PHYSICS OF FLUIDS
VOLUME 13, NUMBER 11
NOVEMBER 2001
Effects of nonreacting solid particle and liquid droplet loading
on an exothermic reacting mixing layer
Richard S. Millera)
Department of Mechanical Engineering, Clemson University, Clemson, South Carolina 29634-0921
共Received 14 September 2000; accepted 30 July 2001兲
Numerical simulations are conducted of two-dimensional 共2D兲 exothermic reacting mixing layers
laden with either solid particles or evaporating liquid droplets. An irreversible reaction of the form
fuel⫹r Oxidizer→共1⫹r兲 Products with exothermic Arrhenius kinetics is considered. The temporally
developing mixing layers are formed by the merging of parallel flowing oxidizer and fuel streams,
each uniformly laden with nonreacting particles or droplets. The gaseous phase is governed by the
compressible form of the Navier–Stokes equations together with transport equations for the fuel,
oxidizer, product, and evaporated vapor species concentrations. Particles and droplets are assumed
smaller than the gas-phase length scales and are tracked individually in the Lagrangian reference
frame. Complete ‘‘two-way’’ couplings of mass, momentum, and energy between phases are
included in the formulation. The simulation parameters are chosen to study the effects of the mass
loading ratio, particle Stokes number, vaporization, flow forcing, and reaction Zeldovich number on
the flame evolutions. Quasi-one-dimensional simulations reveal that the asymptotic state of the
laminar flames is independent of the particle or droplet loading. For forced 2D simulations, both
particles and droplets are preferentially concentrated into the high-strain braid regions of the mixing
layer. Cold solid particles entrained into the mixing zone cool the flame in the braid regions due to
their finite thermal inertia. This results in flame suppression and, under certain conditions, local
flame extinction in the braids. The potential for flame extinction is substantially enhanced by
evaporating droplets through the latent heat, and also by the addition of nonreacting evaporated
vapor which locally dilutes the reactant concentrations. In contrast, combustion proceeds robustly
within vortex cores which have relatively dilute droplet distributions due to preferential
concentration; particularly at moderate Stokes numbers. The extent of flame suppression and local
extinction are increased with increasing reaction activation energy, dispersed phase mass loading,
and also by decreasing particle or droplet Stokes number. © 2001 American Institute of Physics.
关DOI: 10.1063/1.1404137兴
I. INTRODUCTION
structures which play a primary role in the turbulence
dynamics.2,3 Later studies examined the role of these structures in chemically reacting mixing layers through
theoretical,4 experimental,5–7 and numerical8,9 analyses. The
vortical nature of this flow has a substantial impact on dispersed phase dynamics and dispersion. This occurs due to the
‘‘preferential concentration’’ mechanism,10,11 through which
heavy particles migrate away from high-vorticity regions and
towards high-strain regions of the mixing layer.12–20 Particle
dispersion in mixing layers is maximized for ‘‘intermediate’’
Stokes numbers 共ratio of particle time constant to eddy turnover time scale兲 near unity.15,16 In contrast, very small Stokes
number particles closely follow fluid motions, while very
large Stokes number particles remain relatively unaffected by
fluid motions due to their relatively large inertia.
The vast majority of prior two-phase simulations have
been restricted to solid particle laden flows. Generally, the
particles are assumed to be governed by only the drag force
and gravitational settling, under ‘‘one-way’’ coupling conditions. One-way coupling refers to cases with relatively small
particle mass loading ratios for which the presence of the
particles does not significantly affect the surrounding carrier
gas flow.21–23 In this case only a limited number of particles
Multiphase turbulent combustion occurs in a variety of
natural and practical applications; including, liquid fueled
combustion, solid propellant combustion, and fire suppression and control. An important subset of these reacting flows
are those in which the dispersed phase is nonreacting; such
as with fire suppressants. The scope of the present investigation is limited to gaseous nonpremixed turbulent flames
laden with nonreacting discrete spherical solid particles or
liquid droplets smaller than the smallest length scale of the
共corresponding single-phase兲 gas flow. Furthermore, only
relatively small volumetric loadings (⬃10⫺3 ) are considered, for which direct particle–particle collisions may be
neglected.1
The mixing layer formed by the merging of parallel
flowing fluid streams remains one of the classic flow geometries for fundamental studies of turbulent mixing and combustion. Early investigations revealed that these turbulent
shear flows are dominated by large scale organized vortical
a兲
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3303
© 2001 American Institute of Physics
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3304
Phys. Fluids, Vol. 13, No. 11, November 2001
(⬃103 →104 ) are tracked as dictated by the requirements for
obtaining accurate statistics. Such calculations have been
conducted for a variety of flows;11 including homogeneous
turbulence,10,24 –28 mixing layers13,15,16,18,19,29 and channel
flows.30–33 On the other hand, simulations of ‘‘two-way’’
coupled particle laden flows at larger mass loading ratios
have received significantly less attention owing to the combined numerical difficulties of calculating the phase coupling
terms and tracking sufficiently large numbers of particles
(⬃105 →106 ) as required to achieve desired mass loadings.
Two-way coupled solid particle simulations have typically
been performed only for homogeneous turbulence.28,34 –39
One exception to this trend is the recent numerical investigation of the solid particle laden mixing layer by Meiburg
et al.40 Their two-dimensional 共2D兲 simulations showed that
two-way coupling effects are strongest for intermediate
Stokes numbers.
In contrast to solid particle laden flows, evaporating
droplet laden flows must generally be described by a twoway coupled and compressible 共or variable density兲 formulation due to a vaporization source term in the gas-phase
continuity equation. Due to these added complexities, the
extent of simulations of liquid laden flows remain considerably less than for solid particle two-phase flows.
Mashayek41,42 and Jaberi and Mashayek43 conducted simulations of both isotropic and homogeneous shear box turbulence with two-way coupled liquid droplets. Miller et al.44 – 46
extended two-way coupled computations to nonhomogeneous flows with simulations of a three-dimensional 共3D兲
planar mixing layer having one stream laden with evaporating hydrocarbon droplets. Under the conditions of these studies, the droplet laden stream reaches a state of ‘‘saturation’’
共cessation of vaporization兲 due to the combined effects of
gas cooling and growth of the evaporated vapor mass fraction. However, inside the mixing layer droplets mix with
higher temperature fluid from the nonladen stream and
evaporation proceeds to completion. In these regions, initially monodisperse droplet size distributions evolve to polydisperse distributions due to nonuniform rates of evaporation. Preferential concentration also has an important effect
on the evolution of evaporating droplets which have timedependent Stokes numbers. At early times droplets with intermediate Stokes numbers accumulate in the braid regions
of the layer for vaporization rates smaller than the eddy turnover time scale. Maximum vapor concentrations are therefore observed in the braid structures for both 2D and 3D
mixing layers.
The above citations have only addressed nonreacting
flows. In contrast, numerical simulations of two-phase reacting flows have received markedly less attention. Miller and
Bellan44 investigated an infinitely fast adiabatic reaction
based on a passive conserved scalar mixture fraction in the
context of analyzing the validity of the assumed particle distribution function 共PDF兲 method for two-phase combustion.
Mashayek47 extended their earlier computations to simulate
reacting homogeneous shear turbulence. In this case, fuel
droplets are immersed in a carrier gas oxidizer and a simple
chemical reaction of the form Fuel⫹Oxidizer
→ Products was considered. Simulation of two-phase react-
Richard S. Miller
ing non-homogeneous flows remains limited; however, Glaze
and Frankel48 recently simulated one-way coupled solid particle dispersion in a reacting jet using the large eddy simulation 共LES兲 technique.
The primary objective of the present paper is to use numerical simulation to investigate the complex couplings of
turbulent flames with a nonreacting dispersed phase in the
form of either solid particles or evaporating droplets. In particular, the effects of vortical rollup, reaction activation energy, dispersed phase mass loading, and particle or droplet
size are considered. Only nonreacting particles and droplets
are studied; both as a direct interest and as a prelude to future
reacting droplet investigations. In this work 2D temporally
developing mixing layers are considered as a model flow
representative of the physical processes inherent in fully turbulent 3D reacting flows. The primary reason for this limitation is the exceedingly large computational cost of simulating the corresponding 3D flow. Miller and Bellan46 captured
transition to turbulence in their simulations of a 3D temporally developing mixing layer using as many as 18⫻106 grid
points and 5.7⫻106 evaporating droplets. However, Miller
et al.49 were unable to reproduce the transition to turbulence
for similar flow conditions due to the stabilizing effects of
density stratification in their single-phase supercritical mixing layer. Preliminary tests performed for this study confirm
that this situation is maintained for the present flow which
contains stratification as well as exothermicity and particle
loading. Therefore, rather than simulate low-Reynolds number 3D mixing layers with limited vortical development, we
choose to examine 2D flows at relatively large Reynolds
number with multiple vortical pairings. Extension of the
present simulations to 3D domains under conditions conducive to transition would be prohibitively expensive 共e.g., extension of our cases SLD-HIGHc and EVAP-HIGHc described below to 3D would require 91⫻106 grid points and
447⫻106 particles or droplets兲. Despite the fact that many of
the important physical processes found in 3D flows are also
present in the 2D mixing layer 共including vortical structures
and pairing, preferential concentration, particle–flow interaction, and local flame extinction in high-strain flow regions兲,
the reader is cautioned that the results of the present investigation should not be quantitatively extrapolated to the more
complicated 3D turbulent flows.
The paper is organized as follows: The formulation, numerical approach, mixing layer configuration, and flow parameters are described in Sec. II. Results from the simulations are presented in Sec. III, which includes subsections
addressing laminar flame simulations, flow visualizations, as
well as effects of the Zeldovich number, the dispersed phase
mass loading ratio and the initial Stokes number. Conclusions and final discussions are the subject of Sec. IV.
