employ accurate differencing schemes. (second-order accurate); incorporate efficient solution algorithms; incorporate options for. Cartesian, cylindrical, and.
t; 7:> 7!:' ;t
CFD Research 3325
Triana
Blvd.
t
Huntsville,
,
Corporation Alabama
•
35805
Tel:
(205)
•
536-6576
FAX:
(205)
i_
CFD:IC
536-6590
u
COMBUSTION
CHAMBER Final
ANALYSIS
CODE
Report
M u
by
D
A.J.
Przekwas,
Y.G.
Lai,
A. Krishnan,
R.K.
Avva,
and
M.G.
Giridharan
_U
!U i
May
CFDRC
1993
i..,,....
for
National
Aeronautics
George
4
and
C. Marshall Alabama
Space
Administration
Space Flight 35812
Center
i]
_7 i
Contract
D
Number:
NAS8-37824
r_l
NASA
R&D
Services
CORs:
and
Software
G. Wilhold
for
Computational
and
Fluid
K. Tucker
Dynamics
(CFD)
Report:
4105/6
PREFACE
Under
the
CFD
NASA
code:
designed
MSFC
REFLEQS for
reactions.
flow
Particular and
three
REFLEQS-Validation
work.
EQuation
transfer
has
been
problems to
The
and
needs
rocket
results
an
advanced
REFLEQS
with the
in liquid
systems.
developed
Solver).
given
of combustion
has
without of
been
chemical
predicting
engines,
of this
has
flow
using
study
are
LOX/H2
described
in
Report;
3: REFLEQS
authors
CFDRC
volumes:
Final
The
heat
propellant
2:
final
and
efficiency
1:
The
FLow
attention
or LOX/Hydrocarbon following
NAS8-37824,
(REactive
turbulent
environment
the
contract
report
- User's
includes
wish
Applications
descriptions
thanks Kevin
Manual;
and
Manual.
of the
to acknowledge
In particular, Mr.
and
are
Tucker
and
mathematical
thank
due
to:
and
Dr. Paul
and
constructive
all
those
basis
who
of REFLEQS.
have
McConnaughey
contributed
of NASA
to this
MSFC/ED32
!
for
their
guidance
Robert
Williams,
MSFC/ED32, for
their
Mr.
and
feedback
Jack
Hyde
Robert Klaus
Gross,
on code of
suggestions;
Garcia, John
Joe
Ruf
Hutt
and
and Ho
Ted Trinh
Benjamin
of
of MSFC/EP55
performance;
Aerojet
Corporation
for
his
feedback
on
code
performance; Dr.
Ashok
overall Mr.
guidance Clifford
CFDRC Ms.
K. Singhal
for
Jennifer
(President
in code E. Smith,
testing
and
L. Swann
& Technical
development Milind
and
Director project
V. Talpallikar,
validating for preparing
various the
and versions
typescripts
of CFDRC)
for
work; Mark
L.
Ratcliff
of the
code;
of the
reports.
of
and
m i
4105/6
TABLE
°
OF CONTENTS
1
INTRODUCTION
1 1.1
Background
1.2
Project
Objectives
1.3
Salient
Features
1.3.1
Numerical
1.3.2
Physical
1.3.3
Code
Organization
1.3.4
Code
Testing
1
1.4
BASIC
°
PHYSICAL
Approach
of the
3
Code
3
Techniques
4
Outline
,
and
Models
of the
4 4 5
Report
GOVERNING
7
EQUATIONS
L n
3.1
w
3.2
u
3.3
9
MODELS
Turbulence
9
Models
3.1.1
Favre
Averaged
Navier-Stokes
3.1.2
Baldwin-Lomax
3.1.3
Standard
k-e Model
3.1.4
Extended
k-¢ Model
3.1.5
Multiple
3.1.6
Low
Combustion
Models
3.2.1
2-Step
H2-O2
3.2.2
2-Step
CH4
3.2.3
Chemical
13 14
Number
3.3.2
Top-Hat
Reaction
3.3.3
Mean
3.4
Spray
Model
3.5
Radiation
20
Model
22
Model
24
Interaction Scalar
Values
17
Model
- 02 Reaction
Pdf
16
Model
17
Turbulence-Combustion Conserved
15
Model
Equilibrium
3.3.1
9 11
Model
Time-Scale
Reynolds
Equations
25
Equations
25
Model of Thermodynamic
Variables
28 29
B
33
Model
ii
41(]816
TABLE
OF CONTENTS
(Continued)
Pa_a 3.5.1
Governing
GEOMETRY
,
AND
34
Equations
ITS
COMPUTATIONAL
41
REPRESENTATION
41 4.1
Geometry
and
4.2
Body-Fitted
4.3
Blockage
42
Coordinate
(BFC)
System
46
Concept
DISCRETIZATION
,
Grid
48
METHODS
48 F
5.1
Staggered
versus
5.2
A General
5.3
Transient
5.4
Convection
Colocated
Grid
Approach
Convection-Diffusion
49
Equation
51
=
m
| m
Term Term
and
5.4.1
First-Order
5.4.2
Central
5.4.3
Smart
Scheme
5.4.4
Other
High-Order
Different
Upwind Difference
Convection
52
Schemes
53
Scheme
53
Scheme with
Minmod
54
Limiter
55
Schemes
m
5.5.
Diffusion
5.6
Final
5.7
Discretization
56
Terms
57
VELOCITY
.
Linear
Equation
Set
of Mass
PRESSURE
Conservation
and
Mass
Flux
Evaluation
58
62
COL_LING
62
u
°
6.1
SIMPLE
Algorithm
6.2
SIMPLEC
6.3
PISO
64
Algorithm
65
Algorithm
BOUNDARY
67
CONDITIONS
7.1
Inlet
with
Specified
7.2
Exit
7.3
Symmetry
Condition
7.4
Solid
Boundary
7.5
Periodic
Mass
Flow
Rate
(FIXM)
68 68
Boundary 68 Wall
69 71
Boundary
iii
!
410516
TABLE
OF CONTENTS
(Continued)
73 SOLUTION
.
8.1
Overall
8.2
Linear
METHOD
73
Solution
Process
Equation
Solvers
75 8.2.1
Whole
Field
8.2.2
Conjugate
75
Solver
76
.
10.
11.
Gradient
Squared
78
CONCLUSIONS
79
RECOMMENDATIONS
79 10.1
Physical
Model
10.2
Numerical
10.3
Code
Enhancement
Algorithm
Validation
and
80
Improvements Applications
81
82
REFERENCES
iv
m
Solver
410516
LIST OF ILLUSTRATIONS
Figure
1-1.
Figure
3-1.
Figure
4-1.
REFLEQS
Sketch
Code
of the
A Sample
Organization
Control
Volume
38
Two-Dimensional
and
without
An
Illustration
Flow
with
Two
Possible
Grids
with
Blockages
42
I u Figure
4-2.
of the
Transformation
from
Physical
to
Computational
Figure
4-3.
Geometrical
44
Meaning
of Covariant
and
Contravariant
Bases
for
a 2D BFC System
=
Figure
4-4.
Figure
5-1.
46
Illustration
of the
Use
Illustration
of Variable
of Blocked-Off
Regions
47
=
Figure
5-2.
Figure
5-3.
Figure
7-1.
Figure
8-1.
(b) Colocated
Grid
The
Labeling
Scheme
An
Illustration
Grid
Notation
Grid
Node
Locations
for:
(a)
Staggered;
and 49
of the
for
Storage
of a Control
Cell
Periodic
Nomenclature
Face
Volume
Value
Boundary
for
the
V
Evaluation
Conditions
WFS
51
Discussion
53
in the
Z-Direction
72
75
4105/6
1.
INTRODUCTION
1.1
klta;kgr md
Until
recently,
L m
liquid
characteristics
and
prediction by
Dynamics
in the
report
NASA
MSFC,
flows
in liquid
existing
tool
rocket
objective in
A
methodology to
the
were
3D planned
Three
and
of this
and
major
and
approach
would
be
first
Code
activities
supporting
STME
1986).
mid
design
Flight
Center
were Fluid
due
to several
and
(1989-1992),
point
new
annually
Workshops".
predicting
starting
1980's,
Computational
Dynamics
for
of
equations
engine
effort
were code,
engine
to develop
a 3D
flow
thrust
sponsored
two-phase, for
this
CFD
code
environment
chambers.
transient
by
reactive
project
was
the
The
analysis
that in
code
could LOX/H2
should
of
single-phase
in
which
be used and
be capable and
of
two-phase
grids.
was
to
be
implemented
testing,
to be performed
REFLEQS
(1984,
code
predicting
non-orthogonal
generation
Navier-Stokes
three-year
The
was
propellant
block
code.
of
Fluid
CFD
the
methods
Approach
project
steady-state
building
advanced
using
of CFDRC.
liquid
on
the
In
Space
"Computational
of
designed
rocket
Marshall
chambers.
designing
flows
et al
in practical
results
thrust
code
LOX/Hydrocarbon
reactive
NASA
an
Objectives
performing
by
of
the
solutions
Przekwas
way
were
approximations.
on
its
Proceedings
REFLEQS
overall a
found
and
to develop
Project
as
layer
(1986),
presents
chambers
based
coordinated
reported
The
boundary
Liang
(CFD)
initiatives
1.2
thrust
methodologies,
proposed
This
rocket
used in the
2D
code,
verification/validation
the tested
and
most and
advanced,
then
documentation
extended efforts
concurrently.
planned code
tests,
for
this
and
project:
validation,
development and
computational
of
the
new
studies
development.
1
4105/6
The
starting
order
version
numerical
conservation code
of the schemes,
equations
modifications
(see
for
a.
employ
b.
incorporate
C.
incorporate
this
utilized
staggered
SIMPLEC
Yang
1992).
et al,
project
included
efficient
(BFC)
grid
arrangement,
algorithm
and
weak
planned
direction
first form
and
of
selected
following:
schemes
solution
options
The
the
differencing
incorporate
d°
code
interactive
accurate
Coordinate
I
REFLEQS
(second-order
accurate);
algorithms;
for
Cartesian,
cylindrical,
and
Body-Fitted
systems;
advanced
turbulence
models
with
boundary
layer
treatment; develop
e.
