C-type grids around wing-sections. The grid sensi- tivity of the domain with respect to design and grid ..... although acceptable for loosely-coupled systems, is likely to produce sub- optimal results_ ...... .n _091 (r, P1) -_or._al f(t,-oj. 1. 0. _Oyl(r,P ...
NASA-CR-192980 t4 io
£J
DEPARTMENT COLLEGE OLD
OF
MECHANICAL
OF ENGINEERING
DOMINION
NORFOLK,
ENGINERRING AND
AND
MECHANICS
TECHNOLOGY
UNIVERSITY
VIRGINIA
23529
ii
GRID LJ I-
©
SENSITIVITY
FOR
OPTIMIZATION
AND
AERODYNAMIC
FLOW
ANALYSIS
_9
U
._r?,_A,U T--By
/W -v ,2 -C,_Q
I. Sadrehaghighi,
Graduate
Research
Assistant
and 7: F_
>-
S. N. Tiwari,
Principal
p.! v
Investigator
U Progress IJ w
Q (5
Report
For the
period
ended
Prepared
for
National
Aeronautics
Langley
Research
Hampton,
December
1992
and
Administration
Space
Center
VA 23665
!: i
Under Cooperative Dr.
Robert
Agreement E. Smith
ACD - Computer
Jr.,
NCC1 Technical
Applications
(NASA-CR-192980) FOR AERODYNAMIC
L April
1993
- 68 Monitor
Branch GRID SENSITIVITY OPTIMIZATION AND
N93-251"
FLOW ANALYSIS Progress report, period ending Dec. 1992 (Old Dominion Univ.) 120 p
Unclas
G3/02 L
0160302
DEPARTMENT COLLEGE r
OLD
,
OF MECHANICAL OF
ENGINEERING
DOMINION
NORFOLK,
ENGINERRING AND
AND
MECHANICS
TECHNOLOGY
UNIVERSITY
VIRGINIA
23529
t
GRID
L
SENSITIVITY
OPTIMIZATION
FOR AND
FLOW
By I. Sadrehaghighi, r
Research
Assistant
and
:
S. N. Tiwari,
Progress L
Graduate
Investigator
Report
For the
.
Principal
period
ended
Prepared
for
National
Aeronautics
Langley
Research
Hampton,
December
1992
and
Administration
Space
Center
VA 23665
Under Cooperative Dr. r_
A
_w
L-" w
7 =::=-- •
Robert
Agreement E. Smith
ACD
- Computer
April
1993
• -_v
Jr.,
NCC1 Technical
Applications
- 68 Monitor
Branch
AERODYNAMIC ANALYSIS
o
FOREWORD
L=
m
This
is the
progress
Dimensional
report
on the
Navier-Stokes
of the
project,
special
"Grid
Sensitivity
research
Equations
attention
was
project
" Numerical
for Closed-Bluff directed
Solutions
Bodies".
toward
research
Within
of Three-
the guidelines
activities
in the
area
of
w
for Aerodynamic
Optimization
and
Flow
Analysis."
The
period
of
J
performance This -
work
of this specific was supported
research by the
was January
NASA
1, 1992 through
Langley
Research
agreement
was
December
Center
31, 1992.
through
Cooperate
by
Robert
i
Agreement
NCC1-68.
Smith
Jr.
of Analysis
NASA
Langley
Research
The and
cooperate Computation
Center,
Mall
Division Stop
125.
w
m
w
ii
monitored
(Computer
Dr.
Applications
Branch),
E.
ABSTRACT GRID SENSITIVITY OPTIMIZATION
FOR AERODYNAMIC AND FLOW ANALYSIS
Ideen Sadrehaghighi Old Dominion University, 1993 Director: Dr. Surendra N. Tiwari
An design
parameters
cedures
L
algorithm
is developed
for aerodynamic
are developed
rameters
relations
grid
Two
the
sensitivity
grid
as an example.
defining
the
optimization.
for investigating
of a wing-section
(physical)
to obtain
NACA
The
four-digit
first
sensitivity
distinct
with
respect
parameterization
with
procedure
respect
pro-
to design
is based
wing-sections.
to
pa-
on traditional
The
second
is advocat-
functions
such
as NURBS
m
ing
L J
a novel
(Non-Uniform active
w
(geometrical) Rational
algebraic
(TBGG) tivity
with
differentiation
technique,
complex
grids
to design
grid
configurations.
wing-section
geometry.
as Two-Boundary
around
and
equations.
spline
the
known
C-type
respect
of the
using
for defining
to generate
domain
geometrically
B-Splines)
grid generation
is employed
of the
direct
parameterization
grid
Grid
wing-sections. parameters
A hybrid
approach
A comparison
of the
An interGeneration
The
has been
grid
sensi-
obtained
is proposed sensitivity
by
for more coeffÉcients
m
with
those
ity of the
obtained approach.
compressible package surface
using
aerodynamic
two-dimensional
has been using
The
a finite-difference
both
physical
sensitivity
thin-layer
introduced
into and
the
approach
is made
coefficients
Navier-Stokes algorithm
geometric
in order
to verify are
the
obtained
feasibilusing
equations.
An
to optimize
the wing-section
parameterization.
Results
the
optimization
demonstrate
m
substantially
improved
design,
particularly
in the ooo
111
geometric
parameterization
case.
a
__=
TABLE
OF
CONTENTS
page FOREWORD ABSTRACT TABLE
£
OF
.................................................................. .................................................................. CONTENTS
LIST OF TABLES LIST
ii iii
......................................................
iv
...........................................................
OF FIGURES
vii
..........................................................
LIST OF SYMBOLES
viii
........................................................
xi
Chapter 1. INTRODUCTION
..........................................................
1.1 Motivation r
...........................................................
1.2 Literature
Survey
1.3 Objectives
of Present
2. PHYSICAL
z
1
MODEL
..
...................................................... Study
1 5
............................................
.......................................................
7 10
V
2.1 Wing-Section
rz. u,r#
3. GRID
Example
2.1.1
Physical
2.1.2
Geometric
Representation
GENERATION
3.1 Introduction
3.4 Transfinite
..........................................