II. FORMULATION AND APPROACH
The governing equations describe the Lagrangian transport of discrete solid particles or evaporating droplets
through a continuous, calorically perfect and chemically reacting carrier gas flow. Throughout the formulation, the subscripts F, O, P, and V refer to the fuel, oxidizer, product, and
Downloaded 02 Nov 2001 to 130.127.198.18. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 13, No. 11, November 2001
Effects of nonreacting solid particle
evaporated vapor species, respectively; whereas subscript G
refers to the multispecies gas-phase mixture. For the dispersed phase, S denotes solid particles and L denotes liquid
droplets. The compressible form of the governing equations
for the gas-phase include mass, momentum and energy exchange between the gas and the nonreacting dispersed phase
are
⳵␳
⳵
⫹
关 ␳ u j 兴 ⫽S I ,
⳵t ⳵x j
共1兲
⳵
⳵
共 ␳ui兲⫹
关 ␳ u i u j ⫹ P ␦ i j ⫺ ␶ i j 兴 ⫽S II,i ,
⳵t
⳵x j
共2兲
冋
⳵
⳵
⳵T
⫺u i ␶ i j
共 ␳et兲⫹
共 ␳ e t ⫹ P 兲 u j ⫺␭
⳵t
⳵x j
⳵x j
册
⫽S III ⫺ ␻
˙ P ⌬H 0 ,
P⫽ ␳
共3兲
兺␣ 关 R 0 /W ␣ 兴 T,
共4兲
where ␳ is the gas phase density, u i is the gas-phase velocity,
P is the thermodynamic pressure, e t ⫽e⫹u i u i /2 is the total
gas energy, i.e., kinetic energy plus internal energy 关e
⫽⌺ ␣ (Y ␣ C v , ␣ T⫹h V0 ) where the vapor reference enthalpy
(h V0 ) is chosen to be the same for all gas phase species兴, and
Y ␣ , R ␣ ⫽R 0 /W ␣ and C v , ␣ are the mass fraction, gas constant, and constant volume specific heat of species ␣, respectively 共the universal gas constant is R 0 , the molecular weight
is W ␣ and C p, ␣ ⫽R ␣ ⫹C v , ␣ is the constant pressure heat capacity兲. Furthermore, ␶ i j is the Newtonian viscous stress tensor, ␮ is the viscosity, ␭ is the thermal conductivity, and ⌫ is
the Fickian species diffusion coefficient. The right-hand side
共rhs兲 terms S I , S II,i , and S III describing the phase couplings
of mass, momentum, and energy, respectively, are defined
below.
A single-step, irreversible and exothermic reaction of the
form Fuel⫹r Oxidizer→共1⫹r兲 Products is considered
through the solution of transport equations for the mass fractions (Y ␣ ) of the oxidizer ( ␣ ⫽O), fuel ( ␣ ⫽F), and product
( ␣ ⫽ P)
冋
册
⳵
⳵
⳵Y ␣
␳ Y ␣u j⫺ ␳ ⌫
⫽␻
˙␣,
共 ␳Y ␣兲⫹
⳵t
⳵x j
⳵x j
共5兲
and, in the case of liquid droplets, for the nonreacting evaporated vapor
冋
册
⳵
⳵
⳵Y V
␳ Y Vu j⫺ ␳ ⌫
⫽S I .
共 ␳Y V兲⫹
⳵t
⳵x j
⳵x j
共6兲
All species are assumed to diffuse at the same rate, with
diffusivity ⌫. The chemical kinetics are based on the Arrhen˙ ␣ ) kinetics
ius form of the reaction rate ( ␻
冉 冊
冉 冊
␻˙ O ⫽⫺ ␶ ␳ K R 共 T 兲
␻˙ F ⫽⫺ ␳ K R 共 T 兲
WP
Y Y ,
WF O F
WP
Y Y ,
WO O F
共7兲
共8兲
␻˙ P ⫽⫹ 共 1⫹r 兲 ␳ K R 共 T 兲
冉 冊冉 冊
WP
WF
WP
Y Y ,
WO O F
3305
共9兲
with temperature dependent reaction coefficient: K R (T)
⫽A 0 exp关⫺E0/(R0T)兴, where E 0 is the activation energy, and
A 0 is the reaction rate constant 共with inverse units of time兲.
Miller et al.50 compared eight different vaporization
models with experimental data for single droplets. Based on
their findings, a variant of the classical ‘‘D 2 law’’ is chosen
to predict the droplet evolutions. The following model has
previously been employed for calorically perfect species by
Miller and Bellan45 in nonequilibrium form, and by Miller
and Bellan46 in equilibrium form. Nonequilibrium effects
were found to become significant only for very small droplet
sizes 共⬍50 ␮m兲 in high-temperature environments50 and are,
therefore, not included in the present investigation. The modeled Lagrangian equations describing the transient position
(X i ), velocity ( v i ), temperature (T p ), and mass (m p ) of a
single particle 共or droplet兲 are
dX i
⫽vi ,
dt
共10兲
dvi Fi
⫽
,
dt
mp
共11兲
dT p Q⫹ṁ p L V
,
⫽
dt
m pC L
共12兲
冉 冊冉 冊
dm p
1
⫽ṁ p ⫽⫺m p
dt
␶p
Sh
ln关 1⫹B M 兴 ,
3Sc G
共13兲
where F i is the modified Stokes drag force
F i ⫽m p
冉 冊
f1
共 u i⫺ v i 兲,
␶p
共14兲
Q is the heat flux to the surroundings
Q⫽m p
冉 冊冉
f2
␶p
冊
NuC p,G
共 T⫺T p 兲 ,
3PrG
共15兲
the subscript p denotes individual particle conditions, the
particle time constant for Stokes flow is ␶ p ⫽ ␳ L D 2 /(18␮ )
共substitute subscript L→S and dm p /dt⫽0 for solid particles兲, D is the particle diameter, C L is the heat capacity of
the liquid, and the latent heat of evaporation is L V . Additionally, the gas mixture heat capacity is calculated using a mass
averaging; C p,G ⫽⌺ ␣ Y ␣ C p, ␣ 共evaluated at particle locations兲. The gas-phase Prandtl and Schmidt numbers are
Pr G ⫽ ␮ C p,G /␭ and Sc G ⫽ ␮ /( ␳ ⌫), respectively. The evaporation rate is driven by the mass transfer number; B M
⫽(Y s f ⫺Y V )/(1⫺Y s f ) 共subscript sf denotes droplet surface
conditions兲. The semiempirical Ranz–Marshall correlations
are used for the Nusselt 共Nu兲 and Sherwood 共Sh兲 numbers,
whereas f 1 is an empirical correction to Stokes drag accounting for finite droplet Reynolds numbers and evaporation.50
The function f 2 ⫽ ␤ /(e ␤ ⫺1) is an analytical evaporative
heat transfer correction, where the nondimensional evaporation parameter ␤ ⫽⫺1.5 Pr G ␶ p ṁ p /m p is constant for droplets obeying the ‘‘D 2 law.’’ The vapor surface mass fraction
is calculated directly from the surface molar fraction ( ␹ s f )
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3306
Phys. Fluids, Vol. 13, No. 11, November 2001
Richard S. Miller
which is obtained by equating the vapor and liquid fugacities
at the surface 共i.e., ␹ s f P⫽ P sat兲, where the saturation pressure
( P sat) is provided by the Clausius–Clapeyron relation, yielding
Ysf⫽
␹sf
,
␹ s f ⫹ 共 1⫺ ␹ s f 兲 W G /W V
␹sf⫽
LV 1
P atm
1
⫺
exp
P
R V T B,L T p
再 冉
共16兲
冊冎
共17兲
,
where P atm is atmospheric pressure, T B,L is the saturation
temperature at P atm 共i.e., the normal boiling temperature兲.
Finally, the latent heat is a linear function of temperature for
calorically perfect species: L V ⫽h V0 ⫺(C L ⫺C p,V )T p .
Based on the above formulation, the phase coupling
terms are expressed as
S I ⫽⫺
兺␣
⌬x 1 ⌬x 2 ⌬x 3⬘
再
兺再
冋
S II,i ⫽⫺
S III ⫽⫺
再
兺␣
␣
w␣
冎
共18兲
关 ṁ p 兴 ␣ ,
w␣
⌬x 1 ⌬x 2 ⌬x 3⬘
冎
关 F i ⫹ṁ p v i 兴 ␣ ,
共19兲
冎册 冎
共20兲
w␣
⌬x 1 ⌬x 2 ⌬x 3⬘
⫻ v i F i ⫹Q⫹ṁ p
再
v iv i
⫹h V,s f
2
,
␣
where the summations are over local individual droplet contributions, h V,s f ⫽C p,V T p ⫹h V0 is the evaporated vapor enthalpy at the droplet surface, and the single droplet evaporation rate (ṁ p ), drag force (F i ) and heat transfer rate 共Q兲 are
specified by Eqs. 共11兲–共13兲. The local summations are over
all droplets residing within a local numerical discretization
element 共⌬x 1 ⌬x 2 ; for 2D兲 and employ a geometrical
weighting factor, w ␣ , used to distribute the individual droplet contributions to the four nearest neighbor surrounding
grid points 共i.e., corners of the element ⌬x 2 兲. Note that the
source terms are defined on a per unit volume basis since the
particles are modeled as spherical entities, not as cross sections of infinitely long cylinders. In this case, the differential
volume element in the above terms is ⌬x 1 ⌬x 2 ⌬x 3⬘ , where
the characteristic length in the hypothetical x 3 direction
(⌬x 3⬘ ) is defined below. These source terms are then minimally ‘‘smoothed’’ using a conservative operator in order to
retain numerical stability of the Eulerian gas-phase
equations.45 Potential limitations of the modeling of the
source terms has been addressed previously in Ref. 45.
