Wind noise measured at the ground surface

Wind noise measured at the ground surface Jiao Yu Center for Industrial and Medical Ultrasound, Applied Physics Lab, University of Washington, 1013 North East 40th Street, Seattle, Washington 98105-6698 Richard Raspet, Jeremy Webster,a) and JohnPaul Abbott Department of Physics and Astronomy and National Center for Physical Acoustics, University of Mississippi, University, Mississippi 38677 (Received 21 May 2010; revised 16 November 2010; accepted 22 November 2010) Measurements of the wind noise measured at the ground surface outdoors are analyzed using the mirror flow model of anisotropic turbulence by Kraichnan [J. Acoust. Soc. Am. 28(3), 378–390 (1956)]. Predictions of the resulting behavior of the turbulence spectrum with height are developed, as well as predictions of the turbulence-shear interaction pressure at the surface for different wind velocity profiles and microphone mounting geometries are developed. The theoretical results of the behavior of the velocity spectra with height are compared to measurements to demonstrate the applicability of the mirror flow model to outdoor turbulence. The use of a logarithmic wind velocity profile for analysis is tested using meteorological models for wind velocity profiles under different stability conditions. Next, calculations of the turbulence-shear interaction pressure are compared to flush microphone measurements at the surface and microphone measurements with a foam covering flush with the surface. The measurements underneath the thin layers of foam agree closely with the predictions, indicating that the turbulence-shear interaction pressure is the dominant source of wind noise at the surface. The flush microphones measurements are intermittently larger than the predic- tions which may indicate other contributions not accounted for by the turbulence-shear interaction pressure. VC 2011 Acoustical Society of America. [DOI: 10.1121/1.3531809] PACS number(s): 43.28.Ra [JWP] Pages: 622–632 I. INTRODUCTION The research presented in this paper is an extension of earlier research into the calculation of wind noise contribu- tions from the measured atmospheric turbulence spectra and wind velocity. Raspet et al.1 developed theories relating the wind noise measured by screened and unscreened micro- phones to meteorological measurements at the height of the microphone. Reference 1 also established fitting and analysis procedures for the turbulence spectra. The turbulence models were then used by Yu et al.2 in a preliminary analysis of the wind noise spectrum measured in a microphone mounted flush with the ground surface. The analysis adapted a mirror flow model of anisotropic turbulence and mathematical anal- ysis of Kraichnan3 for turbulent boundary layer flow to the problem of wind noise pressure fluctuations measured under atmospheric turbulence. In Ref. 2, predictions of the surface pressure fluctuations were prepared with three models of the wind velocity pro- files: single exponential, double exponential, and logarith- mic. The logarithmic profile was developed since it is often used in meteorology to describe the wind velocity profile under neutral conditions.4 The wind velocity profile was only measured in the first few centimeters above the surface. All measurements were made with the microphone dia- phragm flush mounted in a smooth plate. For some measure- ment sets, the measured and predicted pressure fluctuation spectra agreed closely, and for others the measured levels were considerably higher than the predictions. It was observed that a thin porous layer over the microphone reduced the measured pressure spectral levels to approxi- mately the predicted levels. The research reported, herein, extends and improves the investigation of Ref. 2 in the following ways: (1) Predictions of the behavior of the vertical and longitudi- nal turbulence spectra with height resulting from the mir- ror flow model developed by Kraichnan3 are derived and compared to the measured turbulence spectra to demon- strate that the mirror flow model is realistic for outdoor measurements (Secs. II A and IV A). (2) A new method of integration is developed which elimi- nates convergence problems encountered using the theory of Ref. 2 and is used to derive new expressions for the pressure fluctuation spectrum under logarithmic velocity profiles. This method