Representation
........................................
.......................
.- .............................
Coordinate
Transformation
10 13 22 22
...........................
24
Discritization
...............................................
28
Interpolation
..............................................
31
L w
10
..........................................................
3.2 Boundary-Fitted 3.3 Boundary
................................................
°
iv
3.5 Two-Boundary
Grid
4. THEORETICAL 4.1 Generic
Technique
FORMULATION Sensitivity
4.2 Aerodynamic 4.3 Surface
Generation
40
..........................................
Equation
Parameterization
34
.........................................
Equation
Sensitivity
............................
40
.....................................
41
..............................................
44
_J
5. METHOD
OF SOLUTION
5.1 Introduction
E
.
47
5.2 Grid
Sensitivity
with
Respect
to Design
5.3 Grid
Sensitivity
with
Respect
to Grid
5.4 Flow
Analysis
5.5 Flow
Sensitivity
6. RESULTS
,
6.1 Case
47
..........................................................
5.6 Optimization =
.................................................
AND
and
Boundary
Analysis Problem
DISCUSSION
1: NACA
0012
Parameters Parameters
................
Conditions
..................
...........................
48 51 52
..........................................
54
.............................................
56
.............................................
59
Wing-Section
.....................................
60
6.1.1
Grid
Sensitivity
..................................................
60
6.1.2
Flow
Sensitivity
.................................................
61
6.2 Case
2: NACA
8512
Wing-Section
.....................................
67
p
6.2.1
Grid
Sensitivity
..................................................
67
6.2.2
Flow
Sensitivity
and
68
6.2 Case
3: Generic
Wing-Sectlon
................................
.........................................
6.3.1
Grid
Sensitivity
..................................................
6.3.2
Flow
Sensitivity
and
7. CONCLUSION REFERENCES a
Optimization
AND
Optimization
RECOMMENDATIONS
83 83
................................
"84
..............................
93
...............................................................
95
APPENDICES: A. TRANSFINITE TION
INTERPOLATION
WITH
.........................................................................
LAGRANGIAN
BLENDING
FUNC99
v
_=
,
A.1 Surface
Grid Generation
..........................................
99
A.2 Volume
Grid Generation
.........................................
101
W
=
v
z
-v'
vi
LIST
OF
TABLES
Table 6.1 Litt and
w
r
page drag
sensitivities
with
respect
to design
parameter
T
..............
62
6.2 Lift and drag sensitivities with respect to vector of design parameter Xo ....
71
6.3 Lift and drag sensitivities with respect to vector of grid parameter Xo ......
71
6.4 Comparison of initial and optimized performance variables ..................
71
6.5 Comparison of initial and optimized design parameters ......................
71
6.6 Lift and drag sensitivities with respect to vector of design parameter Xo ....
86
6.7 Comparisonof initial and optimized performance variables ..................
86
6.8 Comparisonof initial and optimized design parameters ......................
86
,u
w
r'!
vii
L
V
LIST
OF
FIGURES
v
Figure
p_ge
2.1 Wing-section
specification
for NACA
2.2 Wing-section
specification
using
four-digit
series
.......................
12
_w
2.3 Quadratic
V
NURBS
(option
of increasing
the number
of control
2.5 Effects
of increasing
the number
of control
2.6 Wing-section
specification
2.7 Seven
points
control
of control
2.9 Cubic
basis
function
and
3.2 Different
mapping
3.3 Dual-block
grid
function
3.6 Hermite
cubic
3.7 Dual-block
(p=3)
types
blending
domain
on camber
points
specification
on quadratic
grid
wing-section basis
function
2) ......................... using
NURBS
(option
for a generic
upper
and
configuration
functions lower
19 3)
....
20
-25
..................................
distribution
18
21
....................................
airplane
18
20
................................................
27 ...............
...........................
boundaries
......................
29 32 36
functions
..........................................
36
decomposition
.........................................
38
3.8 Control domain for lower boundary discretization 3.9 Example
and
17
.........................................
of a wing-section
typical
connecting
points
(option
coordinates
topology from
NURB$
movement
computational
resulting
3.5 Cubic
using
wing-section
point
3.1 Physical
3.4 Grid
16
basis function (p=2) for camber line (option 1) ...................
2.4 Effects
2.8 Effects
1) .........................
for NACA
0012
wing-section
..........................
.................................
38 39
v
3.10 Example grid for NACA 8512 wing-section 4.1 Coe_cient wing-sections
of drag
versus
maximum
thickness
............................................................ Jgm VIII
................................
39
T for symmetrical 46
L
5.1 Design
optimization
strategy
loop ..........................................
58
V
qv
z
6.1 Coordinate
sensitivity
with respect
to maximum
thickness
T (DD)
.........
63
6.2 Coordinate
sensitivity
with respect
to maximum
thickness
T (FD)
..........
64
6.3 Residual
convergence
6.4 Pressure
contours
6.5 Mach
number
6.6 Sensitivity
history for NACA
contours
equation
.............................................. 0012
wing-section
for NACA convergence
0012
.....................
wing-section
history
65 '........
.......................
66
...................................
6.7 Coordinate
sensitivity
with respect
to maximum
thickness
6.8 Coordinate
sensitivity
with respect
to maximum
camber
6.9 Coordinate
sensitivity
with respect
to location
66 T ................
sensitivity
with respect
to grid stretching
6.11 Coordinate
sensitivity
with respect
to grid distribution
6.12
Coordinate
sensitivity
with respect
to grid orthogonality
6.13
Coordinate
sensitivity
with
to outer
6.14
Pressure
6.15
Mach
6.16
Surface
6.17
Sensitivity
6.18
Original
6.19
Design
72
M .................
of maximum
6.10 Coordinate
65
73
camber
parameter
B3
parameter
C .......
74
........
75
B1 ......
parameter
K1 .....
76 77
-qp,
contours
number
for NACA
contours
pressure
respect
wing-section
for NACA
coefficient
equation
8512
8512
for NACA
convergence
boundary
history
L ..........
...........................
wing-section 8512
location
......................
wing-section
..................
..................................