A. Temporally developing mixing layer
The flow geometry under consideration is the twodimensional 共2D兲 temporally developing mixing layer shown
in Fig. 1. The streamwise (x 1 ) and cross stream (x 2 ) domain
lengths are L 1 and L 2 , respectively. Stream 1 (x 2 ⬎0) is
composed of pure oxidizer, whereas Stream 2 (x 2 ⬍0) is
pure fuel. For two-phase simulations, both streams are uniformly laden with particles or droplets. Miller and Bellan45
found that the mass loading ratio 共ratio of liquid mass to
FIG. 1. Schematic of the two-phase temporally developing mixing layer
共boxed region, above兲 including the initial flow profiles. The initial vorticity
thickness ( ␦ ␻ ,0) is not shown to scale.
gaseous mass in each stream兲 is the dominant parameter governing the flow modification in the droplet laden mixing
layer. Therefore, we choose to fix the mass loading ratio to
be the same in each stream; since the gas stream densities are
not equal, the particle or droplet number density is not equal
in the two streams. Boundary conditions are periodic in the
streamwise direction, and nonreflecting outflow for the cross
stream boundaries. The initial vorticity thickness is ␦ ␻ ,0 ,
where ␦ ␻ (t)⫽⌬U 0 / 具 ⳵ u 1 / ⳵ x 2 典 max ; the brackets 具 典 indicate
averaging over the homogeneous x 1 direction and the mean
velocity difference across the layer, ⌬U 0 ⫽U 1 ⫺U 2 , is calculated from a specified value of the convective Mach number (M c ). 51
The base flow mean velocity, mass fraction, temperature
and number density are specified based on an error function
profile; erf(␲1/2x 2 / ␦ ␻ ,0). 52 Initial mean profiles are illustrated in Fig. 1. The temperature is chosen to be constant and
‘‘low’’ in each stream at T⫽T 0 ; however, a temperature
‘‘spike’’ is added at the centerline to ignite the flame. The
peak spiked temperature is denoted T F,0 . The fuel and oxidizer species are kept completely segregated in the initial
profiles through the addition of a thin layer of product species. This prevents an overly fast initial reaction, as the flame
can only begin to burn once diffusional mixing of reactants
has commenced. In all cases, the evaporated vapor mass
fraction, Y V , is initially zero everywhere. For ‘‘forced’’
simulations, perturbations are added to the velocity field in
order to excite the growth of spanwise disturbances 共see below兲.
Finally, we note that several recent investigations53–55
have noted that density stratification across the mixing layer
can result in changes to both the ‘‘relaxed’’ laminar base flow
profiles, as well as the most unstable forcing modes. Both
effects were observed to significantly affect the development
of the mixing layer. On the other hand, for the present reacting flow simulations the most appropriate base flow or forcing profiles are not easily determined due to the timedependent nature of the ensuing ignition and combustion, as
well as to the presence of the dispersed phase. Therefore, the
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Phys. Fluids, Vol. 13, No. 11, November 2001
approach used in this investigation is to fix the density stratification for all simulations, and to use the same error function and forcing used in constant density temporal mixing
layer simulations.52 In order to minimize the effects of the
initial conditions, the streamwise domain length is chosen to
be eight times the smallest disturbance wavelength. As many
as four primary vortex pairing events can occur during a
simulation, thus allowing sufficient time for the mixing layer
to develop in a more natural manner than at early times when
the initial conditions are expected to have a stronger influence on the flow dynamics.
B. Numerical approach
The governing equations are solved numerically using
fourth-order explicit Runge–Kutta temporal integration for
all time derivatives. Spatial derivatives are disctretized using
an explicit eighth order accurate central finite difference
scheme in the streamwise direction (x 1 ), and by a fourthorder accurate tridiagonal compact finite difference in the
cross stream direction (x 2 ). 56 The computational grid is uniformly spaced with constant ⌬x 1 ⫽⌬x 2 . A fourth order
Lagrange interpolation procedure is used to obtain gas-phase
variable values at droplet locations which, in general, do not
reside at grid point locations. Transport equations are solved
for every individual particle or droplet in the flow 共as opposed to stochastic entities representing many real particles兲.
Nonreflecting outflow boundary conditions derived by Poinsot and Lele57 are employed for the x 2 domain boundaries.
Droplets which pass through these boundaries and exit the
domain are removed from the simulation. In the event that
fluid enters the domain from the free stream, no new droplets
are introduced.
All simulations are conducted within a domain having
L 1 ⫽8␭ 1 ⫽58.32␦ ␻ ,0 and L 2 ⫽1.125L 1 , where ␭ 1 is the
smallest forcing wavelength in the x 1 direction. Sinusoidal
velocity perturbations are superimposed on the base flow
profile having wavelengths L 1 , L 1 /2, L 1 /4, and L 1 /8.
For convenience, the disturbances are generated as a
spanwise vorticity distribution of the form ␻ 3
4
⫽⫺⌺ m⫽1
f (x 2 ) 兩 A m sin(2␲x1 /(2m␭1))兩, where the cross
stream weighting is given by f (x 2 )⫽exp(⫺␲x22/␦␻2 ,0), and
the relative amplitudes of the harmonics are A 1 ⫽1, A 2
⫽0.5, and A 3 ⫽A 4 ⫽0.35. The corresponding velocity disturbance distribution is then extracted via the solution of the
appropriate Poisson equation. The nondimensional forcing
* ) is characterized by the spanwise circulation
amplitude (F 2D
of the disturbance relative to the base flow circulation
(␭ 1 ⌬U 0 ). The imposed disturbances instigate the development of eight initial vortices together with four pairing
events prior to the nonphysical intervention of the domain
* ⫽0) cases are
boundaries. Both forced and nonforced (F 2D
considered. The choice of most optimal forcing is particularly complicated for compressible, two-phase, stratified, and
reacting mixing layers. A discussion of this subject pertaining to stratified layers may be found in Ref. 49. The approach chosen for this study is to fix the forcing mode and
amplitude, and to simulate the mixing layer to relatively long
times. Although long time influences of the initial forcing
Effects of nonreacting solid particle
3307
TABLE I. Species property values used in the simulations. The subscripts
denote the solid 共S兲 and liquid 共L兲 phases, and O, F, P, and V denote the
oxidizer, fuel, product, and evaporated vapor species, respectively. All species have equal thermal conductivities and mass diffusivities with
Pr⫽Sc⫽0.697.
Property
WO
WF
WP
WV
C p,O
C p,F
C p, P
C p,V
C S ,C L
␳S ,␳L
T B,L
h 0V
Value
28.97 kg共kg•mole兲⫺1
86.178 kg 共kg•mole兲⫺1
57.574 kg共kg•mole兲⫺1
142.0 kg共kg•mole兲⫺1
1043.8 J.kg⫺1 K⫺1
2251.5 J.kg⫺1 K⫺1
1947.7 J.kg⫺1 K⫺1
2394.5 J.kg⫺1 K⫺1
2520.5 J.kg⫺1 K⫺1
642.0 kg m⫺3
447.7 K
3.36⫻105 J.K⫺1
may exist, the present results are not used to investigate the
asymptotic state of the two-phase mixing layer.
The code is parallelized in both directions using the
Message Passing Interface 共MPI兲 communication routines.
Simulations were conducted on three different computers; a
Silicon Graphics/Cray Origin2000, Hewlett Packard V2500,
and a SUN HPC 6000, using from 8 to 48 CPUs. Computing
times varied accordingly; e.g., the most intensive simulation
共EVAP-HIGHe, see below兲 was conducted using 48 CPUs of
the V2500 and required approximately 6.9 seconds per iteration. The simulation time step (⌬t) is calculated at each
temporal iteration from a specified value of the Courant
number, C 共based on the convective plus acoustic velocities兲.
Both single-phase and solid particle laden simulations have
C⫽0.5, whereas evaporating droplet simulations employ C
⫽0.25 in order to ensure that the transient vaporization is
well resolved temporally.
C. Properties and nondimensional parameters
Table I presents the properties of each of the species
under consideration; fuel, oxidizer, product, evaporated vapor, as well as the solid or liquid species comprising the
dispersed phase. The liquid and solid species have identical
properties; the only distinction between these species is that
the evaporation rate is nulled for solid particles 共i.e., ṁ p
⫽0兲. The species properties used in the investigation were
chosen to model a typical 共though simplified兲 air–
hydrocarbon diffusion flame. The oxidizer species has properties corresponding to those of air, the fuel species corresponds to hexane, the liquid 共or solid兲 and vapor species
have properties of decane, and the product species properties
are determined from mass and energy balances. All species
are assumed to be calorically perfect.
Several nondimensional parameters are introduced in order to simplify the characterization of the mixing layer and
combustion. The flow Reynolds number is Re0
⫽␳⌬U0␦␻,0 / ␮ . The particles or droplets are initially of uniform size specified by the initial Stokes number St0
⫽ ␶ p,0⌬U 0 / ␦ ␻ ,0 . As mentioned previously, the initial mass
loading ratio is prescribed to be the same for each stream.
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3308
Phys. Fluids, Vol. 13, No. 11, November 2001
Richard S. Miller
TABLE II. Initialization parameters 共SP⫽single phase, SLD⫽solid particle, EVAP⫽evaporating droplet,
LAM⫽laminar, LOW⫽low Ze, HIGH⫽high Ze兲. All cases have M c ⫽0.35, Re0⫽450, T 0 ⫽300 K, T F,0
⫽565 K, Da(T F )⫽10, Ce⫽3.0, r⫽1, and P 0 ⫽1 atm. Two phase simulations are initialized with zero slip
velocity ( v i ⫽u i ) and zero slip temperature 关T p ⫽T; except for EVAP cases in which T p ⫽min(T,TB,L)].