Eulerian-Lagrangian
accounting
for
incorporate
fo
spray
turbulence
dynamics
modulation
available
and
effects
finite-rate
for
combustion
reduced
models,
evaporating
systems;
mechanism
combustion
models; incorporate
1 .
include
h.
turbulence-combustion thermal
radiation
interaction
models;
and
model.
m
Early
in
the
methodology type
by
numerical
conservation
Several !
it
new
schemes,
PISO
algorithm
fuel
rich
conditions,
for
As
a result,
Parallel
to
the
code
flows
and
manifold
PISO
to
this
compressible
were
As
code
study,
STME
arrangement,
solution of Cartesian
velocity
project
not
had
development are
continuous
problems
during
nozzle
cooling,
2
vehicle
and
strong
been
form
thoroughly
reduced
of
of new,
tested,
chemistry
development,
testing,
robust
fluid
for and
dynamic
reprogrammed.
support the
TVD-
components.
4-step
completely
active,
numerical
higher-order
procedure,
fundamental
subroutines
transfer
state-of-the-art
Hautmann
result,
to the
development, heat
flows, a
directed
most
incorporate grid
in terms
prior
etc.).
efforts
to
non-staggered
expressed
solvers.
e.g., NLS
decided
non-iterative
methodologies
(e.g.
analyzing
was
implementing
equations
validation
1
project,
STME base
was
provided
development
heating,
for effort,
etc.
410516
1.3
Salient
This for
section
the
turbulent
codes
of two
written
pre-processor
without
modifying
version
of
user
to
and
exits,
the
the
code.
simulate
physical
with
models
are
heat
fully
of the
REFLEQS
transfer
problems
compatible
codes:
code,
specifically
with
and
designed
without
REFLEQS-2D
and
chemical
REFLEQS-3D,
REFLEQSP.
in
modular
allows
the
main
code.
form, user
Geometric
flows
and
features
and
preprocessor,
are
provided
salient
flow
It consists
a common
The
of the Code
outlines
solving
reaction. and
Features
using
to specify Several
internal summarized
77
problem can
supported into
Selected
The input
with
the
configurations,
objects.
language.
in a compact
be
implemented
geometric
solid
the
users
flexibility
in complex
FORTRAN
code
a single allows
multiple
capabilities,
file
flow
the inlets
techniques,
and
below.
m
1.3.1 _z
The
Numerical following
Techniques are numerical
techniques
of the
approach
of
REFLEQS
code.
m
Finite-volume
a°
solving
Favre-averaged
Navier-Stokes
equations.
u
b.
Cartesian,
polar,
C.
Colocated
(non-staggered)
d.
Strong
Stationary
form
or rotating
Body-Fitted
Coordinates
(BFC).
grid.
as dependent
Pressure-based
f.
non-orthogonal
conservative
components e.
and
of
momentum
equations
with
Cartesian
variables.
coordinate
systems.
methodology
for
incompressible
and
compressible
flows.
m
°
h.
Blockage Four
technique
differencing
MUSCL,
and
Steady-state
i.
for flows
with
schemes: third-order
internal
upwind,
solid central
objects. (with
damping
terms),
Osher-Chakravarthy.
and
transient
(second-order
and
conservative
time-accurate)
analysis
capability. °
k.
Fully
implicit
Pressure-based SIMPLEC,
solution and
non-iterative
formulation.
algorithms, PISO
3
including algorithm
SIMPLE, for
all
flow
a variant
of
speeds.
4108/6
Symmetric
.
Whole-Field
Gradient
1.3.2
m.
Multi-component
n.
Eulerian-Lagrangian
physical
L
Equation
of State
C.
Four
be
Tables
for gaseous
for arbitrary
Extended
k-E model
(Y.S. Chen
3.
Multiple
4.
Low-Reynolds
Number
the capability
of resolving
One
Time
finite
model
rate
models
(Kim and
k-¢
Code
Organization under
CFDRC's
conveniently
R&D funds)
of restarting
or PLOT3D some with with
problems,
format), degree
flowchart
screening
of intelligence
NASA's CFDRC's
reading
PLOT3D CFD-VIEW
combustion) JANNAF and
code.
scientific
4
model.
method). two-step
reduced
propellants.
A common
from external
input
by analyzing or FAST
provides
layers.
preprocessor
both 2D and 3D codes.
grids
user's
this
model. model.
of the
supports
1988); and
1982);
and O2/Hydrocarbon
i.
the overall
(Chien,
models
Evaporating reactive spray dynamics Discrete ordinate thermal radiation
1-1 illustrates
Model
(accelerated
for O2/H2
1974);
C.P. Chen,
boundary
chemistry
gas.
and Kim, 1987);
wall functions. (diffusion controlled
chemistry
step
of ideal
and Spalding,
Scale Model
extended reaction
code.
composition
2.
g.
processed
flows.
species.
(Launder
Equilibrium
incorporates
Conjugate
transfer.
for the REFLEQS
k-E model
f.
the flexibility
mass
for two-phase
Standard
e.
(developed format
and
models:
Standard and Instantaneous
h.
Figure
heat
turbulence
mechanism
1.3.3
models
b.
d,
with
approach
Property
2_
_a -
flows
are the physical
.
Z
and
solver
solver.
JANNAF
a.
equation
Models
The following
w
Squared
linear
file input
graphical data
for
files, simple
consistency.
It provides (e.g., EAGLE errors
and
Results
can
postprocessors,
visualization
or more
package.
4105/6
E
4
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•
IUser I
I
Input
F e
I
_,_JREFLEQS-2D
i
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Equation
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Restart File
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m
Figure
1-1.
REFLEQS
Code
Organization
m
m
1.3.4
Code
The
REFLEQS
the
w
following
has
four-step
Check-out
b.
Benchmark
C.
Validation
d.
Field
of
"REFLEQS
1.4
code
a°
Results
the
equations
2
been
checked
validation
problems problems problems
problems
(with
computational
- Validation
Outline
Section
m
Testing
of the
provides embodied
and
out
for a large
of problems
by
using
procedure:
(with
known
(with (with
solutions);
benchmark detailed
global
quality
studies
can
Applications
data);
experiment
performance
be
data);
and
data).
found
in
the
entitled,
volume
Manual".
Report
a general in the
description
REFLEQS
of
codes.
the
Section
governing
partial
3 describes
the
differential
physical
models
available.
5
m
number
410s/6
f1-1
I
r
The
geometry
the
control
and
computational
volume
discretization
including
the
SIMPLEC,
and
A
separate
detailed
PISO,
section
the
for
overall
Conclusions
development
are
and
7)
are
presented
applications
in
and Section
of REFLEQS
discussed
the
to
condition
procedure
explained
solution
in in
Section
Section
4 while 5.
algorithms,
Details SIMPLE,
6.
dedicated
boundary
solution
of
in Section
is
are are
differences
described
each
topologies
techniques
and
(Section
explanation
8 presents code.
similarities
grid
boundary type
linear 9 and
are
given
conditions,
available
to the
equation
solvers
recommendations in Section
providing user. used for
Section in
the
further
10.
m
=
.
m
u
i
6
4108/6
=
=
2.
BASIC
GOVERNING
EQUATIONS
D
In
this
section,
the
presented.
These
momentum,
and
Since
basic equations
energy
a conservative
equations
are
governing are
which
equations derived
can
finite-volume
expressed
In Cartesian
tensor
be expressed
as
be
found
method
in conservative
form,
the
from
mass
0pui
in
for
single-phase
fluid
flow
the
conservation
laws
of
most
is used
fluid
mechanics
in REFLEQS,
all
mass,
textbooks. the
governing
forms.
conservation
3p
3
-_
+G(P
and
momentum
conservation
can
uj) -0
0
0p
0 :ii
+afi
(2.1)
Ot +N(Pue')=-N+- xj
m
where
ui is the
density,
ith
p is the
Cartesian
component
instantaneous
static
are
of the pressure,
instantaneous z_j is the
velocity,
viscous
stress
p is the tensor,
fluid
fi is the
w
body
force.
For
the
Newtonian
fluid,
"_j can be related
to the
velocity
through
(2.2)
where
The
# is the
energy
different enthalpy;
fluid
dynamic
equation classes and
can be expressed
viscosity
can
take
of problems. stagnation
several Two
enthalpy.
and
6_j is the
forms
forms The
are
static
0
_]j
7
lad
and
different
used
in
enthalpy
the
delta.
forms REFLEQS
form
of the
are
good
code: energy
for static
equation
as:
0oh
w
Kronecker
_ui
4105/6
L
Note
that
the
above
equation
is not
strictly
conservative
by
its
nature
and
is
I¢1
recommended
for
enthalpy
form
use
for
of the
is recommended
energy
for
as H = h + v2/2,
incompressible equation,
high
speed
is governed
by
on
and
low
Mach
the
other
hand,
compressible the
flows.
following
m
Law,
L
the
(2.3)
heat
flux
and
(2.4),
is calculated
qj is the
flows.
is fully
The
total
The
total
conservative
enthalpy
H,
and defined
equation
3 [pujH]=_DqJ
In Equations
number
(2.4)
_)
j-component
of the
heat
By using
flux.
Fourier
by
--
-k 3T q.i = _
where
m
The
k is the
species
thermal
conductivity.
conservation
equation
is written
as:
OPYi 3 where
Yi is the
source/sink
For
terms
turbulent
effective effective
mass
flows,
diffusion diffusion
(2.5)
fraction due
O(poGj 3Yil+w
of species
to chemical
the due
diffusion
mass
diffusivity,
and
l_
represents
reactions.
terms
to turbulence.
(or eddy
i, D is the
diffusivities)
in the Section for
above
equations
3 discusses
various
turbulence
are the
replaced
calculation
by
the
of
the
models.
r
8
410516
3.