also allows for the trunca- tion of wind velocity profiles in order to model regions of low wind speed under the roughness length and in foams (Secs. II B 1, II B 3, and IV B). (3) The pressure fluctuation spectra resulting from a multi-ex- ponential model of the wind velocity profile are derived. The multi-exponential model can be fit to arbitrary meas- ured and predicted profiles and is used to investigate whether the assumption of a logarithmic wind velocity profile is sufficient for data analysis of our data sets (Secs. II B 2 and III). (4) Predictions and measurements for the microphones moun- ted under the thin layers of metal foam are investigated in a)Author to whom correspondence should be addressed. Electronic mail: [email protected] 622 J. Acoust. Soc. Am. 129 (2), February 2011 0001-4966/2011/129(2)/622/11/$30.00 VC 2011 Acoustical Society of America Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23 addition to the predictions and measurements for flush microphones (Secs. II B 4 and IV B). (5) The wind velocity profile is measured up to 2.0 m height (Sec. IV B). (6) The effect of different layer thicknesses on the pressure spectra measured under thin layers of foam is investi- gated (Sec. IV C). This research presents a much more complete investiga- tion of the wind noise levels measured at the ground than the preliminary research presented in Ref. 2. II. THEORY The geometry of the measurement and calculation is illus- trated in Fig. 1. The microphones are mounted vertically so the diaphragms are flush in the rigid surface and are either open to the flow or under a thin layer of the open pore metal foam. The rigid surface or the foam surface is flush with the ground surface and the ground surface is level for a long dis- tance. In meteorological terminology, the measurement has good fetch. The vertical wind fluctuations approach zero as the rigid surface is approached, so that the turbulence becomes increasingly anisotropic as the surface is approached. Section II A develops predictions of the behavior of the vertical and longitudinal one-dimensional (1-D) turbulence spectra with height above the surface for comparison with measurements. Section II B derives predictions for the pres- sure fluctuation spectra from two realistic types of wind ve- locity profiles and the measured wind velocity spectra above the surface. A. The vertical and longitudinal spectral model for turbulence height dependence The mirror flow model by Kraichnan simulates the verti- cal anisotropy of the boundary layer turbulence by assuming that the anisotropic velocity field eVað~x; tÞ for the boundary layer flow can be expressed in terms of the velocity field Vað~x; tÞ of homogeneous isotropic turbulence occupying the upper and lower half space as eV1ð~x; tÞ ¼ 2�1=2 ½V1ð~x; tÞ þ V1ð~x �; tÞ� ;eV2ð~x; tÞ ¼ 2�1=2 ½V2ð~x; tÞ � V2ð~x �; tÞ� ;eV3ð~x; tÞ ¼ 2�1=2 ½V3ð~x; tÞ þ V3ð~x �; tÞ� ; (1) where the auxiliary variable~x � satisfies x�1 ¼ x1; x�2 ¼ �x2; x�3 ¼ x3 : Correlation functions of the longitudinal or vertical turbu- lent velocity for the flow above a boundary surface are formed and expressed in terms of the correlation functions for homo- geneous isotropic flows. By applying temporal and spatial Fourier transforms with respect to x01 � x1; x03 � x3; and t0 � t and by applying the symmetry of R11 and R22, we obtain eR11ðx02; x2;~j;xÞ ¼ eU11ðx2; k1Þ ¼ 110 9p Ck4 ð1 0 ð1 0 ðk22 þ k23Þ cos2ðk2x2Þ ½1 þ ðkkÞ2�17=6 dk2dk3; eU22ðx2; k1Þ ¼ 110 9p Ck4 ð1 0 ð1 0 j2 sin2ðk2x2Þ ½1 þ ðkkÞ2�17=6 dk2dk3: (7) When x2 is large, Eq. (7) recovers the spectra for homo- geneous isotropic turbulence. As x2 approaches zero, the hor- izontal spectrum approaches twice the free space value and the vertical spectrum goes to zero. The spectra calculated for the mirror flow model will be compared to measurements in Sec. IV A. B. Turbulence-shear interaction pressure spectra 1. Basic formulation In this section, we apply and improve the method devel- oped by Kraichnan to predict the turbulence-shear interaction pressure at the ground. In Kraichnan3 and other studies7–12 of pressure fluctuations in boundary layer flows, it is assumed that the turbulent-shear interaction pressures are larger than the turbulence-turbulence interaction pressures since the gra- dient of the mean wind speed with height is larger than any other gradient of fluctuating wind velocity