[american institute of aeronautics and astronautics 18th aiaa/ceas aeroacoustics conference (33rd...

American Institute of Aeronautics and Astronautics

1

Sound Radiation from Forward Facing Steps

Stewart A. L. Glegg1 and Benjeman Bryan2 Florida Atlantic University, Boca Raton FL 33431

William Devenport3, and Manuj Awasthi4

Virginia Tech, Blacksburg VA 24061

The sound radiation from a boundary layer flow over a forward facing step is characterized in terms of surface pressure fluctuations. Previous theories have modeled this source as a streamwise dipole whose strength is determined by the pressure fluctuations on the face of the step. However wind tunnel measurements indicate that the far field directionality is almost omnidirectional and does not have the characteristics of a dipole. Recent measurements have shown that the surface pressure fluctuations in the separation zone just downstream of the step are approximately 30dB higher than those just forward of the step and in this paper it is shown that these are the primary source of the radiated sound. The predicted far field sound spectrum agrees well with measured levels.

I. Introduction

Recently there have been some enlightening experimental and numerical studies on the sound radiated by turbulent flow over steps and gaps. These studies are particularly important for the design of quiet marine vehicles where the tolerances of plate joints need to be specified, and on aircraft fuselages where interior noise caused by flow over the fuselage is an issue. The numerical studies have given insight into the sources of the radiated noise. Using Lighthill’s acoustic analogy combined with a "tailored Greens function" Wang (2010) showed that the radiated sound field has the characteristic of a stream-wise dipole and that the source was most effective close to the corner of the step. However the Greens function used by Wang (2010) was a low frequency approximation to the actual Greens function (Howe (1976)) and did not include the possibility of radiation in the direction normal to the flow. Measurements by both Farabee and Cassarella (1986), and Catlett (2010) showed that for a forward facing step there was significant radiation in direction normal to the flow. The purpose of this paper is to investigate this discrepancy and provide an explanation for the differences between the theory based on the low frequency approximation and the measured data. The sound radiation from a forward facing step can also be defined in terms of surface dipole sources. The pressure fluctuation on the forward face represents a stream-wise dipole (equivalent to Wang (2010) and Howe (1976) results), while the pressure fluctuation on the upper surface of the step represents a dipole normal to an infinite rigid surface, which is an effective quadrupole because of the image source below the surface. At low Mach numbers M the power radiated by a dipole is of order M2 less than a quadrupole source and so can usually be ignored. However in the case of a forward facing step Awasthi et al (2011) showed that the measured surface pressure on the top face of the step was approximately 30 dB more than under the undisturbed boundary layer, due to the flow separation over the step. For a flow with a Mach number of 0.1, the acoustic efficiency of a dipole is only 20dB more than a quadrupole, and so, in spite of its quadrupole nature, the pressure fluctuations on the top face of the step may dominate the radiated sound field. In the following we will review the theory for sound radiated from a forward facing step and then use the results presented in Awasthi et al (2011) to predict the far field sound from a forward facing step, and compare the estimated far field spectra with the measurements presented by Catlett (2010). 1 Professor, Department of Ocean and Mechanical Engineering, AIAA Associate Fellow.

2 Graduate Student, Department of Ocean and Mechanical Engineering, AIAA Member 3 Professor, Department of Aerospace and Ocean Engineering, AIAA Associate Fellow.

4 P Graduate Student, Department of Aerospace and Ocean Engineering, AIAA Member.

18th AIAA/CEAS Aeroacoustics Conference (33rd AIAA Aeroacoustics Conference)04 - 06 June 2012, Colorado Springs, CO

AIAA 2012-2050

Copyright © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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II. Theoretical Background Consider a turbulent boundary layer flowing over a forward facing step defined in Cartesian coordinates by a jump from y2=0 to y2=h at y1=0. All the surfaces are rigid and flow noise sources will be determined by the dipole source terms on the surfaces local to the step. In the frequency domain the far field sound pressure p(x,ω) can be described by the surface integral

(1)

p(x,! ) =S! p(y,! ) "G(x | y)

"ndS(y)

where the Greens function G is chosen so that ∂G/∂n=0 on the surface y2=0 upstream of the step. The far field approximation for this Greens function is given as

(2)

G(x | y) = eikro!ikx1y1/ro!ikx3y3/ro

2!ro

"#$

%&'cos(kx2y2 / ro )

where ro is the distance from the inside corner of the step to the observer and the step span is aligned with the y3 axis. The far field sound then is given by the sum of two terms. The first is the contribution from the step face and defined (for an observer in the plane x3=0) as

(3)

po(x,! ) =!ikx1ro

eikro

2"ro 0

h

"!#

#

" p(y,! )cos(ky2x2 / ro )dy2dy3

If the step is acoustically compact then the cosine term can be dropped and the integral represents the net force per unit span on the face of the step. Then modeling the surface pressure as Pu(y3), and assuming that the pressure fluctuations on the forward face are over a wetted height of he, we obtain

(4)

po(x,! ) =!ikhex1ro

eikro

2"ro !"

