# Nonlinear Disturbance Equations

Mechanics

Feb 22, 2014 (4 years and 2 months ago)

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1

A Computational Aeroacoustics Approach to
Trailing Edge Noise Prediction using the

Nonlinear Disturbance Equations

James P. Erwin

Philip J. Morris

Kenneth S. Brentner

Department of Aerospace Engineering

Penn State University

47
th

AIAA

Aerospace Sciences Meeting

January 5, 2009

The offshore wind turbine REpower 5M (rotor diameter: 126 m)
after its successful erection in the Scottish North Sea

2

Outline

Wind turbine acoustics

New trailing edge noise prediction method

The Nonlinear Disturbance Equations (NLDE)

NLDE code

validation

circular cylinder and airfoil test cases

Summary and future work suggestions

3

Acoustic Issues for Wind Turbines

Low frequency sound is relatively unaffected by atmospheric attenuation

can propagate long distances

Blade passage frequency below threshold of human hearing ~15Hz

Broadband noise is probably the dominant noise source (especially
when modulated at blade passage frequency)

Scale of large wind turbines leads to broadband noise at relatively low
frequencies that also propagates long distances

Tower and terrain wake

Nonuniform inflow due to atmospheric boundary layer

4

Self Noise Sources

Ref. Brook, Pope, and Marcolini, 1989

Current methods to predict
-
empirical.

5

Acoustic Issues for Wind Turbines

Large
-
Eddy Simulation (LES) of a complete wind
turbine and all noise sources not feasible in the near
future (especially for design purposes)

Direct computation of broadband noise sources is
possible

if focus is only small noise generating
regions of flow

Must divide the CAA problem into sub
-
parts which
can each be solved in the most efficient way possible

6

Outline

Wind turbine acoustics

New trailing edge noise prediction method

The Nonlinear Disturbance Equations (NLDE)

NLDE code

validation

circular cylinder and airfoil test cases

Summary and future work suggestions

7

Trailing Edge Noise Prediction Method

1.

Obtain mean flow for entire blade

-

RANS solution for “quick” estimate of mean flow

2.
Solve the NLDE on the trailing edge* portion only

-

Use a fine trailing edge grid

-

Solve for time accurate pressure time history

3.

Noise prediction from NLDE solution

-

PSU
-
WOPWOP

Penn State’s noise prediction software

-

Uses NLDE solution to calculate broadband noise and

propagate to observers

* Focus here is on the TE but these tools
will also work for the LE and blade tip (or
other sources)

RANS

8

Outline

Wind turbine acoustics

New trailing edge noise prediction method

The Nonlinear Disturbance Equations (NLDE)

NLDE code

validation

circular cylinder and airfoil test cases

Summary and future work suggestions

9

Nonlinear Disturbance Equations

(NLDE)

Multi
-
level hybrid approach

Use the algorithm best suited to the computation

Steady RANS for mean flow (calculation in entire domain
on relatively coarse grid)

Time accurate solution for disturbances (calculation in
limited region on a refined grid)

Present formulation based on compressible Navier
-
Stokes
equations (ideal for acoustic simulations)

t
x
u
t
x
u
t
x
U
t
x
u
t
x
u
t
x
U
t
x
U
,
"
:
,
~
:
,
,
"
,
~
,
,

simulations

Resolved perturbations

simulated using
time accurate calculations on refined grid

Sub
-
grid scale perturbations

OVERFLOW2

NLDE

modeled

10

Previous Applications

Turbulent boundary layer:

T. Chyczewski, P. Morris, and L. Long, (2000) AIAA Paper 2000
-
2007

Bluff body flows:

R. P. Hansen, L. N. Long, and P. J. Morris, (2000) AIAA Paper
2000
-
1981

High speed jet noise:

Morris, Long, Scheidegger & Boluriaan, (2002)
Int. Journal
Aeroacoustics
, 1(1)

Labourasse & Sagaut (2003)
J. Comp. Phys
.,
182
(L&S, 2003)

11

Nonlinear Disturbance Equations

The traditional compressible Navier Stokes equations can be written as

x
F
t
q

(1D for simplicity)

The NLDE decomposes this into a mean flow and perturbation flow

x
F
t
q
q

)
'
(
0
Since we are solving for the perturbation quantities only,

t
q
x
F
t
q

0
'

NOTE: no subscript ( )
0

or prime ( )ʹ implies an instantaneous quantity

12

Nonlinear Disturbance Equations

The time rate of change of the

mean flow is zero (steady mean flow)

t
q
x
F
t
q

0
'

x
F
t
q

'
The flux vector
F

is

updated at every time step

u
p
E
p
u
u
F
)
(
2

'
0

'
0
u
u
u

'
0
p
p
p

'
0
E
E
E

x
F
t
q

'
Initial

condition?

