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School of Aerospace Engineering
STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN
ROTORCRAFT NOISE PREDICTION
A PhD Thesis DefensePresented to
The Faculty of the Division of Graduate StudiesBy
Gang Wang
Advisor: Dr. Tim C. LieuwenApril 4, 2002
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School of Aerospace Engineering
OUTLINE
Background
LDFV Scheme
Objectives
Results
Contributions & Conclusions
Recommendations
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School of Aerospace Engineering
BACKGROUND
Helicopter has a wide range of military and civil applications.
However, the high noise level associated with it greatly restricts its further applications.
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School of Aerospace Engineering
Rotorcraft Noise
Three categories of rotor noise:Rotational noise: steady or harmonically varying
forces, volume displacements
Broadband noise: random disturbances
Impulsive noiseHigh-Speed-Impulsive (HSI) noiseBlade-Vortex-Interaction (BVI) noise
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School of Aerospace Engineering
Rotorcraft Noise
High-Speed-Impulsive noise
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School of Aerospace Engineering
Rotorcraft Noise
Blade-Vortex-Interaction noise
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School of Aerospace Engineering
Noise Prediction
Great efforts have been spent on quantifying and minimizing rotorcraft noise.
Two noise prediction techniques:Fully computational aerodynamics and
acoustics;High resolution CFD calculation in the near
field coupling with acoustic analogy or Kirchhoff method in the far field;
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School of Aerospace Engineering
Noise Prediction
Blade
Acoustic calculation Region
Far Field
Observer
CFD calculation Region
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School of Aerospace Engineering
Noise Prediction
Near-field CFD calculation is critical to the accuracy of far-field noise prediction.
Much progress has been made on understanding and predicting rotorcraft noise characteristics with the aid of CFD methods.
Special attention is needed to simulate acoustic wave propagation with CFD methods.
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School of Aerospace Engineering
Noise Prediction
Differences between aerodynamic and aeroacoustic problems:Relative Magnitude
Length and Frequency Scale
Dispersion and dissipation errors generated by conventional CFD methods can easily change the observed noise characteristics.
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Noise Prediction
-2
0
2
4
6
8
10
12
-50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100
Dispersion Errors - some waves travel slower than the rest.
T=0
T=50 T=1
00
Magnitude drops as wave propagates…Dissipation
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School of Aerospace Engineering
Noise Prediction
These errors must be reduced to give correct traveling-wave profile.
Dispersion error minimization methods: Dispersion-Relation-Preserving (DRP) Scheme, by Dr. Tam and
Web;
Compact Scheme, by Dr. Lele;
Low Dispersion Finite Volume (LDFV) Scheme, by Dr. Nance and Sankar;
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LDFV SCHEME
The spatial derivative is approximated by the difference of the numerical fluxes evaluated at the adjacent half points.
x j j+1/2 j-1/2
x L R
x
uu
x
u jj
j
2
12
1 2
12
1
2
1 jRLj uuu
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School of Aerospace Engineering
LDFV SCHEME
The general interpolation formulations for the left and right physical fluxes are:
The approximation of spatial derivative becomes:
lj
N
Ml
Rl
R
jlj
M
Nl
Ll
L
juauuau
1
121
21
lj
N
Nll
j
uaxx
u
1
12
1
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School of Aerospace Engineering
LDFV SCHEME
Expand uj+l about xj using classical Taylor series expansion, like:
Specific spatial order scheme can be obtained by comparing corresponding coefficients of each term at both sides of the approximation equation.
1 !m
mm
jjlj xl
m
uuu
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School of Aerospace Engineering
LDFV SCHEME
A further restriction is imposed to match the Fourier transform of the approximation of spatial derivative with its exact transform.
The Fourier transform of the spatial difference approximation is:
F. T.lj
N
Nll
j
uaxx
u
1
12
1
1
12
1 N
Nl
xilklea
ixk
Non-dimensional Wave Number
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School of Aerospace Engineering
LDFV SCHEME
The least square error:
should be minimized with respect to coefficients
and .