78 79 79 80 81
E
and
optimized
wing-section
......................................
82
r
parameters
representation 6.20
=
Cubic
basis
using
seven
of wing-section function
for control
control
points
(option
3) ..................................
points
1 and
5 .............................
87 88
6.21 Coordinate
sensitivity
with
respect
to Y1 ..................................
89
6.22
Coordinate
sensitivity
with
respect
to Y5 ..................................
90
6.23
Optimization
cycle
convergence
6.24
Original
A.1 Grid
and
optimized
on the solid
history
wing-section
(physical)
surfaces
................................... ......................................
......................................
ix
m
NURBS
91 92 103
t i A.2 Domain
decomposition
A.3 Grid on the physical A.4 Grid on the outer A.5 Volume
................................................... and non-physical
boundary
grid (constant-I)
surfaces
...........................
.............................................. surhce
.........................................
:: r
=
103
•
r
Ne_
w_ = =
f
X
104 105 106
E
LIST
Of
SYMBOLS
=
v
v
_J i
a
= local
speed
of sound
B_
= stretching
C
= location
Co
= drag
CI
= skin friction
CL
= lift coefficient
Cp
= pressure
Cx, Cv
= force
D_
= NURBS
e
= energy
F
= physical
/
= objective
G
= dependent
F,G
= inviscid
fluxes
= viscous
flux vector
parameter of maximum
camber
coefficient coefficient
coefficient
coefficients
in x and
control per unit
point
y directions
coordinate
volume
model
function parameter
_
g
= optimization
constraints
H
= independent
parameter
J
= jacobian
K1, If_
= magnitude
L
= far-field
M
= maximum
of transformation of orthogonality boundary
vectors
location
camber
r
xi
%/
M
= banded
part
m
= number
of knots
Moo
= free-stream = B-spline
of coefficient
of a NURBS
Mach basis
matrix curve
number
function
v
N
= off-banded
ft
= number
n
= unit
P
-- vector
Pi
= local
P
= degree
Pr
-" Prandtl
Q
= vector
Q.
= steady-state
R(r)
= bottom
R
= steady-state
part
of coefficient
of control
normal
points
matrix
of NURBS
vector
of independent
parameters
pressure
w
F
of a NURBS
curve
number of field variables field variables
boundary
= NURBS
orthogonality
residual basis
function
R•
= Riemann
Rcoo
= free-stream
L
r
= bottom
boundary
w
r
= uniform
knot
s
= top
boundary
grid
= top
boundary
orthogonality
.r7
= vector
vector
invariants Reynolds grid
vector
of search
number distribution
of NURBS distribution vector
direction
T
= maximum
thickness
t
= grid
U
= horizontal
interpolant
uoo
= free-stream
velocity
stretching
parameter
component
xii
= contravariantvelocityvectors
U,V lt)
V)
W
= velocitycomponents
in physicaldomain
V
= verticalinterpolant
X
= vector of physicalcoordinates
XB
= vector of surface coordinates
Xo
= vector of design parameters
Xo
= vector of grid parameters
X,,Y,
= control point coordinates
X,y,Z
= physical coordinates
v
V
Xl,
Yl
= surface coordinates
xa,
Y2
= far-field boundary coordinates
Yc,
YT
= camber and thickness curve ordinates
--2
Greek
r_ _=
Symbols
a
= angle
of attack
ct 7
= x-direction
blending
parameter
3_
= y-direction
blending
parameter
7
= scalar
/f
= Kronecker
0
= surface
_, O, (
= computational
p
= density
poo
= free-stream
move
parameter delta
slope coordinates
density
= summation rl
= local
shear
-r
= viscous
wi
= NURBS
stress
stress
term
174
curve
weighting
parameter
Xln
v
Chapter
1
INTRODUCTION
1.1 Motivation Integrated mary can
objective
Speed
Civil
Transport
(HSCT)
aircraft
interactions
are
ciplines
a wide
are:
confined
range
models
interconnected
and
affect
A Multidisciplinary i
•
with
sufficient
required
and
(NASP) flight
The Each
High
conditions,
process
requires
analysis
is based
with
the primary
propulsion.
and
a discipline.
engineering
These
dis-
disciplines
are
other.
Design
information
materials
laws associated
conditions,
control,
interest
of extreme
disciplines.
a pri-
The sudden
Plane
important.
physical
flight
structures, each
because
of engineering describing
has become
composite
AeroSpace
particularly
to atmospheric
aerodynamics,
and
as National where,
components
community.
of complex
such
mathematical
For a vehicle
introduction
vehicles,
over
of airplane
in aerodynamic
aerospace
analyses
on solving
and optimization
researchers
to the
interdisciplinary
many
v
for most
be attributed
by advanced
the
design
Optimization
to predict
(MDO)
the influence
would
of a design
provide
the designer
parameter
on all rel-
v
evant
disciplines.
optimization formation
w
by each discipline from
process,
although
optimal
results_
proach
The traditional
the preceding acceptable
the
to MDO
in a sequential analysis
manner
of another
which
analysis
are
and
more
is to perform where
discipline.
for loosely-coupled
For systems
is to perform
approach
systems, tightly
optimization
one discipline This tedious is likely
coupled, at each
the analysis
uses the inand
lengthy
to produce
a relatively discipline
and
sub-
new
concurrently.
ap-
2
As opposed order
to a sequential
(i.e.,
a design
derivative)
change
are achieved which
on
by a system
On
disciplines
system
local
level,
technique
thus
enables
involved.
of equations
the
the
this
information, all the
communicate
ciplines.
approach,
known
response
within
the
him
to to predict
The
interaction
as Global
due
each
supplies
with
the
influence
among
Sensitivity
to design
discipline,
designer
the
Local
of
disciplines
Equations
perturbations
first
(GSE),
among
Sensitivity
all dis-
Equations
Y
(LSE)
are responsible
braic,
regardless
with
each
for similar
of the
is still
wing
the
capabilities appreciated
exhaust
the
m
entire The
The
system
order The
governing
are linear equations
must
MDO
usually
first direction
storage
a relatively direct even
cost
all the
airplane
associated
supercomputers.
