Dimensional analysis of windscreen noise

Dimensional analysis of windscreen noise M. $trasber• David W. Taylor Research Center, Bethesda, Maryland20084-5000

(Received 2 April 1987; accepted for publication 6 October 1987)

It is shown that data on wind noise in spherical and cylindrical windscreens may be represented by a single universal curve if the data are plotted in nondimensional form. Appropriate dimensionless variables are a dimensionless frequency ( fD /V) plotted against a sound pressure coefficient (•b 1/3/P V2) or a dimensionless spectral density ( fS/p2 V 4), where f is frequency, D is screen diameter, V is wind speed, •b 1 ]3 is sound pressure of the wind noise in a 1/3-octave band, p is fluid density, and S is spectral density of the wind noise at frequencyf Data obtained from various disparate sources form a single curve with surprisingly small scatter for values of dimensionless frequency ( fD /V) up to 5.

PACS numbers: 43.28.Ra, 43.88.Kb

INTRODUCTION

The wind noise sensed by a microphone inside a windscreen is a complicated aerodynamic noise phenomenon that has resisted theoretical analysis. Accordingly, estimates of windscreen noise are presently based on measurements made with the particular screen of interest at specific wind speeds. If it is assumed that the noise is due to the response of the microphone to random pressure fluctuations associated with turbulence generated by the flow around the screen, then it should be possible to apply the principles of dimensional analysis to data obtained at specific speeds and screen dimensions so that they become applicable to a range of these variables. The purpose of this article is to show that mea- sured values of windscreen noise obtained from disparate sources for spherical and cylindrical screens do indeed con- form to the scaling laws of dimensional analysis. According- ly, it is possible to deduce an approximate universal empiri- cal relation between noise level, frequency, screen diameter, and wind speed generally applicable to the lower portion of the audio-frequency range.

I. DIMENSIONAL ANALYSIS

Consideration of the various physical variables that might influence the turbulence and associated windscreen noise indicates that the noise in a given frequency band should depend on the frequency and bandwidth, on the size and shape of the windscreen and its porosity and details of construction, on the wind speed, on the density of the fluid medium and, perhaps, on the viscosity and compressibility of the fluid. The effects of all these variables might be complicated and not independent of each other. The question arises as to how the effects of individual variables can be

displayed in the simplest and most direct manner. Dimen- sional analysis provides a procedure for doing this.

The use of dimensional analysis is commonplace in fluid dynamics; see, e.g., Streeter's handbook. • For the present, it is sufficient to summarize the procedure as follows. First, the various physical quantities that might influence the phenomenon under investigation are identified, as has already been done above. Second, the dimensions of each quantity

are expressed in terms of the fundamental dimensions of length, time, and mass (L,T,M) or, alternatively, length, time, and force (L, T,F). For example, pressure has the dimensions ML -1T- 2 or FL -2. Third, the physical quantities are transformed into dimensionless variables by multi- plying or dividing each quantity by an appropriate combination of other relevent quantities, the combination being chosen so that the resultant is dimensionless. Finally, the dependence of the various quantities on each other is expressed in functional form as a relation between the dimensionless variables. The functional form of the relation

between variables, expressed in dimensionless form, no long- er depends on specific values of the physical quantities. The application of these procedural steps to windscreen noise is described in the paragraphs that follow.

The data on windscreen noise can be put into nondimensional form in the following way. The spectrum of the noise is assumed to be given as an rms sound pressure•b in a relatively narrow band of width Afcentered at frequencyf This sound pressure is expressed in terms of a dimensionless sound pressure coefficient (•/p V 2 ), where V is the wind speed and p the density of the fluid medium (note that p V 2 has the dimensions of pressure). The bandwidth is expressed as a bandwidth ratio (Af/f) and the center frequency as a dimensionless frequency parameter (fD/V), where D is a characteristic dimension of the screen, e.g., the diameter for cylinders and spheres. The viscosity of the medium is expressed as a dimensionless ratio (D V/v), where v is the kinematic viscosity (v equals •0.15 cm •/s for air at 20 øC). The compressibility is expressed as a ratio (V/c), where c is the velocity of sound in the fluid (note that the sound velocity and bulk modulus Yare related by c • = Y/p). The last three dimensionless parameters have the well-known names $trouhal number (fD/V), Reynolds number (DV/v), and Mach number (V/c).

The sound pressure coefficient is the unknown quantity that is a function of the other dimensionless parameters. In symbolic form,

(•/pV •) = •-[ ( fD /V),(Af /f ),(DV/v),( V/c) ]. (1)

544 J. Acoust. Soc. Am. 83 (2), February 1988 544

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The form of the function may depend on the shape of the screen, its porosity, and details of construction; but the dependence on the various quantitative variables is contained in the four dimensionless arguments within the brackets. Note that although seven independent physical variables are involved, their effects are taken into account by only four dimensionless parameters. One of the advantages of dimensional analysis is that it reduces the number of variables re- quiring consideration by at least two.

