School of Aerospace Engineering
A Thesis Proposal
by
Ebru Usta
Advisor: Dr.L.N.SANKAR
APPLICATION OF A SYMMETRIC
TOTAL VARIATION DIMINISHING SCHEME
TO AERODYNAMICS AND
AEROACOUSTICS OF ROTORS
Supported by the National Rotorcraft Technology Center(NRTC)
School of Aerospace Engineering
Overview
•
Motivation and Objectives
•
Background
•
Mathematical and Numerical Formulation
•
Symmetric TVD Scheme (STVD)
•
Validation with 1

D and 2

D Wave
Problem
•
Results and Discussion
•
Shock Noise Prediction for the UH

1H
rotor
•
Tip Vortex Structure and Hover
Performance of the UH

60A rotor
•
Proposed Work
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MOTIVATION and OBJECTIVES
•
Helicopter rotor’s flowfield is dominated by compressibility effects,
a complex vortex wake structure and viscous effects.
•
Accurate prediction of the aerodynamic flowfield and aeroacoustics
of a helicopter rotor is a challenging problem in rotorcraft CFD.
•
Existing methods for tip vortex and noise prediction suffer from
numerous errors.
•
As a result, accurate aerodynamics and aeroacoustics prediction
methods are
urgently needed.
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PROBLEMS WITH THE CFD METHODS
I. DISSIPATION ERRORS
•
Numerical dissipation
–
Dissipation
causes a gradual decrease in
the amplitude of an acoustic wave or the
magnitude of the tip vortex as it propagates
away from the blade surface.
–
The computed vortical wake, in particular,
diffuses very rapidly due to numerical
dissipation
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II. DISPERSION ERRORS
•
Numerical dispersion
–
Dispersion
causes waves of different
wavelengths originating at the blade
surface to incorrectly propagate at different
speeds.
–
Because of dispersion errors, the waves
may distort in nonphysical manner as they
propagate away from the blade surface.
School of Aerospace Engineering
RECENT PROGRESS IN REDUCING
DISPERSION ERRORS
•
Tam and his coworkers recently developed a
low dispersion numerical scheme called the
Dispersion

Relation

Preserving (DRP) finite
difference scheme(1996).
•
Nance et. al. extended the DRP ideas to
curvilinear grids(GT thesis 1997).
•
Other works include: Carpenter, Baeder,
Ekaterinaris, Smith et al. and CAA Workshops
I and II.
School of Aerospace Engineering
RECENT PROGRESS (continued)
•
Wang, Sankar and Tadghighi implemented
Nance's Low Dispersion Finite Volume
(LDFV) ideas into TURNS and studied shock
noise and hover performance of
rotorcraft(1998).
–
A
side benefit of the high order accuracy
LDFV and DRP schemes is their
reduced
dissipation
or numerical viscosity
.
–
These schemes have numerical viscosity
that is typically
proportional to
D
5
where
D
is
the grid spacing.
School of Aerospace Engineering
RECENT PROGRESS IN
REDUCING
DISSIPATION ERRORS
•
The easiest way to reduce dissipation errors is
to increase the formal accuracy of the upwind
scheme.
–
Third order schemes in TURNS and
OVERFLOW generate errors proportional to
D
3
.
–
Fourth order operator compact implicit
schemes (OCI) have been studied by
M.Smith (GT, 1994) and Ekaterinaris (Nielsen
Eng.,1999)
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RECENT PROGRESS (continued)
–
Hariharan and Sankar have explored 5
th
order and 7
th
order upwind schemes with
dissipation errors proportional to
D
5
and
D
7
respectively
(GT thesis 1995).
–
Wake studied the evaluation of a line
vortex in space and time using 6
th
order
spatially accurate scheme and have
presented 9
th
order results in fixed wing
mode(1995).
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RECENT PROGRESS (cont’d)
GRID CLUSTERING EFFECTS
•
Numerical errors may also be reduced by use
of a fine grid, and/or grid clustering.
–
Tang et. al. recently have developed a grid
redistribution method that clusters the grid
points near the tip vortices and reduces the
numerical diffusion of vorticity(1999).
–
Strawn et. al. used high density embedded
grids(CHIMERA) for improving the wake

