration [cm]
SMR [µm]
50.0
40.0
30.0
20.0
fine
medium
coarse
3.0
2.0
1.0
10.0
0.0
0.0
49
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.12: Sauter mean radius for
three spatial resolutions (VFC-method
with Y void = 0.06).
0.0
0.0
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.13: Spray penetration for
three spatial resolutions (VFC-method
with Y void = 0.06).
Using Y void = 0.06, the VFC-method performs very well in the sense that
the Sauter mean radius shown in Figure 4.12 is not grid dependent any more.
Moreover, the results agree well with the ones obtained by the NTC-algorithm.
This indicates that the method compensates the error in the collision calculations
correctly. However, the penetration of the spray is still grid dependent (cf. Figure
4.13). Surprisingly the general behavior is reversed when compared with the
O’Rourke collision model (cf. Figure 4.9). The deeper spray penetration in the
coarser mesh is the result of the number of its azimuthal sections and its influence
on the aerodynamic drag (Recall that the liquid-gas phase transfer is switched
off). In each azimuthal section, droplets form a separate jet due to the collisions.
This effect was already discussed in the case of Cartesian meshes in section 4.1.
Since a coarser mesh contains a less number of azimuthal sections, the spray has
fewer jets and hence experiences less aerodynamic drag.
4.3 The Effect of the Drop Collision Model
50
Note that the mesh dependent penetration in the case of the O’Rourke model
is dominated by the size of the droplets and not by the number of azimuthal
sections. This assertion is supported by the fact that the mesh dependence of the
O’Rourke model is much stronger.
When the average void fraction Y void is evaluated at every single timestep the
VFC-method is less effective, as shown in Figure 4.14 and Figure 4.15 . This
is a result of incorrect values of the average void fractions (cf. Figure 4.11).
As noticeable in Figure 4.14, the compensation factor is too small in the early
stage of the injection. But the mesh dependence is still reduced, as is shown in
Figures 4.14 and 4.15.
5.0
60.0
4.0
Penetration [cm]
SMR [µm]
50.0
fine
medium
coarse
40.0
30.0
20.0
3.0
2.0
1.0
10.0
0.0
0.0
fine
medium
coarse
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.14: Sauter mean radius for
three spatial resolutions (VFC-method
with time dependent Y void ).
0.0
0.0
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.15: Spray penetration for
three spatial resolutions (VFC-method
with time dependent Y void ).
Note that the mesh dependence of the penetration correlates with the drop
sizes in the early injection phase. This is another indication that the drop size
dominates the penetration mesh dependence over the number of azimuthal sections, as discussed before. These observations suggest that, in absence of the other
4.3 The Effect of the Drop Collision Model
51
spray models, the main influence of the collision model on the mesh dependence
is the drop size and its consequences on the penetration.
4.3.2
The Real Spray Case
In the real spray case, the mesh dependence computed with the O’Rourke collision model is significant, as is shown in Figures 4.16 and 4.18. The penetration
obtained by the medium mesh agrees well with the real spray data of Hiroyasu
& Kadota [8]. But the computation using the fine mesh drastically over-predicts
the penetration leading to spray impingement at the bottom of the cylinder.
The NTC-algorithm reduces the mesh dependence of the Sauter mean radius,
but not of the penetration, as is illustrated in Figures 4.17 and 4.19, respectively.
This leads to the conclusion that the size of the droplets is not the only property
that is responsible for the length of the penetration.
250
200
SMR [µm]
SMR [µm]
200
250
fine
medium
coarse
150
100
50
0
0.0
fine
medium
coarse
150
100
50
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.16: Sauter mean radius for
three spatial resolutions (O’Rourke collision model).
0
0.0
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.17:
Sauter mean radius
for three spatial resolutions (NTCalgorithm).
4.3 The Effect of the Drop Collision Model
10.0
Penetration [cm]
8.0
7.0
10.0
fine
medium
coarse
Hiroyasu
8.0
6.0
5.0
4.0
3.0
7.0
6.0
5.0
4.0
3.0
2.0
2.0
1.0
1.0
0.0
0.0
0.5
1.0
1.5
Time [ms]
fine
medium
coarse
Hiroyasu
9.0
Penetration [cm]
9.0
52
2.0
0.0
0.0
2.5
Figure 4.18: Spray penetration for
three spatial resolutions (O’Rourke collision model).