PHYSICAL
MODELS
3.1
Turbulence
Models
In
this
section,
followed
by
REFLEQS eddy i L.
code.
implemented
basic 2.
steady
or
with
analytical
of
solutions
Hence
one
wide
variety
tions
for
are,
The
complexity
for
is forced
time-averaging
models
available
and
describe
the
the
in the
Boussinesq
turbulence
already
of
the
models
Navier-Stokes makes
number
of flows
Numerical
equations,
the
(DNS)
under or
coupled to
of engineering
to solve
in
transitional
methods.
Simulation
flow
it impossible
or numerical
can be applied
introduced fluid
laminar,
conditions,
a limited
been
to Newtonian
compressible,
to approximate
Direct
have
applicable
or
techniques
flows,
Favre
in general,
boundary
all but
to resort
laminar
turbulence
equations
nonlinearity
the
of numerical
various
is outlined
Equations
incompressible
conditions.
modeling
code.
dynamics
equations
turbulence
3.1.2-3.1.7
Navier-Stokes
transient,
the
of the
Section
fluid
behind
introduces
REFLEQS
Averaged
These
turbulent
3.1.1
concept.
governing
Section
framework
description
Section
in the
Favre
The
theoretical
a detailed
viscosity
3.1.1 JL_ L
the
obtain interest.
Even
though
Navier-Stokes of
a
equa-
turbulent
flows
is
t L feasible
only
unsteady these
at
and
scales
they
As
for
direct
equations
a wide
numbers.
range
short
time
simulations
applications are
flows
to
usually yield
q_ is decomposed
of averaging
Reynolds
very
engineering
equivalent quantity
Reynolds
of time
steps are
and
too
flows
are
and
length
scales,
fine
grids.
The
CPU
the
fastest
large
are
(or
generally
time)
only
require
averaged
over
averaged
even
for
in to mean
time-mean time
equations. and
In
fluctuating
and
inherently resolution
and
of
memory
and
largest
quantities,
or the
ensemble averaging
parts.
The
the of
Navier-
statistically
process, following
a flow two
types
used.
Averaging:
•
_ = _ + qb where
9
b
Turbulent
day computers.
most
Stokes
low
contain
requires
requirements present
very
qb -- (1/T)
ft ,d to
a+ T
qJdt
4105/6
Favre
(or
density)
Averaging:
Note
that
overbar
denotes
be
compared
T should
large
stationary
over
quantities
can
vary
Applying
the
Favre
one
obtains,
detailed
Reynolds
a number
the
qb = _ + q_"where
the
on
time
It also much
procedure
Favre-averaged see
while
fluctuation
a scale
averaging
derivation,
averaging
of samples.
in time
_ = p--_/p
and
scale
must
denotes
so that
be borne
larger
than
Smith,
(FANS)
Favre mean
averaging.
quantities
in mind
that
the
are mean
T.
to Navier-Stokes
Navier-Stokes
Cebeci
tilde
equations equations
(Equation
2.1),
below.
(For
given
1974.)
J
(3.1)
The
FANS
additional w
equations
contain
unknown
terms
between
the
modeled
to achieve
All
the
eddy function
fluctuating
turbulence
viscosity of the
-_
uiu C
information
models
concept
of the
strain
the
arise FANS
available
in which
than
called
components closure
mean
less
full
Reynolds
in the
NS
equations,
but
These
correlations
stresses.
averaging
process,
and
need
have
to be
equations.
in REFLEQS
the
the
Reynolds
employ
stress
the
generalized
-_
uiu;
3 3Xm 8ij --3
8q
is treated
Boussinesq
as a linear
rate
:_.._
--P Ui Uj = "'
+ 3xi
10
(3.2)
411_/6
Here
k is half
viscosity.
the
trace
Following
product
the
of a velocity
of the
Reynolds
kinetic
theory
scale
q and
stress
tensor
of gases,
a length
scale
the
and/zt
eddy
is the
viscosity
turbulent
eddy
is modeled
as the
L
t.tt = C p q e
where way
C is a constant q and
estimated.
/.t and
p, and
3.1.2
Baldwin-Lomax
This
belongs
length
tilde
are
class
model
modeling
and
Like
the
Lomax
for
are
from
the
inner
function
are
velocity
(1978)
models
differ
in
the
of models,
the
overbar
form
because
the
velocity
and
referred
to as
convenience.
layer,
developed
inner
Prandtl's to estimate
turbulence
models
relations.
It is also
Prandtl's
commonly
mixing-length
hypothesis
in
scales.
model
#t in the
used
for
it employs
of and
this Cebeci outer
model and
parts
length
Smith of the
mixing-length the
primarily
for
(1974), boundary
wall-bounded it
flows.
employs
different
layer.
(3.4)
for yy >_Ycrossover < YcrossoverlI for
_tt P't inner outer
I'tt =
In
turbulence
description
dropped
algebraic
because
mixing-length
expressions
following
of algebraic
obtained
length
Various
Model
a mixing-length
and
In the
for u, v, etc.
to the
scales
Baldwin
n
e are
of proportionality.
(3.3)
model
and
the
Van
Driest's
damping
scale,
(3.5)
f =tcy[1-exp(-y+/A+)]
where wall
A+ is the units,
Van
is defined
Driest's
damping
constant.
ur=
11
w
.
the
distance
from
the
wall
in
as
y+=yur/v,
r
y+,
(3.6)
4108/6
In the preceding
expression u_is commonly
known
as the friction velocity with
_'w
being the shear stress at the wall.
The
velocity
scale
q is modeled
as
the
product
of
and
the
root
mean
square
vorticity,
,O),=I(_____-
Using
the
preceding
_X)2+(__
expressions,
the
_)2+(_
eddy
(3.7)
__)2]1/2
viscosity
in the
inner
layer
is obtained
as
_tin,_ = p e2lcol The
outer
layer
eddy
viscosity
is determined
_touter
where
K is the
Clauser
constant,
The
quantities
ymax and
YmaxFmax,
following
expression:
(3.9)
(y)
additional
constant,
and
Cwk Yma x _
Fmax are determined
_C w
the
= K Ccp p FwakeFkleb
Ccp is an
f wake = min
from
(3.8)
from
the
(3.10)
function
F(y)= y Ic01[1-exp(-y+/A+)]
(3.11)
7 1
The
quantity
layer
and
Fkleb(Y)
Y_x
is the
Fmax
is the
is the
Klebanoff
value
maximum
value
of y at which
intermittency
of the
factor
F(y)
that
maximum
given
occurs
within
the
boundary
occurs.
by
1
12
410516
Fkl_b(y)=
The
quantity
velocity
Udif
in the
is
the
boundary
layer
Udif
The
second
The
values
term
difference
= _/
for
the
A÷=26
1 +5.5
(u2
between
the
constants
maximum
and
minimum
total
stationary
W2)min
+ V2 +
- %/( u2
(3.13)
walls.
appearing
Ccp=l.6
(3.12)
x station)
+ v 2 + W2)max
for
-1
Ymax J]
(i.e., at a fixed
in Uaif is zero
used
[
in the
Ck_b=0.3
preceding
Cwk =0.25
expressions
Jc=0.4
are
K=0.0168
w
3.1.3
Standard
Several
k-cModel
versions
REFLEQS differential commonly equations dissipation
of
is based
k-e
on Launder
equations known which rate
models
to as
the
govern
are and
in
Spalding
estimate
respectively.
transport The
today, (1974).
the
two-equation the
use
velocity model.
of the
modeled
with
the
production
P defined
model
employed
in
model
employs
two
partial
and
length
scales
and
hence
equations
are
turbulent
kinetic
energy
the
k-
(TKE)
and and
eits
are
pc2
= C_, --X-- - C_, T
the
This
These
equations
pp¢
but
3
+
(3.14)
as
13
4105/6
=
=
F
P = vt
+ 3x---_ 3 3x m
3xj
3 k 3x m
(3.15)
F
The
square
obtained
root
of k is taken
to
be
the
velocity
scale
while
the
length
scale
from
r3/4 k3/2 _= _ ¢
The L
is
expression
for
eddy
viscosity
(3.16)
is
£
C"k2 Vt="
w
The
five
constants
used
in the
model
(3.17)
E
are:
C. = 0.09 ; Ca_ = 1.44 ; C_2 = 1.92 ; crk = 1.0 ; era= 1.3
Wall
functions
in Section
3.1.4 This
model
range
was
standard k-¢ time
for
this
model.
A description
of wall
functions
is provided
k-eModel
to make
standard
used
7.4.
Extended
model the
are
developed the
k-e
dissipation model.
model. scale,
by
k/P
In
Chert rate
The addition
is introduced
and
Kim
more
in the
by
responsive
k-equation to the
(1987)
retains dissipation E-equation
modifying to the
the rate
same time
as shown
mean
the
strain
form scale,
standard
as k/e,
k-E
rate that
than of
the
a production
below.
(3.18)
14
41(]6/6
The
net
effect
mean
strain
weak.
The
of
this
is strong model
formulation and
is
to enhance
to suppress
constants
the
the
development
generation
of
of
e when
the
8 when
mean
the
strain
is
are:
C)_ = 0.09 ; C¢I = 1.15 ; C_2 = 1.9 ; C_3 = 0.25 ; crk = 0.75 ; cy_= 1.15
3.1.5
Multiple
Time-Scale
Most
turbulence
models,
for
both
the
ments
and
most
of the
place.
on
scales
source
of the of the
energy.
two
kp,
intensity,
when
kt, ¢p and
Simulation
(DNS)
eddies)
Hanjalic
et al., 1980;
being
based
production at, however
wave
numbers
these
equations
model
carrying of the
shown
that
eddies)
and
dissipation
range
kp and
dissipation
Chen
(1988)
have
of the
differ
from
each
other
a
turbulent depends rate.
is high, a transport
range
and
dissipation
production uses
suggested
of
solution
and
takes
developed
partitioning
transfer
The
have
scale experi-
have
variable
when
time
et al., 1981)
as a part
energy
vanishes.
most
Fabris
and
on
is determined production,
higher
Kim
one However,
(energy
where
production k.
assume
of turbulence
scales
(dissipative
into
model,
of turbulence.
at large
model
partitioning
k-¢
rates
occurs
quantities
into
standard
dissipation
spectrum
turbulence
The
is moved numbers
(e.g.,
energy
turbulence
partition
each
authors
time-scale
local
wave
Numerical
to smaller
sum
kinetic
m
Direct
the
the
production
partitioning
multiple
and
turbulence
Several
kt, the
including
production the
is cascaded
Model
and equation only
The to low for in the
terms.