components. The source equation for the pressure fluctuations due to the inter- action of the turbulence with a shear is then given by r2pð~x; tÞ ¼ � 2qsðx2Þ@ eV2=@x1; (8) where p is the pressure fluctuation, s(x2)¼ dU1/dx2 is the vertical gradient of the mean longitudinal velocity, eV2 is the vertical turbulent velocity component, and q¼ 1.2 kg/m3 is the density of air. For the mirror flow model given in Sec. II A, and under the boundary conditions that the normal pressure gradient is zero at the rigid boundary and the pressure is bounded at in- finity, Kraichnan3 derives a formula for the pressure fluctua- tion spectrum at the surface: pð0;~j;xÞj j2 ¼ 4ð2pÞ�3=2q2k21j�2 ð1 0 ð1 0 e�jðx2þx 0 2 Þ � sðx02Þsðx2Þ½ This form guarantees the mean longitudinal velocity is zero at and below the roughness length and is U at large heights. The velocity gradient is then sðx2Þ ¼ Xn j¼0 sje �bjðx2�x0Þ; x2 � x0; 0 0 � x2 < x0 ; (13) 8>: where sj ¼ UbjAj; j ¼ 0; 1; :::; n � 1; Ubn 1 � Xn�1 i¼0 Ai ! ; j ¼ n: 8>: (14) Eq. (11) becomes pð0;k1Þj j2 ¼ 440Ck4q2k21 9p Xi¼n;j¼n i¼0;j¼0 sisje ðbiþbjÞx0 � ð1 0 ð1 0 dk2dk3 ½1þ ðkkÞ2�17=6 ð1 x0 e�ðjþbiÞx 0 2 � sinðk2x02Þdx02 ð1 x0 e�ðjþbjÞx2 sinðk2x2Þdx2: (15) In Sec. III, the multiple exponential model is used to investigate the effect of the variation in wind velocity profile with atmospheric stability conditions on the predicted pres- sure fluctuation spectrum. 3. Application to a logarithmic profile The wind velocity outdoors often varies approximately logarithmically with height in the surface layer. Under the roughness length, the average velocity is assumed to be zero. The logarithmic profile also approximates the wind velocity profile for unstable but mechanically dominated turbulence near the ground surface.4 For a logarithmic profile, the mean velocity has the form U1ðx2Þ ¼ a ln x2x0 � � ; x2 � x0; 0 0 � x2 < x0; (16) ( which gives a mean velocity gradient of sðx2Þ ¼ a x2 ; x2 � x0; 0 0 � x2 < x0: (17) � Eq. (11) becomes pð0; k1Þj j2 ¼ 440a2q2k21Ck 4 9p ð1 0 ð1 0 dk2dk3 ½1 þ ðkkÞ2�17=6 � ð1 x0 e�jx2 sinðk2x2Þ x2 dx2 � � ð1 x0 e�jx 0 2 sinðk2x02Þ x02 dx02 � : (18) 4. Predictions for microphones under porous foam layers The normal pressure derivative is assumed to be zero at the rigid plane surface. Figure 1 displays the coordinate sys- tem for the measurement and calculation with the foam. The plane where the microphone is mounted is chosen as the zero height (x2, x 0 2¼ 0). The wind profile above the foam is assumed to be identical to the measured profile above the grass. For open celled foam, it is reasonable to assume that there are vertical turbulent velocity fluctuations inside the foam but that the horizontal velocity will be small and negli- gible. The vertical component of the turbulent velocity goes to zero on the rigid plane surface (a flat plate) where the microphone is flush mounted (see Fig. 1). The wind velocity profile is set to zero at the roughness height above the foam surface. This modifies the definitions of the velocity profiles and gradients by the foam thickness. We do not expect this to realistically model the details of the wind velocity profile near the surface. However, at moderately low wave numbers, the dominant source region is found to be several centi- meters to several meters above the surface.13 III. EFFECTS OF ATMOSPHERIC STABILITY The wind velocity profile measurements used in this pa- per are limited in height to about 2.0 m. Our measurement setup is portable and the measurements are made at an active airport, which limits the height of the measurement. Close to the ground, the measured profiles typically follow a logarith- mic behavior down to the roughness length with small devia- tion, although higher measurements might display systematic variations due to the changing stability conditions. In this section, we investigate the sensitivity of the pressure fluctua- tion spectrum to deviations of the wind velocity profile from logarithmic due to atmospheric stability effects. We base this study on measured values for the wind velocity profile and turbulence spectrum from the ground to 2.0 m. The details of the measurement are described in Sec. IV. We will fit the measured profiles to standard forms of the wind velocity pro- file under differing stability conditions consistent with the measurement days in this study. The atmospheric modeling is based on