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.19: Spray penetration for
three spatial resolutions (NTCalgorithm).
Table 4.4: Relation between gas velocity and penetration.
coarse mesh
medium mesh
fine mesh
(impingement)
Max. gas velocity [cm/s]
Max. spray penetration [cm]
6060
8141
9462
7.3
9.0
10.0
The main reason can be found in the insufficient resolution of the momentum
exchange between the droplets and the gas, as discussed earlier in section 3.1.
In the coarser mesh the spray is poorly resolved and, therefore, the total volume
of all cells which contain droplets is much larger than the actual spray volume.
The result is that the spay has to accelerate more gas. This leads to a lower gas
velocity in a coarser mesh. This statement can be confirmed by comparing the
velocity and penetration data in Table 4.4. In the considered case, the injection
velocity of the particle is 14000 cm/s.
4.3 The Effect of the Drop Collision Model
53
This observation is also in agreement with a study by Abraham [1]. His
conclusion was that if the computational grid size is greater than the orifice size,
it can be shown that the axial gas momentum component is not conserved. The
momentum balance is achieved by an increase on pressure in the computational
cell adjacent to the orifice.
With this knowledge in mind, it is not surprising that the VFC-method as
well as the NTC-algorithm does not reduce the mesh dependence of the spray
penetration significantly, as is shown in Figures 4.22 and 4.23. The drop size
remains mesh dependent to a certain degree (cf. Figure 4.20 and 4.21). The
VFC-method with an estimated average void fraction Y void performs as well as
the grid independent NTC-algorithm (cf. Figure 4.19), whereas the alternative
VFC-method shows shortcomings, especially in the first half of the simulation.
Note that the drop size is also affected by other models which may still depend on
the grid. For example, the liquid-gas momentum exchange determines the relative
drop-gas velocity and hence the drop breakup. Another aspect is that the spray
is not globally uniformly distributed. The liquid density varies from region to
region as illustrated in section 3.1. Therefore, the liquid fraction averaged over
the entire spray provides a compensation factor which can only be considered as
an estimate.
4.3 The Effect of the Drop Collision Model
250
250
fine
medium
coarse
200
SMR [µm]
SMR [µm]
200
150
100
50
0
0.0
0.5
1.0
1.5
Time [ms]
2.0
100
1.0
1.5
Time [ms]
2.0
2.5
10.0
fine
medium
coarse
Hiroyasu
9.0
8.0
6.0
5.0
4.0
3.0
7.0
5.0
4.0
3.0
2.0
1.0
1.0
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.22: Spray penetration for
three spatial resolutions (VFC-method
with Y void = 0.06).
fine
medium
coarse
Hiroyasu
6.0
2.0
0.0
0.0
0.5
Figure 4.21: Sauter mean radius for
three spatial resolutions (VFC-method
with time dependent Y void ).
Penetration [cm]
Penetration [cm]
7.0
150
0
0.0
2.5
10.0
8.0
fine
medium
coarse
50
Figure 4.20: Sauter mean radius for
three spatial resolutions (VFC-method
with Y void = 0.01).
9.0
54
0.0
0.0
0.5
1.0
1.5
Time [ms]
2.0
2.5
Figure 4.23: Spray penetration for
three spatial resolutions (VFC-method
with time dependent Y void ).
The VFC-method is successful in reducing the mesh dependence in the collision model. Using a suitable average value for Y void , the performance is comparable with the grid independent NTC-algorithm. The determination of the average
4.4 The Effect of the Evaporation Model
55
liquid fraction fails when not enough spray cells are available. Therefore, the
use of the VFC-method with the time dependent value for Y void yields the best
results when applied to the later period of the injection.
4.4
The Effect of the Evaporation Model
In order to investigate the effect of the evaporation model, only cases without
chemical reactions are considered. As for the non-evaporating case, a separate
validation case and a real spray case are under consideration. The conditions
and parameters are summarized in Tables 4.2 and 4.3. The VFC-method with a
variable Y void is tested for its influence on the evaporation rate.