Case
SP-LOW-LAM
SP-HIGH-LAM
SP-LOW
SP-HIGH
SLD-HIGH-LAM
SLD-LOWa
SLD-LOWb
SLD-HIGHa
SLD-HIGHb
SLD-HIGHc
SLD-HIGHd
EVAP-HIGH-LAM
EVAP-HIGHa
EVAP-HIGHb
EVAP-HIGHc
EVAP-HIGHd
N 1 ⫻N 2
Np
M L0
St 0
Ze
*
F 2D
⌬x/D 0
␶ p,0 /⌬t
10⫻576
10⫻576
512⫻576
512⫻576
10⫻576
512⫻576
512⫻576
512⫻576
512⫻576
512⫻576
512⫻576
10⫻576
512⫻576
512⫻576
512⫻576
512⫻576
0
0
0
0
3542
1.81⫻105
2.88⫻105
1.81⫻105
2.88⫻105
1.45⫻106
3.02⫻104
3542
1.81⫻105
2.88⫻105
1.45⫻106
3.02⫻104
¯
¯
¯
¯
0.25
0.25
0.5
0.25
0.5
0.25
0.25
0.25
0.25
0.5
0.25
0.25
¯
¯
¯
¯
2.0
2.0
2.0
2.0
2.0
0.25
12.0
2.0
2.0
2.0
0.25
12.0
9.21
15.35
9.21
15.35
15.35
9.21
9.21
15.35
15.35
15.35
15.35
15.35
15.35
15.35
15.35
15.35
0
0
0.1
0.1
0
0.1
0.1
0.1
0.1
0.1
0.1
0
0.1
0.1
0.1
0.1
¯
¯
¯
¯
6.67
6.67
6.67
6.67
6.67
18.9
2.72
6.67
6.67
6.67
18.9
2.72
¯
¯
¯
¯
90.5
90.5
90.5
90.5
90.5
11.3
543
181
181
181
22.6
1086
For constant density streams with uniform mass particles, the
mass loading is defined as
M L 0⫽
N p, ␣ m p
⬘ 共 ␣ 兲 /2
␳ L 1 L 2 L 3,
,
共21兲
for either individual stream 共N p, ␣ is the number of particles
in stream ␣兲. In the above, a reference length in the x 3 direction must be specified to define a meaningful mass loading for a 2D flow 共the particles are spherical, not cross sections of infinitely long cylinders兲. Assuming that the particles
are initially distributed at the ‘‘nodes’’of a hypothetical Cartesian grid in the 2D plane, then the corresponding isotropic
3D distribution would have the same particle separation in
⬘ ␣ ) is dethe third direction. The spanwise length scale L 3,(
fined to be this hypothetical separation distance, which is
determined by equating the assumed Cartesian particle grid
area to the total area of either stream
⬘ 2共 ␣ 兲 ⫽
N p, ␣ L 3,
L 1L 2
.
2
共22兲
For simplicity, we define the characteristic x 3 length scale for
the phase coupling source terms, Eqs. 共18兲–共20兲, to be ⌬x 3⬘
⬘ ⫹L 3,(2)
⬘ )/2 关this does not affect the physical ‘‘cor⫽(L 3,(1)
rectness’’ of the approach, but does mean that the effective
mass loading ratios are slightly altered from the definition in
Eq. 共21兲兴. It is noted that the above defined mass loading
parameter should not be quantitatively correlated to that of
3D flows. It does, however, provide a direct measure of the
effect of increasing the total particle mass acting on the Eulerian gas-phase flow.
The chemical reaction is also specified by nondimensional parameters. The Zeldovich number parameterizes the
reaction activation energy: Ze⫽E 0 /(R 0 T 0 ), and the heat release parameter Ce specifies the heat of reaction ⌬H 0 :Ce
⫽⫺⌬H 0 /(C p,F T 0 ). With the above notation, the adiabatic
flame temperature (T F ) is approximated by T F ⫽T 0 (1
⫹Ce), where T 0 is the initial temperature of both streams in
the mixing layer. Furthermore, the Damkohler number 共Da兲
is defined as the ratio of the characteristic time scale of the
flow to the characteristic reaction time scale: Da(T)
⫽K R (T) ␦ ␻ ,0 /⌬U 0 . The Damkohler number is defined as a
function of the temperature in order to simplify the following
analyses.
III. RESULTS
The simulations conducted for this study are summarized in Table II which provides the designation, grid resolution, total number of particles (N p ), mass loading ratio,
particle Stokes number, Zeldovich number, forcing amplitude, and relative initial particle diameters and particle time
constants. The following conventions are used in naming
each simulations: SP⫽single phase, SLD⫽solid particle
laden, EVAP⫽evaporating droplet laden, LAM⫽laminar
flow, LOW⫽low-activation energy, and HIGH⫽highactivation energy. The unforced ‘‘LAM’’ simulations are
conducted with L 1 ⫽10⌬x 1 and correspond to quasi-onedimensional 共1D兲 laminar flames. The final two columns in
Table II show that the particle sizes are all smaller than the
grid spacing, and that the particle temporal evolutions are
well resolved.
In order to simplify the analysis, the remaining initialization parameters are fixed for all simulations. The flow
Reynolds number is Re0⫽450, the convective Mach number
is M c ⫽0.35, the pressure is initially constant and standard
( P 0 ⫽ P atm), the uniform temperature in both streams is T 0
⫽300 K, and the spiked ignition temperature is chosen to be
T F,0⫽565 K. For the reaction, the Damkohler number evaluated at the adiabatic flame temperature is in all cases
Da(T F )⫽10, the reaction coefficient is r⫽1, and the heat
release parameter is Ce⫽3 (T F ⫽1200 K). Finally, the particles or droplets are initially randomly dispersed throughout
either stream, with specified size, zero slip velocity ( v i
⫽u i ), and zero slip temperature (T p ⫽T). However, droplet
temperatures cannot exceed the boiling temperature; there-
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Phys. Fluids, Vol. 13, No. 11, November 2001
FIG. 2. Gas-phase temperature dependence of 共a兲 the reaction Damkohler
number, and 共b兲 the time to complete evaporation for a single droplet in an
infinite quiescent environment.
fore, for droplets T p ⫽min(T,TB,L). All simulations are conducted until a nondimensional time t * ⫽t⌬U 0 / ␦ ␻ ,0⫽160 at
which point the final pairing has been nearly completed.
A. Effect of combustion and droplet parameters
A justification for the choices of several of the above
parameters is provided in Fig. 2. Figure 2共a兲 depicts the reaction rate Damkohler number as a function of temperature
for the two values of the Zeldovich number considered in
this study. In both cases, Da(T F )⫽10 is specified; therefore,
the reaction rate constant is also a function of Ze 共A 0
⫽5.17⫻106 s⫺1 for Ze⫽9.21, and A 0 ⫽2.40⫻107 s⫺1 for
Ze⫽15.35兲. In this manner, only the relative rate of the reaction at lower temperatures is affected by changes to the
Zeldovich number. The Damkohler numbers at the initial
spiked ignition temperature are Da(T F,0)⫽0.75, and
Da(T F,0)⫽0.13 for Ze⫽9.21 and Ze⫽15.35, respectively.
Therefore, LOW cases burn relatively vigorously early in the
simulation, whereas HIGH cases will require a substantially
longer time to commence vigorous combustion. Furthermore, the reaction rate is essentially negligible for both Zeldovich numbers at the ambient temperature T⫽300 K;
thereby allowing for varying degrees of local flame extinction in the mixing layer simulations that follow.
The droplet properties and time scales were chosen to
mimic the behavior of typical species used in fire suppres-
Effects of nonreacting solid particle
3309
FIG. 3. Mean cross stream profiles for laminar flame simulations at time
t * ⫽160; 共a兲 temperature, and 共b兲 droplet number density.
sion 共e.g., water; decane properties are used instead of actual
water properties in order to simplify comparisons with a concurrent study of reacting droplets兲 in that evaporation occurs
rapidly in flame regions, but very slowly under ambient conditions. The relative evaporation time (t E ) of single droplets
is examined in Fig. 2共b兲. Droplets cannot be simulated to
arbitrarily small sizes due to an eventual inadequate temporal
resolution of the particle evolutions 共i.e., ␶ p will become too
small relative to ⌬t兲. For the present investigation, droplets
are assumed to be completely evaporated once their individual Stokes number reaches St⫽0.075, which was found to
have an adequate temporal resolution. With this final Stokes
number, all but 5.0⫻10⫺4 , 7.3⫻10⫺3 , and 1.6⫻10⫺1 of the
initial mass is allowed to evaporate for droplets with St0⫽12,
St0⫽2.0, and St0⫽0.25, respectively. The results in Fig. 2共b兲
correspond to the total time required to reach this size limit
for single isolated droplets in infinite quiescent media of either pure fuel or oxidizer as a function of the ambient temperature. These curves are therefore solutions of only Eqs.
共12兲 and 共13兲 with u i ⫽ v i ⫽0. Each curve represents the results of 40 such single droplet simulations. Although only an
approximation to what occurs in the more complicated 2D
simulations, Fig. 2共b兲 provides an estimate of the droplet life
time relative to the flow time scale for the range of gas temperatures that may be encountered by droplets in the flame.
In all cases, the droplet life time is smaller than the total
simulation time (t * ⫽160) for temperatures above T F,0 . In
contrast, droplets in the free streams away from the combus-
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3310
Phys. Fluids, Vol. 13, No. 11, November 2001
Richard S. Miller
FIG. 4. Temporal evolution of flow statistics for laminar flame simulations; 共a兲 momentum thickness, 共b兲 product thickness, 共c兲 maximum temperature, and
共d兲 maximum reaction rate.
tion region encounter cooler fluid (T 0 ⫽300 K), and act essentially as nonevaporating particles at these temperatures.
Note also that Fig. 2共b兲 shows that droplets evaporate faster
in pure oxidizer than in pure fuel at high temperatures. This
can result in a bias in the vaporization rates for droplets
depending on which stream of the mixing layer they
reside in.
B. Laminar flame evolution
Quasi-1D laminar flames are examined as an aid to understanding the 2D mixing layer flows. Simulations SPLOW-LAM, SP-HIGH-LAM, SLD-HIGH-LAM, and EVAPHIGH-LAM 共Table II兲 elucidate the effects of the Zeldovich
number 共for SP flames兲, as well as either solid particle or
liquid droplet loading (M L 0 ⫽0.25) on the laminar flame.
All of these simulations are conducted with resolutions of
10⫻512 grid points. The complete set of 2D governing
equations is solved for these flows; however, no forcing is
used and the single-phase 共SP兲 cases are perfectly 1D solutions. On the other hand, minor x 1 variations of flow variables exist for the two-phase cases due to the phase coupling
source terms in the Eulerian gas phase equations. Nevertheless, these spatial variations are negligible to the overall flow
and the results are essentially 1D. Each of these flows are
initialized in the same manner as the 2D mixing layers and
are also allowed to evolve until a nondimensional time t *
⫽160. The final time mean temperature and mean number
density profiles across the laminar flames are shown in Fig.