(.+., 1 m
E
=
0
pp2
0
= C p,
pp_p
+ Gp, kp
p8
2
0
I_t
+ _t
OSp)
m
w
15
41o516
W
3
3
Pd _2
PPer' et
% k-i-+G_,- ,_
(p_)+_ (p,,,/_,): c,,, kt +c_- kt The
eddy
viscosity
is defined
(3.19)
J
as
vt = C_ k2/ep
where
k, the
total
TKE,
is calculated
(3.20)
as (3.21)
k = kp + kt
u
The
model
Ca
= 0.09;
constants
are:
w
m
= w
= 0.21;
C¢pl
3.1.6 All
cr_ = era = 0.75;
Cqa2
Low
Rey-nolds
three
model,
models and
the
therefore
require
functions
may
transfer, reduced
Number
multiple the be
by the
modified
use
and as
shown
above, time
used
scale
of wall
accurate
the
Standard
model,
equations
are
functions.
in flows etc.
below
Cet2
viz.
This
of low-Reynolds k-E
= 0.29;
-" 1.28;
Ca3 = 1.66
Model
described
not
= cra = 1.15
C_3 = 1.84;Cal
relaminarization,
momentum
wall
= 1.24;
cr_
with
large
difficulty
to include
the the
effect
the
separation, with
models
which
to
the of
the
wall. molecular
extended
number
wall
The
used blowing,
functions
permit k-e
viscosity
k-e
form
commonly suction,
associated k-e
way
model,
of high-Reynolds However,
number all
k-e
integration equations in the
and wall heat can
be of are
near
regions.
16
410816
(3.22) _t = C_f _ pk2 e
Several
versions
careful
analysis
Stowers N
W
low-Reynolds of the
(1988),
Avva
implementation equations
in
number
comparative
k-¢
studies
et al (1990), REFLEQS.
the The
models
done
low-Re
by
Patel
model
model
are
available et al. (1985),
of Chien
parameters
today. Brankovic
(1982)
appearing
After
was in
a and
selected
the
for
preceding
are:
m
C,
= 0.09 ; C_1 = 1.35 ; C_2 = 1.8 ; crt = 1.0 ; cr_ = 1.3
r_--_-_xp(_.01_5 y*),h--_.0;h= _-0.22_xp[_R;6)_] D- 2_k/_; E--2@_)_xp(-05y+) 3.2
Combustion
The
combustion
Models
models
instantaneous, rate
reactions
implemented
equilibrium,
model
in the such
(3.23)
and
REFLEQS
that
finite
code
a particular
rate
was fuel
in
the
chemistry
restricted species
REFLEQS models.
to one-step
may
appear
code
include:
Previously, or multi-step
as a reactant
the
finite-
irreversible in one
reaction
R
step
only.
capability n
In
this
of the
the
same
Air
and
fuel
study,
efforts
REFLEQS
code
(or oxidizer)
2-Step
The
combustion
and
Chinitz
Hz-O_
model
for
to
dependent,
can
exist
in more
are
considered.
reaction
is based
combustions
R¢0¢tion
made
to include
species
Hydrocarbon-Air
3.2.1
were
extend
the
chemistry
multi-step than
one
reactions step.
The
modeling in which cases
of H2-
Model I-I2-O2
on
the
2-step
model
of Rogers
m
(1983).
17
4105/6
(3.24a)
H2 + 02 = 2 OH
Further in
simplifications
equilibrium,
are and
implementation
of
made
the
this
H2
+ 20H
by
assuming
second
that
step,
reaction
(3.24b)
= 21-120
the
Equation
mechanism
into
first
step,
(3.24b), the
Equation is
REFLEQS
(3.24a),
irreversible. code
is The
is discussed
below.
2
=:
L
Basic
Equations:
The
solve
the
fractions
then
derive
mixture the
conservation.
=
mass For
finite-rate
basic
approach and
fractions this
chemistry,
the
the (or
2-step
of REFLEQS "fuel"
concentrations)
H2-C)2
basic
species
reaction are
the
chemical
of a finite-rate of all
of
equations
in solving
species
mixed
reaction according
equilibrium
tranport
species,
equations
first,
is to and
to element chemistry of
and
conserved
=
mixture
fractions
F (fi) = O,
i = 1, 2 .... , NCMPS
- 1
(3.25)
m
where Em I
fuel
NCMPS species
is the
[H2]
number
in Equation
of mixture (3.24b)
is given
F _H2_
The
extra
relations
include
compositions.
The
equation
of
as:
= _ Kfd_Iao[H2] [OH] 2
equilibrium
transport
Equation
of (3.24a),
(3.26)
i.e.,
(3.27) =
=
and
element
conservation
[OH]
+ 2[ 103
CD = 0.44
Substituting
The
Equation
(3.68),
dv 18 f (u - v)- --+g vp
__= dt
where
Pd is the
determined
droplet
from
the
liquid
density
velocities
(3.71)
Pd
p d _,_
and
f = CoRe24.
The
droplet
locations
are
as:
dR dt
--
= v
(3.72)
__o
u
where
The
R is the
mass
position
conservation
vector
of the
equation
for
drna dt where gas.
Sh xv and
is the x_. are
Sherwood the
vapor
number mass
=-
droplet.
the
droplet
Sh (pD)_
and
is given
(xv-x.)
D is the
fractions
at the
30
by:
mass droplet
(3.73)
diffusion surface
coefficient and
the
free
for
the
stream,
410516
U N
:___z W
respectively.
In this
Sherwood
number
model,
xv is calculated
is obtained
from
the
from
the
following
liquid
saturation
pressure.
expression
(Crowe
The
et al, 1977):
Sh = 2 + 0.6Re °.5 Sc 0333
where
Sc is the
droplet
Schmidt
is rewritten
number.
in terms
Since
(3.74)
m a = pdrd3/6,
the
mass
equation
for
the
of its diameter:
.
. Xv - X,,.
--_"=d(d)-2Sh{pDJ
Pad (3.75)
The
energy
equation
for
the
droplet
is written
,. dTa m at.. ,t _
where
_t is
the
sensible
heat
as:
ddmal = ?t + _'-_1
transferred
to
(3.76)
the
droplet
and
Td
is
the
droplet
m
temperature. specific
L is the
heat
of the
latent
droplet.
heat
of vaporization
Clis calculated
for
the
droplet
fluid
and
Ca is the
as:
n
= Nu_kd
(Tg-
T,i)
(3.77)
B
where
k and
respectively.
Tg The
are Nusselt
the
thermal number,
Nu
where (3.76),
Pr is the the
energy
Prandtl equation
conductivity
number.
Nu,
=
is obtained
from
temperature the
following
of
Substituting
is rewritten
the
above
the
gas,
correlation:
= 2 + 0.6Re °.5 Pr 0"333
dTa dt =
and
(3.78)
expressions
into
Equation
as:
=.Tg-
Ta 0
QL 0
(3.79)
where
u
31
w
4105/6
w
L Sh (pD) (xv - x_) Nuk
QL=
Pdd2C
(3.80)
d
0=- 6Nuk
If the
droplet
temperature
mass
conservation
is below
equation.
conservation
equation
zero,
droplet
its
Once
is obtained
boiling
the by
(3.81)
point,
droplet
setting
Equation
attains
the
left
its
hand
(3.75) boiling
side
is used point,
as the
the
of Equation
mass
(3.76)
to
w
i.e., the
temperature
cannot
exceed
point.
Therefore,
_q__" L
dm_= dt =
its boiling
(3.82)
,
This the
equation
implies
that
to the
droplet
all the
heat
transferred
from
the
gas
is used
to vaporize
droplet.
lml
The
solutions
conservation Eulerian F L
gas
The
phase)
source/sinks
The u
equations.
equations
in
provide
sources/sinks
which
the
sources/sink
are droplet
assigned is
terms to the
located.
The
to the
particular
gas
phase
cell
calculation
(in
the
of
the
is as follows.
mass
of
source/sink
the
term
droplet for
a cell
is
continuously
is given
monitored
along
its
trajectory.
The
as:
,)io-(i)out
3
= nPa_
3
_
6
(3.83)
i s
the
summation
takes
place
over
all
the
droplets
located
in
the
given
cell.
The
w
subscripts density w
evaluated
"in"
and
of droplets
"out"
refer
in a given
to cell parcel.
inlet
and
Similarly
outlet the
conditions, momentum
r_ is the source/sink
number term
is
as:
w
32
41o516
_=_ u
3
w
3
6
i
This
includes
source
the
term
effects
is a vector.
of The
frictional energy
(3.84)
interactions
with
source/sink
term
the
The
gas.
is given
momentum
as:
= araby- nZ, ,
(3.85)
i
where heat
hv is the transfer
3.5
A
from
Radiation
large
listed
Hottel's
transfer
at the
droplet
vaporization
(calculated
techniques
of thermal
Hottel's
b.
Flux
G
P-N
d.
Monte-Carlo
e.
Discrete-transfer
f.