Since it is assumed that the windscreen noise has a con-

tinuous spectrum, the dependence on bandwidth is known and may be placed outside the function, viz.,

(.•/pV •) = x/Af /f •[(fD/•,(DV/•),(V/c) ]. (2) Furthermore, if all the noise data are given in 1/3-octave bands, the dependence on bandwidth need not be shown ex- plicitly and Eq. (2) may be written

(.•,/3/pV •) = •-,/3[ ( fD /V),(DV/v),( V/c) ], (3) where •b•/3 is the rms sound pressure in a 1/3-0ctave band.

Alternatively, the sound pressure may be expressed in terms of a nondimensional spectral density fS(f)/p • V 4, where $(f) is the spectral density at frequencyf The prod- uctfS(f ) may be considered to be the mean-square sound pressure in a band whose width is equal to its center frequencyf.

Of the three arguments determining the value of the function in Eq. (3), it is assumed that the Strouhal number ( fD /V) is the primary one. This assumption is reasonable because the specific value of the Reynolds number usually has little effect on turbulent flow once it exceeds a critical

value, which is the case for the data to be considered here. The noise should also be relatively independent of fluid compressibility, and therefore of Mach number, for wind speeds much below the speed of sound and for frequencies corre- sponding to acoustic wavelengths several times larger than the screen dimensions; e.g., frequencies below 1000 Hz for a 10-cm screen. This is so because the fluid around the screen

behaves as though it were incompressible under such cir- cumstances; see, e.g., Lamb.'-

The remainder of this article will be concerned with ex-

amining data from various sources to determine the dependence of the sound pressure coefficient on the Strouhal number, and the extent to which this dependence is influenced by variations in Reynolds and Mach numbers.

II. DATA ANALYSIS

A typical contemporary set of data on windscreen noise is shown in Fig. 1, reproduced from Fig. 14(d) of Hosier and Donavan. 3 This figure shows 1/3-octave band levels as a function of center frequency at various wind speeds, for an open-cell foam plastic spherical screen 9.5 cm in diameter surrounding a •-in. microphone. Some of their data for two screen diameters and three speeds are replotted in nondimensional form in Fig. 2 using logarithmic scales. Also plotted in this figure are data from several other sources identified in the table at the top of the figure. The Blomquist data 4 were reported in 1973, also for open-cell spherical screens. The Dyer data • were reported in 1954 for cylindrical windscreens. The van Leeuwen data 6 were reported in 1960 for a

loo ..................... &' '2' 'M/'$ ..... + 4

x 6

90 ß 10 • 12 z 14

8O

• 6o

• so

+

3o

• ......... • ß . • , ...... •, FREOUENCY, HZ

FIG. 1. Typical spectra of windscreen noise at various wind speeds for a 9.5-cm spherical screen (from Hosier and Donavan • ).

wire mesh screen roughly spherical in shape about 8 cm in diameter. (Incidentally, van Leeuwen discussed dimensional analysis but neglected the screen dimensions in his data reduction. )

The data from these disparate sources are in relatively good agreement with each other when plotted in this dimensionless form up to a dimensionless frequency of about 5, except for the high-frequency portion of the Blomquist data with the symbol 7. A dimensionless frequency of 5 corresponds to a frequency of 1000 Hz for a 5-cm-diam screen at a speed of 10 m/s ( -• 20 kn). The relatively small scatter of the data points is somewhat surprising; the mean difference of an individual data point from the dashed line, about q- 3 dB, is probably not much larger than the repeatability of measurements with a particular screen on different occasions. It is also noteworthy that the data for cylinders do not differ significantly from those of spheres when nondimension- alized in terms of cylinder diameter; apparently, the height of the cylinder is not significant.

The dashed straight line drawn through the data points provides a reasonably good fit up to a dimensionless frequency of about 5. This line can be represented by the equation

20 log(•l/3/pV 2) = -- 81 -- 23 log(fD/V), (4) where logarithms are to the base 10. For convenience, this equation may be rewritten as a numerical relation between the sound pressure level and the other variables expressed in commonly used units'

L1/3 = 61 + 63 log _V-- 23 log_f-- 23 log D_, (5)