capture (1999)
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SCOPE OF THE PRESENT WORK
•
The main purpose of this study is to develop
and validate the spatially higher order
accurate methods for modeling rotors in
hover and forward flight.
•
As the formal order of accuracy increases, it
becomes more and more difficult to
simultaneously reduce dispersion, dissipation
and truncation errors.
•
Are there better schemes available?
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SCOPE OF THE PRESENT WORK
•
Use Yee's symmetric TVD scheme to
accurately model
tip vortex structure
and
shock noise phenomena
of rotors.
•
Yee’s idea: High order central difference
schemes can be coupled to lower order
dissipation terms to yield accurate results.
•
For this purpose, a version of the NASA Ames
code TURNS, referred to here as TURNS

STVDx (x=4,6,8), has been developed.
School of Aerospace Engineering
WHAT IS A TVD SCHEME?
•
For a TVD scheme, Sum of slopes always
decreases, ensuring no new maxima occur.
n
t
x
u
Sum of slopes =
n
x
u


New Maxima
l
n
t
+
Sum of slopes =
l
n
x
u
+


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Symmetric TVD Scheme
0
+
x
F
t
q
0
2
/
1
2
/
1
D
+
+
x
F
F
dt
dq
i
i
The semi

discrete form at a typical node 'i' is:
School of Aerospace Engineering
Symmetric TVD Scheme (continued)
0
)
(


)
(


2
1
2
/
1
1
2
/
1
1
1
D
+
D
+
+
+
+
x
q
q
A
q
q
A
x
F
F
dt
dq
i
i
i
i
i
i
i
i
•
Dr. Helen Yee recommends the following
second order form:
where computed using “Roe averages” of q at
adjacent points.


q
F
A
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STVD (cont’d)
Second order STVD scheme:
L
R
i
i
i
q
q
A
q
F
q
F
F
+
+
+
2
1
)
(
)
(
2
1
1
2
/
1
This part is used to control
dispersion and truncation errors
This part is used to
control dissipation
errors
•
Dispersion and dissipation errors may be
independently controlled.
School of Aerospace Engineering
Fourth
order
STVD
scheme
:
L
R
i
i
i
i
i
q
q
A
F
F
F
F
F
+
+
+
+
+
2
1
7
7
12
1
1
1
2
2
/
1
STVD (cont’d)
and : MUSCL interpolation with a suitable limiter.
L
q
R
q
Sixth order STVD scheme:
L
R
i
i
i
i
i
i
i
q
q
A
F
F
F
F
F
F
F
+
+
+
+
+
+
+
2
1
8
37
37
8
30
1
2
1
1
2
3
2
/
1
School of Aerospace Engineering
STVD (cont’d)
Eighth
order STVD scheme on Non

Uniform Grids:
L
R
i
i
i
i
i
i
i
i
i
q
q
A
hF
gF
fF
eF
dF
cF
bF
aF
F
+
+
+
+
+
+
+
+
+
+
+
+
2
1
)
(
4
3
2
1
1
2
3
2
/
1
distance along the coordinate line
2
1
i
i
x
x
2
/
1
+
i
x
School of Aerospace Engineering
STVD (cont’d)
–
Where a,b,c,d,e,f,
g,h are coefficients of
the related fluxes.
•
Note that this scheme also accounts for the
non

uniform grid spacing.
4
,..,
2
,
3
)
(
)
(
4
3
3
4
3
2
/
1
+
+
+
+
i
i
i
k
x
x
x
x
a
i
k
l
i
l
l
i
i
k
l
i
l
l
i
k
School of Aerospace Engineering
CONSTRUCTION OF and
•
and were found using third order MUSCL
interpolations.
•
Koren Limiter, and a LDFV Limiter were explored.
•
In some sample bench mark cases, and
were found using higher order (4th, 6th and 8th)
dissipation terms with no limiters.
L
q
R
q
R
q
L
q
R
q
L
q
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1