3. Note that the cross stream coordinate x 2 has been normalized by the instantaneous momentum thickness defined by46
␦ m⫽
1
关 具 ␳ u 1典 2⫺ 具 ␳ u 1典 1 兴 2
冕
⫹L 2 /2
⫺L 2 /2
兵 关 具 ␳ u 1 典 2 ⫺ 具 ␳ u 1 典 共 x 2 兲兴
⫻ 关 具 ␳ u 1 典 共 x 2 兲 ⫺ 具 ␳ u 1 典 1 兴 其 dx 2 ,
共23兲
where 具 ␳ u 1 典 ␣ is the mean momentum evaluated at the x 2
boundary of stream ␣. The particle number density 共n兲 is
calculated in a similar manner to the phase coupling terms,
n⫽ 兺 ␣ 兵 w ␣ 其 , and is interpreted as the number of particles per
grid point. For evaporating droplets, any individual droplet
which has reached the minimum size St⫽0.075 is removed
from the simulation, and has no contribution to the summation.
The results of Fig. 3 indicate that the long time laminar
flames, which are nonsymmetric across the layer due to density and property stratification, are essentially independent of
the Zeldovich number and the dispersed phase loading. At
the observed laminar flame temperatures 关Fig. 3共a兲兴, the reaction rates are much faster than the characteristic flow time
scale 关see Fig. 2共a兲兴. This means that all of the flames, regardless of the Zeldovich number, are diffusion controlled;
i.e., limited by diffusion as opposed to being limited by
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Phys. Fluids, Vol. 13, No. 11, November 2001
Effects of nonreacting solid particle
3311
FIG. 5. Temperature contours for simulation SP-LOW; 共a兲 t * ⫽40, 共b兲 t * ⫽80, 共c兲 t * ⫽120, and 共d兲 t * ⫽160.
chemical kinetics. This accounts for the similarity in the SPLOW-LAM and the SP-HIGH-LAM flow profiles. On the
other hand, the observed similarity in the particle and droplet
laden flame profiles is attributed to the fact that the solid
particles achieve an equilibrium temperature with the surrounding flame, whereas droplets are completely evaporated
within the flame at long times. Once the liquid within the
flame has evaporated, the gaseous vapor can diffuse away
from the combustion zone, and the reaction continues in a
similar manner to single-phase cases. This is clarified in Fig.
3共b兲 which shows a zero droplet number density in the approximate region ⫺2⭐x 2 / ␦ m ⭐3 due to the completion of
vaporization. As a final note, we observe that the laminar
flame temperatures 关Fig. 3共a兲兴 are substantially lower than
the adiabatic flame temperature (T F ⫽1200 K) for all 1D
simulations due to diffusional losses.
Despite the observed long time similarities of the laminar flames, substantial differences in their evolutions are apparent at earlier times. Figure 4 presents the temporal development of the momentum thickness, product thickness
␦ P⫽
冕 冕
⫹L 2 /2
⫺L 2 /2
L1
0
␳ Y P dx 1 dx 2 ,
共24兲
maximum temperature, and maximum reaction rate for the
laminar flames. These results reveal that the diffusion limited
asymptotic flame behavior occurs only after t * ⬇125. Prior
to this time the flames develop quite differently, particularly
as a function of the Zeldovich number. For LOW cases
(Ze⫽9.21), the reaction rate is relatively fast immediately,
as noted by the early and rapid rise of the maximum temperature and reaction rate 关Figs. 4共c兲 and 4共d兲兴. This is accompanied by an early development of linear momentum
and product thickness growth rates for times t * ⬎50 关Figs.
4共a兲 and 4共b兲兴. On the other hand, for the HIGH cases
(Ze⫽15.35), the reaction rate is relatively slow at the initial
temperature (T F,0⫽565 K) and the maximum temperature
decreases substantially until a sufficient mass of reactants
have diffused into the flame region to ignite the flame at time
t * ⬇50. Solid particles have a finite thermal inertia and,
therefore, have the potential to delay ignition due to the energy lost to heating the particles. However, any changes depicted in the results of Fig. 4 due to solid particles are too
small to draw conclusions in this regard. The characteristic
time scale for particle heating is proportional to the particle
Stokes number, in this case St⫽2.0, in units of the eddy
turnover time 关see Eq. 共12兲兴. Evaporating droplets cause substantial additional losses of thermal energy from the flame
due to latent heat effects 关note that the droplets cannot exceed the liquid boiling temperature, T B,L ⫽447.7 K, which
results in large convective heat flux from the higher temperature flame; see Eq. 共12兲兴. However, the droplets reach complete evaporation relatively quickly at these temperatures
关see Fig. 2共b兲兴, and are rapidly destroyed within the reaction
zone.
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3312
Phys. Fluids, Vol. 13, No. 11, November 2001
Richard S. Miller
FIG. 6. Temperature contours for simulation SP-HIGH; 共a兲 t * ⫽40, 共b兲 t * ⫽80, 共c兲 t * ⫽120, and 共d兲 t * ⫽160.
C. Mixing layer flow visualization
As will be shown below, the qualitative and quantitative
behaviors of the 2D forced mixing layer simulations can be
markedly more complex than observed in the much simpler
laminar flame simulations. The mixing layer analysis is begun in this section with a qualitative exposition of the flame
development aided by flow visualizations. The typical evolution of a single-phase low activation energy flame is illustrated in Fig. 5 which depicts contours of the gas temperature
for case SP-LOW at times t * ⫽40, 80, 120, and 160. Hereinafter, all gray scale contour plots indicate maximum domain values by the maximum labeled contour. At t * ⫽40 the
flame is nonsymmetric about the centerline due to the density
and property stratification across the mixing layer. At this
time, the first pairings of the eight initial vortices have commenced. By time t * ⫽80 another set of pairings is occurring
leaving two distinct spanwise vortices. The braid regions are
highly strained, thereby greatly enhancing both thermal and
mass diffusion of the hot product species into the surrounding cooler free steam fluid 共temperature maxima are within
the vortices兲. However, the reaction is sufficiently strong at
this Zeldovich number to prevent local flame extinction in
the braids. In contrast, the evolution of the temperature contours is substantially altered at the higher Zeldovich number
as observed in Fig. 6 for case SP-HIGH. In this case, the
reaction rate is insufficient at lower temperatures to overcome the diffusional losses within the braid regions of the
mixing layer. Clear evidence of local flame extinction is seen
FIG. 7. Temperature contours for simulation SLD-LOWb; 共a兲 t * ⫽80, and
共b兲 t * ⫽160.
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Phys. Fluids, Vol. 13, No. 11, November 2001
Effects of nonreacting solid particle
3313
FIG. 8. Contours for simulation SLD-HIGHb; 共a兲 temperature at time t * ⫽80, 共b兲 temperature at time t * ⫽160, 共c兲 particle number density at time t *
⫽80, and 共d兲 particle number density at time t * ⫽160.
in the contours for times t * ⫽40 and 80. However, at later
times the burning vortices eventually merge and re-ignite the
extinguished braid fluid. The ‘‘thicker’’ nature of the final
time flame in Fig. 6共d兲 as compared to Fig. 5共d兲 will be
discussed below 共note that only a portion of the x 2 length of
the domain is shown in these contour plots; the actual domain size is ⫺33⭐x 2 / ␦ ␻ ,0⭐⫹33兲.
Analogous temperature contours for solid particle laden
flames are presented in Figs. 7 and 8, corresponding to simulations SLD-LOWb and SLD-HIGHb 共with M L 0 ⫽0.5 and
St0⫽2.0兲. The contours are depicted for times t * ⫽80 and
160. Contours of the particle number density 共n兲 are also
provided for simulation SLD-HIGHb 关Figs. 8共c兲 and 8共d兲兴.
As noted in previous investigations of two-way coupled solid
particle17 and liquid droplet45,46 laden mixing layers, the
presence of the dispersed phase suppresses the development
of vortical structures through an effective dissipation of kinetic energy. A comparison of the vortices depicted in Figs. 7
and 8 with the single-phase results of Figs. 5 and 6 shows a
qualitative consistency with this effect. Note that the more
‘‘jagged’’ nature of the contours in Figs. 7 and 8 is a physical
effect associated with the spatial distribution of particles and
the resulting coupling source terms in the Eulerian governing
equations.45 The development of the two-phase flames is also
qualitatively similar to the single-phase trends discussed
above. However, for these flows the solid particles tend to
preferentially concentrate in the braid regions 关Figs. 8共c兲 and
8共d兲兴. Particles entrained into the mixing layer from the free
stream are therefore drawn into the braids with lower temperature than the local gas. The primary effect of this process
is a further reduction to the flame temperature in the braids,
and therefore, an additional tendency towards local extinction. This is evident for both Zeldovich numbers under consideration. For the lower activation energy 共Fig. 7兲, no complete extinction in the braids is observed; however, the braid
flame remains substantially thinner and less developed than
the corresponding single-phase flow 共Fig. 5兲. This effect is
magnified for larger activation energy 共Fig. 8兲. In this case,
local flame extinction in the braids is clearly evident at time
t * ⫽80. Even at the final time, t * ⫽160 关Figs. 8共b兲 and 8共d兲兴,
the presence of particles results in a very thin flame structure
within the braids that was not observed for the single-phase
flame. The extent of this local flame suppression and/or extinction will be quantified below.
The last flow configuration for consideration in this section is the liquid droplet laden mixing layer. Contours of
temperature, droplet number density and evaporated vapor
mass fraction are provided in Fig. 9 for simulation EVAPHIGHb 共with M L 0 ⫽0.5 and St0⫽2.0兲 at times t * ⫽120 and
160. Differences in the behavior of the evaporating liquid
laden flow are clearly evident; particularly in regards to extinction. At time t * ⫽120 only two distinct ‘‘flame balls’’ are
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3314
Phys. Fluids, Vol. 13, No. 11, November 2001
Richard S. Miller
FIG. 9. Contours for simulation EVAP-HIGHb; 共a兲 temperature at time t * ⫽120, 共b兲 temperature at time t * ⫽160, 共c兲 droplet number density
at time t * ⫽120, 共d兲 droplet number density at time t * ⫽160, 共e兲 evaporated vapor mass fraction at time t * ⫽120, and 共f兲 evaporated vapor mass fraction
at time t * ⫽160.
found, with a large extent of cold fluid between them 关Fig.
