Peter D. Lysak, William K. Bonness, and John B. FahnlineApplied Research Laboratory, Penn State University

2nd FLINOVIA SymposiumApril 27-28, 2017

COMPARISON OF LOW WAVENUMBER MODELSFOR TURBULENT BOUNDARY LAYER EXCITATION

Fluid Dynamics and Acoustics Office

Outline

• Review of TBL excitation of a structure– Modal analysis approach– Small correlation area simplification– Transformation between spatial and wavenumber coordinates– Low wavenumber spectrum

• Differences in TBL wavenumber-frequency spectrum models for low wavenumbers

– Wavenumber white model (Corcos)– k2 model (Chase)– Simplified modal force for k2 spectrum

• Proposed experimental configuration to distinguish between wavenumber white and k2 low wavenumber spectrum

General Formulation of the Vibration Response

• Modal expansion for vibration response due to a stochastic forcing function:

• Modal frequency response function:– Mode natural frequency n

– Loss factor n

• Modal force is found by integrating the cross-spectrum of the driving pressure fluctuations:

)()()()()(),,( * mnnm n

mnmaa HH yxyx

Cross-spectrum of acceleration between point x and point y

Mass-normalized mode shapes

Modal frequency response functions

Modal force matrix

nn

nn i

H

2

2

/1/)(

yxyxyx 22 dd),,()()()( ppnmmn

Reference: Bonness, W. K., Fahnline, J. B., Lysak, P. D., and Shepherd, M. R. (2017). Modal Forcing Functions for Structural Vibration from Turbulent Boundary Layer Flow. Journal of Sound and Vibration, Vol. 395, pp. 224-239.

TBL Pressure Cross-Spectrum (Corcos Model)

• Assume the cross-spectrum only depends on the distance r = y − x• Write as the product of the point pressure spectrum and the spatial

coherence cUri

ccpppp e

Ur

Ur /3

31

11expexp)(),(

r

Point pressure spectrum

Streamwisedecay

Cross-stream decay

Streamwiseconvection

Small Correlation Area Simplification

• Modal force equation:

• If the mode shape varies slowly compared to the spatial decorrelation of the TBL pressure, the modal force integrals can be separated:

rrxxx 22 d),(d)()()( ppnmmn

Mode ShapeTBL Cross-Spectrum

Low frequency

Medium frequency

High frequency

Direction of flow

Point pressure × Effective correlation area

xrrrxx 22 dd),()()()( ppnmmn

Constant

Small Correlation Area Simplification

• Modal force simplification:

• Equivalent to driving the structure with uncorrelated point drives distributed over the surface (rain on the roof)

• Each point drive includes the effective correlation area surrounding the point

• For Corcos model, get

rr 2d),()( ppmnmn C

)(1

d),(31

21

212

pp

ccpp

UU

rr

Effective correlation area

Example: Small Correlation Area Simplification

• Example calculations for a rib-stiffened plate show that small correlation area simplification gives the high frequency asymptote of modal force

• Whether this is relevant depends on the non-dimensional resonance frequency (L/Uc) – in water it is often in the high frequency region

Non-dimensional Frequency Non-dimensional Frequency

Wavenumber Representation

• To provide additional insight, transform from the spatial domain to the wavenumber domain

• The equivalent expressions for the model force in wavenumber space are:

• Correlation area is related to the zero-wavenumber level of the TBL pressure wavenumber-frequency spectrum

kkkk 2*4 d),()()()2()( PFF nmmn xrrrxx 22 dd),()()()( ppnmmn

Spatial DomainMode ShapeTBL Pressure Cross-Spectrum

Wavenumber DomainWavenumber Sensitivity FunctionWavenumber-Frequency Spectrum),( rpp

)(kF

),( kP)(x

rrxxx 22 d),(d)()()( ppnmmn ),0()2(d)()()2()( 22*2 PFF nmmn kkk

Full:

Simplified:

Equivalent

• Plot the TBL wavenumber-frequency spectrum (at a fixed frequency and with k3=0) on a log k1 scale

Wavenumber White Models

Convective ridgeStreamwise

decay• There are only two

possibilities for the low wavenumber region:

1. The spectrum crosses k1 = 0 at a finite value, giving the appearance of a wavenumber white low frequency spectrum

2. The spectrum decays to zero as k1 = 0

k1 / kc

Chase TBL Model

• The Chase model probably has the strongest theoretical basis, but it is not wavenumber white

• The small correlation area simplification cannot be used, as it gives a modal force of zero

• The spectrum goes as k2

for low wavenumbers

• The transition to the low wavenumber region depends on the boundary layer thickness

),0()2(d)()()2()( 22*2 PFF nmmn kkk

Simplified Modal Force for k2 Spectrum

• Note that the mode shapes act as “wavenumber filters” for the TBL wavenumber-frequency spectrum

• Idealizing as low pass filters with a cutoff wavenumber ko and assuming radial symmetry, get simplified result:

• Substituting the Chase k2 low wavenumber model gives

kkkk 2*4 d),()()()2()( PFF nmmn

2||

2

22*2d),(

)2(d)()()2()(o

knmmn k

PFF o

k

kkkkk

)()(const)( 2

2

ppomnmn kUC

(Similar to small correlation area simplification, but has ko factor)

Comparison of Results and Proposed Experiment

• Wavenumber white model

• k2 model

• Example structure where these differences would be noticeable – array of panels connected to a rigid frame

• Flat plate boundary layer growth:– Friction velocity slowly decreases downstream

– Boundary layer thickness grows steadily downstream

)(const)(

2

ppc

mnmnUC

)()(const)( 2

2

ppomnmn kUC

Consider only the first mode of each panel and measure the acceleration at the center of each panel

1.0

17.0

xUUU

14.0

16.0

xUx

Predictions for Panel Array

• At high enough frequencies, the point pressure spectrum is independent of boundary layer thickness

• Predicted vibration of panel array due to developing boundary layer:

42

)( Upp

• 5 cm square panels

• Aluminum• 1 mm thick• 5 m/s flow in water

Wavenumber white model decreases due to decreasing friction velocity

k2 model increases due to increasing boundary layer thickness

Simplified modal force for k2 spectrum is not very accurate, but does show trend

Level of first resonance peak for each panel

Conclusion

• It might be possible to measure this trend in an experiment, even if the absolute level of the low wavenumber spectrum is difficult to obtain

• Detailed experimental design still needs to be performed – at this stage it is just an observation based on analysis of the models

• Two types of low wavenumber models predict opposite trends for the vibration of an array of panels