"

# Pu(y3)dy3

The second contribution to the far field comes from the top surface of the step and is defined as

(5)

pt (x,! ) =!kx2ro

eikro

2"rosin(khx2 / ro )

0

"

#!"

"

# p(y,! )dy1dy3

We will assume that the pressure on the top face of the step is given by the function

(6)

p ~ Pu(y3)ei! y1/Uc + Pd (y3)e

i! y1/Uc

(a2 + (y1 / h)2 )1/2

where Pu is the pressure induced by the undisturbed boundary layer and Pd is the disturbance induced by the separated flow over the step that has been modeled as having a linear decay at large distances downstream. Assuming that the separated flow term dominates the surface pressure leads to the far field

(7)

pt (x,! ) =!kx2ro

eikro

4"rosin(khx2 / ro )

!"

"

# Pd (y3)0

"

#e!iky1x1/ro+i! y1/Uc

(a2 + (y1 / h)2 )1/2

dy1dy3

Evaluating the integrals gives

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(8)

pt (x,! ) = !khx2

ro

eikro

2"rosin(khx2 / ro )

!"

"

# Pd (y3)F(aµh)dy3

F(aµh) = 1h 0

"

#e!iµy1

(a2 + (y1 / h)2 )1/2 dy1 µ=kx1 / ro +! /Uc

We then calculate the far field sound pressure spectrum Spp(ω)=(π/T)E(|p(ω)|2) and assume that the spanwise integration can be represented by

(9)

!"

"

#!"

"

#!TE[Pu(y3)Pu

*( $y3)]dy3$dy3 = L!uSuu(" )

!"

"

#!"

"

#!TE[Pd (y3)Pd

*( $y3)]dy3$dy3 = L!dSdd (" )

where the spanwise integration is carried out assuming a span of length L and a spanwise correlation length scale of l. In this formula Suu represents the spectrum of the undisturbed boundary layer and Sdd the surface pressure spectrum induced by the separated flow on the top face of the step. The far field sound spectrum then follows as

(10)

Spp(x,! ) =khex12"ro

2

!"#

$%&

2

L!uSuu(! )+ F2 (aµh)sin2(khx2 / ro )

khx22"ro

2

!"#

$%&

2

L!dSdd (! )

The first term represents the dipole sound from the face of the step while the second term is the contribution from the top surface of the step. Typically we are interested in values of aµh>>1 and so the function F~i/aµh and for small Mach numbers µ~ω/Uc so we can approximate

(11)

Spp(x,! ) =khex12"ro

2

!"#

$%&

2

L!uSuu(! )+ sin2(khx2 / ro )

Mcx2a2"ro

2

!"#

$%&

2

L!dSdd (! )

For low Mach number flows (Mc~0.01) the second term in this equation is usually considered to be small compared to the first term. However for flows where Mc~0.2 and Sdd is more than 30 dB greater than Suu, this assumption cannot necessarily be made.

III. Experimental Results

Awasthi et al (2011) carried out measurements of the flow over a series of forward facing steps immersed in a thick high Reynolds number turbulent boundary layer. The step sizes were close to 3.5%, 15% and 60% of the boundary layer thickness and, with two flow speeds, step-size Reynolds numbers from 6640 to 213000 were achieved. Detailed measurements of surface pressure fluctuation spectra were made from 5 boundary layer thicknesses upstream to 22 boundary layer thicknesses downstream of the step. Spectra measured downstream of the step are shown in figure 1 for the three different step heights at a flow speed of 30 m/s. The important observations from these measurements is that the spectral level measured downstream of the step is up to 30 dB greater than the spectral level of the undisturbed boundary layer and that the level never drops below the spectral level of the undisturbed boundary layer. This latter result suggests that the additional pressure fluctuations generated by the step can be usefully thought of as being super-imposed upon the undisturbed boundary, and thus the latter can be subtracted out to reveal the exclusive influence of the step. Furthermore, when this is done, the step disturbance spectrum is found to be self similar and

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collapses to a single curve for all cases and locations when normalized on the peak spectral level and peak frequency, as shown for example in figure 2-6 (see Aswathi et al(2011)).