and
alge-
associated
with
The
such
magnitude
design
can
analysis
physics
analysis
and
as well.
these leads
components
are relatively difficulties
toward
cheap solvers
and
have
modifying reliable
with
for 2D applications.
such small
been
and proposed
would
problem, implicit
solvers.
equations,
Two
by different computational
require
the
or wing-section.
manageable.
for design
advantages,
Clearly
materials
governing
as a wing
the existing
technique
all their
mostly
of the
underly-
involved.
aero-elastic
using
coupling
analysis The
of composite
solved
for such
individual
discipline
For a simple
be simultaneously demand
use
easily can
cost
non-linear
can
as a
of this problem
size supercomputer. for each
disci-
such
analysis
of a medium
of the
relevant
component
capability
operations
to overcome
favored
for an isolated
using
or structural
to only
of optimization
to obtain
even
analysis
aerodynamic
structural
limit
The
computer
and LSE
a discrete
computational
eral directions groups.
when
matrix
extensive
cost
GSE
of the
optimization
task
of current
involve
non-linear
will likely v
and
is the expensive
aerodynamics
require
nature
computational
computational
ing problem
;
The
be best
the
design
a formidable
or fuselage.
strain
mathematical
Both
discipline. A complete
plines
response.
and
gen-
research tools
in
optimization.
extremely
large
t
Recent
efforts
iterative
techniques
efficient
matrix,
to affect
the
The second parallel
concentrated
on development
and improvement
resulting
from
convergence direction
processing
linearization
criteria
points
of existing
and
implementation
ones.
of the
the
The
computing
of the
equations,
of error through
of next generation
Parallel
of emcient
conditioning
governing
propagation
to the advent
capabilities.
and
would
are prone the system.
of supercomputers be ideal
co-
for MDO
with analysis
v
where
each discipline
Consequently,
could
the problem
ment)
should
need,
the High Performance
been
change
established
is focused
be assigned
to a particular
formulation
and
in order to adapt
on developing
the
algorithm
to confront
technology
(i.e.,
this
(HPCCP)
has
Program
computing,
develop-
Recognizing
such problems.
for TeraFLOP
efficiency.
software
architecture.
and Communication
government
for greater
design
to new computer
Computing
by the federal
processor
This program an improvement
2
of almost
1000 times For the
and
present,
optimization
being
over current
model
an important
a more
technology. realistic
for simple
component
task
would
be to consider
configurations.
of MDO,
The
has become
a discrete
aerodynamic
an area
design
optimization,
of interest
for many
v
v
researchers.
An essential
is acquiring
the sensitivity
rameters. are
Several
currently
entiation
element
of functions
methods
available.
(DD),
in design
Adjoint
Differentiation
(AD),
and
characteristics.
The Direct
optimization
of CFD
concerning Among
and
the
solutions
of aerodynamic with respect
the
derivation
of sensitivity
most
frequently
mentioned
Variable
(AV),
Symbolic
Finite
Difference
(FD).
Each
to design
equations are
Differentiation
pa-
(LSE)
Direct
(SD),
technique
surfaces
Differ-
Automatic
has its own unique
z
being exact,
due to direct
parameters.
The
Adjoint
impressive
results.
equally
as MACSYMA
Differentiation, differentiation Variable,
to carry
in this study,
of governing
equations
having
For Symbolic
can be used
adopted
out
its roots
Differentiation, these
has the advantage with respect
in structural
analysis,
a symbolic
differentiations.
to design produces
manipulator
Automatic
of
such
Differen-
tiation,
still
computed
at preliminary
easily
stages,
for all elementary
finite
difference
approach,
finite
difference
approximation
ter is perturbed between
from
the
new
force technique
cially
when
be the
major
metric,
flow-dependent,
value,
the
recently
involved
functions.
popular,
and
be The
is based
a design
is obtained, the
can
upon
parame-
the difference
sensitivity
derivatives.
This
computationally
intensive,
espe-
is large.
according
influence
to optimization
derivatives
In this approach,
of being
can be classified
contributors
the most
to obtain
disadvantage
exact
and intrinsic
a new solution
is used
parameters
that
operations
until
of parameters
design
fact
of the derivatives.
parameters
Uncoupled
and
old solution
number
Design pled.
simple
has
the
arithmetic
the nominal
and
brute
the
exploits
the
to whether
solution
process.
or not they
independently
These
are cou-
and would
parameters
could
be geo-
I
the primary usually The the
shape
free-stream
interior
and
conditions
process.
boundary
parameters
parameters,
grid
which,
affect
aerodynamic with
of the
grid with
respect
with
relation. of the
respect These
coefficient
to the
systems matrix.
with
They
to the
to the design
design
This
state
direct
are:
the
There
variables,
and
can be solved inversion
the
affect optimiza-
most
affiuhnt
to other
design
to geometric grid
are two basic
and
de-
the
field
components
in
sensitivity
(2) obtaining
of the gov-
the
sensitivity
of the state
vari-
by a set of linear-algebraic
directly
procedure
the
respect
The sensitivity
are described
of attack.
and
respect
surface
(1) obtaining
parameters.
parameters
of equations
affect
solution
considered
For optimization
are
optimization,
the
with
solution.
or angle
in aerodynamic
are
specify
parameters
number
influencing
in parameters
parameters
Flow-dependent
optimization
the flow-field
respect
design
Mach
parameters
although,
sensitivity.
equations
new
therefore,
geometric
a perturbation
in turn,
as free-stream
respectability.
erning
ables
grids;
geometric
surface.
relatively
optimization,
is gaining
sign
obtaining
such
Traditionally,
The
aerodynamic
parameters,
in aerodynamic
2
of a typical
grid-dependent
tion
-......a
or grid-dependent.
by a LU
becomes
decomposition
extremely
expert-
sive as the matrix
ffi
problem
solver
overcome
dimension
with
influence
increases. of off-diagonal
The
literature
pioneering
bieski
[1,2]'
sensitivity using
approach
elements
of an efficient
iterated
can
banded
be implemented
to
this difficulty.