13

Outline

Wind turbine acoustics

New trailing edge noise prediction method

The Nonlinear Disturbance Equations (NLDE)

NLDE code

validation

circular cylinder and airfoil test cases

Summary and future work suggestions

14

NLDE Code

Code features:

Compressible, 3
-
D structured grid Navier
-
Stokes solver

Fortran 90 language

MPI (Message Passing Interface) parallel code

Code structure allows for easy addition and removal of features

Boundary conditions tailored for CAA

4
th

order accurate 5 stage LDDRK time integration
[10]

Low
-
Dissipation and Dispersion Runge Kutta

4
th

order accurate DRP finite differencing
[11]

Dispersion
-
Relation
-
Preserving

Explicit low pass filtering
[12]

15

Code Validation

2
-
D Gaussian Pulse

Mach 0.5 background flow (rightward)

Mach 0.0 and 0.5 background flow

201 x 201 grid

No artificial damping

No low pass filtering

3
9
2
ln
9
2
ln
2
2
2
2
012
.
'
1010
'
m
kg
e
Pa
e
p
y
x
y
x

X
p
'
(
P
a
)
-
5
0
0
5
0
0
2
0
0
4
0
0
6
0
0
8
0
0
1
0
0
0
Initial condition

16

Code Validation

2
-
D Gaussian Pulse

Mach 0.5 background flow

t = 0.02 seconds

t = 0.06 seconds

t = 0.2 seconds

t = 0.4 seconds

17

61 x 61 x 61 Cartesian grid

Zero background flow

No artificial damping

No low pass filtering

3
9
2
ln
9
2
ln
2
2
2
2
2
2
012
.
'
1010
'
m
kg
e
Pa
e
p
z
y
x
z
y
x

Initial acoustic pressure pulse

Code Validation

3
-
D Gaussian Pulse

18

Code Validation

3
-
D Gaussian Pulse

19

Code Validation

Tam and Dong pressure contours

Equivalent NLDE code contours

Mach 0.5 background flow

Tam and Dong, 1993 [14]

20

Outline

Wind turbine acoustics

New trailing edge noise prediction method

The Nonlinear Disturbance Equations (NLDE)

NLDE code

validation

circular cylinder and airfoil test cases

Summary and future work suggestions

21

Circular Cylinder Flow

2
-
D

X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
Coarse grid

100 points circumferentially

5% wall spacing

Fine grid

301 points circumferentially

0.5% wall spacing

Hyperbolic tangent stretching

22

Circular Cylinder Flow

2
-
D

Re
d

= 90,000 (based on diameter)

Uniform Mach 0.2 (rightward) mean flow

Radiation condition applied at far field boundaries

Instantaneous no slip condition

at surface is enforced by

specifying u
´

=
-
u
0
, v
´

=
-
v
0
, and w
´

=
-
w
0

X
Y
Z
X
Y
Z
Mean flow (initial condition)

Instantaneous flow

(shortly after no slip condition is applied)

23

Circular Cylinder Flow

2
-
D

coarse cylinder grid

fine cylinder grid

24

T
i
m
e
L
i
f
t
c
o
e
f
f
i
c
i
e
n
t
5
1
0
1
5
2
0
-
1
-
0
.
5
0
0
.
5
1
Circular Cylinder Flow

2
-
D

U
fL
St

f

shedding frequency

L

length scale (diameter)

U

flow velocity

204
.
0
64
.
68
)
002
.
0
(
7000

St
.025

.030

.035

.040

Time (seconds)

25

Circular Cylinder Flow

2
-
D

Acoustic data surfaces provide PSU
-
WOPWOP with

ρ,ρu,ρ
v,
ρw,p
´

Acoustic data surfaces can be placed anywhere in the flow field

but they must enclose the body of interest

NLDE grid

acoustic data

26

Circular Cylinder Flow

2
-
D

PSU
-
WOPWOP calculates the acoustic pressure and

sound pressure level for any combination of observer positions

27

Circular Cylinder Flow

2
-
D

90
°

observer

directivity

Fine grid

Mach 0.2

28

Apply tools developed previously to real

Initiate the simulation with assumed
uniform mean flow

Study the noise characteristics of different
trailing edges

NACA series airfoils

Increase resolution in areas of interest

Trailing edges

etc

X
Y
Z
X
Y
Z
29

NACA 0012 Airfoil

0.1% trailing edge thickness (relative to chord)