IV. Empirical Fit to the Spectra

Figure 2 shows the spectrum of the undisturbed boundary layer with a curve fit based on (12)

4!Suu(" )Ue

(#Ue2 )2$

= A( f$ /Ue )%o e!& ( f$ /Ue )2

B + ( f$ /Ue )3( )'o

A = 5.25x10!6 B = 0.1 'o = 0.355 %o = 0.1242 & =0.0286

Similarly the normalized spectra downstream of the step shown in figure 3 can be modeled by

(13) 4!Sdd (" )

!R

= A1( f / fR )#1

B1 + ( f / fR )2( )$1

A1 = 6 B1 =1.75 $1 =1.71 #1 =1.28

The variation of the peak spectral level downstream of the step was found to be modeled by

(14) !R

(!Ue2 )2h

= 1a2 + (y1 / h)

2

where a=√75, and this curve fit is shown in figure 4. The peak frequency required in equation (13) is shown in figure 5. Using these results we can then predict the far field sound providing that the spanwise correlation length scale l is known. In the absence of a measured value this has been estimated using the analysis specified in the appendix as l~2πb2/(1+(ωb1/Uc)2)1/2 where b1 and b2 are the lengthscales in the axial and transverse directions.

V. Far Field Sound Levels

Catlett (2010) presented measurements of sound radiated by a forward-facing step in a wall jet flow in an anechoic wind tunnel. The wall jet has a jet exit velocity of 60 m/s, and this is matched to an equivalent boundary layer flow with an outer flow speed of 21 m/s (M=0.06). The far field sound spectra were obtained at four different angles to the flow over a frequency range of 500Hz to 15 kHz. A number of different step heights were considered but we will only present results for h=11.7mm. The width of the step was 1.208m. For this case the effective boundary layer thickness was 17.7 mm, the integral length scale was chosen as δ/3 for the upstream flow and h for the flow downstream of the step based on a review of the measurements by Aswathi et al(2011). Measurements were made at angles of 51.5˚,74˚,97.5˚,and 123.5˚ at a distance of 0.557m from the step. The procedure described above can be used to estimate the sound levels measured in Catlett’s (2010) experiment by combining the estimated spectra from Awathi et al (2011) measurements (using a different facility) and the theory given in section 2 above. The wetted face of the step was chosen as he=h/4 , based on RANS calculations of flow over the step, and the peak spectral frequency was estimated from figure 5 as fr=0.2U/δ. The resulting predictions of the far field sound are compared to the measurements in figure 6 and show very good agreement. It is interesting to compare the relative contributions of the streamwise and normal dipole in these predictions as shown in figure 7. We conclude that the far field sound is dominated by the normal dipole source caused by the fluctuations on the upper surface of the step at lower frequencies. This is an important result because this term has previously been assumed to be negligible compared to the streamwise dipole.

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VI. Conclusion

This paper provides an approach for calculating the sound radiation from a forward facing step based on a surface pressure formulation and the measurements of Awasthi et al. (2011). The results show that the surface pressure fluctuations on the surface just downstream of the step are the dominant source of sound generation even for flows with Mach numbers of order 0.05. Previously it had been assumed that at low Mach number the sound radiation would be dominated by a streamwise dipole but this is shown not to be the case because the separated flow downstream of the step causes surface pressure fluctuations that are ~30dB more than those for the undisturbed boundary layer.

Acknowledgement

The authors would like to thank the Office of Naval Research, in particular Dr. Ron Joslin, for their support under grants N00014-12-1-0373, N00014-12-1-0374 and N00014-09-1-0315. We would also like to acknowledge Dr. Bill Blake for his helpful suggestions and insight.

References Awasthi M., Forest J. B., Morton M. A., Devenport W., Glegg S., 2011, “The Disturbance of a High Reynolds

Number Turbulent Boundary Layer by Small Forward Steps”, 17th AIAA/CEAS Aeroacoustics Conference, June 6th-8th, Portland OR, AIAA-2011-2777.

Catlett, M.R., 2010, “Flow Induced Noise from Turbulent Flow over Steps and Gaps”, Master’s Thesis, AOE Department, Virginia Tech. Avail: http://scholar.lib.vt.edu/theses/available/etd-05172010-192806/.

Catlett M, Devenport W and Glegg S, 2010, “Sound from boundary layer flow over steps and gaps”, AIAA 16th AIAA/CEAS Aeroacoustics Conference, June 7th-9th 2010, Stockholm, Sweden. AIAA Paper 2010-3774.

