Dec 23, 1993 - Figure. 3.1. Domain partitioning into a discretization, a) hexahedra on R3, b) composite hexahedra ...... Figure. 7.20b) presents in perspective.
//f
dE- I;;-9/Y AN ARBITRARY GRID CFD ALGORITHM FOR CONFIG URA TION AERODYNAMICS ANALYSIS Vol. Prepared by: A. J. Baker, G. S. Iannelli, COMPUTATIONAL
1.
P.D. Manhardt
MECHANICS
Theory
and
Validations
and J. A. Orzechowski
CORPORATION
,,t" o', r_ I
Field,
0
U r-"
o', Z
Prepared for: NASA Ames Research Moffett
m
0 0
Center
California
r7
FINAL REPORT FOR SBIR PHASE CONTRACT NO.: NAS2-12568
II
ac
0
>-0
a_
¢1[ tJ.. Z_Z
""0
Report: CMC TR2.1 December 1993
COMPUTATIONAL 601 Concord Knoxville,
Street, TN
Suite
37919-3382
Phone:
(615)
FAX:
(615)
emaih
ajbaker
- 94
MECHANICS
546-3664 546.7463 @comco3.akcess.com
'_[ 0
u'_ "r tr_
.--
_
>
r-
r..:OE (:3 _ ua
0
CORPORATION
116 U.S.A.
i ¢_
,_
@,
Table
of Contents
Abstract iv
Nomenclature Figures
vi
and Tables
ix Page
INTRODUCTION
lo
o
G
.
.
1
THE AERODYNAMICS
PROBLEM
2.1
Synopsis
2.2
Conservation
2.3
Turbulence,
2.4
Non-dimensionalizafion
2.5
Canonical
2.6
Well-posed
5 5
law
systems
5
Reynolds-averaging
8 9
form
10
boundary
APPROXIMATION,
conditions
ERROR
3.1
Overview
3.2
Approximation,
3.3
Error extremization,
3.4
Spatial
3.5
Fully
3.6
Summary
14
CONSTRAINT
15 15
measure
of error
the weak
semi-discrefization, discrete
form,
finite
algebraic
15
statement volume,
16 finite
statement
element
17 21 23
WELL-POSEDNESS,
STABILITY,
4.1
Overview
4.2
WeU-posedness,
boundary
4.3
Stability,
dissipation
4.4
Accuracy,
4.5
Stability,
4.6
Summary
THE
STATEMENT
CONVERGENCE
25 25
Taylor
asymptotic artificial
REM/AERO
5.1
Synopsis
5.2
Finite
element
5.3
REMI
algorithm
conditions
25 29
convergence
32
dissipation
33 39
CFD
ALGORITHM
43 43
TWS h algorithm matrix
statement
nomenclature illustrations
43 51
Page
5.4 The REMI 5.5
6.
algorithm
56
Summary
64
AUXILIARY 6.1
RaNS/E
PROCEDURES,
LINEAR
ALGEBRA
65
Synopsis
65
6.2
Initial
condition
generation
6.3
Implicit
6.4
Equilibrium
6.5
Tensor
6.6
Summary
7.
DISCUSSION
7.1
Synopsis
7.2
Subsonic
7.3
Transonic
7.4
Supersonic
inviscid
7.5
Hypersonic
Euler
7.6
Viscous
8.
SUMMARY
Runge-Kutta reacting
matrix
65
algorithm
67
air algorithm
70
factorization
71
product
77
AND
RESULTS
79 79
inviscid inviscid
transonic
AND
verifications,
d=2,3
verifications,
79
benchmarks,
verifications,
d=2
88
d=2
93
verification,
validation,
d=2 axisymmetric
100
benchmark,
validation,
d=2
104
CONCLUSIONS
111
References
117
Appendices A.
AKCESS.AERO
B. TWS h FE REMI C. AKCESS.AERO D.
AKCESS.AERO
REMI
template,
algorithm, template, REMI
d=2, Newton
122
d= 1,2,3
133 137
d=2, TP quasi-Newton template,
°°.