4.4.1
The Validation Case
In order to measure the mesh dependence of the evaporation model, it is reasonable to consider the evolution of the vapor mass or the liquid fuel in the cylinder.
In a finer mesh, the evaporation rate is expected to be lower because in a smaller
cell less energy in the form of heat is available. In this study, the mesh does
not seem to affect the evaporation process very much. Figure 4.24 shows the
total injected fuel mass versus the amount that is present in the cylinder for the
three different meshes under consideration. The evaporation rate depends only
slightly on the resolution of the mesh. In a coarser mesh, the amount of liquid
fuel is hardly any less, i.e. the evaporation rate is hardly any higher. If the VFCmethod is applied, the situation remains almost the same. There is no noticeable
difference between the results shown in Figures 4.24 and 4.25.
4.4 The Effect of the Evaporation Model
0.010
Fuel mass [g]
0.008
0.007
0.010
injected fuel
fine
medium
coarse
0.009
0.008
Fuel mass [g]
0.009
0.006
0.005
0.004
0.003
0.007
0.005
0.004
0.003
0.002
0.001
0.001
0.001
Time [s]
0.002
Figure 4.24: Liquid fuel mass for three
spatial resolutions (KIVA).
injected fuel
fine
medium
coarse
0.006
0.002
0.000
0.000
56
0.000
0.000
0.001
Time [s]
0.002
Figure 4.25: Liquid fuel mass for three
spatial resolutions (KIVA with VFCmethod).
The VFC-method reduces the energy available for the droplets to evaporate if
the energy is over-predicted due to the size of the cell. The temperature seems to
be so high that the vaporization process is not limited by the amount of energy
available in a cell in these cases. Another factor must be responsible for the weak
dependence on the mesh. The reason might be that vapor mass fraction Y1 =
ρ1
ρ
tends to be larger in a finer mesh and thus repress the evaporation according to
equation (2.24) and (2.26). Each of the droplets shown in Figure 4.26 is located
in a cell free of other liquid. Suppose the small cell provides enough energy that
the amount of vapor produced in one timestep is the same in both cases, then the
vapor mass fraction after one timestep is significantly larger in the smaller cell.
In the following timestep, the evaporation rate in the small cell is reduced. This
effect may be weakened by diffusion of the vapor but a weak mesh dependence is
still noticeable in Figure 4.24.
4.4 The Effect of the Evaporation Model
57
Figure 4.26: When a droplet evaporates, a finer coarse provides a higher ambient
vapor mass fraction.
Applying the compensation factor cV F C to manipulate the vapor mass fraction
has been shown to be a poor approach because it increased the mesh dependence
and, therefore, it is not pursued further in this study.
The liquid penetration in Figure 4.27 is in accordance with the vapor penetration shown in Figure 4.28 because the quiescent gas used in these computations
cannot carry the vapor away. The mesh dependence seems to be more significant for penetration rather than for the vapor mass. The reason is that for the
coarser mesh the evaporation is slightly enhanced (cf. Figure 4.24). Therefore,
the droplets evaporate faster (i.e. disintegrate faster), resulting in a reduced
penetration.
4.4 The Effect of the Evaporation Model
10.0
8.0
10.0
fine
medium
coarse
9.0
Vapor penetration [cm]
Liquid penetration [cm]
9.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
0.0
8.0
fine
medium
coarse
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.5
1.0
Time [ms]
1.5
0.0
0.0
2.0
Figure 4.27: Liquid penetration for
three spatial resolutions.
4.4.2
58
0.5
1.0
Time [ms]
1.5
2.0
Figure 4.28: Vapor penetration for
three spatial resolutions.
The Real Spray Case
The real spray case features a weak mesh dependence in terms of the amount
of liquid fuel (cf. Figure 4.29). In this respect, the mesh is not relevant for the
evaporating process as already observed in the validation case.
0.010
0.009
Fuel mass [g]
0.008
0.007
injected fuel
fine
medium
coarse
0.006
0.005
0.004
0.003
0.002
0.001
0.000
0.000
0.001
Time [s]
0.002
Figure 4.29: Liquid fuel mass with experimental data for three spatial resolutions.
4.4 The Effect of the Evaporation Model
59
But the behavior is different for the penetration of the liquid and the vapor (cf.