1.2
The
A hybrid
work
sensitivity
unsteady
rotating
propfan
transport
wing and
researchers
been
structure
focus
model
on more
and
complex
with can
be used
design.
interactions Elbanna
wing-section
an analytical
approach
lifting-surface
theory
as a benchmark
criteria
technique,
aeroelastic
analysis
et al.
[5], where
a coupled
et al.
and Carlson
[6] and
sensitivity
and for a aero-
a few other
of active
[7] developed
us-
wing-section
Some
Livne
So-
to include
such as inclusion
aerodynamic
from
capabilities
to an isolated
[4].
extensive.
a plea
A semi-analytical
by Grossman the
with
linearized
applied
Kaza
influences
process.
for evaluating
been
started
[3] developed
methods.
has
is quite
their present
Yates
This
inve.stigated
optimization
for MDO
in combination
by Murthy
the overM1optimization technique
forces.
of approximate
blades
and
for extending
aerodynamics,
has
analysis
coefficients.
the accuracy
ing linear
Survey
analysis
community
differentiation
the
for assessing
dynamic
on sensitivity
of aerodynamic
an implicit
to evaluate
sensitivity
on
to the CFD analysis
Literature
controls
on
a quasi-analytical
coefficients
in transonic
\__ and
supersonic
equations cients.
flight
using
regimes.
the symbolic
The procedure
wing-sections
[8].
Later,
manipulator
was applied
For non-linear
on involvement
of CFD
[9,10]
an aerodynamic
presented
equations. 1Numbers
The
for both
procedure
in brackets
indicate
they extended
was
MACSYMA
to a ONERA aerodynamics, flow and design
applied
references
the technique
M6 wing most
sensitivity
strategy to design
to obtain
using
to 3D full potential the sensitivity
planform
of the analyses. direct
efforts
with
NACA
1406
are concentrated
Baysal
and
differentiation
a scramjet-afterbody
coeffi-
Eleshaky of Euler
configuration
for an optimized
t
composition complex
volving
et al.
[12] conducted
equations.
The method nozzle
Navier-Stokese
having
analysis
a feasibility was
a optimizer.
wing-section
inlet.
(ANSERS),
developed
costs
The
with
thin-layer
derivatives
and
authors,
flow
an
have
and
were
external
flow
with
flow
been
are
to geometric
were optimized
The
in-
problems,
to include
sensitivity
geometries
with
derivatives
respect
nozzle
de-
analysis
sensitivity
Aerodynamic
by these
associated
to two test
the formulation
performance.
domain
of sensitivity
equations
[13]. Both
improved
to include
applied
a double-throat
a significantly
module
study
successfully
of governing
flow through
four-digit
extended
computational
later expanded
and
internal
the
a supersonic
The authors
for an
over a NACA
and
differentiation
equations
was later
to reduce
Taylor
parameters.
obtained
scheme
[11].
by direct
design
in order
a subsonic
obtained
This
configurations
Euler
design
thrust.
capabilities
including
F
axial
new
sensitivity
implemented
in
"2"
this
study.
Burgreen
timization
et al.
on two fronts.
point-based
approach
instead
of familiar
calculate
the
flow solutions.
sign strategies,
developed
preliminary-design wing namic
configuration. design
The scheme functions
ical parameterization
°
.
The et al.
of wing-sections
to define
camber
notable
by Hutchison
Verhoff
also utilizes
Alternating
Other
approaches.
using
Chebyshev and
of surface,
with
improvement
expensive
includes Direction
schemes et al.
strategy [17,18]
of an aerodynamic
involves
parameterization
The second and
the efficiency
The first improvement
for surface
mial parameterization. g-_2
[14] improved
polynomials
together
distribution produces
(ADI)
variable-
technique
the
for optimal
of wing-section.
and HSCT aerody-
sensitivities.
parametric
an efficient
de-
conceptional
aerodynamic with
to
complexity
to optimize
a method
computed
the package
method
used
analytically
thickness
the use of Newton's
to combine
been
grid polyno-
include
developed
a previously
op-
a Bezier-Bernstein
Implicit
[15,16], has
replacing
shape
Due optimal
stretching to analytresults.
7 k
1.3
Objectives
After reviewing
relevant
L
namic
sensitivity
analysis,
v
sively.
The grid sensitivity
tural design models.
fecting
gradient
the overall optimization
one aspect
of aerody-
has not been investigated
sufficient
Careless
the sensitivity
[19]. Development
exten-
are based on struc-
for preliminary
design analysis.
errors within
process
that
in most of these studies
although
for detailed
Study
it is apparent
grid sensitivity,
algorithms
Such models,
would introduce
Present
literature,
namely
design, are not acceptable uations,
of
or conceptional
grid sensitivity module,
eval-
therefore,
in-
of an efficient and reliable
=
grid sensitivity
module
with special emphasis
on aerodynamic
applications
appears
essential. Unlike
aerodynamic
used on structural
design
sitivity
can be thought
natural
frequency,
considerations,
models
of years.
of structural
to finite element
loads,
grid point
have been cited for grid sensitivity
derivatives.
as implicit
differentiation,
differentiation
The other,
known as variational
is based on implicit
develop a fast and inexpensive aerodynamic
optimization
Among tial), algebraic The explicit tiation
method
grid sen-
such as displacement
of discretlzed
of continuum
for grid sensitivity
or
[20]. Two basic known
finite ele-
equations,
The main objective
classes of grid generation
grid generation
of grid coordinates
approach.
In this context,
is
here is to
to be used on an automated
cycle.
two major
formulation,
derivative
has been
The first approach,
which is based on the variation
or material
analysis
locations
approaches
ment system.
systems
resulting
are ideally suited
in a fast and suitable
with respect
effort here is to avoid the time
L_
for a number
as perturbation
with respect
the grid sensitivity
systems
and costly
Differen-
for achieving
this objective.
grid, enables
direct
to design parameters
consuming
(Algebraic,
differen-
[21,22]. The underlying
numerical
differentiation.
In
addition,
the
analytical
ysis.
