study of a low-dispersion finite volume scheme in rotorcraft noise prediction

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


OUTLINE

Background

LDFV Scheme

Objectives

Results

Contributions & Conclusions

Recommendations


BACKGROUND

Helicopter has a wide range of military and civil applications.

However, the high noise level associated with it greatly restricts its further applications.


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


Rotorcraft Noise

High-Speed-Impulsive noise


Rotorcraft Noise

Blade-Vortex-Interaction noise


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;


Noise Prediction

Blade

Acoustic calculation Region

Far Field

Observer

CFD calculation Region


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.


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.


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


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;


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


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


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


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


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


OBJECTIVES

Study dispersion and dissipation characteristics of LDFV scheme

Apply LDFV scheme to rotorcraft noise prediction


RESULTS

Analysis of dispersion and dissipation characteristics of LDFV scheme

High-Speed-Impulsive or shock noise prediction

Spherically symmetric wave propagation study


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



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

,



The phase velocity of full-discretization:

The amplitude error:

c

CFLxk

kzkz

kc

Re

Imarctan*

1ImRe 22 kzkz



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


c phase/c




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


|z(

)|-1


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


|z(

)|-1




The phase difference between predicted results and exact solution:

The predicted amplitude:

c

ckcttkckctΦ

** 1

N

jkN

jk kzuu 0,,



-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



-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


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.


Shock Noise Prediction

1/7 scale model of untwisted rectangular UH-1H blades in hover condition

NACA0012 airfoil

Non-lifting case



Blade

C-Grid in Cylindrical Planes

H-Grid in Rotor Disk Plane

H-Grid in Span-Wise Planes

C-H Computational Grid



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



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


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


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


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


Experimental Data



-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)




Experimental Data



MUSCL and LDFV scheme results are very similar.

Two factors to be investigated:Flux Limiter

Error generation


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.



-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


Experimental Data


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?


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



-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


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



-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)


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



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


(Without Limiter)

Phase Difference at r=20.0m (Without Limiter)


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


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



-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


(With Limiter)

Phase Difference at r=3.229m (With Limiter)


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



-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)


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


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.


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.


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.


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.


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.

study of a low-dispersion finite volume scheme in rotorcraft noise prediction

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