Farabee, T. M., and Cassarella, M. J., 1986, “Measurements of Fluctuating Wall Pressure for Separated/Reattached Boundary Layer Flows”, ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 108, 1986, pp. 301-307.

Ji, M., and Wang, M., 2010, “Sound generation by turbulent boundary-layer flow over small steps”, Journal of Fluid Mechanics, vol. 443, pp197-229.

Appendix

To estimate the spanwise correlation lengthscale we will assume that the normalized surface pressure correlation function can be defined as exp(-((Uτ/b1)2+(Δz/b2)2)1/2) where τ is the time delay, bi are length scales and Δz is the spanwise displacement. The spanwise correlation lengthscale is defined as

! =20

!

"0

!

" e#! cos("# )d#d($z)

0

!

" e#U# /b1 cos("# )d#

where ξ= ((Uτ/b1)2+(Δz/b2)2)1/2. Specifying Uτ/b1=ξcosβ and Δz/b2=ξsinβ so dτd(Δz)=(b1b2/U)ξdξdβ and evaluating the integral on the bottom line gives

! = (1+!o2 )2b2

0

!

"0

2"

" e## cos(!o# cos$ )#d#d$

where ωο=ωb1/U. The integral over β gives 2πJo(ωοξ) where Jo is a Bessel function of the first kind, and the integral over ξ can be obtained from tables of Laplace transforms giving

! = 4!b2(1+!o

2 )1/2

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Figure 1. The surface pressure spectra measured downstream of a forward facing step for three different step heights at a flow speed of 30 m/s

10-2 10-1 100 101 10210-10

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10-13.65mm step, Ue=30m/s

f!/Ue

"ppUe/(#U

e2 )2 !

x/h=24.7x/h=36.4x/h=59x/h=91.2x/h=134.2x/h=146x/h=168.6x/h=219.2x/h=292.3x/h=438.4x/h=525.6x/h=572.6x/h=584.4x/h=607Smooth w all

!"#$%&'

!"#$()&'!"#$&*+'

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f!/Ue

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e2 )2 !


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f!/Ue

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e2 )2 !


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f!/Ue

"ppUe/(#U

e2 )2 !

x/h=5.7x/h=6.2x/h=8x/h=9.1x/h=14.8x/h=22.8x/h=33.1x/h=33.6x/h=35.4x/h=36.5x/h=42.2x/h=54.8x/h=73.1x/h=109.6x/h=131.4x/h=142.7x/h=148.4x/h=149.5x/h=151.3x/h=151.7Smooth w all

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f!/Ue

"ppUe/(#U

e2 )2 !


!"#$(,-'

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f!/Ue

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e2 )2 !

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f!/Ue

"ppUe/(#U

e2 )2 !

x/h=1.4x/h=1.5x/h=2x/h=2.3x/h=3.7x/h=8.3x/h=8.4x/h=8.8x/h=9.1x/h=10.5x/h=35.7x/h=35.8x/h=36.2x/h=36.5Smooth w all

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f!/Ue

"ppUe/(#U

e2 )2 !

x/h=1.4x/h=1.5x/h=2x/h=2.3x/h=3.7x/h=8.3x/h=8.4x/h=8.8x/h=9.1x/h=10.5x/h=35.7x/h=35.8x/h=36.2x/h=36.5Smooth w all

f./U!

#!."#$%&'/0#"(()*'' #!."+,&-'/0#".((**'' #!."(*&-'/0#"+*(***''

/0!#'

!" 22 )( #

#$UUpp

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Figure 2: The surface pressure for an undisturbed boundary layer and the empirical fit given by equation (12) (blue line)

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Figure 3: The normalized surface pressure spectra for the flow downstream of a forward facing step and the empirical fit given by equation (13)

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Figure 4: The peak surface pressure spectral level as a function of distance downstream of a forward facing step, and the empirical fit given by equation (14).

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Figure 5: The non dimensional peak frequency fδ/U∞ of the surface pressure spectra as a function of distance downstream of a forward facing step.

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Figure 6: The predicted far field sound spectra (smooth lines) for a forward facing step compared to the measurements (jagged lines) of Catlett (2010) at angles θ to the direction of the flow.

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Figure 7: Comparison of the streamwise dipole far field sound levels (solid lines) with the normal dipole far field sound levels (dash dot) for observer angles θ=123.5º (blue), 97.5º(green), 74º(red), 51.5º(black)

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