Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
PO Box 30State College, PA 16804
Efficient Modeling of Structural Vibration and Noise from Turbulent Boundary Layer Excitation
W.K. Bonness, J.B. Fahnline, P.D. Lysak
M.R. Shepherd
Generalization of Results for Arbitrary Flow Speed
Center for Acoustics and Vibration WorkshopFlow-Induced Noise Session
April 29, 2013
Outline
• Introduction– Problem statement and objective– Fundamental equations and transformation to modal space
• Modal Force– TBL wall pressure cross-spectrum– TBL low wavenumber approximation– Small TBL correlation length (high frequency) approximation
• Examples Comparing Solution Methods– Simple plate– Complex rib-stiffened plate
• Generalization of Results for Arbitrary Flow Speed
2
Turbulent Boundary Layer (TBL)
TBL Excited Structures
Cross-spectrum of structural response due to TBL excitation:
Outputcross-spectrum
matrix
Inputcross-spectrum
matrix
Transfer Functionmatrix
Transfer Functionmatrix
[out × out] [in × in] [in × out][out × in]
*G H G HTXX FF
3
x1
0U Turbulent eddies
Area of correlated pressurex3
x2
Turbulent Boundary Layer (TBL)
TBL Excited Structures
• Problem:– Modeling the TBL excitation of a large practical structure makes the
numerical problem too big
• Objective:– Identify a modeling approach which extends the numerical analysis to
higher frequencies of interest
4
Low Frequency Mid Frequency High Frequency
Fundamental Equations
Displacementcross-spectrum
matrix
Forcecross-spectrum
matrix
Transfer Functionmatrix
Transfer Functionmatrix
*G H G HTXX FF
5
comes from the equations of motion H
2K B M A C X Fi Stiffness Damping Mass Acoustic
couplingStructuralcoupling
FEM BEM Specified
1H 1 FXH
Transformation to Modal Space
6
Cast equations into modal space by assuming physical response can be written as a summation of modes
ΨxX
xψψψX NM...21
Pre and post multiply fundamental equations by and , respectively. TΨ Ψ
1 FXH
*G H G HTXX FF hΦh T*Ξ
1 fxh
Physical Space Modal Space
Φ FFG
Modal Force
Matrix Form(Discrete System) GT
FF
7
Scalar Form(Continuous System)
( ) ( ) ( ) ( , , )ij i j ppS S
x x r x x r dr dx
Pressurecross-spectrum
Interpolated mode shapes
Modal force cross-spectrum for modes i and j
Position (x) and separation vector (r) on surface
Representative TBL pressure cross-spectrum model , Corcos (JASA, 1963)
1 3| | | |( , , ) ( )exp exp exppp pp
c c c
iU U U
cU15
-20 -10 0 10 20
0
0.2
0.4
0.6
0.8
1
xi3*kc
Nor
mal
ized
Pre
ssur
e Sp
ectr
um
-80 -60 -40 -20 0 20 40 60 80
-0.5
0
0.5
1
xi1*kc
Nor
mal
ized
Pre
ssur
e Sp
ectr
um
Correlation distance (and integration limits)
( , )pp x x r
TBL Spatial Correlation Functions
Streamwise Direction Cross-flow Direction
8
10.7 cUe
10.11 cUe
CorcosTBL Model
( , )pp x x r
CorcosTBL Model
Integration Limits
Integration Limits
cU cU
cU40 7TBL
10-3 10-2 10-1 100-70
-60
-50
-40
-30
-20
-10
0
10
k1/kc
Nor
mal
ized
Pre
ssur
e Sp
ectr
um, d
B
CorcosChaseMellenKoSmolyakov
EstablishLow-wavenumber level consistent with Mellen Model
- IFT to transform low-wavenumber level and integration limit into spatial domain
- Apply “low-wavenumber” TBL model in the same manner as the full cross-spectrum approach
Low Wavenumber Approximation
9
Upper wavenumber limit at kmax = 0.1kc
,,,, 31 kkPpp
pp
kkP
,, 31Spatial FT
Small TBL Correlation Length Approximation
• Start with full equation for modal force
• Simplify for high frequencies
x
y
a
b
At high frequencies, pp(,,) is nearly zero outside a small region surrounding point (x,y)
( , )x yFlow