Lla
Rla
Optimization Equations
2
2
21
12
1
xkdeai
xkEN
Nl
xilkl
0,0
Rl
Ll a
E
a
E
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School of Aerospace Engineering
OBJECTIVES
Study dispersion and dissipation characteristics of LDFV scheme
Apply LDFV scheme to rotorcraft noise prediction
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School of Aerospace Engineering
RESULTS
Analysis of dispersion and dissipation characteristics of LDFV scheme
High-Speed-Impulsive or shock noise prediction
Spherically symmetric wave propagation study
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Analysis of LDFV Scheme
The one-dimensional advection equation:
Semi-discretization of advection equation:
Discrete operator A is of the form:
0
x
uc
t
u
jj
ut
u
A
1
12
N
Nl
lla
x
cEA
j
llj uu E
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School of Aerospace Engineering
Analysis of LDFV Scheme
Full-discretization of advection equation written in operator notation:
Z is time shift operator:
Define an amplification factor:
which satisfies characteristic equation
nn utu ZAM1
nn uu Z1
0,ˆ kzkAtkz M
njk
njk ukzu ,1
,
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School of Aerospace Engineering
Analysis of LDFV Scheme
The phase velocity of full-discretization:
The amplitude error:
c
CFLxk
kzkz
kc
Re
Imarctan*
1ImRe 22 kzkz
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Analysis of LDFV Scheme
Phase Velocity, CFL=0.01
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
Non-Dimensional Wavenumber(kx)
c phase/c
MUSCL Scheme LDFV-3 SchemeLDFV-6 Scheme
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.99
0.992
0.994
0.996
0.998
1
1.002
1.004
1.006
1.008
1.01
Non-Dimensional Wavenumber(kx)
c phase/c
MUSCL Scheme LDFV-3 SchemeLDFV-6 Scheme
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School of Aerospace Engineering
Analysis of LDFV Scheme
Amplitude Error, CFL=0.01
0 0.5 1 1.5 2 2.5 3 3.5-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
Non-Dimensional Wavenumber(kx)
|z(
)|-1
MUSCL Scheme LDFV-3 SchemeLDFV-6 Scheme
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1x 10
-3
Non-Dimensional Wavenumber(kx)
|z(
)|-1
MUSCL Scheme LDFV-3 SchemeLDFV-6 Scheme
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School of Aerospace Engineering
Analysis of LDFV Scheme
The phase difference between predicted results and exact solution:
The predicted amplitude:
c
ckcttkckctΦ
** 1
N
jkN
jk kzuu 0,,
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Analysis of LDFV Scheme
-10
-8
-6
-4
-2
0
2
0 0.2 0.4 0.6 0.8 1 1.2
kdx
log(
|u(k
dx)|)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1 1.2
kdx
Ph
ase
Dif
fere
nce
(D
egre
e)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Fourier Transform Comparison at t=50
Phase Difference Comparison at t=50
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Analysis of LDFV Scheme
-10
-8
-6
-4
-2
0
2
0 0.2 0.4 0.6 0.8 1 1.2
kdx
log(
|u(k
dx)|)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1 1.2
kdx
Phas
e D
iffe
renc
e (D
egre
e)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Fourier Transform Comparison at t=400
Phase Difference Comparison at t=400
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Comments on LDFV Scheme Analysis
Both MUSCL and LDFV schemes can accurately capture low non-dimensional wave number components.
LDFV scheme generates lower numerical errors for high non-dimensional wave number components than MUSCL scheme.
The above observations are still kept as propagation distance increases although low error generation ranges are reduced.