545 J. Acoust. Soc. Am., Vol. 83, No. 2, February 1988 M. Strasberg: Dimensional analysis of windscreen noise 545


-5O

m

•'• -60

-7O

o

LU

• -90

o -100

z

-110

Frequency in Hz• For V : 10 m/s and D : 10 cm - --

20 50 1 O0 200 500 1000 I i I i I I I I

Symbol Speed Di am Source m/sec cm -

2 6 6.2• 3 14 6 2 Hosier . 5 10 9.5 Donavan 6 14 9.5 7 11.1 10

8 11.1 18 Blomquist 9 11.1 25 -

A 1 to 3 ~8 van Leeu'wen D 22.9 2 Dyer E 11.5 5 - F 21.3 5

Y7 7 I

'-,. ,

-- 55 =

o

o

-- 35

0.1 .2 .5 2 5 0 20

DIMENSIONLESS FREQUENCY fD/V

FIG. 2. Windscreen noise spectra from four sources plotted in dimensionless form. The scales along the top and right side are specific to a 10-cm screen at a wind speed of 10 m/s.

where L1/3 is the sound pressure level in a 1/3-octave band re: 2 >< 10 -s N/m 2 , f is the frequency in Hz, _V is the wind speed in m/s, D is the screen diameter in cm, and the density has been taken as that of air at 20 *C (1.3 >< 10 -3 g/cm 3). The underlining of symbols is intended as a reminder that the formula is not dimensionally homogenous and the quantities must be expressed in the units specified above.

Dimensional scales are shown along the top and right side of Fig. 2 to indicate specific values of sound pressure level versus frequency for a 10-m/s (20-kn) wind passing a 10-cm (4-in.) screen. Levels for other speeds and screen dimensions can be determined by adding 40 times the logarithm of the speed ratio to the level scale, i.e., 12 dB per double speed, while shifting the frequency scale in propor- tion to the speed ratio and inversely with the diameter ratio.

If it is desired to express the noise as a spectrum level, Eq. (5) may be rewritten as

Ls = 67 + 63 log _V-- 33 log _f-- 23 log D_, (6) where Ls is the spectrum level re: 2 >< 10- s N/m 2 in a 1-Hz band and the other quantities have the units specified for Eq. (5).

It is worthy of note that the data for cylinders do not indicate any significant peak at the vortex shedding frequency, which corresponds to a value of dimensionless frequency (fD/V) of about 0.2. That this is as it should be can be understood from the following argument. When a cylinder sheds vortices, they are shed alternately from opposite sides. The so-called frequency of vortex shedding is the frequency

of shedding from one side. The shedding of vortices results in an alternating side force on the cylinder at the same frequency as the vortex shedding, since a side force is developed in one direction when a vortex leaves one side and in the opposite direction when leaving the other side. However, an omnidirectional pressure sensor at the center of the cylinder does not distinguish one side from the other, so it senses the same increment of pressure (say, positive) regardless of the side from which the vortex is shed. Accordingly, the lowest frequency sensed by an omnidirectional pressure sensor at the center of the cylinder is twice the vortex shedding frequency.

Ill. HYDROPHONE FLOW NOISE

Although hydrophones differ from microphone windscreens in their construction, it seems reasonable to assume that the noise generated by fluid flowing past them is, in both cases, due to random pressure fluctuations on the hydrophone or screen surface associated with turbulence generated by the flow around them. Accordingly, it would be expected that hydrophone flow noise should depend on the same dimensionless variables as those used for windscreen

noise. More specifically, values of the dimensionless pressure coefficient should be about equal for small hydrophones and windscreens at the same dimensionless frequency. The large difference in the densities of water and air should be accounted for entirely by the density factor p in the denomi- nator of the pressure coefficient •]/3/P V2.



The only direct measurements I have found of self-noise due to water flowing past a small hydrophone are those reported by McGrath et al. 7 Their measurements were made with a 7-cm-diam cylindrical hydrophone at low frequencies, below 20 Hz, and low water speeds, below • kn, corre- sponding to natttral currents existing in the sea. This combination of frequencies, speeds, and hydrophone dimensions result, by coincidence, in values of dimensionless frequency within the range of the windscreen data, although the actual frequencies and speeds are quite different.

The hydrophone flow noise spectra shown in McGrath's Fig. 8 for three water speeds are replotted here in Fig. 3 in terms of dimensionless variables. The use of these variables does put the 0.25- and 0.45-kn spectra onto a com- mon curve, but the 0.4-kn spectrum is several dB lower. Also plotted in Fig. 3, for comparison, is the straight-line average of windscreen noise taken from Fig. 2. The hydrophone levels are significantly higher, some 15 to 20 dB higher.

The differences in level are not likely to be caused by viscous effects. The Reynolds numbers of the flow are, by coincidence, in about the same range for the water and air measurements. The kinematic viscosity of air is about 15 times that of water, so a speed of 10 m/s in air corresponds to the same Reynolds number as a speed of 0.7 m/s ( 1.4 kn) in water for any given body size. Compressibility effects are also expected to be negligible at the low speeds and low frequencies involved, as discussed previously.