0 0( ) ( , ) ( , ) ( , , )
a b a x b y
ij i j ppx yx y x y d d dy dx
0 0( ) ( , ) ( , ) ( , , )
a b
ij i j ppx y x y dy dx d d
Assume:• Mode shape is constant over small
correlated region• Limits of separation vector can be
extended to infinity, since pp goes to zero
10
Evaluation ofHigh Frequency Limit
• With high frequency approximation, two area integrals become independent of each other
• In general, express the solution of the second integral as
0 0( ) ( , ) ( , ) ( , , )
a b
ij i j ppx y x y dy dx d d
Evaluate using finite element model
(frequency independent)
Evaluate analytically(gives frequency dependence)
1 3( ) ( , , ) ( ) 2 ( ) 2 ( )pp ppF d d
Point Pressure Spectrum
StreamwiseCorrelation Length
SpanwiseCorrelation Length
11
Examples Comparing Solution Methods
1) Full TBL Cross-spectrum Integration Method
2) TBL Low Wavenumber Approximation Methodintegrate full cross-spectrum
3) Small TBL Correlation Length Approximation Methodhigh frequency
12
0 200 400 600 800 1000-40
-30
-20
-10
0
10
20
30
40
Frequency [Hz]
Acc
eler
ance
[dB
re (m
/s2 /N
)2 /Hz]
m=1m=2m=3m=4m=5
0.7m 1.0m
Analytical Model- Simply Supported- Aluminum Plate- 10 mm thick
m=1, n=1
m=2, n=1
m=1, n=2
Accuracy Check using Simple Plate
13
H
NMψψψ ...21
0 10 20 30 40 50 60-80
-70
-60
-50
-40
-30
-20
-10
0
Frequency [Hz]
Mod
al F
orce
[dB
re N
2 /Hz]
n=1n=2n=3n=4n=5
m=1
Grid: 40x28
Ko – 16.7 kts
Grid: 40x28
Analytical (Matlab) results Numerical (NASTRAN) results
m=1 n=1n=2n=3n=4n=5n=6
- Confirms numerical calculations are correct for more complex geometries
Ko – 16.7 kts
Computed Modal Force: Analytical vs Numerical
14
( ) ( ) ( ) ( , , )ij i j ppS S
x x r x x r dr dx
Modal ForceSimply Supported Plate
15
Numerical (NASTRAN) results
Numerical error from integrationof cross-spectrum
100
101
102
103
-20
-10
0
10
20
30
40
50
60
Frequency
Rad
iate
d Po
wer
Per
Uni
t TB
L Pr
essu
re (d
B)
Simply Supported full modelSimply Supported HF limitSimply Supported low-k
Radiated PowerSimply-Supported Plate
16
Full Cross-SpectrumLow Wavenumber Approx.High Frequency Asymptote
Complex Rib-Stiffened PlateConvergence Test
2540 Elements 9184 Elements
19932 Elements Flow Direction
Simply-Supported BCs
~ 4” x 4” ~ 2” x 2”
~ 1.3” x 1.3”
2c
c c
kU
10 kts @ 10 Hz
12c in
convective wavelength
Valid TBL freq: 5 Hz
Valid TBL freq: 10 Hz
Valid TBL freq: 15 Hz
17
First Six Mode Shapes
• The modes quickly change from being global to being localized to only a few “panels”
13.9 Hz 37.6 Hz 38.0 Hz
54.5 Hz 62.0 Hz 62.5 Hz
18
Results – Mode 1 Self Term
F1 = 3.0 Hz in water
Numerical error from integrationof cross-spectrum
Upper frequency
19
Flow Direction
Results – Mode 30 Self Term
f30 = 58.4 Hz in water
20
Flow Direction
Generalization of Results for Arbitrary Flow Speed
4c
upperUf
L
, 2c upperkL
Maximum analysis frequency for a given mesh size and convective speed
Integration error (insufficient mesh
resolution)
Full integration can be intractable
1c
m
kk
or 2
c cm m
U
Depends on the mode of interest, convective speed
and frequency
When is high frequency approximation valid?
0.76c
b
kk
2.1c
b
kk
When is high frequency approximation valid?
Difference is less than 1 dB when kc/kb > 7
When is high frequency approximation valid?
When is high frequency approximation valid?
A hybrid approach is recommended
( ) ( ) ( ) ( , , )ij i j ppS S
x x r x x r dr dx
Full integration below kc/kb = 7
High frequency approximation for kc/kb > 7
21
21 1 3
2 21
c cij
U UC
No restriction on flow speed or finite element mesh size