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22 Φεβ 2014 (πριν από 4 χρόνια και 2 μήνες)

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School of Aerospace Engineering

A Thesis Proposal

by

Ebru Usta

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

the amplitude of an acoustic wave or the
magnitude of the tip vortex as it propagates

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
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.

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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.

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RECENT PROGRESS (continued)

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.

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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.

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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:

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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

|
|
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.

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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
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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
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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
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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

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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
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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

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-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.

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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

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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:

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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

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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
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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
4.663E-05
8.028E-04
5.081E-04
4.523E-04
2.188E-04
1.822E-04
2.029E-04
MAC4
1.516E-04
3.479E-04
1.565E-04
1.282E-04
6.481E-05
2.139E-05
3.214E-05
STVD6
1.368E-04
2.704E-04
1.277E-04
1.110E-04
5.587E-05
2.715E-05
6.224E-05
STVD8
1.461E-04
2.718E-04
1.232E-04
1.046E-04
4.541E-05
2.522E-05
4.939E-05
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
-
-
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).

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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
TURNS-STVD4
-5536
12.15
TURNS-STVD6
-5612
10.94
TURNS-STVD8
-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
TURNS-STVD4
-1190
14.00
TURNS-STVD6
-1235
10.76
TURNS-STVD8
-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
TURNS-STVD4
-487
22.32
TURNS-STVD6
-378
39.71
TURNS-STVD8
-408
34.92
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PLANFORM OF THE UH
-
60A MODEL ROTOR

-
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.

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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
TURNS-STVD4
Experiment
TURNS-STVD6
TURNS-STVD8
TURNS
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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
TURNS-STVD4
Experiment
TURNS-STVD8
TURNS-STVD6
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
TURNS-STVD4
TURNS-STVD6
TURNS-STVD8
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
TURNS-STVD4
TURNS-STVD6
TURNS-STVD8
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
TURNS-STVD4
TURNS-STVD6
TURNS-STVD8
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

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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

The STVDx schemes require little or no
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,
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.