III
quasi-Newton,
d=3
151
Abstract
The
solicitation
evaluation
of
aerodynamics fitted to
for a
CFD
derivation
law
Phase
applicable
an "arbitrary
weak
possessing
completion of all theoretical calculus and vector field theory
•
intrinsic
embedding
of
methodologies
a continuum
Galerkin
approximate
solution
I project
(TWS)"
CFD
weak
in
the
capturing
statement
algorithm,
applicable
published
numerical
stability law
specification
useable
•
a fully discrete theory algebraic quasi-Newton iteration method stationary relaxation solvers
system eligible for any appropriate using sparse, block-banded and/or
Phase
the
I project
finite
conservation
•
any time
discretization,
results
provided
implicit,
element)
explicit
theoretical
spatial
Euler,
laminar
conservation
•
a diagonal construction
•
well-posed statement stability
flow
and
semi-
or multi-step
foundation
specific options for coding and verification in a Phase II project. theoretically independent option was selected as follows: •
for The
selection
level
Navier-Stokes
Reynolds-averaged
law systems
scalar
simplification
boundary
to the
conditions
for
Euler/Navier-Stokes methodology
TWS
numerical
continuum
constructions
iv
to
Navier-Stokes
employing
extremizing
volume,
contributed
amenable discretization
The
(finite
completion
body-
continuum
and
error for any approximation
any
a well-ordered,
attributes:
previously
for shock
critical
configuration
Reynolds-averaged
details
sixteen
requiring
Phase
and
and
•
with
to
The
design
three-dimensional
not
the following
•
•
grid
Euler
requested
to
statement
potential,
systems,
dissipation
I project
for robustness."
a "Taylor
transonic
conservation
using
system of
SBIR
algorithm
analysis,
coordinate
unsteady
the
dissipation
Taylor using
weak
Lyapunov
of
for each
a finite
element
spatial
quadrilateral distortion
•
single step 0-implicit time discretizations
•
a block-banded, matrix
original
verification
proposed
achieving
this goal
details
and
years,
from
for
extensible
to
the
the given The
associated
with
(only)
research
code
details
and to establish
the
cost, new
(FEMNAS)
was
Newton
proved
the
product arbitrary
Runge-Kutta
iteration
verification,
algorithm
via
benchmark
prototype
Phase
project
and
three-dimensional
II project
a practical
flows evolved,
starting
point,
was
In
necessity,
difficult
to
accurately
out
a
to confirm
especially
apparent
develop
the
necessary
the
two
myriad
operational The restriction
tensor
for
for
two-dimensional
results.
since
that period,
continued
extension,
and validation
and
performance
to thrash
and
fighter
very
formally
this
developed
benchmark
became
needed
coding
a generic
in the two-year
time
algorithm.
anticipated
about it
therefore
to provide
was
verification,
to two-dimensions
of
project
at no added
jacobian
implicit
quasi-Newton
three-dimensional
As
impossibility.
additional
tensor
potentially
two-stage
strategy
scope
for turbulent configuration.
an
a B-stable,
linear
of
geometries.
lex-delta
was
mesh
problems,
aerodynamics
The
and
using
elements
products
block
validation
hexahedral)
mesh-sweeping
tensor
algebraic,
@
semi-discretization
(hence
product
quasi-
arbitrarily-distorted
meshings. The
results
of these
fundamental
hypersonic
inviscid
and
verifications
reported
herein.
Newton
tensor
product
verification
and
operational
readiness
and
2-D
simulations
laminar-viscous Additionally,
jacobian
benchmark in the
flow
tests
are
the
fully
results.
production
test
cases
and
presented
herein,
along
3-D
theory code,
supersonic
constitutes
algorithm
AKCESS.,,
V
transonic,
3-D
The
detailed.
for
is only which
the
associated with now
and major quasi-
very
basic
approaching
is briefly
described
a
NOMENCLATURE a
A Ak B C CpCv
d D e
eh E
scalar
convection
element
matrix
kinetic
flux vector matrix
Courant
number,
specific
heat
dimension
mass
volume
4
kinetic
mesh
weak
measure,
turbulent dissipation
Im
mixing
mi
momentum
P
pressure
Pe
Peclet
kinetic
energy,
degree
conductivity
length
scale
differential
mesh
finite
polynomial
length
m
N
turbulent
length
reference
normal
space
basis eddy
ID
(resolution)
enthalpy
conductivity,
I'1
error
statement
thermal
Mach
data
factor
function
mass
element
(resolution)
Sobolev
Mk
domain
flux vector
trial space
M
d=3
total energy
flux vector
Galerkin
partial
prefix,
number
amplification
Z(.)