Discrete-ordinate
Zone
temperature
according
and
to Equation
qi is the (3.77)).
are
radiation.
available
The
most
for widely
solving
the
used
equations
among
them
are
approximation
for
The
equation.
method
was
in the
the
first
use
of
suited
methods This
detailed
The for
method
been
factors
widely
which
differential
is computationally
used
of
the
needs
of this
configurations
33
procedure
knowledge
disadvantage
complex
involve
numerical
It has
of exchange
geometries. not
general
equation.
absence
the
complex
flux
method
transfer
involves
and
method
method
method
transfer
method
method
method
radiative
uneconomical
transfer
to the
numerical
a.
the
advance
media.
of
the
radiation This
gas
vapor
Model
zone
handle w
of the
below.
w
-_
the
number
governing
=
enthalpy
to estimate
the
to be
and
and worked
is that
strongly
approximation efficient
to
flow
method
with
developed
reaction. out
in
it is very
participating
to
the
radiative
for
this
reason
it
410516
i
has
been
suffers
widely
from
to this
all
Carlo
the
releases
method,
the
medium
combustor
After
method
is
can
for
this
the
combustion
droplets
be
soot
particles
for
Montechosen
their
lifetimes.
discrete-transfer
in a straightforward
cannot and
the
randomly
The
however,
geometries.
transfer, the
applied
method
accurate.
chamber
can
method, and
heat
method,
expensive. and
this
to complex is more
radiative
In
This
review
is well are
suited
several
be easily
in-scattering
imposed
more
and
at inlets
with
CFD and
and
codes
the
it has
treat
hence
Firstly,
the
is not
geometries.
simple.
Also,
solution
on
the
procedure
that
scattering suitable
for
discretechambers.
discrete-ordinate
method
the
boundary
conditions
method
is accurate
this control for
the
combustion
Secondly,
the
Finally,
based
found
the
complex
exits.
been
in rocket
method.
is relatively
accurately
equations
polar
methods, prediction
of this
for term
the
radiation
advantages
coupled
governing
of all
for
extended
of the
easily
method
method.
is economical
containing
to extend
computationally
geometries.
comprehensive
There
be
through
However,
applications.
ordinate w
difficulty
treatments
accurate
hand,
the
problems.
approximation
tracked
other
to complex
P-N
numerical
are
the
combustion and
the
most
this
on
manner by
method,
is the
Unfortunately,
general
accuracy
existing
method
energy
in
inadequate
Compared
Among
used
volume
this
evaluation can and
can
approach.
method
are
be
The
described
below. I
3.5.1
Governing
Equations
The
integro-differential
and
scattering
gray
radiative medium
(_ . V) I(r, f)) =-(k
n
where intensity absorption
.(2 is the which and
transfer
can be written
equation
is a function
of propagation of both coefficients
for
as (Fiveland,
o f, + o) I(r, fa) + k Ib (r) + -_-" _ ,
direction
scattering
heat
of the position respectively.
34
an
emitting-absorbing
1984)
I(r,f)_¢(f)'-_f))d
radiation
(r) and
beam,
direction
Ib is the
(3.86)
f2"
I is the (_),
intensity
k and
radiation cr are
of black
the
body
410516
7. =
; =
W
4
radiation
at
the
transfer
from
temperature
of
the
medium,
¢ is the
phase
function
of the
energy
W
the
left
The
hand
three
incoming
a'2" direction
to the
outgoing
side
represents
the
of the
intensity
terms
absorption
The
the
on
and
the
right
hand
out-scattering,
boundary
condition
gradient side
emission
for
solving
represent and
the
where
I is the
intensity
is the
surface
at the
boundary
In the
discrete-ordinate
set
emissivity,
above
hand
ordinate.
for
side The
energy
p is the
the
may
Equations
(3.86)
on
direction
intensity
be written
£2.
due
to
as
ff)dff
(3.87)
a surface
reflectivity,
in
term
respectively.
[n.fflI(r,
surface
The
specified
changes
equation
leaving
£2.
and
at a boundary n is the
unit
location, normal
e
vector
location.
of equations
right
of radiant
in the
in-scattering
p£ .=',o I (r, f2) = e I b (r) + -_
m
direction
method, a finite
of the
number
above
of ordinate
equations
discrete-ordinate
and
are
equations
m-a7 + m-a7 =-(k
(3.87)
directions. approximated
may
then
are
replaced
The
integral
by
summation
be written
by
a discrete
terms
on
over
the each
as,
+g +NEw=',='mg ";m=
(3.88)
m
For
cylindrical
polar
geometries,
the
discrete-ordinate
equation
may
be written
as,
w
m._ !
_m a (r In) 1 -73-----7----7
3 01mlm) 3I m a4t +bt,n____=_k+,,,.m+._v+_-___w,dqjm,m_m.;rn=I,...Mt ,_T l-t. Y.
1
(3.89)
rtl'
where
the
second
redistribution.
term
It accounts
on for
the the
left
hand
change
side in the
represents direction
the cosines
angular as a beam
intensity travels
u
in a straight
line.
35
410516
-q
The
different
Wall
boundary
conditions
required
to solve
this
equation
are
given
as,
Boundary
Im = alb + P _ win" I'trd Ira"
(3.90)
m
Symmetry
Boundary (3.91)
Inlet
and
Exit
Boundary Im = 0
iIn
the
above
respectively. i
W
direction partial
L
$4
equations, For
cosines
a particular along
differential
of the
m'
x and
equations
for
i.e., direction
12
denote
the
direction,
the
approximation,
values
m and
the
and
m intensities,
m
n
w
w
Cosines Direction Number
and
Their
are
and
m,
the
equations Ira.
associated
Table Direction
by These
directions the
outgoing
denoted
y directions.
ordinate
cosines
(3.92)
In the
values
directions,
/.t and
represent present
considered.
_ are
m coupled code,
Table
the
3-1
only
the
lists
the
weights.
3-1.
Associated
Weights
for
Direction Cosines l_m
incoming
_m
the
$4 Approximation Weight
_Im
Wm
1 2
-0.908248 -0.908248
-0.295876 " 0.295876
.295876 .295876
/rJ3 _J3
3 4 5 6
-0.295876 -0.295876 -0.295876 -0.295876
-0.908248 -0.295876 0.295876 0.908248
.295876 .908248 .908248 .295876
_3 _3 _3 _3
7 8 9 10
0.295876 0.295876 0.295876 0.295876
-0.908248 -0.295876 0.295876 0.908248
.908248 .908248 .908248 .295876
_3 _3 _3 _3
11 12
0.908248 0.908248
-0.295876 0.295876
.295876 .295876
_3 _3 4105/6
t3-1
u
36
w
410516
Equation system
(3.88)
can
be cast
in a fully
conservative
form
a general
BFC
coordinate
as
O-_[gmlmJFlx+_mlmJFly]+
= l
where
for
x'
coordinate
and
y"
are
the
transformation
[
-(k
local
+8)I
grid
_tmlmJF2x+_mlmJF2y]
°
m + kI b +"_-_
2 Wm'(_m'mlm
coordinate
directions,
] J is the
(3.93)
Jacobian
of the
and 3x Fix = -_ Ox Fly = --_
i
J
F_
= "_
w
F2y =
(3.94)
Equation
(3.89)
can
be
integrated
over
the
control
volume,
shown
in Figure
3-1,
to
obtain
gm (Ae lm'-
Aw lm'_) + _m (An lm" - As ir_) + (An - As ) [_/m+ 1/2 Ip, m+ 1/2 -wmTm- l/2 Ip, m-1/2]
= - (k+ _) V I,,,p+ kV G + V4-ff_ w,,,.O,,,'m I,,.p (3.95) where
the
coefficients
y are evaluated
from
the
recurrence
relation
u
37
w
4105/6
(3.96)
"_m+ I/2 = 'Ym- 1/2 - _m Wm
In
the
and
above
equation
subscripts
respectively.
A
denotes
the
cell
w, e, s, n, P denote Figure
3-1
shows
face
the
a sketch
area
and
west,
east,
of the
control
V denotes
south,
the
north,
cell
and
volume
cell
center,
volume.
r_
I n
W
E
u
I
Figure
The
in-scattering
function the
term
¢ which
medium.
For
can
on
3-1.
the
Sketch
right
be prescribed
a linear
of the
hand for
anisotropic
Control
side both
of
Volume
this
isotropic
scattering,
the
_z
r4-1
410_
equation and
phase
contains
anisotropic function
the
phase
scattering may
by
be written
i
as,
i
_m'm= l . 0 + ao[_.m_.m" + _m _m' +'rlm _m "]
(3.97)
_=m
where backward,
ao is the
asymmetry
isotropic
and
factor forward
that
iies
between
scattering,
-1 and
1.
The
values
-1,0,1
denote
respectively.
i
38
m
=__
w
410516
t]
Equations
(3.95) involves 5 intensities out of which
the boundary
conditions.
The other cell face intensities can be eliminated
of the cell center intensity by a spatially weighted the east and north cell face intensitiesare unknown, of cell
center
intensity
by
Ir%-(1-OO 0_
In, =
where
¢z is the
yields
a central
Equation N
in terms
For example,
they can be eliminated
if
in terms
Ira, (3.98a)
Imp-(1- )Im.
w
w
approximation.
from
using
Ira. =
2
2 intensities are known
equation
(3.98) for
the
weighting
factor,
differencing into cell
the
cz=l yields or
the
integrated
center
o_
(3.98b)
a upwind
"diamond
differencing
differencing"
discrete-ordinate
equation
scheme
and
(x=1/2
scheme.
Substituting
results
in a algebraic
intensity,
u
gtmAIm +_mBIm Imp = gtmAN + _mAs
+SI-S2
+ (x(k + or) V
(3.99)
where
A = A e (1 - (z) + A w o¢ B =An(1-(z)+AsO_ Sl = O_k'V' Ibp u
S2 = o_v TE Wr; %"' Ir;l' n
Equation
(3.99)
is appropriate
direction
of integration
when
both
the
direction
cosines
are
positive
and
the
N
proceeds
in a direction
39
of increasing
x and
y.