D WAVE PROBLEM
•
The initial solution at t=0 is given by
•
The exact solution is
0
+
x
u
t
u
16
2
)
0
,
(
x
e
t
x
u
16
2
)
,
(
t
x
e
t
x
u
School of Aerospace Engineering
1

D WAVE PROBLEM (continued)
•
The accuracy of the schemes is assessed by
computing the of the error
calculated as:
IMAX
u
u
Error
Average
exact
2
)
(
IMAX : The maximum number of grid points
norm
L
2
School of Aerospace Engineering
1

D WAVE PROBLEM (cont’d)
•
1

D wave equation is solved explicitly using
second order Runge Kutta method as
follows:
n
l
l
l
n
x
n
i
p
i
x
u
x
u
t
u
u
)
/
(
D
D
p
l
l
l
p
x
n
i
p
i
n
i
x
u
x
u
t
u
u
u
)
/
(
5
.
0
5
.
0
)
(
5
.
0
1
D
D
+
+
l
: Formal accuracy of the scheme
School of Aerospace Engineering
6
5.5
5
4.5
4
3.5
3
2.5
2
3.2
3
2.8
2.6
2.4
LOG(1/N)
LOG (L2NORM)
stvd4
stvd6
stvd8
•
Higher order schemes, e.g. STVD8, consistently
produces lowest errors on all grids.
•
For STVD8, the slope is the steepest, indicating that
the errors decrease quickly with refinement.
School of Aerospace Engineering
2

D Problem:
Pulse interacting with uniform flow
and solid wall.
CAA workshop test Problem organized by
Prof. Chris Tam (FSU)
0
)
(
)
(
'
'
'
+
+
y
Bq
x
Aq
t
q
t=0
+
V
School of Aerospace Engineering
•
Several baseline solutions (6th order
MacCormack, 3rd order Upwind) are available
for comparison.
•
Exact solutions are also available for
comparison(Nance, Ph.D Dissertation)
•
At boundaries, non

reflective boundary
conditions were used.
•
In this study,STVD4, STVD6 and STVD8
solutions were obtained. Only the 8th order
results are shown here.
Approach:
School of Aerospace Engineering
BOUNDARY CONDITIONS
To avoid entropy layers,
to preserve total enthalpy, h
0
)
(
'
q
A
x
+
)
(
)
(
'
'
q
B
q
B
y
y
+
+
0
)
(
)
(
'
'
'
+
+
+
y
Bq
x
q
A
t
q
(No vorticity)
0
)
'
(
0
)
'
(
0
0
y
y
p
y
u
v
School of Aerospace Engineering
TIME HISTORY OF PRESSURE AT THE WALL
T=15
0.01
0
0.01
0.02
0.03
0.04
0.05
100
50
0
50
100
mac4
upwind3
stvd8
exact
T=30
0.1
0
0.1
0.2
0.3
100
50
0
50
100
mac4
upwind3
stvd8
exact
T=45
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25
100
50
0
50
100
mac4
upwind3
stvd8
exact
T=60
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25
100
50
0
50
100
mac4
upwind3
stvd8
exact
School of Aerospace Engineering
T=75
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
100
50
0
50
100
X
P
mac4
upwind3
stvd8
exact
T=100
0.1
0.05
0
0.05
0.1
0.15
0.2
100
50
0
50
100
mac4
upwind3
stvd8
exact
T=150
0.1
0.05
0
0.05
0.1
0.15
100
50
0
50
100
mac4
upwind3
stvd8
exact
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PRESSURE CONTOURS
T=75
T=100
T=150
Oscillations due to
no dissipation term
T=75
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PRESSURE CONTOURS
Oscillations due to
no dissipation term
With dissipation term
T=75
T=75
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PRESSURE CONTOURS(cont’d)
T=100
T=100
With dissipation
OSCILLATIONS
T=100
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PRESSURE CONTOURS(cont’d)
T=150
T=150
With
dissipation
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TRUNCATION ERROR ASSESMENT
T=15
T=30
T=45
T=60
T=75
T=100
T=150
UPW3
4.663E05
8.028E04
5.081E04
4.523E04
2.188E04
1.822E04
2.029E04
MAC4
1.516E04
3.479E04
1.565E04
1.282E04
6.481E05
2.139E05
3.214E05
STVD6
1.368E04
2.704E04
1.277E04
1.110E04
5.587E05
2.715E05
6.224E05
STVD8
1.461E04
2.718E04
1.232E04
1.046E04
4.541E05
2.522E05
4.939E05
Scheme
CPU Time
UPW3
16
’
:24
MAC4
12
’
:12
STVD6
11
’
:43
STVD8
12
’
:15
CPU TIME:
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RESULTS and DISCUSSION
•
4th,6th and 8th order Symmetric TVD schemes
have been applied to model helicopter rotor
shock noise for UH