matrix
number
GWS
L
element
total energy,
specific
g
kt
d=2
approximation
dissipative
k
prefix,
matrix
semi-discrete
Euler
jacobian
capacities
specific
Eckert
d=l
of problem
diffusion
Eu
HP
prefix,
element
Ec
h
speed,
measure
equation
function vector
matrix,
(resolution)
collision
reference
factor,
number
coordinate element
basis
number
vi
function
molecular
mass
Pr
R R
RQ Re
Prandtl
number
polytropic
gas law
universal
gas law
residual
weak
Reynolds
source term
St
Stanton
t
time
ui U V
constant
statement
number
$
T
constant
number
temperature velocity
vector
reference scalar
(resolution)
velocity
speed
V
convection
INS
weak
wi
expansion
coefficient
xi
coordinate
system
Y
wall normal
Yi
species
-PUiUj
Reynolds
stress
Uiuj
kinematic
Reynolds
matrix
statement set (resolution)
coordinate
mass
fraction tensor
vii
stress
tensor
Ct
A
8V 82 V2 Vh E
artificial
dissipation
parameter
artificial
dissipation
parameter
central
difference
Kronecker central
difference
laplacian discrete
dissipation
alternator
0
implicitness
tt
absolute
¢a
potential
vi
operator level
wavelength
viscosity,
artificial
viscosity
density function
test
space
function
set
trial
space
function
set
domain boundary (t}
II
vorticity, column
i} T
row
[] FJ
wave matrix
number
matrix
square diagonal
operator
parameter
mode
P
derivative
tensor
kinematic eddy
operator
viscosity
Fourier
vt
second
divergence
error,
derivative
operator
Eijk
V
first
delta
matrix matrix
viii
dissipation
parameter
List
of
Figures
Figure 3.1
Domain
partitioning
into
Page 2O
a discretization,
a) hexahedra on R 3, b) composite hexahedra and subdivision into five tetrahedra 3.2
Tensor
product
dispositions, 4.1
Amplification
element
and phase
various weak from Chaffin 5.1
finite
domains
a) two-dimensional,
with
and node
eight
nodes
coordinate
20
b) three-dimensional
velocity
error distributions,
41
statement algorithms, and Baker(1994)
Finite
element
space
(_e)
5.2
Gauss
symmetric
5.3
AKCESS.AERO
REMI
template
for {FR}e
54
5.4
AKCESS.AERO
REMI
template
for {FM1}e
57
6.1
AKCESS.AERO
REMI
template
jacobian
6.2
AKCESS.AERO
REMI
template,
7.1
Converging subsonic b) d=3, 4:1 area ratio
7.2
AKCESS.,
domains
for tensor
in physical product
basis
quadrature
template
duct
space
46
for d=2
47
form
coordinates
[RE,E]e,
TP jacobian
verification,
for REMI
(D.e) and transform
76
d=2
[RE, ETP],
a) d=2, 2:1 area
d=2
77
ratio,
80
81
d=2 IC generation,
a) nodal density, element-averaged metric data, b) average density, Gauss quadrature element matrices. 7.3
AKCESS.,, template for REMI d=3 IC generation, density, Gauss quadrature element matrices.
7.4
Converging
duct
a) modestly
non-cartesian
7.5a
IC algorithm Main
validation
TWS h density
-- 0.2, averaged
metric
check mesh
case,
d=2,
A, b) highly
solution,
84 distorted
d=2 converging
template,
ix
82
averaged
a) mesh
A,
mesh, duct,
b) mesh
B. 85
B
7.5b
IC algorithm Main
7.6
TWS h density
=_0.2, Gauss
REMI
FE
7.8
duct,
Total
pressure
duct,
Main
duct
nozzle
verification
REMI
deLaval
nodal
nozzle,
algorithm,
REMI
B, d=2,
at=O.O05,
B.
solutions,
87
B, d=2, a) [5=0.3, b) _=0.2 89
converging
a) _=0.3,
b) _=0.2.
--- 0.2, 4:1 area
ratio,
solution
velocity
with
a) cross-section
for axial
REMI
pressure
solutions,
pressure
solution
A,
9O
distributions,
91
momentum.