An important
most
general
derivatives
ingredient
parameter.
putational
cost.
to avoid
a surge
alleviate
that
a desirable
of grid sensitivity
parameterization
as a design
are exact,
would
This,
It is essential
every
convenient,
to keep
on computational
is the surface
be to specify
although
the
feature
grid
point
analytical
on
due
of parameters
An
anal-
parameterization.
is unacceptable
number
expenses.
for sensitivity
The
the
surface
to high
com-
as low' as possible
parameterization,
may
-v
using
spline
function
problem functions
to represent
but
it suffers
such
as a Bezier
the
surface
from
lack of generality.
or Non-Uniform
[23,24].
In this
A compromise
Rational
manner,
would
B-Spline
most
be
(NURBS)
aerodynamically
in-
t-
clined
surfaces
can
Another
be represented important
to grid parameters, mization,
q*
eral the
choice
able
grid
and
is no longer
tives,
grid
an abrupt
stability
in the
finite-difference Also,
5
the
grid clustering
faster
only that
in grid
flow solution. computations and
in regions
far-field
boundary
solution
for a fixed
that
a valid
and
prompt
near
efficiency
of most
of high
gradients
location
has been
initial
conditions.
grid,
by
a suit-
also the
play
lines
are
layer,
shocks,
previous
factor
are
property and
an essential
are
as a dominant
objec-
location
schemes
boundary
related
those
boundary
of grid
computational
For example,
influenced
ill-conditioning,
can
orthogonality
identified
gen-
is an important
inaccuracy,
(e.g.,
and
Among
far-field
factor
stability,
but
opti-
resulting
of generating
objectives.
orthogonality
where
problem
respect
to grid
cycle,
be strongly
grid smoothness
size may
with
leading
accuracy,
may the
certain
clustering,
The
concept,
The
problem
implies
For example,
sensitivity
of grid in optimization rates.
to achieve
parameters.
grid
This
to generating
orthogonality,
change
quality
for any
This
grid
(design)
attention.
convergence
restricted
significant.
accuracy
the
of grid.
smoothness, most
more
flow solvers
quality
of manipulating
since
T
of most
issue
considered
and
few control
of grid sensitivity,
to enhance
flow analysis
reliability
aspect
also deserves
can be used
in better
with
in-
role
in
desirable.
enhanced
by
etc.).
The
in influencing
the
investigations
indicate
-
z
that
for a symmetrical
dependency
r._
niques,
on the
representations
z
t
s_
IIF'
L
i\
_
the error in lift coei]icient
extent
[33].
As required
of the grid with respect
to those
has an inverse
by most
radial
optimization
parameters
influencing
techthese
is required. The
the
boundary
the sensitivity
objectives
rithm
wing-section,
and
organization of a typical
boundary
theoretical
of this study
is provided
Finally,
some
are
grid distribution
formulation
solution
model
and
in Chap.
concluding
is as follows.
derived
are developed
aerodynamic
5. The results
remarks
in Chap.
are provided
The
2. The grid
in Chap.
sensitivity are presented in Chap.
physical
7.-
geometric
generation
3. Chapter
equation. and
and
The
discussed
algo-
4 discusses method
of
in Chap.
6.
Chapter
2
PHYSICAL
MODEL
P
2.1 Wing-Section
Example
il_
The since
much
design
physical research
is essential
supersonic
model has
considered
been
devoted
Other
study
is an isolated
to its development
for the performance
speeds.
for this
of an advanced
applications
could
and
aircraft
be helicopter
wing-section
representation. for both
rotor
This
subsonic
blades,
and
and
high
h,
performance sections.
fans. The first
classical
NACA
The
thickness
line
Families
of wing
of the
chosen
to generate
analytical)
The
second
wing-sections
maximum
ber,
C -
wing sections sections
the
desired
representation approach
using
wing-
resulting
is a geometric
in (i.e.,
NURBS.
described
for grid-generation
by combining
expressions
possess
the concise
description
of a wing
25 provides
distribution thickness,
chordwise
are
are examined
resultant
Reference
a thickness
four-digit
The
mainly
T -= the
NACA
(i.e.,
wing-sections.
four-digit
parameters.
and
been
is a physical
NACA
distribution.
and
have
Representation
suit the problem, design
approach
representation
Physical
eterization.
approaches
four-digit
approximative)
2.1.1
Two
about M ---- the
position
wing-section
the general the
maximum
of maximum
is based
mean
on the l0
the
line.
ordinate. geometry
section
The
ordinate
a mean
necessary
equations
features
which
of the
of the
line and
in terms
design
The
param-
a mean
parameters
numbering section.
that
of several
define
mean
a
are:
line or camsystem The
first
for and
11 second
integers
represent
T.
represent
Symmetrical
as in the case of NACA tion definition. covering
both
in the next
M and sections
are
the top and The
mean
into
the chord
M y0(_)= _(2c_-
Details
x2),
(1 -c) 2
section
Yr(X)
The section
thickness
is given
= A(0-2969x½
coordinates
fourth
integers
a schematic line x -
of mapping
x(r)
of the sec= x(fl({))
will be discussed
is
x < C
yo(_) = M(I - 2C+ 2C_- _) The
and
for the first two integers,
2.1 provides
of the section. line equation
the third
by zeros
Figure
is mapped
bottom
while
designated
0012 wing-section.
The _-coordinate
chapter.
C respectively,
(2.1)
' _ > c.
(2.2)
by
-- 0.126x
-- 0.3516z
_ + 0.2843x 3 -- 0.1015x4).
(2.3)
are
(2.4) where
= = _
o
=
P_ represents
the
vector
of independent
parameters
to be defined
later.
12 r_
YT
0.5
2
3
4
y = T (0.2969X - 0.126 X- 0.3516 X + 0.2843 X - 0.I015 X ) T 0.2
1
_'_--X
(a) Thickness distribution
Y C
I |
7_
'2cx-x2' 2 x
C
o_c)2
o[,_
IM
"_----
C ----_
1
(b) Camber line Y ....
v--_+__
Camber
line
Surface
definition
r
°
°
°
"
_X
(c) Schematic
Fig.2.1 Wing-section $
=
.r L--
of wing-section
specification
for NACA
four-digit
series.