1H rotor and tip vortex
structure of UH

60A rotor.
•
The following results are presented:
–
Original TURNS code (3rd order MUSCL scheme)
–
Modified flow solver TURNS

STVDx (x=4,6,8)
–
Comparison with experimental data for UH

60A and
UH

1H rotor.
•
All rotor calculations were done on identical
grids, to eliminate grid differences from skewing
the interpretation of results.
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SHOCK NOISE PREDICTION OF UH

1H
ROTOR
•
Calculations have been performed for a two

bladed UH

1H rotor in hover.
•
The blades are untwisted and have a
rectangular planform with NACA 0012 airfoil
sections and an aspect ratio of 13.7133.
•
The sound pressure levels have been
compared to the experimental data for a 1/7
scale model (Purcell,1989).
School of Aerospace Engineering
Shock Noise Prediction, r/R=1.111, Tip Mach =0.90,
Grid Size 75x45x31
7000
6000
5000
4000
3000
2000
1000
0
1000
2000
0
0.5
1
1.5
2
Time (msec.)
Pressure(Pa)
stvd6
exp
muscl
stvd4
stvd8
SCHEME
(P
a
P
)
(Pascal)
% Error
Experiment
6302
0.00
Baseline TURNS
5523
12.30
TURNSSTVD4
5536
12.15
TURNSSTVD6
5612
10.94
TURNSSTVD8
6311
0.14
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Shock Noise Prediction, r/R=1.78, Tip Mach= 0.90,
Grid Size 75x45x31
1600
1400
1200
1000
800
600
400
200
0
200
400
0
0.5
1
1.5
2
Time(msec.)
Pressure(Pa)
stvd6
exp
muscl
stvd4
stvd8
SCHEME
(P
a
P
)
(Pascal)
% Error
Experiment
1384
0.00
Baseline TURNS
977
29.40
TURNSSTVD4
1190
14.00
TURNSSTVD6
1235
10.76
TURNSSTVD8
1234
10.84
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Shock Noise Prediction,r/R=3.09, Tip Mach =0.90,
Grid Size 75x45x31
700
600
500
400
300
200
100
0
100
200
0
0.5
1
1.5
2
Time(msec.)
Pressure(Pa)
stvd6
exp
stvd4
muscl
stvd8
SCHEME
(P
a
P
)
(Pascal)
% Error
Experiment
627
0.00
Baseline TURNS
320
48.96
TURNSSTVD4
487
22.32
TURNSSTVD6
378
39.71
TURNSSTVD8
408
34.92
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PLANFORM OF THE UH

60A MODEL ROTOR
•
Four blades, a non

linear twist, and no taper.
•
20 degrees of rearward sweep that begins at r/R=0.93.
•
The aspect ratio and Solidity Factor 15.3 and 0.0825.
School of Aerospace Engineering
PRESSURE DISTRIBUTION ALONG THE SURFACE
OF UH