TWS h solution a) t=0.4,
number,
92
15% parabolic
94
for Mach
b) t=l.0,
c) t=1.2,
e) t=2.8
algorithm
arc, scalar
TWS h Euler
_=0.2
c) entropy, REMI
Euler
problem,
unsteady
d) mesh
d=3, Main
b) REMI
deLaval
d) d=1.8,
7.12
solution, [5=0.3.
IRK ODE
7.11
verification,
scalar
c) mesh
_, Mesh
_ d=2, mesh
overlay,
duct,
steady-state
= 0.2, scalar
---0.2, scalar
Converging
b) steady 7.10
Main
Euler
86
d=2 converging
template,
loss error for REMI
a) IC density
7.9
quadrature
TWS h algorithm
converging 7.7
solution,
d)
solution,
{1} T a) 65x35 Mach
algorithm
mesh,
number,
e) axial
TWS h Euler
solution,
arc, Main
= 0.68,
scalar
of a) axial
momentum,
_=0.2,
steady-state, b) velocity
steady-state,
{1} T perspective
b) transverse
vector
field,
momentum. 15% parabolic
and contour
graphs
c) Mach
number,
momentum,
95
d) pressure. 7.13
REMI
algorithm
TWS h Euler
solution,
0 = 20 °, [5 = 0.3 {1}T, a) initial isoclines, 7.14
REMI
c)
1st adapted
algorithm
mesh,
TWS h Euler
0=20 °, [5 = 0.3 {1} T, a) final contour 7.15
REMI
and perspective algorithm
65x35
[5 = 0.3 {1}T, solution-adapted isocline distributions.
uniform
d) resultant solution,
adapted
graphs,
TWS h Euler
supersonic
mesh,
65x35
X
mesh,
b) density
supersonic meshing,
96
isoclines. wedge
number,
flow,
b) density
density
supersonic
c) Mach
solution,
wedge
flow,
98
isoclines;
d) pressure. shock
resultant
reflection, density
99
7.16
REMI
algorithm
TWS h Euler
_a = 0.3 {1}T final mesh, a) pressure, b) entropy. 7.17
REMI
algorithm
blunt-body
surface 7.18
algorithm
REMI
arc,
7.21 7.22
algorithm
_=0.2,
boundary
algorithm
shock-boundary separation
Sub-grid wave, k=l;
region
viscous
Burgers
shoch
trailing
4% parabolic
of a) axial
107
shock
a) density,
laminar
Main
resolution;
comparisons
symbols
are data from
problems,
= 2.15,
inviscid
k=l or 2, b) p-embedded Re=105,
xi
108
b) Mach
velocity
simulation,
momentum,
problem.
to 103, d) p-embedded
106
edge.
interaction,
FE verification
105
momentum,
layer
WS n solution,
or 2, solution, converged verged to 10 -9 .
4% parabolic
b) axial
laminar,
solution,
c) skin friction,
p-embeddin_ a) standard
mesh,
mass
space.
validation
TWS h Navier-Stokes
b) surface pressures et al (1987) 8.1
layer
surface
presentations
near
layer, Re=105, 13= 0.3,{1}T c) axial momentum.
Supersonic a) REMI
closeup
103
species
viscous,
solution
perspective
mesh,
Ma_=8,
real-air
laminar,
in nodal
a)
102
density.
streamline/body
b) companion
101
perspective
steady-state,
TWS h Navier-Stokes
c) pressure
Shock-laminar
boundary number, 7.23
plotted
Re=4.0xl06,
REMI
stagnation
d)
_=0.2{ 1 }T, a) non-uniform
momentum
b) pressure,
solution,
hypersonic
Ma_=6.5, and
number,
TWS h Navier-Stokes,
arc, Re=4.0xl06,
REMI
meshes,
reflection,
graphs,
Euler solutions,
quad
Mach
of temperature,
algorithm
c) axial
of c)
real-air
state
shock
perspective
= 8.0, contour
TWS h Euler
and
distributions fractions
7.20
Ma_
supersonic
and
65x35
distribution;
a) ideal-air
7.19
adapted
distributions
REMI
contour
TWS h steady
flow
b) density
solutions,
c) standard solution,
Re=105,
109
on Degrez
square solution, WS h, k=l k=l,
con-
116
List of Tables Table Page
2.1 4.1
Euler-admissable Dirichlet boundary conditions Summary of CFD algorithms from Baker and Kim (1987) Gauss
quadrature
FE k=l basis
within
coordinates
interpolation
Taylor
and weights,
matrix
[M200]
xii
(BC)
weak
statement,
14 40
d=2
47
for d=1,2,3
48
I. INTRODUCTION
The work and
plan
broad-range
efficient
validation
on
regions, with
bounded
the
of
appeared
development
projects
of the
iteration
Both
was
with
relative
the MacCormack
diffusion,
viewed
genuine
viscous
splitting
methods
dissipative
as
combination
of Riemann
solvers,
However,
differencing,
and
order
These
spatial
nominally-uniform
aerodynamics
accuracy
various
volume
following
a variety
a
CFD
detraction This
boundary-fitted geometry.