For
negative
41os16
=
=
N
direction started
cosine, from
cosines,
the direction
the appropriate
the
intensities
appropriate
direction.
in-scattering
term
thus
decoupling
Assuming
of integration
local
corner
at all
of the domain. cell
This procedure is evaluated
the m ordinate
radiative
i.e. the net radiative
the
heat
centers
explicitely
by
and Thus,
are
is repeated
the integration for each
obtained
by
the
previous
is
set of direction marching
for all the ordinate using
sweep in
directions. iteration
the The
values,
equations.
equilibrium, flux,
is reversed
the source
is given
term
for each
computational
cell
by,
s= v k G Ir_win- V 4rIkX b
(3.100)
m
w
This
source
term
and
the radiative
is added transfer
to the gas phase equations
energy
are solved
equation. in an iterative
The gas phase
equations
manner.
m
m
m
w
m
40
410618
4.
GEOMETRY
The
REFLEQS
feature
of
code
the
blockages. In
Coordinate
The
geometries
section,
and
discrete
structured
grid
a non-orthogonal flow
a brief and
REPRESENTATION
the
body-fitted
problems
with
description internal
approach.
of
blockage
grid
complex
the
One with
internal
geometries
geometry,
concepts
important
the
are
can
be
Body-Fitted
presented.
Grid
code
as long
of
many
system,
Geometry.
domain
use
result,
this
REFLEQS
a single
is the
a
(BFC)
ITS COMPUTATIONAL
uses
code
As
handled.
4.1
AND
can
be
as the
representation
used
flow
to
solve
domain
of the
flow
any
fluid
is covered
flow
with
geometry
and
exists
three
problems
with
a structured
grid.
computational
complex A grid
domain.
is a
A grid
is
i
considered
to be
identify lines.
D
three
l a and are
a single Since
structured
limitations.
An two
grid
in Figures
Body-Fitted-Coordlnate
In the
REFLEQS
system;
2D
system.
The
with
simple
is able
BFC grid.
Figure
and
face
three
different polar
and
code most
In general,
the
a 3D
volume
to identify
flow flow
with
(for
of a control
allows
a cell blockages,
problems
despite
and
problem)
is on
internal
problem
blockage
two
grid
or a point the
of
above
its geometric
is shown
another
to
in Figure
without
4-
blockage,
System
coordinate system;
polar and
lines
4-1c.
(BFC)
geometries time.
one
grid
can be used
to cover
layouts,
axisymmetric
execution
index
REFLEQS
4-1b
4.2
Cartesian
any
of a two-dimensional
grid
code,
and
(I,J,K)
the
example
possible
displayed
if there directions
words, grid.
mentioned
i
distinctive
In other
a structured
structured
they
systems
and
systems
are
could
REFLEQS
an
used:
arbitrary
specialized
offer code
are
some works
Cartesian
grid
Body-Fitted-Coordinate grid
savings with
the
the
systems of
the
for
problems
storage
BFC
system
and
of such
a grid
system.
and on
a
=
u
4-1b
and
4-1c
are
just
two
simple
41
examples
4105/6
I
_/////////////Z/Z///_"/Z///_"/Z_Z/////_
outlet
inlet
"/,
_////.////./////////.
(a)
lilII
II II
k._
I I I
• m
I
I
I I
! I
IIII
E m
(b) m
w
i
(c) m
Figure
4-1.
A Sample
without
Two-Dimensional
Flow
u
Two
Possible
Grids
with
and
Blockages 42
w
with
410_/6
Mathematically, physical
a BFC
domain
problem.
to
The
correlation.
_ and
vl
is
can
be
viewed
coordinate
always
each
This
each
triplet
Therefore,
on
is a function the
Cartesian
point
on
as
coordinates.
rectangular
domain
For
reference,
future
code
as shown
the
J indices
I and
physical in
Figure
domain,
Figure
works that
can
be
_ or is associated on
a BFC
a physical
in Figure
a brief
in
from
4-2
for
have
a
a base
4-2a.
As
a
a 2D direct
Cartesian result,
the
= 4"(x,y,z)or y= 43 z=zccn, )
domain
of _, vl and
Note
illustrated
the
shown
transformation
as
a physical
REFLEQS
and
point
used
be expressed
as
lines
=n(x,y,z),
Therefore,
as a coordinate
domain
to identify
system
transformation
can
computational
In order
coordinate
system
identified with
grid
domain
by
(4.1)
a triplet
a particular
system is
y, z).
(i, j, k) index.
expressed
always
(x,
on
a base
transformed
to
a
4-2.
introduction
of
the
BFC
coordinate
system
is given
w
below. work
of
replace are
A comprehensive Thompson (_, rl, 4).
defined
description et al,
In a BFC
(1985). coordinate
of BFC For
coordinate
derivation system,
system
convenience, (_1, _2, _3), the
can
be
found
in the
(_1 _2, _3) is used covariant
base
to
vectors
as
w
(4.2)
m
m
w
43
b
410516
w
,,_ell, (l,J)
4105/6
(a) physical
domain
..,
..
f4-2
_
X
u
3 2 J=l I=
2 (b)
Figure
4-2.
An Illustration
..
computational
of the Transformation
domain from
Physical
to Computational
Domain
44
410516
where base
r is the vectors
displacement
are
defined
vector
and
_2 X _3
"2
_3 X _1
!
;a -
!
J is the
coordinate
Jacobian
line
-_k(i_je
k).
defined
and
But note
to x [ + y f + z/_
, and
contravariant
as
-'I
where
is equal
as J= (sl
eJ is a base that
_j and
vector
sJ are
not
_1 X ¢2
-'3
;_ =---7--
x _2)" _3" Basically, normal unit
(4.3)
to the
bases
and
_j is a base surface unit
bases
vector
formed can
along by
si
_j and
be defined
as (4.4)
hi-- V/NJ and (4.5)
_J
= w
where
hj is usually
called
scale
(no summation
on j)
factors.
u
It is easy
to show
that
the
covariant
and
contravariant
base
vectors
satisfy
the
basic
relationship
_i" _j=
8ij
(4.6)
w
where
8ij is the
In
BFC
the
expressed
Kronecker
system,
the
in conservative
delta.
gradient, form
divergence,
curl,
and
Laplacian
operators
can
be
as:
m
Gradient: m
3
L
m
Divergence:
45
41O516
Curl:
Laplacian:
The
geometric
point
meaning
P curvilinear
simply
the
Figure
4-3.
Vx_'-I
3_3
g2f=j
32
of the
above
coordinates
volume
of each
(J_'k x V-)J
.
quantities
shown
can
in Figure
be easily
4-3.
For
explained
a BFC
on
grid,
a 2D
local
the
Jacobian
Bases
for
is
cell.
m
u
=
i
Geometrical BFC
Meaning
of Covariant
and
Contravariant
a 2D
System
z
4.3
L_ m
In
Blockage
many
and
engineering
there
single concept.
Concept
can
domain With
be
problems internal
structured blockage
flow grid, capability,
the
boundaries
obstacles. the
of the To
REFLEQS several
facilitate code
parts
46
domain the
employs
of the
flow
are
very
use
and
the
so called
domain
complicated
generation
can
of a
"blockage" be real
solids
4108/6
H
or
imaginary
illustrated used.
As
blockage
dead in Figure
4-4
it can
seen,
be
capability
off
regions
will
will
have
solid
discussed
regions
here
and
for
and
are
a 2D
flow
blocked-off. where
a structured
blockage
participate wall
which
in the
interested
real
is very
convenience
grid
properties.
both
grid
offers
Such
The
should
difficult
and
numerical
readers
and
in grid
generation
blocked-off
numerical
the
imaginary to
book
blocked-
process
blockages
by Patankar
is
without
The
solution of the
are
blockage
generate
generation.
treatment
consult
regions
but is not
(1980).
_\\\\\\\\\\NN\\\\\\'_
rob.._ y
y
,_-.,\\\\\\\\\\\\\\%_%%_ 410_6f4-4 ,%NNNNN%N\\\NNNNN'K\\N\\_
%
t m
×-._\%\__x
i J_
u
v
:
:
I
I
I
I
I
!!!!
II
!:::
II II
:
:
:
:
I
"
"
"
:
I
I
I
5 .... ii?!;@_!@ :_:_:_._ 11.63)
the
velocity
(7.5)
variation
may
be
u_ fn (Ey +) u=--_
E = 9.70and
described
by
a
i.e.:
¢r,,,a
where
a
boundary
u =l--d- )
In
and
adiabatic
y+ -
In
is
variables,
temperature
with
no-slip
a priori,
dependent
constant flows
the
out.
knowledge
wall.
functions,
of the
either
turbulent
wall
wall
For
a combination
is carried
the
at the used.
equation, For
using
a result,
pressure
cells
flux,
procedure
and
near
the
without
walls.
hand,
a constant for
flows
solid
other
from
example,
exists
at the
the
value,
turbulent
_c= 0.4034
are
experimentally
69
(7.6)
determined
constants.
41(_16
w
In both
the viscous
calculated
from
(y+ < 11.63) and
the product
internal
of effective
(y+ > 11.63)
viscosity/,/fly
sublayers, and
normal
the shear
stress
is
velocity
gradient
&t / o_y, i.e.:
U
3u zw =_q_-
(7.7)
where
fla. for y+ < 1 1
.63
_ef/= _ _tturb for y+ > 11.63
(7.8)
m
Near
the wall,
balance
the transport
between
equation
for the turbulent
the local production
and dissipation
kinetic
energy,
k, reduces
to a
of k to give:
= m
_t
= pe
(7.9)
=__
The
velocity
gradient
may
be replaced
from
Equation
(7.7) and
the dissipation
rate
from:
k2 ixt = C. p -_-
(7.10)
to give: 7/
2
_'w= C_'2 P k = p u_
m
Hence,
it follows
from
Equation
(7.11)
(7.5)
w
70
410616
1/4
1/2-
pC_
k
u
I gn(Ey+)
7.5
Periodic
The the
Boundary.
periodic two
ends
(cyclic)
boundary
of the
calculation
occur
domain
can
from
0 to 360 ° is to be considered
flow
pattern
The
general
In this
in the
rule
z, the
boundaries
in
conditions
This
z = last
are
a polar-coordinate
angular
is that
finite
flux,
is the
convective
direction
with
whole
one
if
another.
angular
extent
is present
in the
7-1b).
are
to be expected
is to be expected
at that
at z = o and
surface,
then
the
cyclic.
circumstance,
indices
the
up
"repetition"
(Figure
conditions
sides,
join
or when
direction
on both
circumferential
in which
7-1a)
identical
in the
z-direction
direction
coordinate
equal
in the
(Figure
whenever
appear
the
boundary
conditions
can be specified
_LB = #N " qJHB= qJ1 " and
where
(7.12)
HB and flux
LB denote in the
high
boundary
as follows:
C_ = CZHB
(k = N)
and
(7.13)
low
boundary
(k = 1), Cz
z-direction.