the material from Ref. 4. The mean velocity profile for unstable air satisfies U1ðx2Þ ¼ a ln x2 x0 � wm x2 L � �� � ; (19) where x0 is the roughness length, x2 is the height, and L is the Monin-Obukhov length which is negative for unstable air. The more negative x2/L is, the greater the contribution of convective turbulence. For the commonly used Businger- Dyer form in unstable air, the expression for wm is wm ¼ ln 1 þ x2 2 � 1 þ x 2 � 2" # � 2 arctan x þ p 2 ; (20) where x¼ (1� 16x2/L)1/4. When L¼�1, the neutral condi- tion is recovered. J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface 625 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23 The Monin-Obukhov length, L, depends on wind speed, heat flux, and roughness. The combined effect of wind and heat flux can be described by the Turner Classes, which can be estimated by Table 6.4 of Panofsky and Dutton.4 Turner Classes 2 and 3 are appropriate for most of our measure- ments. The roughness length of the grass at the measurement site is estimated to be z0¼ 0.01 m. Figure 6.6 of Panofsky and Dutton provides an approximate range for L between �9.0 and �40.0 m. We take L¼�10.0 m, L¼�40.0 m, and L¼�1 m as examples in our analysis. Figure 2(a) displays measured wind velocity profile data and fits from the ground up to 2.0 m using Eqs. (19) and (20) for L¼�10.0 m, L¼�40.0 m, and L¼�1 m. Table I lists the values of a and x0 obtained from the fits displayed in Fig. 2(a). The similarity of the fits in Fig. 2(a) indicates that the atmospheric stability condition has little effect on the wind velocity profile in the first few meters above the ground. Figure 2(b) displays the three profile fits extended up to 100.0 m height. As the profile is extended to greater heights, the difference between the three cases becomes obvious. The neutral condition has larger velocities and velocity gradients than the unstable conditions above 5.0 m. The roughness lengths are 0.020, 0.017, and 0.016 m for L¼�10.0 m, L¼�40.0 m, and L¼�1 m, respectively. The three profiles with zero mean velocity under the roughness length are then fit with the multiple exponential profile from the roughness length up to 100.0 m height with [bj]j¼0¼ 10j�3, n¼ 6. Setting the power of the exponential function in Eq. (12) to powers of ten and using U and the A’s as the fitting parameters is an effective method of fitting a given measured profile to the multiple exponential form. A few terms obtain a good fit to a variety of profiles. Table II lists the fitting parameters obtained for L¼�10.0 m, L¼�40.0 m, and L¼�1 m, respectively. Figure 3 shows the pressure spectra predicted from the three different stability conditions using the multiple expo- nential profile fits. All the predictions use the C and k values derived from measurement: C¼ 7.638 and k¼ 7.419. The two unstable condition pressure spectra are a little bit lower than the neutral atmospheric condition pressure spectra at very low wave number. This is consistent with Fig. 2(b), because the pressure fluctuation source is propor- tional to the velocity gradient. The stability conditions for the expected range of Monin-Obukhov lengths do not have a large effect even at moderately low wave number. The maxi- mum spectral level difference between the unstable condi- tion and neutral condition is about 5 dB. At high wave number, the slight difference between the three predictions is due to the variation of the roughness length values, x0, obtained from the fits. The difference between the unstable condition and neutral condition on the turbulence-shear interaction pressure spectrum is minor. We have not taken measurements under stable condi- tions—all the measurements have been taken during the day- time, under windy conditions. The range modeled here (L¼�10.0 m to L¼�1 m) should encompass all our data. At the lower limit of measured wave number in this paper (0.1 to 0.2 m�1), the difference between neutral conditions and unstable conditions is on the order of 2 to 3 dB. Hence, FIG. 2. (a) The measured profile and the fits for L¼�10.0 m, L¼�40.0 m, and L¼�1 m. (b) The three profiles extended up to 100.0 m height. The dotted, dashed, and solid lines are the fits for L¼�10.0 m, L¼�40.0 m, and L¼�1 m, respectively. The black squares are the measured profile in the first 2.0 m. TABLE I. Values of a and x0 obtained from the fits to the wind velocity profile