Several tentative explanations of the differences in level come to mind. Significant differences in the construction may influence the level. For example, a hydrophone may be more sensitive to vibratory accelerations induced by the flow than is a microphone. Another difference is that the pressure-sensitive element of the hydrophone may be relatively closer to the surface of the hydrophone than the sensitive element of the microphone is to the windscreen surface. It is known that the response of a sensitive element to random and partially uncorrelated pressure fluctuations on a surface decreases as the distance between the surface and the ele-

ment is increased. 8 If this phenomenon has a significant ef-

• -70 --

• -•o- ""-ø'•.•S•~or s -

• -90 --

o 100-- • - -110 J I I I I Ill I I I 1 I I III _

• 0.2 0.5 1 2 5 10 20 '-' D[HENS[ONLESS FREQUENCY FD/V

FIG. 3. Hydrophone flow noise compared with windscreen noise plotted in dimensionless form. (Hydrophone noise from McGrath et al.7 )

fect on the noise, then a dimensionless parameter (h/D) or ( fh / I x) would be important in addition to ( fD / Ix), where h is the average distance of the sensitive element from the surface of the screen. The dependence on h may not show up in the windscreen data because the value of (h/D) is approxi- mately the same for all the screens (h/D = •2).

IV. DISCUSSION

I have not been tempted to conjecture on the signifi- cance of the observed nearly linear relation between the windscreen noise level and the logarithm of the dimensionless frequency. Nor do I conjecture on the departure from that simple relation which begins at a dimensionless frequency above about 5. This departure is associated with a second- ary peak in the 1/3-octave spectra occurring in the Blom- quist spectra and some of the Hosier and Donavan spectra, in the range 2 to 10 kHz. The frequency of this peak occurs at dimensionless frequencies (fD /I x) between 3 and 4 for the Hosier data, and at apparently erratic frequencies in the Blomquist data. Perhaps the details of the windscreen construction become significant at higher frequencies. Hosier and Donavan 3 suggest that flow through the screen pores may generate high-frequency noise.

This analysis is oversimplified, admittedly, in its lack of consideration of several factors that might influence the noise. The possible influence of the distance between the sensitive portion of the microphone and the windscreen surface has already been discussed in the previous section. More- over, Blomquist 4 and others have reported that the porosity of the screen affects the wind noise. (The Blomquist data used here are for his 800-pores/m screens only, since these are the only screens for which he reported spectra.) Never- theless, it is believed to be useful to plot windscreen noise data in dimensionless form even when these difficult-to-

quantify factors are investigated, in order to highlight the extent of the departure from simple scaling caused by them.

Finally, it should be noted that all the data used here were obtained from measurements made in the laboratory, by moving the microphones through substantially quiet air. In this situation, the noise is associated with turbulence generated by motion of the microphone. A naturally occurring wind may be sufficiently turbulent, however, that noise will be generated by interaction between the microphone and this pre-existing turbulence. If the wind turbulence is strong enough, the interaction noise may exceed the self-generated noise, as is discussed elsewhere. 9

1Handbook of Fluid Dynamics, edited by V. L. Streeter (McGraw-Hill, New York, 1961 ), see Chap. 15 by M. Holt.

2H. Lamb, Hydrodynamics (Cambridge U. P., London, 1953); see discussion of Eq. 19 in Art. 290.

3R. N. Hosier and P. R. Donavan, "Microphone Windscreen Perfor- mance," National Bureau of Standards Report NBSIR 79-1599, Jan. 1979.

4D. J. Blomquist, "An Experimental Investigation of Foam Windscreens," Proc. InterNoise 73, 589-593 (1973).

547 J. Acoust. Soc. Am., Vol. 83, No. 2, February 1988 M. Strasberg: Dimensi•)nal analysis of wind•icreen noise 547


5I. Dyer, "Self Noise of Cylindrical Wind-screens" Bolt Bernaek and New- man Report 225, Job 637, Aug. 1954.

6F. J. van Leeuwen, "Windscreens and 'Anti-Plop' Screens for Micro- phones" (in Dutch), Omroep-Techische Mededelingen 2(4), 168-172 (Nov. 1960).

7j. M. McGrath, O. M. Griffin, and R. A. Finger, "Infrasonic Flow-Noise

Measurements Using an H-58 Cylindrical Hydrophone," J. Acoust. Soc. Am. 61, 390-395 (1977).

8G. Maidanik and W. T. Reader, "Filtering Action of a Blanket Dome," J. Acoust. Soc. Am. 44, 497-502 (1968).

øM. Strasberg, "Nonacoustic Noise Interference in Measurements of Infra- sonic Ambient Noise," J. Acoust. Soc. Am. 66, 1487-1493 (1979).



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Dimensional analysis of windscreen noise