71
41Q5/6
L w
====
•
360 ° 0°
__4
CYCLIC
CONDITIONS
FOR
_=
120°SECTOR
1
180 ° 1
(b)
(a) Figure
7-1.
Grid
Notation
for
Periodic
Boundary
Conditions
in the
Z-Direction
1
72
41(]616
8.
In
SOLUTION
the
METHOD
previous
sections,
correction
equation,
described.
The
describe
how
8.1
Overall
The
Navier-Stokes
equations.
linear iterative each
the
section
are
are
Due
described
to solve
a set
dependent
derivation
boundary all
are
is well
of
the
pressure-
conditions
the
have
elements
been
together
and
solved.
the
transient
above
procedure, and
can
and
coupled
equations,
it
iteration
delayed
approach the be
is called
segregated
solution
process
Navier-Stokes
initial
is one
the
defined
overall
non-linear
governing
variables
approach
following,
an
the
de-coupling
This
Prescribe
of highly
Therefore
to the
solver.
In the
equations
of
linearized.
approach.
of the
is to assemble
governing
discretization
equation
equation
the
Process
different
governing
a.
this
method,
implementation
equations
method.
linear
of
discretization
discretized
among terms
the
Solution
In
coupling
and task
the
the
is
final
is
seen
that
and
usually
linear
solved
differential the
the
non-
called
equation
separately
an
set using
for any
or equation-by-equation of the
segregated
method
is
of
the
equations.
fluid
flow
field
{q
0} at
the
beginning
w
computation b.
Make
(to).
one
{¢n+1}.
increment
For
this
in time
purpose,
(tn+l = tn + At)
an
iterative
and
method
we
try
is used.
to get
solution
Therefore,
set
w
L
c.
With
do
The
known
{_n*l}k,
all the
link
coefficients
are
solved
Anb are
evaluated.
[un÷n,"
.
the mass
momentum known
equations pressure
_n+l}k.
This
first velocity
to
obtain
field
will
_ i not
Jk+l satisfy
using the
conservation.
L_ w
73
!
410516
J
Pressure
e.
new
correction
velocities
satisfy
the
Any
f.
are
other
scalar are
Steps
c-f
equation
solved
are
until
b-g
can
be
pressure
• This satisfy
new
_p'_+l}k+i
velocity
momentum
as energy,
all
and
field
may
equation.
turbulence,
equations
criterion
step
new
combustion
(¢n+l}t+l.
convergence time
not
_, such
to get
the
Jk+l
but
repeated
at n+l
Steps
h.
as _ z
equations,
predetermined solution
to obtain
obtained
continuity
equations °
is solved
is
are
met.
satisfied
At
this
or
a
point,
the
any
time
is calculated.
used
to
obtain
a
transient
solution
at
locations.
8.2
Linear
Equation
the
overal
solution
equation
set
In
usually
from
execution Elimination
execution
time.
the
selection
is an
for
In the
60's
solving and
popular.
Despite
iteration,
ADI
method
Therefore,
ADI
is also
developed. Strongly
storage
It is rarely
candidate
solver.
each fraction
time,
Guassian
Implicit
fact
is the
of cpu
time
of linear and
(SIP).
but
very
prohibitively candidate.
The
to
WFS,
(WFS) unlike
SIP,
the
large
two
advanced
is an
The
very
time
per
system. solvers
enhanced
from
based was
solve
as and
iteration
cpu
is free
such
storage
and
which
of
a trade-off
method
storage
In REFLEQS,
Solver
amounts
approach.
is the (ADI)
solver
solver
iterative
small slow
large
direct
ours
linear
equation
prohibitive
Implicit
the
is always
A
segregated
Direction
linear
solvers
like
solve
require
requires
matrix
it requires
Field
may
needed.
for
banded
Alternating
Whole
and
accuracy
is to
This
equation
solver
a good
component
equations.
a useful
that
Procedure
governing
method
becomes not
important
accurate
sparse
70's
the
One
one
a large
Therefore,
among
best
process,
resulted
consumes
memory.
Solvers
any
are
Stone's adjustable
w
0_- parameter
and
(CGS)
which
with the
solver reasonable
following
is symmetric. may
storage
Another
provide and
an
execution
solver accurate time
is the solution
cost.
The
Conjugate for two
Gradient
difficult solvers
linear are
Squared system
described
in
sections.
74
410516
[I N
8.2.1
Whole
Figure
8-1
for
WFS
the
Field
Solver
illustrates
the
backsubstitution
node
arrangement
and
the
nomenclature
discussion.
PN j+l
x - substituted node O" implicit n°de El- cancelation node
_jN
_1
I
I
I
SH
Figure
With
the
i,j,k
8-1.
indices
SL
Grid
Node
omitted
the
Nomenclature
for
finite-difference
(FD)
x
the
WFS
SH
Discussion
equation
can
be expressed
as:
(8.1) The
difference
symmetric
between 3D
WFS
for
WFS
and
SIP
the
system
solvers
of
the
is the FD
solution
equations
algorithm can
be
of
summarized
the
zas
follows.
a°
In SIP while
b.
only
in WFS
No
extra
are
arrangement
(SH,
treated
implicitly
in backsubstitution
(P, H, N)
(P, L, H, N).
Nonsymmetric in WFS
C.
3 points
SN)
memory
symmetric
storage
of cancellation arrangement
is needed
75
nodes is retained,
in WFS
compared
is used (SL,
in SIP
while
SH).
to SIP.
4106/6
8.2.2
.Gg__aj_ugate Gradient
Conjugate often
gradient
provide
the
no
various
CGS
solvers
a faster
(vectorization, Among
type
Squared
user
Solver
have
convergence
rate.
specified
parameters,
nonsymmetric
and
has
found
algorithm
implement
in the
expressed
as follows:
been
CFD
become
code.
Initialization
The
very
popular
They
also etc.)
nonpositive to
have over
faster
years
many
conjugate
applied
methods.
gradient and
to the
they
advantages
iterative
convergence
algorithm
since
other
classic
definite
yield
CGS
in recent
solvers,
is simpler
system
AX
to = b is
ro = b - AXo
(n=0),
qo = p-1 = O; P-1
1
=
jOn
P n = 7oT r.;fin= n-'--'_1 P u n = rn + fin qn m
v n = Apn m
Pn
- T
(Yn= ro vn;an=-_. Interaction
(n>O),
qn+l = Un - C_nVn
rn+l=rn._A(u.+q.÷l)
The
convergence
radius
of the
the
system.
one
whose
obtained by
rate
a direct
._X = p-I AX
conjugate
coefficient
matrix
To a matrix,
the
coefficient by
of
using method,
an
and linear
matrix
has
approximate and
gradient can be
effectively
system
AX
a
lower
matrix
applying
= p-_ b = b" In practice,
algorithms
the each
76
this
transformed
in a CGS
to iteration
the one
the
spectral
preconditioning
into
condition
algorithm
on by
= b is transformed
p, and
time
accelerated
spectral
CGS
depends
an
equivalent
number.
This
system
is solvable
equivalent has
is
system
to compute
a
410516
matrix-vector computed
by
product
of the
solving
a linear
decomposition
is used
preconditioned
CGS
as
is very
form
y =._X,
system
with/7=
of the
a preconditioner efficient
and
form in
the
p-lA. py = AX. code.
The
vector Incomplete
It has
been
y is actually Cholesky found
that
robust.
m
w
77
410516
tl i
i
¢a,a
9.
As
CONCLUSIONS
a result
of this
developed art
for
were
To
and
used
to ensure
simulate
were
in
of
flow
incorporated
turbulence
step
and
PDF
based
was
problems
to
boundary
condition
and
ensure
application
Manual.
The
several Materials
ordinate
has
including
thermal
turbomachinery Tester
(BSMT)
the
grid
code
for
arrangement,
numerical
engines,
several
standard
k-e,
schemes
physical
models
been
radiation
applications,
k-e,
low
(equilibrium,
and
one-
dynamics
verified to
models; model;
and
model.
against
coordinate etc.
in the
successfully AMCC,
models
spray
conditions,
presented
extended
models);
interaction
extensively
initial
flow
State-of-theinto
order
been
models);
invariance
SSME,
high
combustion
combustion
and
also
and
multiple-scale
mechanism
are
incorporated Colocated
rocket
Eulerian-Lagrangian
studies
configurations
and
solution
combustors.
has
accuracy.
turbulence
order,
code
equations
hydrocarbon/O2
systematically
were
(Baldwin-Lomax,
reduced
discrete
in rocket
REFLEQS,
including:
number, and
two-phase
code
code
code,
geometries.
in liquid
models
Reynolds
The
complex
solution
the
flows
algorithms
combustion
into
H2/O2
spray
momentum
good
and
CFD-combustion
solution
simulations
conservation
advanced
two-phase
schemes
flow
strong
an
analyzing
numerical
accurate
study,
STME,
REFLEQS:
including
and
number
orientation,
Results
applied
a large
of the Valiation
for several OMV.
NASA
It has MSFC's
grid
skewness,
testing,
validation,
and
Applications
liquid also
of test
rocket been
Bearing
engine used
Seals
for and
problems.
w
78
410516
10.
RECOMMENDATIONS
Based
on
trends The
Physical
b.
Numerical
C.
Code
resolve
Multi-Step,
scale
=
is much walls
speed
flow
be
mechanisms
Supercritical point.
high the
flow
chemistry form
O2-H2
Model:
The
Most
conditions.