9共a兲兴. At this Zeldovich number (Ze⫽15.35), the reaction
rate is too slow to overcome the cooling of the gas phase
caused by droplets preferentially concentrating into the
braids of the mixing layer 关Figs. 9共c兲 and 9共d兲兴. Furthermore,
any droplets which cannot escape the burning vortices are
completely evaporated in a relatively short time 关see Fig.
2共b兲兴. This results in the distinctly droplet devoid regions in
the number density contours. Once the droplets in these vortex structures are evaporated, the flame development within
these regions continues unimpeded by droplets. In addition,
the earlier presence of droplets has already allowed a significant extent of diffusional mixing of reactants; therefore, the
later developing flames burn rather robustly in the vortices.
For example, the peak temperature at time t * ⫽160 is
T MAX⫽1243 K 关Fig. 9共b兲兴 which is higher than the peak
laminar flame temperature of T LAMINAR⬇900 K 关Fig. 3共a兲兴.
Despite local extinction and preferential concentration in the
braids, vortex pairing provides another route by which droplets are entrained into the burning vortices. Consider the final
time 关Fig. 9共b兲兴, during which a thin but well defined layer of
extinguished fluid resides between the two primary vortices.
As with earlier vortex pairings 共not shown兲, the two distinct
combustion zones will eventually merge into a single vortex
of burning reactants, rapidly vaporizing any droplets which
become engulfed within the combustion zone.
An interesting feature of the evaporated vapor mass fraction contours in Figs. 9共e兲 and 9共f兲 is that the vapor resides in
the vortices, and not in the braid regions where preferential
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Phys. Fluids, Vol. 13, No. 11, November 2001
FIG. 10. Temporal evolution of 共a兲 the maximum temperature, and 共b兲 the
extinction factor.
concentration accumulates droplets. This is in contrast to the
droplet laden 共nonreacting兲 mixing layer studied by Miller
and Bellan,45,46 who observed a much more uniform distribution of vapor, with maximal concentrations within the
braids. This is explained as follows. At very early times
droplets residing within the initial preheat zone begin to
evaporate, resulting in a thin layer of vapor mass fraction
across the entire streamwise length of the layer. As the spanwise vortex rollup commences, the droplets accumulate
in the braids and extinguish the flame due primarily to latent
heat effects. As the braid fluid is cooled to near ambient
temperatures, the vapor concentration gradient is enhanced
by the straining of the braid structures, and the remaining vapor diffuses into the surrounding fluid. In contrast,
droplets residing within the newly formed vortices rapidly
evaporate, and more easily avoid being preferentially
concentrated to the braids as they become smaller and more
closely follow fluid motions. In this manner, the vapor frac-
Effects of nonreacting solid particle
3315
FIG. 11. Temporal evolution of 共a兲 the momentum thickness, and 共b兲 the
product thickness.
tion inside the vortices is increased early in the simulation.
The vapor concentration gradients within the vortices are
also smaller than those found in the braids; resulting in a
reduced rate of diffusional mixing of the vapor into the surrounding fluid.
D. Quantitative mixing layer analysis
In the previous section, qualitative flow visualizations
were used to illustrate several differences between the behavior of the ‘‘turbulent’’ 2D mixing layer and laminar quasi-1D
flames. In particular, the phenomena of local flame extinction
and its relation to the preferential concentration of particles
or droplets in high-strain regions was discussed. The analysis
is now turned towards a quantitative investigation of the effects of particles or droplets on the reacting mixing layer. In
order to aid in measuring the extent of local extinction during the following analysis, we introduce a new parameter,
␣ F ( ␹ ), denoted the ‘‘extinction factor:’’
⫹L /2 L
␣ F共 ␹ 兲 ⫽
兰 ⫺L 2 /2兰 0 1 H 共 ␹ ⬘ 共 x兲 ⫺ ␹ 兲 H 共 Y O Y F 共 x兲 ⫺5.0⫻10⫺5 兲 dx 1 dx 2
2
⫹L /2 L
兰 ⫺L 2 /2兰 0 1 H 共 Y O Y F 共 x兲 ⫺5.0⫻10⫺5 兲 dx 1 dx 2
,
共25兲
2
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3316
Phys. Fluids, Vol. 13, No. 11, November 2001
FIG. 12. Temporal evolution of 共a兲 the momentum thickness, and 共b兲 the
product thickness.
where H is the Heaviside function, ␹ ⫽(T⫺T 0 )/(T F ⫺T 0 ) is
the relative gas temperature between the ambient and the
adiabatic flame temperature, and the product Y O Y F indicates
the potential for chemical reactions to occur 关see Eqs. 共7兲–
共9兲兴. Therefore, ␣ F ( ␹ ) is the relative fraction of the domain
with ‘‘premixed’’ reactants having a nondimensional temperature greater than ␹. The extinction factor is bounded by
0⭐ ␣ F ( ␹ )⭐1, where ␣ F ⫽0 indicates that no portion of the
domain with mixed reactants is hotter than ␹, and ␣ F ⫽1
indicates that the entire portion of the domain with mixed
reactants is hotter than ␹. By choosing an appropriate value
for ␹ within the range of essentially nonreacting temperatures 关such as ␹ ⫽0.1 (T⫽390 K); see Fig. 2共a兲兴, the fraction
of extinguished reactants is approximately determined. In
what follows, the extinction factor, as well as other statistical
parameters, is used to quantify the effects of the Zeldovich
number, mass loading ratio, and the initial Stokes number on
the development of the reacting mixing layer.
1. Effect of the Zeldovich number
The effects of the Zeldovich number on the reacting
mixing layer are examined for three flow configurations: 共1兲
Single-phase flow 共cases SP-LOW and SP-HIGH兲, 共2兲 solid
particle loading with M L 0 ⫽0.25 共cases SLD-LOWa and
SLD-HIGHa兲, and 共3兲 solid particle loading with M L 0 ⫽0.5
共cases SLD-LOWb and SLD-HIGHb兲. Note that evaporating
droplets are not included in this stage of the discussion in
order to simplify the analysis. The temporal development of
Richard S. Miller
FIG. 13. Temporal evolution of 共a兲 the maximum temperature, and 共b兲 the
extinction factor.
the maximum temperature and the extinction factor ␣ F ( ␹
⫽0.1) are presented in Fig. 10 for all three sets of the simulations. At early times, and relatively low temperatures, the
peak temperature is observed to initially decrease for approximately the first 20 eddy turnover times for the HIGH
Zeldovich number cases 共dashed curves兲. In contrast, the
LOW Zeldovich number cases have a substantially larger
reaction rate at the initial preheat temperature 关see Fig. 2共a兲兴,
and the maximum temperature in these flows begins to increase much sooner. However, at long times the trends are
reversed and the largest flame temperatures are found in
HIGH Zeldovich number cases. This occurs because the initial delay in ignition for HIGH Ze mixing layers allows time
for reactants to become relatively well mixed inside of the
vortices. Once ignition does finally occur, the concentrations
of mixed reactants are larger than can occur in the LOW
cases, and a rapid and robust ignition at high-temperatures
follows. Nevertheless, the extinction factors presented in Fig.
10共b兲 indicate that a significantly smaller portion of the
mixed reactants are actually burning for HIGH Ze flows due
to enhanced extinction in the braid regions. For the HIGH
flows, only approximately 50% of the regions containing
mixed reactants are sufficiently hot to burn robustly, even at
the final simulation time.
This same delay in ignition for larger Zeldovich number
flows causes an enhancement in the long time ‘‘thickness’’ of
the mixing layer which was observed previously in the flow
visualizations 关see Figs. 5共d兲 and 6共d兲兴. Figure 11 quantifies
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Phys. Fluids, Vol. 13, No. 11, November 2001
Effects of nonreacting solid particle
3317
FIG. 14. Particle and droplet number density contours at time t * ⫽160 for simulations; 共a兲 SLD-HIGHc, 共b兲 EVAP-HIGHc, 共c兲 SLD-HIGHd, and 共d兲
EVAP-HIGHd.
this effect through the depiction of the momentum thickness
and product thickness evolutions for the same simulations.
For both single-phase and solid particle laden flows, the
HIGH Ze cases exhibit a larger momentum thickness and
product thickness at long times than the corresponding LOW
cases. This feature can be attributed to the delay for ignition
in the higher Zeldovich number flames. During early times,
these flames behave more like nonreacting mixing layers in
comparison to the lower Ze flows. McMurtry et al.9 conducted one of the first numerical studies of exothermic reacting mixing layers. Their results show that chemical heat release slows the development of large-scale structures and
reduces the entrainment of reactants into the layer. Since the
lower Zeldovich number flames are characterized by larger
early heat release, these flames also exhibit more reduced
growth and entrainment rates than the higher Ze flames. In
contrast, ignition is delayed for high Zeldovich numbers,
thereby allowing a faster early growth of the layer with significant mixing of reactants prior to ignition. This premixing
of reactants allows for a more ‘‘explosive’’ growth of the
product thickness once ignition finally occurs. A corresponding rapid increase in the momentum thickness also occurs
corresponding to the delayed ignition due to thermal expansion effects.58
2. Effect of the mass loading ratio
The results presented in Figs. 10 and 11 also elucidate
the effect of the solid particle mass loading ratio for values
M L 0 ⫽0, 0.25, and 0.5 at both Zeldovich numbers. In agreement with the previous nonreacting mixing layer results of
Miller and Bellan,45,46 increasing liquid mass loading ratios
result in reduced development of the momentum thickness of
the pre-transitional layer 共note that Miller and Bellan46 observed a reverse in this trend for post-transitional flow; however, no similar conclusions can be drawn from the present
2D stratified and reacting flow results兲. Meiburg et al.40 observed a similar phenomena for the 2D solid particle laden
mixing layer which they attributed to a reduction in the
transport of vorticity from the braids to the vortex cores.