Mach 0.2 (Re
c

= 4.5 million), 0
°

aoa

Representative of the tip of a 9 meter turbine blade

30

NACA 0012 Airfoil

Laminar Boundary Layer

Vortex Shedding Noise

31

NACA 0012 Airfoil

90
°

observer

directivity

Observers placed on a circle with a radius of 5 chords centered at TE

32

Blade Systems Design Study (BSDS) wind turbine rotor

Grid provided by Sandia National Laboratories

“Flatback” airfoil design for structural strength at root of blade

How does this affect airfoil performance and noise?

5.5% trailing edge thickness

(relative to chord)

33

34

35

90
°

observer

directivity

36

Outline

Wind turbine acoustics

New trailing edge noise prediction method

The Nonlinear Disturbance Equations (NLDE)

NLDE code

validation

circular cylinder and airfoil test cases

Summary and future work suggestions

37

Summary

New CAA method for trailing edge noise prediction

NLDE flow solver is based on first principles methods for broadband
noise prediction

Coupled with OVERFLOW2 and PSU
-
WOPWOP

NLDE code

Validated with exact solutions

Tested with circular cylinder flow and first airfoil attempts

PSU
-
WOPWOP support

Noise prediction of any area of interest

Acquiring good RANS solution is not critical

The NLDE solution provides correction to mean flow (faster convergence
with better RANS solution)

using uniform mean flow for code
development

38

Future work suggestions

Triggering flow unsteadiness for realistic TBL
-
TE noise
calculations

Same issue with LES or DES simulations

(L&S, 2003) used random, divergence free initialization

Use of recycling in initial upstream region

Accurate turbulence characteristics needed for accurate broadband
noise prediction

Multistep method to decrease runtime of compressible viscous
calculations

Airfoil calculations take days to simulate sufficient time length

Compare noise of different blade sections

Develop thorough and well defined test cases to properly analyze
blade sections of interest, like the flatback BSDS sections.

39

Acknowledgement

This research was supported by Sandia National Laboratories,
Purchase Order No. A0342 677302,
Dale Berg and Matthew Barone, Technical Monitors.

40

References

41

Review:

Nonlinear Disturbance Equations

1.

What are they?

-

The complete set of compressible Navier
-
Stokes

equations separated into an
assumed mean flow
component and a

perturbation

component

-

The NLDE solve for the perturbation component about an estimated mean

2.

What are the benefits?

-

Resolve different flow scales

-

Allows simple application of detailed CAA boundary conditions

-

Mean flow is assumed to already satisfy BCs

-

NLDE equations only need to be solved in small region of

flow that generates noise

3.

How are they solved?

-

Same way as the traditional N
-
S equations

-

Mean flow is treated as a known source term

-

Only the perturbation variables are numerically integrated for a

time
-
accurate solution

of acoustic pressure

42

X
p
'
(
P
a
)
-
5
0
0
5
0
0
2
0
0
4
0
0
6
0
0
8
0
0
1
0
0
0
Code Validation

2
-
D Gaussian Pulse

Mach 0.0 and 0.5 background flow

201 x 201 grid

No artificial damping

No low pass filtering

3
9
2
ln
9
2
ln
2
2
2
2
012
.
'
1010
'
m
kg
e
Pa
e
p
y
x
y
x

X
Y
-
1
0
0
-
5
0
0
5
0
1
0
0
-
1
0
0
-
5
0
0
5
0
1
0
0
1
0
1
0
9
0
9
8
0
8
7
0
7
6
0
6
5
0
5
4
0
4
3
0
3
2
0
2
1
0
1
0
p
'
(
P
a
)
43

Code Validation

2
-
D Gaussian Pulse

Mach 0.5 background flow (rightward)

44

Circular Cylinder Flow

2
-
D

coarse cylinder grid

45

Circular Cylinder Flow

2
-
D

fine cylinder grid

46

NACA 0012 Airfoil

47

NACA 0012 airfoil

Laminar Boundary Layer

Vortex Shedding Noise

48