60A AT r/R=0.920
r/R=0.920
1.5
1
0.5
0
0.5
1
1.5
0
0.2
0.4
0.6
0.8
1
Chord
Cp
TURNSSTVD4
Experiment
TURNSSTVD6
TURNSSTVD8
TURNS
School of Aerospace Engineering
PRESSURE DISTRIBUTION ALONG THE SURFACE
OF UH

60A AT r/R=0.99
r/R=0.99
2
1.5
1
0.5
0
0.5
1
1.5
0
0.2
0.4
0.6
0.8
1
Chord
Cp
TURNSSTVD4
Experiment
TURNSSTVD8
TURNSSTVD6
TURNS
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PERFORMANCE OF THE UH

60A ROTOR
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0
2
4
6
8
10
12
Collective Pitch(deg.)
CT/solidity
EXPERIMENT
TURNSSTVD4
TURNSSTVD6
TURNSSTVD8
TURNS
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PERFORMANCE OF THE UH

60A ROTOR
0
0.002
0.004
0.006
0.008
0.01
0.012
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
CT/solidity
CQ/solidity
EXPERIMENT
TURNSSTVD4
TURNSSTVD6
TURNSSTVD8
TURNS
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PERFORMANCE OF THE UH

60A ROTOR
VISCOUS RESULTS for 149x89x61 GRID SIZE
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0
0.02
0.04
0.06
0.08
0.1
0.12
CT/solidity
FM
experiment
TURNSSTVD4
TURNSSTVD6
TURNSSTVD8
TURNS
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CONVERGENCE HISTORY FOR TURNS

STVD8
FOR UH

60A ROTOR
0.004
0.0045
0.005
0.0055
0.006
0.0065
0.007
0.0075
0
5000
10000
15000
20000
Iteration Number
CT
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VISCOUS CALCULATIONS DONE IN COLLABORATION
WITH UTRC AT UTRC ON A 181x75 x49 FINER GRID OF
UH

60A ROTOR
Blade Loading vs. collective pitch
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Torque versus Blade Loading
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Figure of Merit versus Blade Loading
Error of 0.01

0.02 in FM; well
within 100 lb. or 200 lb. error
in thrust; considered very
good by industry.
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CONCLUDING REMARKS
•
The accuracy characteristics of the STVDx
schemes have been systematically investigated
in 1

D and 2

D problems where exact solutions
exist.
•
Several high order Symmetric TVD schemes
have been implemented in the TURNS code .
•
The tip vortex structure of UH

60A rotor and
shock noise phenomena for UH

1H rotor are
accurately modeled with these high order
schemes compared to the baseline third order
MUSCL scheme.
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CONCLUDING REMARKS(cont’d)
•
The eighth order STVD scheme is found to
give the best thrust predictions for the UH

60A rotor, even on a coarse grid.
•
The shock noise predictions were also, in
general, better with the higher order
schemes in spite of having loss in accuracy
when a high scheme is used on a very
coarse grid, 3 radii away.
•
The STVDx schemes require little or no
additional computational time, compared to
the MUSCL scheme.
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CONCLUDING REMARKS(cont’d)
•
Many existing CFD solvers may easily be
retrofitted with the symmetric TVD scheme.
•
UTRC Viscous results compare very well with
the model test.
•
The Figure of Merit is generally 1

2 points
under the experimental data which is
considered very good.
•
These results are much better than using
baseline TURNS.
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PROPOSED WORK
Perfecting the Hover Code:
•
Increase formal accuracy of metrics, Jacobian,
time, boundary conditions, load integration
schemes.
•
Additional validations for another rotor, to be
chosen in consultation with industry and thesis
committee.
•
Study of Vortex Ring State and climb using GT
experimental data
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PROPOSED WORK (continued)
IF TIME PERMITS
,
•
Use embedded adaptive grid for improved
wake capturing
•
Use of Spalart

Allmaras turbulence model for
hover prediction.
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