Furthermore, the present results also show that both the
maximum temperature and the extinction factor decrease
with the addition of particles due to their finite thermal inertia and corresponding tendency to cool the surrounding fluid.
Further studies of transitional mixing layers are required to
determine the ultimate fate of the growth rate and peak temperature statistics in fully turbulent flows.
The effects of the mass loading ratio are further investigated in Figs. 12 and 13 for both solid particles and evapo-
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3318
Phys. Fluids, Vol. 13, No. 11, November 2001
FIG. 15. Temporal evolution of 共a兲 the momentum thickness, and 共b兲 the
product thickness.
rating droplets. In both cases, mass loadings of M L 0 ⫽0,
0.25, and 0.5 are considered for a fixed Zeldovich number
Ze⫽15.35 共HIGH兲. Evaporating droplets are also observed
to reduce the momentum thickness and product thickness of
the layer with increased liquid loading. In addition, the
evaporating droplets substantially amplify the effects on the
mixing layer in comparison to solid particles. For example,
Fig. 13共a兲 reveals that the time required for the peak temperature to begin to grow 共ignition兲 is nearly twice as long
for evaporating droplet laden flows due to their additional
latent heat effects, as well as to the dilution of reactant concentrations by evaporated vapor. In addition, the extinction
factor is approximately halved for the droplets 关Fig. 13共b兲兴.
As noted previously, this is due to the droplets concentrating
in the braid regions of the mixing layer and locally extinguishing the flame.
3. Effect of the Stokes number
The last subject for investigation is the effect of the initial particle or droplet size, quantified in this case by the
initial Stokes number, St0. To this point, only particles with
‘‘moderate’’ Stokes numbers St0⫽2.0 have been discussed;
however, the simulations described in Table II also address
‘‘small’’ particles with St0⫽0.25, and ‘‘large’’ particles with
St0⫽12.0. These are compared at fixed mass loading ratio
(M L 0 ⫽0.25) and Zeldovich number 共Ze⫽15.35; HIGH兲.
Contours of the final time (t * ⫽160) number density distri-
Richard S. Miller
FIG. 16. Temporal evolution of 共a兲 the maximum temperature, and 共b兲 the
extinction factor.
butions are presented for both the small and large particles
and droplets in Fig. 14. As discussed in the introduction, it is
well known that very small particles are able to closely follow fluid motions, and therefore, display relatively little preferential concentration. This is clearly evident in Fig. 14共a兲
which confirms that solid particles are indeed well distributed throughout both the vortex cores and the braid regions
of the reacting mixing layer. On the other hand, the number
density distribution for St0⫽0.25 evaporating droplets 关Fig.
14共b兲兴 is qualitatively very different. In this case, the droplets
appear to exhibit very little preferential concentration in
braid regions. However, the vortex regions are nearly perfectly devoid of droplets. Whereas the small droplets do indeed closely follow the fluid motions, the life time of small
droplets in heated gas is very small compared to the characteristic flow time scales 关see Fig. 2共b兲兴. Therefore, any droplets following fluid motions into the flame regions are rapidly
vaporized to completion and no longer contribute to the
number density contours.
In contrast to small particles, very large particles are also
known to display reduced preferential concentration due to a
reduced response to fluid motions. Figures 14共c兲 and 14共d兲
present the corresponding number density contours for cases
SLD-HIGHd and EVAP-HIGHd which have St0⫽12. Although some reduction in the preferential concentration is
observed in Fig. 14共c兲 for solid particles, the Stokes number
is not large enough to show a dramatic decrease in comparison to St0⫽2.0 共simulation of larger St0 is not feasible at this
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Phys. Fluids, Vol. 13, No. 11, November 2001
mass loading ratio since sufficiently large numbers of particles must be included to maintain relatively smooth phase
coupling terms兲. Also, in contrast with the smaller evaporating droplets, some of the larger droplets are found inside of
the combustion region in Fig. 14共d兲. Although these droplets
do evaporate relatively quickly in the heated gas, their life
times are much longer than for small droplets 关Fig. 2共b兲兴.
Quantitative comparisons of the effect of the particle or
droplet size are provided in Figs. 15 and 16. These figures
show the momentum thickness, product thickness, maximum
temperature, and extinction factor for both particles and
droplets with St0⫽0.25, 2.0, and 12.0; all at fixed M L 0
⫽0.25 and Ze⫽15.35. In agreement with previous nonreacting two-phase mixing layer results,45 changes to the initial
Stokes number 共at fixed mass loading兲 do not significantly
alter the growth rate of the layer as measured by the momentum thickness. However, the present results show that
changes in the preferential concentration as a function of the
Stokes number do result in a markedly increased product
thickness for the largest particles and droplets 关Fig. 15共b兲兴.
Since these particles do not closely follow fluid motions,
fluid is constantly being swept past the droplets, and therefore, has less exposure time to be cooled by direct contact
with particles or by liquid droplet latent heat effects. This
allows for larger maximum temperatures 关Fig. 16共a兲兴 and a
substantially reduced extent of local flame extinction 关Fig.
16共b兲兴 for large particles or droplets. For the liquid laden
flames this effect is further augmented by the relatively
longer life times of large droplets 关Fig. 2共b兲兴.
IV. CONCLUSIONS
Numerical simulations have been conducted of both
solid particle and liquid droplet laden diffusion flames. The
simulations were conducted for both quasi-1D laminar
flames and for forced 2D mixing layers. A single-step irreversible and exothermic reaction of the form Fuel
⫹r Oxidizer→共1⫹r兲Products was considered. The particles
共or droplets兲 were assumed to be spherical, smaller than the
gas phase length scales, and to obey a modified Stokes drag
law. Droplet vaporization was assumed to be governed by the
classical rapid mixing model in a form appropriate for calorically perfect species. Neither the particles or the droplets
contribute to the reaction; droplets evaporate to a nonreacting vapor species different from either the fuel or oxidizer
species. The primary parameters of interest to the study were
the reaction Zeldovich number, the initial particle Stokes
number and the dispersed phase mass loading ratio.
The simulation results yielded both a qualitative and
quantitative description of the two-phase reacting mixing
layer. Laminar diffusion flames were observed to evolve to
long time distributions which are essentially independent of
the Zeldovich number, as well as the particle or droplet mass
loading. Solid particles reach an equilibrium temperature
with the flame at these times, whereas droplets are completely evaporated within the laminar flame region. In contrast, for 2D mixing layers the rollup of spanwise vortices
creates high-strain braid regions in which local extinction of
the reaction can occur for sufficiently large Zeldovich num-
Effects of nonreacting solid particle
3319
bers. Local flame extinction can also be significantly enhanced by the presence of either solid particles or evaporating droplets. Solid particles are entrained into the mixing
layer from the relatively cold free streams. Their finite thermal inertia, therefore, has the effect of further cooling the
braid regions into which they are preferentially concentrated
for moderate Stokes numbers. Evaporating droplets exhibit a
more enhanced potential for local flame quenching due to the
latent heat of vaporization, as well as through the diluting
effects of the added nonreacting gaseous vapor species. In
general, local flame extinction is maximized for large activation energies, large mass loading ratios, and by small particle
or droplet sizes. These observations have direct consequences for practical flame suppressant delivery; suggesting
that atomization of a large liquid mass to very small droplet
sizes will optimize both local and global flame extinction
through a more uniform delivery to both braids and vortices.
ACKNOWLEDGMENTS
This research was supported by the National Science
Foundation through the Faculty Early Career Development
Program; Grant No. CTS-9983762. Computational support
was provided by the National Computational Science Alliance 共NCSA兲 under Grant No. CTS990040N and utilized the
NCSA SGI/Cray Origin2000. Additional computational support was provided by the California Institute of Technology’s
Center for Advanced Computing Research 共CACR兲 utilizing
the Hewlett-Packard V2500, and by Clemson University’s
Division of Computing and Information Technology utilizing
a SUN HPC 6000.
1
C. T. Crowe, T. R. Troutt, and J. N. Chung, ‘‘Numerical models for twophase turbulent flows,’’ Annu. Rev. Fluid Mech. 28, 11 共1996兲.
2
G. L. Brown and A. Roshko, ‘‘On density effects and large scale structure
in turbulent mixing layers,’’ J. Fluid Mech. 64, 775 共1974兲.
3
C. D. Winant and F. K. Browand, ‘‘Vortex pairing: The mechanisms of
turbulent mixing layer growth at moderate Reynolds number,’’ J. Fluid
Mech. 63, 237 共1974兲.
4
D. S. Shin and J. H. Ferziger, ‘‘Linear stability of the reacting mixing
layer,’’ AIAA J. 29, 1634 共1991兲.
5
M. F. Miller, T. C. Island, B. Yip, C. T. Bowman, M. G. Mungal, and R. K.
Hanson, ‘‘An experimental study of the structure of a compressible, reacting mixing layer,’’ AIAA Pap. 93-0354 共1993兲.
6
C. M. Coats and A. P. Richardson, ‘‘Nonpremixed combustion in turbulent
mixing layers Part 1: Flame characteristics,’’ Combust. Flame 122, 253
共2000兲.
7
C. M. Coats, A. P. Richardson, and S. Wang, ‘‘Nonpremixed combustion
in turbulent mixing layers Part 2: Mixing and flame stabilization,’’ Combust. Flame 122, 271 共2000兲.
8
J. J. Riley, R. W. Metcalfe, and S. A. Orszag, ‘‘Direct numerical simulations of chemical reacting turbulent mixing layers,’’ Phys. Fluids 29, 406
共1986兲.
9
P. A. McMurtry, J. J. Riley, and R. W. Metcalfe, ‘‘Effects of heat release on
the large-scale structure in turbulent mixing layers,’’ J. Fluid Mech. 199,
297 共1989兲.
10
K. D. Squires and J. K. Eaton, ‘‘Preferential concentration of particles by
turbulence,’’ Phys. Fluids A 3, 1169 共1991兲.
11
J. K. Eaton and J. R. Fessler, ‘‘Preferential concentration of particles by
turbulence,’’ Int. J. Multiphase Flow 20, 169 共1994兲.
