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