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Shock Noise Prediction
1/7 scale model of untwisted rectangular UH-1H blades in hover condition
NACA0012 airfoil
Non-lifting case
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Shock Noise Prediction
Blade
C-Grid in Cylindrical Planes
H-Grid in Rotor Disk Plane
H-Grid in Span-Wise Planes
C-H Computational Grid
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Shock Noise Prediction
Wrap-Around Direction
Normal Direction
Normal Direction
Root
Tip
Span-Wise Direction
Typical grid: 133 in Wrap-around direction
55 in Span-wise direction
35 in Normal direction
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Shock Noise Prediction
r/R=1/MTip
r/R=1.78
R
Microphone
Shock Wave
r/R=2.18
r/R=3.09
Shock Noise Measurement Locations and Method
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Grid Independence Study
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0 0.5 1 1.5 2
Time(msec)
Aco
ustic
Pre
ssur
e(P
a)
100*55*35
133*55*35
250*55*35
Experimental Data
Acoustic Pressure Time History, MTip=0.90, r/R=3.09
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Grid Independence Study
0
0.005
0.01
0.015
0.02
0.025
0.03
0 50 100 150 200 250 300
Grid Size in Wrap-Around Direction
Der
ivat
ion
79*55*35
100*55*35133*55*35
150*55*35
175*55*35
200*55*35 250*55*35
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Acoustic Pressure Time HistoryMTip = 0.9, r/R=1.111
-7
-6
-5
-4
-3
-2
-1
0
1
2
0 0.5 1 1.5 2
Time(msec)
Aco
ust
ic P
ress
ure(
kP
a)
MUSCL+SuperBee Limiter
LDFV-3+SuperBee Limiter
LDFV-6+SuperBee Limiter
Experimental Data
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Acoustic Pressure Time HistoryMTip = 0.9, r/R=3.09
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0 0.5 1 1.5 2
Time(msec)
Aco
usti
c Pr
essu
re(k
Pa)
MUSCL+SuperBee Limiter
LDFV-3+SuperBee Limiter
LDFV-6+SuperBee Limiter
Experimental Data
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Shock Noise Prediction
MUSCL and LDFV scheme results are very similar.
Two factors to be investigated:Flux Limiter
Error generation
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Flux Limiter Effects
Flux limiter switches higher order schemes back to first order scheme in the large flux gradient regions.
Different flux limiters will give different results in those regions.
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Acoustic Pressure Time HistoryMTip = 0.9, r/R=3.09
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0 0.5 1 1.5 2
Time(msec)
Aco
ustic
Pre
ssur
e(kP
a)
MUSCL+Dif ferential Limiter
MUSCL+SuperBee Limiter
Experimental Data
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Error Generation Issues
What’s the non-dimensional wave number range of shock noise signal?
Is the simulated domain large enough to see difference between the results of two schemes?
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Spherically Symmetric Wave Propagation Study
The spherically symmetric wave equation:
Shock noise signal at sonic cylinder is chosen as initial condition.
0
2
22
2
2
r
rpc
t
rp
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Spherically Symmetric Wave Propagation Study
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
r (m)
p(r)
(kP
a)
Initial Acoustic Signal Distribution
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
k dr
|p(k
)|
kr=0.0614
Spectrum of Initial Signal
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Wave Distribution at r=3.229m(Without Limiter)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
2.85 2.95 3.05 3.15 3.25 3.35 3.45 3.55
r (m)
p(r)
(kP
a)
MUSCL Schem e
LDFV-3 Schem e
LDFV-6 Schem e
Exact Solution
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Spherically Symmetric Wave Propagation Study
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
0 0.2 0.4 0.6 0.8 1 1.2
k dr
log(
|p(k
)|)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
0
20
40
60
80
100
120
140
160
180
200
0 0.2 0.4 0.6 0.8 1 1.2
k dr
Pha
se D
iffe
renc
e (D
egre
e)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Fourier Transform of Signal Distributions at r=3.229m
(Without Limiter)
Phase Difference at r=3.229m (Without Limiter)
![Page 45: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/45.jpg)
School of Aerospace Engineering
Wave Distribution at r=20.0m(Without Limiter)
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
19.7 19.8 19.9 20 20.1 20.2 20.3
r (m)
p(r)
(kP
a)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
![Page 46: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/46.jpg)
School of Aerospace Engineering