Advanced
University
and
categories:
engines,
rocket
the
following
and
equilibrium
must
be taken
thrust
such
as those
the
of low
temperature
of
chemical
account.
The
(near
for
established
fluids
change
drastically
operate
under
developed
or high
model
- O2).
being
time
is no longer
CnHm
chambers
under
stoichiometric)
assumption into
in the
only
which
demonstrated
multi-step
is valid
is far from
the
properties
for
equilibrium
distributions
analysis
In regions
ratio
chemical
temperature
this
scale.
and
engine
models,
and
The
conditions
mixture
region),
general
Model:
species
time
the
nozzle rate
in rocket
However,
temperature
as 7-step
Spray
three
current
REFLEQS.
and
Kinetics
nozzle.
where
the
(such
following
dynamics
to predict
the
than
of finite in
of for
enhancement:
used
and
in the
formulated
assessment
recommended
Applications.
Chemical
and
or regions
the effect
critical
further
smaller
(i.e.,
in the
combustion
widely
number
cooled
and
spray
chamber
Mach
and
are
Enhancement
need
has been
project
Improvement; and
Finite-Rate,
combustion
w
Validation
models
this
Enhancement;
Method
Model
during
improvements
are classified
Model
complex
analysis
work
several
enhancements
Physical
low
of
needs,
a.
physical
i
experience
future
suggested
10.1
To
the
and
valid should
reaction
near
the
supercritical
at Pennsylvania
State
w
incorporated process
i
Coupled require
University
to include
under
these
Atomization droplet
size
of
the
effects
Tennessee
Space
of supercritical
Institute phenomena
(UTSI), on
the
should
be
vaporization
conditions.
and
Spray
distribution
Dynamics and
Model:
spatial
79
Existing
resolution
for
spray droplet
dynamics formation
models rate.
4108/6
This
information,
Recently, have
advanced
been
suitable
w
be
assumed
models
Heat
The with
in
spray
and
multi-injector
for
dynamics
Embedding
Giridharan
and
results.
VOF
methods,
et al, 1992).
The
interactions
most should
shear
coaxial,
shear/swirl,
heat
transfer
model
the
design
a
large
enhancements
Multi-Domain be
grid
flux
greatly
It
and
the
transfer injector
transfer
between face
model
various
to the
needs
should
engine
incoming
to
be
be
cryogenic
incorporated
in
capability.
Improvements
methodology of
indicate
component
number
of
The
cells
useful
purpose
for
the
for
To into
and
of blocked
refinement
multi-domain
component
in
capability
multi-media achieve the
exterior
of multi-domain
number
grid
needed.
interior
use
analyzing
incorporation
of
the
limits
from
Multi-domain,
(10 6) are
analysis by
a shift
interactions.
recommended
and
A general
heat
across
optimized
is also
of
heat
are
generation
domain.
gradients.
effects
heat
Capability:.
can
A conjugate
multi-domain
a system
with
following
the
code.
flow
in
rocket
capability.
It allows
computational
cells
localized should
this,
be
regions
of
in
high
incorporated
into
code.
Adaptive solution regions
Gridding of
the
of high
gradients. flow
on
and
optimized
conjugate
the
to
simulation
the
on Jet
injector
Model:
Algorithm
trends
performance
the
as the
Numerical
efficient
numerically
assess
(such
conjunction
engines
based
et al, 1991
of single
Transfer
to
propellants).
=
influence
injectors.
components
Current
great
models,
analysis
and
implemented
10.2
has
(Giridharan
for
incorporated
Conjugate
n
atomization
developed
impinging
a priori,
This
domain.
gridding
feature
Capability:. problem. flow
A moving into
the
feature
Therefore,
gradients
facilitates
This
the grid
and
the
allows
grid
become
points
relatively
efficient
utilization
capability
should
the are
grid
be used
adapt
automatically
coarser of the
to
in regions
computational to implement
itself
to the
clustered
in
with
small
cells
in the
the
adaptive
code.
m
80
410816
N
Convergence
Enhancement
coupled-variable
conjugate
computational
efficiency
In addition,
a second-order
method
should
to accelerate
be
10.3
Code
The
REFLEQS
Even
though
problems
(as
additional
detailed
turbulence,
u=
and
code
described
experimental efforts
of fine
can
without
the
data on
be
large
should
systems rate
and
assessed.
grid
solutions.
applied
heat
such
as multi-grid
be implemented of linear
scheme
and
such
These
has in
are
been the
available. coupled
to a wide
transfer
validation
flows
solvers
methods
to improve
nonlinear
as the
features
and the
equations.
Newton
iteration
are especially
useful
and Applications
code
and
Advanced
convergence
implemented
Validation
with
gradient of solving
convergence
flows,
Capability:
and
variety
REFLEQS:
with
checked Validation
need
More
subsonic,
and
supersonic
combustion.
extensively
studies
of,
out
for
and
Applications
to be performed
emphasis
complex
should
physics
for be
(spray,
a large
cases
placed
on
combustion,
number
of
Manual), where 3D
reliable validation radiation,
etc.).
:
w
81
m
41o816
11.
REFERENCES
Avva,
R.K.,
Smith,
Low
Reynolds
Sciences
Baldwin,
B.S.
Brankovic,
and
Singhal,
Number
A.K.,
(1990),
Versions
of
k-e
(1978),
"Thin
"Comparative Models,"
Study
of High
AIAA-90-0246,
and
Aerospace
Meeting.
and
Model
-
C.E.,
Lomax,
H.
for Separated
A.
and
Flows,"
Stowers,
AIAA
S.T.
Layer
Paper
(1988),
Approximation
and
Algebraic
78-257.
"'Review
of
Low-Reynolds
Number
_=_
Turbulence
Cebeci,
T.
and
Academic
Models
for Complex
Smith,
A.M.O.
Press,
New
Kim.
S-W.
Internal
(1974),
Separated
Analysis
of
Flows,"
AIAA-88-3006.
Turbulent
Boundary
Layers,
York.
w
Chen,
Y-S.
and
Extended
k-¢
(1987),
Turbulence
"'Computation
Closure
of
Model,"
Turbulent
NASA
Flows
CR-179204,
Using
NASA
an
Marshall
*ll*
Space
lli
Chien,
Flight
Center,
K. Y. (1982),
"Predictions
Reynolds-Number
r_
Crowe,
C.T.,
Fabris,
G.,
for
M.P.,
of Channel
and
Gas-Droplet
Harsha,
Modeling
P.T., of
AL.
Turbulence
Sharma,
Model
Huntsville,
Model,"
Stock,
Flows,
and
Layer
Boundary-Layer
Flows
J., Vol.
33-38.
AIAA
D.E.,
"The
J. Fluids.
Eng.,
Edelman,
Boundary
and
R.B.,
Flows
for
20, pp.
Partical-Source-in-Cell pp.
325-332,
(1981),
Low °
(PSI-CELL)
1977.
"Multiple-Scale
Scramjet
with
Turbulence
Applications,"
NASA
CR-
3433.
Fiveland,
W.A., for
Giridharan,
"Discrete-Ordinates
Rectangular
M.G.,
Przekwas,
Solutions
Enclosures,"
Lee, A.J.,
J.G., (1991)
J. of Heat
Krishnan, "A
A.,
of
Transfer,
Yang,
Computer
the
Radiative vol.
H.Q.,
Model
106,
Ibrahim, for
Liquid
Transport kpp.
699-706,
Equation 1984.
E., Chuech, Jet
Atomization
S., and in
r
82
=
Illl
4105/6
_=
Rocket
Thrust
Center,
Report
Giridharan,
M.G.,
Liquid
K., Launder,
Springer
Thrust
F.H., Viscous
and
A.,
for
NASA
Przekwas,
Between
Marshall
B.E.,
and
Space
New
Welch,
and
Gross, and
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rw
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u
m w
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b
410516
-
N/L A Nab0n_
I.
Aeronaut's
Report
Report
2. Government
No.
4. Title and
Documentation
Page
an_
Accession
No.
I
3. Recipient's
5. Report
Subtitle
Ccmbustion
Chamber
Analysis
Code
- Final
No.
Date
May
Report
Catalog
1993
6. Performing
Organization
Code
8. Performing
Organization
Report
No.
7. Author(a}
A.J. Przekwas, Y.G. and M.G. Giridharan 9. Performing
Organization
Name
Lai,
A.
Krishnan,
R.K. Avva 10. Work
Unit
and Address 11. Contract
CFD Research 3325 Trianan
Corporation Blvd.
12. Sponsoring
r AL
Agency
Name
15. Supplementary
No.
of Report
and Period
Covered
35805 Final Report Dec, 1989 - June
and Address
NASA Marsha]l Space Alabama 35812
w
or Grant
NAS8-37824 13. Type
Huntsville
No.
Fl_ght
14. Sponsoring
Center
Agency
1'993
Code
Notes
16. Abstract
A three-dimensional, Navier-Stokes code
time dependent, has been developed
Favre to
averaged, finite model compressible
volume and
incompressible flows (with and without chemical reactions) in liquid rocket engines. The code has a non-staggered formulation with generalized body-fitted-coordinates (BFC) capability. Higher order differencing methodologies such as MUSCL and Osher-Chakravarthy schemes are available. Turbulent flows can be modeled using any of the five turbulent models present in the code. A two-phase, two-liquid, Lagrangian spray model has been incorporated into the code. Chemical equilibrium and finite rate reaction models are available to model chemically reacting flows. The discrete ordinate method is used to model effects of thermal radiation. The code has been validated extensively against benchmark several
experimental data and has propulsion system components
been applied of the SSME 18. Di=¢_ibution
17. Key Words
(Suggested
Classif.
model the
NASA
FORM
(of this report)
1628 OCT 86
in
Statement
20. Security
Unclassified
Classif.
(of this page}
21.
No.
of pages
88 ""
flows STME.
by Author(s})
Cctnputational Fluid Dynamics, Reacting Flows, Two-Phase Sprays, Thermal Radiation
19, Security
to and
22.
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