13
w
2.1.2
Geometric
Representation
Another L
function
to approximate
sentation L_
approach
for representing
the surface.
is the Non- Uniform
vide a powerful
geometric
a wlng-sectlon
The most
Rational
commonly
B-Spline
used
(NURBS)
tool for representing
model
both
is using
a spline
approximative
repre-
function.
analytic
The NURBS
shapes
(conics,
pro-
quadrics,
m
surfaces curve
of revolution,
etc.)
and free-form
surfaces
[26-28].
The relation
for a NURBS
is
--i
x(r) =
E '=0
(
X(r)
x(r) = {_ y(r))
E'_=oN,,,(r)w,
D'={
Xi }Yi
(2.5)
m_ r_
where
X(r)
trol points B-Spline
is the vector (forming
basis
valued
a control
function
surface
polygon),
defined
-
rri+p
The
ri are the so-called
knots
ri --
=
in the r-direction,
w_ are weights,
recursively
Ni,0(r) Ni,p(r)
coordinate
01
Ni,p_,(r) r i
where number
the
end
knots
of knots,
a and
m + 1, and
b are
are the p-th
degree
ri otherwise _< r _< ri+l } +
ri+p+x -
a uniform
--
knot
p+l "_.._,
N_.p(r)
the con-
as
ri+p+l
forming
and
Di are
r
Ni+,,p_,(r).
(2.6)
ri+l
vector
p+l
repeated
number
with
of control
multiplicity points,
m = n + p + 1.
p + 1.
The
n + 1, are related
degree,
p,
by
(2.8)
For most
w
practical
is defined
applications
on the interval
the knot
X(r) Ri,n(r)
among
many
are the Rational others
found
Basis
(2.5)
and
can
Functions,
the
basis
be rewritten
Ni'v(r)°°' = _=o Ni,v(r)wi
Ri'v(r)
i---0
where
is normalized
(a = O, b = 1). Equation
'_ = _ R/,p(r)Di
y
vector
function
as
(2.9) i = O, .... , n
satisfying
the the following
properties
in [22]
n
y_ R,,n(r)
awtt
= 1
Ri,_,(r)
> 0.
(2.10)
i=0
Three gorithm. three
options
are
available
first
option,
the
In the control
tracted
points.
to the
The
camber.
to define camber
thickness
The
first
a wing-section
line
is defined
distribution, and
last
using
by
Eq.(2.3),
control
a NURBS is then
points
the
are
NURBS curve
added
fixed
alusing
and
for the
sub-
section
=_ E
i
m
chord.
The
point,
its
shows
the
(i.e.,
design weight,
and
the
corresponding The
of control
The
second
using
NURBS
Both
camber
choice
This
approach,
option
thickness
although
parameters The
points
third
basis
(p=2,n=2)
with
of control
2.4 and
on camber,
wing-section,
The
new
distribution
as shown
both
more
wing-section
2.6.
is to bypass
the
and and
the effect the
basis
using it also
and
set
2.3 to 1
betwe.en
of increasing
function.
distribution using
three increases
thickness
control
Figure
weights
can be obtained
control,
camber
2.2.
is a trade-off
thickness
are defined
design
in Fig.
points
2.5 illustrate
camber
curves
of the middle
in Fig.
function
of number
location
T as shown
[4]. Figures
promises
option
are the
thickness
is to define
representation. and
this option
maximum
and functionality
number
of design
using
quadratic
wi = 1, i = 0, 2).
complexity the
parameters
curves Eq.
(2.4).
control
points.
the
number
distributions
com-
E
t
pletely
and
control
Figure
2.7 illustrates
points
at the leading
the wing-section a seven and
trailing
directly
control edges
point
with
NURBS
control
representation
are fixed.
Two control
points
and
of a wing-section. points
weights. The
at the 0% chord
15
are
used
shown N
w
w
function
to affect
the bluntness
in Fig 2.8 creates (p=3)
using
of the
the effect
this approach
section.
of camber is shown
The in the in Fig.
movement
of control
wing-section. 2.9 with weights
points
The
cubic
set
to 1.
as
basis
16
L_
YT
_ T (0.2969
Y- 0.-5
O.5 X - 0.126 X- 0.3516
2 X + 0.2843
3 X - 0.1015
L E_
l
0
(a)
Thickness
distribution
X
Y
C I
--7_=. E
Control points Control polygon Camber line
....
!
i__
l g....
(b) Camber
X
line
Y Y = Yc +
YT
....
Camber
line
_
Surface
definition
= L r_
(c) Schematic
Fig. 2.2
Wing-section
of wing-section
specification
using
NURBS
(option
1).
a X )
17
r
i
v
Z-z
L
R (r) 2,p
R (r) O,p R R (r) i,p
(r) 1,p
0 L
0
r(_) Fig. 2.3
Quadratic
basis function (p=2) for camber line (option
1).
18 w
Y
•
-- -.... _
l
L
Control
II ........
o_-
points
Camber line Control polygon Wing-section
"_
X
L w
Fig. 2.4
Effects of increasing
the number
of control
points
on camber and wing-section.
_ = ..__. m
=
=
r_
r,A R
R
(r) 0,p
R 3,p
(r) 1,p
(r) i,p
R (r) 2,p
w
o Fig. 2.5
r_
Effects
r(_) of increasing
the number
of control
l points
on a quadratic
basis function.
== 19
YT i |
_a
//
Control
points
....
Control polygon
--
Thickness
distribution
" ---...
0
(a)
Thickness
distribution
Yt
1
I |
F
X
Control
points
....
Control
polygon
_
Camber
line
I
(b) Camber
X
line
Y ....
Y=V_+_YT
Camber
line
Surface
definition
w
(c) Schematic
Fig. 2.6
Wing-section
of wing-section
specification
using NURBS
(option
2).
2O w
Y Control
points
Control polygon
ilwm.
Wing-section
J w
X w
Fig. 2.7
Seven
control points wing-section
specification
using
NURBS
i
!
•
Y • .....
Control
points
Control polygon Wing-section
w
=
,,i, o
•
X
m
Fig. 2.8
Effect of control
point movement.