Spherically Symmetric Wave Propagation Study
0
20
40
60
80
100
120
140
160
180
200
0 0.2 0.4 0.6 0.8 1 1.2
k dr
Phas
e D
iffer
ence
(Deg
ree)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
0 0.2 0.4 0.6 0.8 1 1.2
k dr
log(
|p(k
)|)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
Fourier Transform of Signal Distributions at r=20.0m
(Without Limiter)
Phase Difference at r=20.0m (Without Limiter)
![Page 47: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/47.jpg)
School of Aerospace Engineering
Negative Pressure Peak Under-prediction Ratio (Without Limiter)
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25r (m)
Neg
ativ
e Pr
essu
re P
eak
Rat
io
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
![Page 48: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/48.jpg)
School of Aerospace Engineering
Wave Distribution at r=3.229m(With Limiter)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
2.85 2.95 3.05 3.15 3.25 3.35 3.45 3.55
r (m)
p(r)
(kP
a)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
![Page 49: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/49.jpg)
School of Aerospace Engineering
Spherically Symmetric Wave Propagation Study
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0 0.2 0.4 0.6 0.8 1 1.2
k dr
log(
|p(k
)|)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
0
20
40
60
80
100
120
140
160
180
200
0 0.2 0.4 0.6 0.8 1 1.2
k dr
Phas
e D
iffer
ence
(Deg
ree)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Fourier Transform of Signal Distributions at r=3.229m
(With Limiter)
Phase Difference at r=3.229m (With Limiter)
![Page 50: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/50.jpg)
School of Aerospace Engineering
Wave Distribution at r=20.0m(With Limiter)
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
19.7 19.8 19.9 20 20.1 20.2 20.3
r (m)
p(r)
(kPa
)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
![Page 51: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/51.jpg)
School of Aerospace Engineering
Spherically Symmetric Wave Propagation Study
-6
-5
-4
-3
-2
-1
0
0 0.2 0.4 0.6 0.8 1 1.2
k dr
log(
|p(k
)|)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Exact Solution
0
20
40
60
80
100
120
140
160
180
200
0 0.2 0.4 0.6 0.8 1 1.2
k dr
Phas
e D
iffer
ence
(D
egre
e)
MUSCL Scheme
LDFV-3 Scheme
LDFV-6 Scheme
Fourier Transform of Signal Distributions at r=20.0m (With Limiter)
Phase Difference at r=20.0m (With Limiter)
![Page 52: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/52.jpg)
School of Aerospace Engineering
Negative Pressure Peak Under-prediction Ratio (With Limiter)
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25r (m)
Neg
ativ
e Pr
essu
re P
eak
Rat
io
MUSCL Result
LDFV-3 Result
LDFV-6 Result
Exact Solution
![Page 53: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/53.jpg)
School of Aerospace Engineering
Comments on Spherically Symmetric Wave Propagation Study
Characteristic components of signal are captured accurately by both MUSCL and LDFV schemes.
Errors generated by high non-dimensional wave number components need long propagation distance (r/R>20) to have effects on predicted results.
![Page 54: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/54.jpg)
School of Aerospace Engineering
Comments on Spherically Symmetric Wave Propagation Study
Two radiuses away from blade tip are not long enough for current shock noise signals to develop big difference between the results of MUSCL and LDFV schemes.
Flux limiter reduces the difference between MUSCL and LDFV schemes.
![Page 55: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/55.jpg)
School of Aerospace Engineering
CONTRIBUTIONS
Investigated dispersion and dissipation characteristics of LDFV scheme.
Applied LDFV scheme to rotorcraft noise prediction.
Demonstrated the effects of flux limiter on numerical simulation.
![Page 56: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/56.jpg)
School of Aerospace Engineering
CONCLUSIONS
LDFV scheme has low dispersion and dissipation errors in a wider non-dimensional wave number range than MUSCL scheme.
Large propagation distance (r/R>20) is needed to show difference between MUSCL and LDFV scheme results.
Flux limiter has large influence on prediction results.
![Page 57: STUDY OF A LOW-DISPERSION FINITE VOLUME SCHEME IN ROTORCRAFT NOISE PREDICTION](https://reader035.vdocuments.us/reader035/viewer/2022062217/56814558550346895db2278c/html5/thumbnails/57.jpg)
School of Aerospace Engineering
RECOMMENDATIONS
Develop new flux limiter to keep low dispersion features of LDFV scheme;
Combine low dispersion time marching scheme with current spatial low dispersion scheme.