12
B. J. Lazaro and J. C. Lasheras, ‘‘Particle dispersion in a turbulent, plane,
free shear layer,’’ Phys. Fluids A 1, 1035 共1989兲.
13
M. Samimy and S. K. Lele, ‘‘Motion of particles with inertia in a compressible free shear layer,’’ Phys. Fluids A 3, 1915 共1991兲.
14
F. Wen, N. Kamalu, J. N. Chung, C. T. Crowe, and T. R. Troutt, ‘‘Particle
Downloaded 02 Nov 2001 to 130.127.198.18. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
3320
Phys. Fluids, Vol. 13, No. 11, November 2001
dispersion by vortex structures in plane mixing layers,’’ J. Fluids Eng. 114,
657 共1992兲.
15
J. E. Martin and E. Meiburg, ‘‘The accumulation and dispersion of heavy
particles in forced two-dimensional mixing layers. I. The fundamental and
subharmonic cases,’’ Phys. Fluids 6, 1116 共1994兲.
16
N. Raju and E. Meiburg, ‘‘The accumulation and dispersion of heavy
particles in forced two-dimensional mixing layers. Part 2: The effect of
gravity,’’ Phys. Fluids 7, 1241 共1995兲.
17
K. T. Kiger and J. C. Lasheras, ‘‘Dissipation due to particle/turbulence
interaction in a two-phase, turbulent, shear layer,’’ Phys. Fluids 9, 3005
共1997兲.
18
W. Ling, J. N. Chung, T. R. Troutt, and C. T. Crowe, ‘‘Direct numerical
simulation of a three-dimensional temporal mixing layer with particle dispersion,’’ J. Fluid Mech. 358, 61 共1998兲.
19
Q. Wang, K. D. Squires, and L. P. Wang, ‘‘On the effects of nonuniform
seeding on particle dispersion in two-dimensional mixing layers,’’ Phys.
Fluids 10, 1700 共1998兲.
20
M. C. Soteriou and X. Yang, ‘‘Particle dispersion in variable density and
viscosity shear flows,’’ Phys. Fluids 11, 1373 共1999兲.
21
R. A. Gore and C. T. Crowe, ‘‘Effect of particle size on modulating turbulent intensity,’’ Int. J. Multiphase Flow 15, 279 共1989兲.
22
S. Elghobashi, ‘‘On predicting particle-laden turbulent flows,’’ Appl. Sci.
Res. 52, 309 共1994兲.
23
V. M. Kenning and C. T. Crowe, ‘‘On the effect of particles on carrier
phase turbulence in gas-particle flows,’’ Int. J. Multiphase Flow 23, 403
共1997兲.
24
J. J. Riley and G. S. Patterson, ‘‘Diffusion experiments with numerically
integrated isotropic turbulence,’’ Phys. Fluids 17, 292 共1974兲.
25
K. D. Squires and J. K. Eaton, ‘‘Measurements of particle dispersion obtained from direct numerical simulations of isotropic turbulence,’’ J. Fluid
Mech. 226, 1 共1991兲.
26
S. Elghobashi and G. C. Truesdell, ‘‘Direct simulation of particle dispersion in decaying isotropic turbulence,’’ J. Fluid Mech. 242, 655 共1992兲.
27
L. P. Wang and M. R. Maxey, ‘‘Settling velocity and concentration distribution of heavy particles in isotropic turbulence,’’ J. Fluid Mech. 256, 27
共1993兲.
28
M. Boivin, O. Simonin, and K. D. Squires, ‘‘Direct numerical simulation
of turbulence modulation by particles in isotropic turbulence,’’ J. Fluid
Mech. 375, 235 共1998兲.
29
F. Wen, N. Kamalu, J. N. Chung, and C. T. Crowe, ‘‘Particle dispersion by
vortex structures in plane mixing layers,’’ ASME J. Fluids Eng. 114, 657
共1992兲.
30
S. Pedinotti, G. Mariotti, and S. Banerjee, ‘‘Direct numerical simulation of
particle behavior in the wall region of turbulent flows in horizontal channels,’’ Int. J. Multiphase Flow 18, 927 共1992兲.
31
H. Ounis, G. Ahmadi, and J. B. McLaughlin, ‘‘Brownian particle deposition in a directly simulated turbulent channel flow,’’ Phys. Fluids A 5, 1427
共1993兲.
32
M. Chen and J. B. McLaughlin, ‘‘A new correlation for the aerosol deposition rate in vertical ducts,’’ Colloid and Interface Sci. 169, 437 共1995兲.
33
D. W. I. Rouson, J. K. Eaton, and S. D. Abrahamson, ‘‘A direct numerical
simulation of a particle-laden turbulent channel flow,’’ Technical Report,
Department of Mechanical Engineering Report No. TSD-101, Stanford
University, Stanford, CA, 1997.
34
K. D. Squires and J. K. Eaton, ‘‘Particle response and turbulence modification in isotropic turbulence,’’ Phys. Fluids A 2, 1191 共1990兲.
35
S. Elghobashi and G. C. Truesdell, ‘‘On the two-way interaction between
homogeneous turbulence and dispersed solid particles. I: Turbulence
modification,’’ Phys. Fluids A 5, 1790 共1993兲.
36
K. D. Squires and J. K. Eaton, ‘‘Effect of selective modification of turbulence on two-equation models for particle-laden turbulent flows,’’ J. Fluids
Eng. 116, 778 共1994兲.
Richard S. Miller
37
G. C. Truesdell and S. Elghobashi, ‘‘On the two-way interaction between
homogeneous turbulence and dispersed solid particles. II: Particle dispersion,’’ Phys. Fluids 6, 1405 共1994兲.
38
M. R. Maxey, B. K. Patel, E. J. Chang, and L. P. Wang, ‘‘Simulations of
dispersed turbulent multiphase flow,’’ Fluid Dyn. Res. 20, 143 共1997兲.
39
S. Sundaram and L. R. Collins, ‘‘A numerical study of the modulation of
isotropic turbulence by suspended particles,’’ J. Fluid Mech. 379, 105
共1999兲.
40
E. Meiburg, E. Wallner, A. Pagella, A. Riaz, C. Hartel, and F. Necker,
‘‘Vorticity dynamics of dilute two-way coupled particle laden mixing layers,’’ J. Fluid Mech. 421, 185 共2000兲.
41
F. Mashayek, ‘‘Droplet-turbulence interactions in low Mach number homogeneous shear two-phase flows,’’ J. Fluid Mech. 367, 163 共1998兲.
42
F. Mashayek, ‘‘Direct numerical simulations of evaporating droplet dispersion in forced low Mach number turbulence,’’ Int. J. Heat Mass Transf. 41,
2601 共1998兲.
43
F. A. Jaberi and F. Mashayek, ‘‘Temperature decay in two-phase turbulent
flows,’’ Int. J. Heat Mass Transf. 43, 993 共2000兲.
44
R. S. Miller and J. Bellan, ‘‘On the validity of the assumed pdf method for
modeling binary mixing/reaction of evaporated vapor in gas/liquid-droplet
turbulent shear flow,’’ Proceedings of the 27th Symposium 共International兲
on Combustion 共1998兲, pp. 1065–1072.
45
R. S. Miller and J. Bellan, ‘‘Direct numerical simulation of a confined
three-dimensional gas mixing layer with one evaporating hydrocarbondroplet laden stream,’’ J. Fluid Mech. 384, 293 共1999兲.
46
R. S. Miller and J. Bellan, ‘‘Direct numerical simulation and subgrid
analysis of a transitional droplet laden mixing layer,’’ Phys. Fluids 12, 650
共2000兲.
47
F. Mashayek, ‘‘Numerical investigation of reacting droplets in homogeneous shear turbulence,’’ J. Fluid Mech. 405, 1 共2000兲.
48
D. J. Glaze and S. H. Frankel, ‘‘Effect of dispersion characteristics on
particle temperature in an idealized nonpremixed reacting jet,’’ Int. J. Multiphase Flow 26, 609 共2000兲.
49
R. S. Miller, K. G. Harstad, and J. Bellan, ‘‘Direct numerical simulation of
supercritical fluid mixing layers applied to heptane-nitrogen,’’ J. Fluid
Mech. 436, 1 共2001兲.
50
R. S. Miller, K. Harstad, and J. Bellan, ‘‘Evaluation of equilibrium and
non-equilibrium evaporation models for many-droplet gas-liquid flow
simulations,’’ Int. J. Multiphase Flow 24, 1025 共1998兲.
51
D. Papamoschou and A. Roshko, ‘‘The compressible turbulent mixing
layer: An experimental study,’’ J. Fluid Mech. 197, 453 共1988兲.
52
R. D. Moser and M. M. Rogers, ‘‘Mixing transition and the cascade to
small scales in a plane mixing layer,’’ Phys. Fluids A 3, 1128 共1991兲.
53
F. Kozusko, C. E. Grosch, T. L. Jackson, C. A. Kennedy, and T. B. Gatski,
‘‘The structure of variable property, compressible mixing layers in binary
gas mixtures,’’ Phys. Fluids 8, 1945 共1996兲.
54
F. Kozusko, D. G. Lasseigne, C. E. Grosch, and T. L. Jackson, ‘‘The
stability of compressible mixing layers in binary gases,’’ Phys. Fluids 8,
1954 共1996兲.
55
C. A. Kennedy and J. H. Chen, ‘‘Mean flow effects on the linear stability
of compressible planar jets,’’ Phys. Fluids 10, 615 共1998兲.
56
C. A. Kennedy and M. H. Carpenter, ‘‘Several new numerical methods for
compressible shear-layer simulations,’’ Appl. Numer. Math. 14, 397
共1994兲.
57
T. J. Poinsot and S. K. Lele, ‘‘Boundary conditions for direct numerical
simulations of compressible viscous flows,’’ J. Comput. Phys. 101, 104
共1992兲.
58
R. S. Miller, C. K. Madnia, and P. Givi, ‘‘Structure of a turbulent reacting
mixing layer,’’ Combust. Sci. Technol. 99, 1 共1994兲.
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