law
was
were
meshings
CFD
algorithm,
to
to
the
Euler which
meshing-induced
total
vectorizability.)
1970s,
leading
algorithm,
to
designed
employed
for
development
of flux
with
Roe
extension were
(1981),
via e.g.,
some
differencing.
specifically-added
expansion
vector
developed,
upwind
to
to
to RaNS
all employing
directional
intrinsic
artificial
compromise
(1978),
of
added
a potential
variants
absence
artificial
characteristic-direction
were
developed
using
finite
to avoid
differencing. code-implemented The on
abiding
the
coordinate
An alternative
historic
middle
Many
was
The
to its
theories and
and
exceptional
stiffness.
systems,
for stencil
upwind
applied
due
the
prompted
Osher
semi-discretizations. cartesian
commensurate
computer,
factored
CFD
diffusion
algorithms
in
implicit
averaged-states
of switches
of direct
using (RaNS)
an
explicit
due
to parasitic
(1982),
numerical
to
witnessed
applications
initiated
(1976)
feature
and
(1969)
differencing.
vanLeer
versatility
has
a rebirth,
conservation
distinguishing
diffusion.
enjoyed
analysis.
central
application
of three-dimensional
requirement.
RaNS
a theoretical
(1978),
common
analysis
insensitivity
for hyperbolic
Steger-Warming
personnel
and Beam-Warming
both
and
Navier-Stokes
meshing
technical
thereby
aerodynamics
flux-vector
with
community
Beam-Warming
on
accurate
geometries
of discretizations
CFD
for
were
was
three-dimensional
the MacCormack
since
focus
stable,
coding
itself.
CFD
(It has
research
efficient
both
for derivation,
Reynolds-averaged
surfaces,
the
called
CFD algorithm, The
generation
inappropriate
stiffness.
Several
finite
decades
development
originally
low
two
aerodynamics
aerodynamics
parasitic
was
aerodynamics
resources,
configuration
and
of a CFD algorithm
past
expenditure
the
by
the capabilities In
The
to this
element
for general
laminar,
Adjunct
project
meshings.
analyses
(Euler),
simulations.
finite
arbitrary
aerodynamics flow
II contractual
of a new
absolutely
configuration inviscid
of this SBIR Phase
character
transformed
transformation spatial
semi-discretization
was
difference
use of structured,
computational from
the procedure
domain, transformed emerged
or
in
the
1980's,
using
in the
physical
codes
were
domain, time
specifically
"hp")
accuracy
with
This memory extra
solution
steps
Without
exception
cartesian
mesh
splitting
implicit
to compute the
triangle/tetrahedra
From
with
the
Phase
algorithm
that
mathematical
extension
to (meshing
hopefully
leading
weak
have
to attainment
statement
We the
goal,
moved and
to use
stiffness. to
as large and
structured
of operator-
Coincidentally,
finite
volume
CFD
to derive,
code
and validate
mesh
versatility
and
arbitrary
quality genuine
(Euler)
shock-capturing,
RaNS
applications.
of approximation
flux vector
linear basis, tensor of quads/hexahedra
product
tensor product iteration block
mesh,
implicit
manipulation
two step
finite
derived
meshing
the above
solution
and The
the
direct
decisions
error to produce
conservation law systems with intrinsic boundary conditions, suitable for Euler
i.e., an arbitrary
such
procedures
at least,
sought
exhibited
for extremization
and
considered
enhanced
of this goal were:
0-implicit one step discretizations
algebraic,
for
issues
has
amenable
enforceable via weak statement generated ideal and real-gas equations of state
matrix algebra
FE basis
refinement/de-refinement.
become
11 project
for)
Euler/RaNS
continuum well-posed
for mesh
parasitic
robustness,
requirements
Taylor-series
and
estimation.