Figures 4.30 and 4.31). The penetration is deeper in the finer mesh. As already
mentioned in the non-evaporating case of section 4.3.2, the gas velocity has a
big influence on the penetration. Table 4.5 shows that there is also a correlation
between the gas velocity and the penetration in the evaporating case. If the gas
which surrounds the spray moves faster with the droplets, a smaller aerodynamic
drag makes the droplets penetrate further. Since the gas carries the vapor, the
gas velocity is most responsible for the penetration of the vapor cloud.
Observe that in contrast to the non-evaporating case, the experimental penetration is best reproduced with the fine mesh.
10.0
8.0
7.0
10.0
fine
medium
coarse
Koss
9.0
Vapor penetration [cm]
Liquid penetration [cm]
9.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
0.0
8.0
7.0
fine
medium
coarse
Koss
6.0
5.0
4.0
3.0
2.0
1.0
0.5
1.0
Time [ms]
1.5
2.0
Figure 4.30: Liquid penetration with
experimental data for three spatial resolutions.
0.0
0.0
0.5
1.0
Time [ms]
1.5
2.0
Figure 4.31: Vapor penetration with
experimental data for three spatial resolutions.
4.4 The Effect of the Evaporation Model
60
Table 4.5: Relation between gas velocity and penetration.
coarse mesh
medium mesh
fine mesh
6653
9587
13923
Max. liquid penetration [cm]
1.5
1.9
2.7
Max. vapor penetration [cm]
4.9
5.5
6.5
Max. gas velocity [cm/s]
61
Chapter 5
Conclusions
The mesh dependence for the stochastic discrete droplet model has been investigated for high-velocity dense sprays injected into a cylindrical vessel using a
KIVA3-based code. The VFC-method has been developed in order to reduce the
grid dependence for the collision and evaporation models. The main idea behind
the method is the compensation for the lack of spatial resolution by correcting
the droplet density in each cell according to a predetermined average liquid void
fraction. The method has been implemented in a KIVA-based code and tested under idealized and realistic conditions for non-evaporating and evaporating sprays
using coarse, medium and fine meshes. In addition, the effect of the VFC-method
on the collision model has been tested against the NTC-algorithm whose collision
model is grid independent. The evaluation criteria used to judge the model performances include the Sauter mean radius of the spray, its penetration and the
evaporation rate when applicable.
Non-evaporating sprays have been validated under idealized conditions where
most other models had been switched off. In this case, the Sauter mean radii were
62
virtually grid independent and compared well with the ones from the computations with the NTC-algorithm. The penetrations obtained by the VFC-method
still showed grid dependence due to the formation of individual jets caused by
the ”clover leaf” artifact in the azimuthal sections.
The non-evaporating real spray cases were considered for the VFC-method
with an estimated comparison value and the NTC-algorithm. Both, the NTCalgorithm and the VFC-method, showed little mesh dependence for the Sauter
mean radius. However, both algorithms showed a strong mesh dependence for
the penetration. It can be concluded that the mesh dependence is not a result
of the collision model but mainly due to insufficient resolution of the liquid-gas
momentum transfer. In both cases, the medium mesh yielded best agreement
with the experimental data.
The evaporation model for the validation case proved to be almost mesh independent. The effect on the penetration however was still noticeable, and the
VFC evaporation correction did not influence the result. Therefore, the real evaporating sprays have been computed without the VFC-method. Comparisons with
experimental results of the liquid and vapor penetration showed the best agreements for the fine mesh. The mesh dependence in this case is again a consequence
of the insufficient resolution of the liquid-gas momentum transfer.
The VFC-method suffers mainly from an exact determination of the average
void fraction Y void which is used as a comparison value. The determination
fails when not enough cells are uniformly filled with droplets especially in the
initial phase of the spray. The strongest mesh dependence is attributed to the
insufficient liquid-gas momentum transfer, as has been confirmed by investigating
the spray-induced gas velocities for the different meshes. Therefore, more research
63
needs to be done in order to extend the VFC-method to the liquid-gas momentum
exchange and thus reduce the mesh dependence effectively in spray computations.
BIBLIOGRAPHY
64
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