(option
3).
L
W
=
.
R (r)
(r)
!
R (r)
3,p
2,p
R (r) 4,p
R (r) i,p =
w
0
r(_) Fig. 2.9
L
_--÷_
Cubic basis
function(p=3).
-
z
t
J
Chapter GRID
3
GENERATION
3.1 Introduction In order tem
to study
of nonlinear
the flow-field
partial
differential
around
any aerodynamic
equations
must
be solved
configuration, over
a sys-
a highly
complex
L_ K3
geometry
[29]. The
an implied
rule
domain
specifies
of interest the
should
connectivity
be descretized
of the
points.
into a set of points This
where
discretization,
known
L-
as grid generation, topology grid
of the
with
of the
is constrained region
respect
true
where
to any
by underlying
the
of the
solution
above
physics,
is desired
constraints,
surface
[30-32].
may
geometry,
A poorly
fail to reveal
and
the
constructed
critical
aspects
solution.
The
discretization
of the
field
requires
some
organization
in order
for the
w
B
4
solution
to be efficient.
The
location
of outer
boundaries,
solution
[33,34].
Furthermore,
logistic
structure
and the
of the
data
orthogonallty
such
as grid
can influence
the
spacing, nature
the of the
w
the
discretization
must
conform
to the
boundaries
of
2
the
region
This
in such
organization
for alignment system
with
the
coordinate
all boundaries.
accuracy,
the
boundary
can be provided
for rectangular
curvilinear with
a way that
grid
system
is reflected
coordinate
in routine
cylindrical
coordinate
covers
field
To minimize spacing
can be accurately
by a curvilinear
boundary region,
condition
should
the
the
number
be smooth,
w
22
and
choice
of grid with
system
concentration
lines
required
the
need
coordinate
region,
coordinate points
where
of cartesian
for circular has
represented.J35].
etc.
This
coincident
for a desired
in regions
of high
--
23
solution
w
gradients.
or corners), = i
These
regions
compressibility
shear
layers).
scales,
and
be the result
(entropy
A complex
often
may
and shock
flow may
of unknown
contain
geometry (large surface
of
layers),
a variety
and
viscosity
slopes
(boundary
of such regions
of various
and length
location.
w
Two primary tified.
w
There
W
are algebraic
tems
are mainly
[36],
Multi-Surface
[38].
The
categories
for arbitrary
systems
composed
and
basic
[37],
mathematical
tion of the field values
partial
of interpolative
Interpolation
and
the
generation
differential
schemes
structure
from
coordinate
systems.
such
boundary.
iden-
The algebraic
sys-
Interpolation
Interpolation
methods
For partial
been
as _ransfinite
Two-Boundary of these
have
techniques
are based
differential
on
interpola-
equation
systems,
r
a set of partial differential z
differential
methods
equations
may
must
be elliptic,
be solved
parabolic,
to obtain
or hyperbolic,
the field values.
depending
The
on the bound-
£
ary specification advantages L
of the problem.
and
drawbacks
Each
of these
depending
grid
on geometry
generation and
systems
application
has
its own
of the
problem.
_c
Algebraic
m
N
generating
systems
control
of the physical
grid
skewed
grids
ferential
z_
systems,
computer
although
intensive,
practice
in recent
then smooth I-a
for boundaries
successful
years,
array
shape
effective of general
speed and
and
grid
with
strong
offer
relatively
simplicity spacing.
curvature smooth
to originate
a differential for most
providing
However,
grids cases.
the grid
system.
while
or slope
for three-dimensional
has been
the field using
and cost An
E_
specially
offer
Such
they
an explicit
might
discontinuity. for most An
using
Partial
applications,
alternative,
difare
a common
an algebraic
hybrid
produce
system
approach
proven
and to be
applications.
purpose
grid
generation
softwares
have
been
emerged
w
over
past
few years.
Among
GLE of Thompson widely lizes
w
=
.
used. a novel
The
[40], and GRAPE2D
approach
many
others,
GRIDGEN solves
for determination
the GRAPE2D by Stelnbrenner
Poisson's of the
equation boundary
of Sorenson et al.
[39], the EA-
[41], are
the
in two-dimension control
functions.
and
most utiThe
--
24
EAGLE
D
code
dimensional with
both
ment.
combines
in surface
field grid generation. algebraic
Another
full Computer provides
techniques
and
The GRIDGEN
differential
generation
new
arrival,
called
Aided
Design
system
an efficient
grid generation
and also quick
series
as well as two or three-
is a more
capabilities has
the
(CAD),
grid
generation
procedure
to reflect
appearance
on an interactive
ICEM/CFD, with
recent
capability
the
environ-
of combining module
CAD
a
[42]. This,
model
changes
on
u
grids.
Most of these
However, versed r i
=-
packages
intelligent
in current
furnish
a host
use of the majority grid
generation
of options
of these
with
options
a high degree requires
of flexibility.
the user
to be well
techniques.
•
Due
to directness
der of this chapter
and
would
relative
be devoted
simplicity to their
of algebraic
development.
systems,
the remain-
The relevant
aspects
of
m
algebraic
generation
boundary
system
discretization,
such
and
as boundary
surface
coordinate
transformation,
mapping,
discussed
following
grid generation
are
in the
Coordinate
Transformation
l
sections. w
3.2 Boundary-Fitted Structured
algebraic
grid
generation
techniques
can
be thought
of as trans-
D
formation domain
from a rectangular as shown
parameters,
in Fig.
P, and
computational
domain
3.1 [43]. The transformation
can be expressed
to an arbitrarily is governed
-shaped by vector
physical of control
as
almm
(3.1)
= { • ff, ,i, P) } r/, P)
w_ r_ i
where w
0_2 are obtained
guaranteeing
variation
< I
associated spacing.
exponential
to the
has been
when
the
with
the
A large
function,
solution
the
choice
usage
variwould
inaccuracies
[35].
of hyperbolic
sine
as
= Y,_, + [Y,- Y,_,]sinh[Bi_,(_)]
for
X,_,
< _ < Xi
(3.3)
sinh(Bi_a)
V
where 0