to handle
since
Galerkin applications
meshing
computational
regions,
employed
et al (1991).
in 1987,
of a CFD
adaptability
methods
refined
promising
applications
layer
application
Taylor
generation/adaptation data
RaNS
in boundary
meshings
view
mesh
measure to
integration
c.f., Barth, this
ingredients
extension
embedding time
error
and
elements
and
the
Locally
error
by associated
stiffness,
from
appeared
via local
The
simulations
demonstrated,
efficiency
parasitic
Euler
method.
methodology
of finite
et al (1986).
derived
were
is moderated
requirements,
constructions,
tetrahedra
adaptive
promise
to
dissipation
degree-of-freedom
bright
restricted
meshings"
Oden,
of the Lax-Wendroff
and
solution
non-structured
et al (1984),
generally
1984)
triangles
(termed
Loehner,
numerical
(Donea
nested
"absolutely
e.g.,
explicit,
added
generalization using
potentially
surface
combination,
element
spatial
to solution
if successful,
configuration
aerodynamics
2
integrals
Runge-Kutta
quasi-Newton
amenable
dissipation and RaNS,
time
semi-discretization
block-banded
linear
adaptivity
would CFD
lead
to attainment
algorithm,
applicable
of
f
to both
Euler
and
RaNS
circumvented
some
mathematical
robustness,
However, goal
was
the
conservation
detractions
severely
after
progress
included
practical
difficulties and
factors
also
spanning
in the
procedures
on "arbitrary"
additional
key areas
were
contributing of numerical
issues
contributed
years.
During
this period,
to the
both
and
Benchmarks
necessity.
The
Navier-Stokes generated
CFD
limited
approximation block
all
subsonic
tensor
this
slow
(6 months)
and
computer
(before
and code
practice
clean
theory
proved
product
matrix
algebra
lead
a no-cost
theoretical
effort
to resolution and
were
by Iannelli,
hypersonic
issues,
executed
form
as
Euler
and validations
over
algorithmic
to two-dimensional and
extension
pursued
of many
validations
supersonic
viable,
stiffly-stable Euler
stiffness inverse
family.
generated Reynolds
constitute
a comprehensive,
element
to two-dimensions,
and
in transonic,
by
his
a practical
and
reported
The
The
laminar
herein
and
second
It is useful via boundary number.
and definition
resultant
evaluations
order
were
for shock
The
meshing range
of the developed
the
replacement
of
with
transverse
these
results
arbitrary
3
as well
grid
finite
algorithm..
form
(ENO)
is achieved, for the
_term
non-rectangular
computational
weak
time-marching
capturing
flows
are highly
the
of the
non-oscillatory
diagonal
leave on
accurate
REMI
hypersonic
meshings
flows
Runge-Kutta
layer
and
assessment
CFD
essentially
a simplified
far-field
appropriate
implicit
quality
with
positive
statement
supersonic
remeshing
supersonic
via
weak
high
TWS h theory.
oscillation, integrals.
derivation
central
to request
for benchmark
finite
solution-adaptive
surface
of
results
to shocks
underlying
without
step
statements
need
limited
transonic,
arbitrary-grid,
Although
and
of
problem
reported
developed
of the
range
code,
to
project-
by this code.
The
using
computer
and
to the
workstation
elegantly
of the
two-dimensional
Theoretical
as the
a dedicated
(1991),
FEMNAS
available).
capability
accrued
Ames
for
meshings.
his dissertation practical.
potential
contributing
3300
selections
meshings.
a fledgling,
factors
dissipation
distorted)
exhibiting
meshings
NASA
role,
these
use of "arbitrary"
only
regular
on the
view,
the verification
of our SGI Model
speed
(highly
that
Practical
as he developed
developed
and
to achieve
years.
several
theoretical
while
efficiency
operation
a central
constructions,
on rather
T-1 communication
played
In our
to the extent
delivery
in remote
descriptions.
required
two
delayed
incomplete
These
of detail
capability
a period
NASNET
operating
under-estimated,
verification/benchmark end
of previous
code
volume
law
domain
statement-generated
algorithm
appears
for the
0-implicit
as handling resolution are
discussed
element
CFD
the
a truly single parasitic
on the
order
following algorithm.
As these operational
advances
in
our
became
achieved,
emerging
the
AKCESS.,
software
previously
established
"research"
codes.
algorithm
is thoroughly
detailed
herein,
quasi-Newton generated move to
jacobian. by AKCESS.*
rapidly
3-D,
as
to recovery
Only
modest
to date
are
of reported
AKCESS.*
moves
to
three-dimensional
The
platform,
resultant
including
the
and
FEMNAS
3-D
readiness
to
TWS h CFD
"REMr'
FE
product
its
results
we
as benchmark in
factorized
numerical
However,
as well
become
"code"
Euler
included.
has
successor
tensor
3-D
tests,
operational
the
3-D
verification-level available
algorithm
expect
to
extensions
parallel-processing
implementation. The
near-term
to greatly
shorten
we
warrants
hope
project required
completion. to convert
emergence the time
of this versatile
software
to implement/validate
the
significant
We
have
CFD theory
theoretical
government
certainly
learned
to genuine,
and
specifically
and/or
personal
a lasting
robust
4
platform,
lesson
and convergent
designed
practical
resources
committed
on estimating code
musings,
the
practice.
to effort
2. THE AERODYNAMICS
PROBLEM STATEMENT
2.1 Synopsis The goal and
is to establish
Navier-Stokes
a robust,
conservation
accurate
law
systems
configuration
aerodynamics
problem
statement.
mathematical
descriptions,
including
closure
and dissipation leads 2.2
mechanisms.
to mathematical
The
Following
law
basic
assumption
is
and
energy,
dissipative
and
modeled-turbulence
variable,
Denote as
q=q(x,t),
that
set
"q"
(PDE)
differential
and
in
equation,
spanning
a region
For where Therein, where
u i is termed Continuing
is the array,
included
state the
in
a
associated
thermodynamics
an eigenvalue
conservation
law
analysis
systems.
statements
the
variables,
the
expressed
as
desired usually
denote
i.e.,
the
state array
density,
L(-)
an
and is the
of
Euclidean
and/or
in (2.1), for laminar
variable
the
the
(matrix)
and
the
scalar
state
Then,
the
non-linear,
partial
x
corresponding
flow
specific
dependent
with
denotes
scalar
the
Finally,
is
vector
{p.m, EIT" transpose.
energy. f, while
s is a source
fjv term
Both
functional
tensor
real
m i = Pui,
modeling.
of state
cartesian
its
resolution
total internal
fv.
positive
q(x,t),=
kinetic flux
equation
heat transfer,
R + the
as
on closure
description,
a
t denotes
with
flux vector
of
9_d (1< d < 3), and
"T"
of the
form
system
selected
vector
resolution
Navier-Stokes viscous
the
coordinate
superscript
E=pe is the volume
as needed,
and
and
called
homogeneous
(global)
space
is usually
of the dissipative
for generality
mass,
mathematical
array.
governing
denotes
resolution
m is the momentum
"velocity,"
for
for thermodynamics,
yields
to
system,
following,
resolution
For the Euler
flow
for
domain of definition of (2.1) is R + x f2 with by I and Dc _d _ (x.lxl,,
+
_
_
_
#
=
I +
I
I
I
_
°
_
÷
I
.c:
I i ,4_
N I
A
O
.... I
_
=
I
_
=== I
.=,.¢
_3
L)
0
_
0
_
0
O ;>-.
33
¢=
.c
"_
.u L3
o
"
i O_GtNAL
PAGE
OF POO_ _m/UJl_ 4O
F_
0.1
I
-0.I
I
l
FD,All FE,Beta=0 Linear FE, Beta=. 1 Linear FE,Beta=.2 QUICK-5 FV QUICK-3 FV
I
..... - m--. ---x...... -_--.o-\
"i1:_
-0.2
C=I/2 -0.3
8=-I/2
-0.4 w
I
I
I
I
16
8
4
3
Modal
I
Wavelenmh
l
2
(_=nA x)
I
I
C=1/2 0=-112
o
.W •o ooO o° oO s
..4} .-
.-°"
._
°-
.-
! 16
8
i 3
4
Modal Wavelength Figure
4.1
Amplification statement
and
phase
algorithms,
velocity Chaffin 41
2
(X_-nA x)
error
distributions,
& Baker
(1994)
various
weak
5. 5.1
The
REMI
and
FE TWS h CFD
theoretically
algorithm
described.
corresponding
is coined Momentum
"REMI," the acronym _}T for 1< I _