models, Eqs. (19) and (20). L (m) a x0 (m) �10.0 1.30 0.020 �40.0 1.18 0.017 �1 1.13 0.016 TABLE II. Fit parameters of the multiple exponential form, Eq. (12), to the three extended profiles from the roughness length up to 100.0 m height. L (m) U (m/s) A0 A1 A2 A3 A4 A5 �10.0 11.7 0.360 0.032 0.142 0.220 0.203 0.057 �40.0 14.8 0.476 0.013 0.143 0.179 0.152 0.040 �1 14.4 0.277 0.176 0.178 0.179 0.150 0.045 626 J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23 the logarithmic fit of our measured profile near the ground can be used to provide a reasonable prediction of the pres- sure fluctuation spectrum over most of the wave number range. IV. MEASUREMENT AND ANALYSIS Three types of measurements have been performed. The first series of measurements was taken to investigate whether the mirror flow model accurately simulates the behavior of the vertical spectrum of the turbulence as the ground surface is approached. We also measure and present the behavior of the longitudinal turbulence spectra as a function of height above the ground. The second series of measurements was taken to investigate the pressure fluctuation levels measured by the flush mounted microphones at the ground surface simultaneously with the pressure levels measured under 2.54 cm of the porous metal foam. The third series of measure- ments was taken to investigate the effect of foam thickness on the measured pressure fluctuation spectral level. All data presented were taken at Clegg Field in Oxford, MS, from February to July in 2009. Clegg Field is a large flat open area with mowed grass about 4–6 cm deep. Panofsky and Dutton4 indicate such a surface has a roughness length corre- sponding to approximately 0.01 m. A. Investigation of the turbulence spectrum model In order to investigate whether the mirror flow model is a valid model for the behavior of the vertical and longitudinal turbulence spectra with height, two sets of turbulence spectra data were taken. In the first, a reference longitudinal wind ve- locity spectrum was taken using a Gill Instruments R3A-100 Ultrasonic Research Anemometer mounted at 2.0 m above the ground surface. The spectrum measured at 2.0 m is approximately homogeneous and isotropic in the wave num- ber range of interest. The Gill Anemometer (Gill Instruments Ltd., Hampshire, England) has an internal sampling rate of 100 Hz. Since Eq. (5) requires a measurement of the wind velocity spectrum in the direction of flow, the anemometer is aligned so that the u1 direction is the mean direction of the incident wind. If the wind shifts during the course of a mea- surement, we perform a coordinate transform on the data to assure that the mean of the u3 component of the velocity is zero. Then the spectrum of the u1 component is fit to Eq. (5) in order to get the C and k values used in calculating the lon- gitudinal and vertical turbulence spectra for flow at lower heights. The fitting method is described in Ref. 1. The second set of turbulence spectrum measurements are of the spectra at different heights using six Applied Tech- nologies Anemometers mounted at 10, 50, 90, 110, 150, and 190 cm above the ground. The internal sampling rate of the anemometers is 10 Hz. For the longitudinal or vertical turbu- lence spectrum measurements, the measurement axes of all the six anemometers were placed along the wind direction to measure the longitudinal turbulence velocity or along the direction perpendicular to the ground to measure the vertical turbulence velocity. All velocity data were acquired simultaneously for 1800 s (30 min). The data from the Gill anemometer and the six 1-D anemometers were collected at sampling rates of 100 and 10 Hz, respectively. All of the sensors were connected to a National Instruments data acquisition card controlled by a pro- gram written in LABVIEW. After acquisition, all data analysis was done with MATLAB, except for the graphical fittings done using ORIGIN. The wind velocity power spectra were generated following the method of Ref. 1. For the investigation of the height de- pendence of the vertical and longitudinal turbulence spectra, each run was broken into non-overlapping blocks of 256 sam- ples as a compromise between good averaging and good resolu- tion. Each block was detrended and windowed before its power spectral density (PSD) was calculated. The average of the block PSDs was calculated and converted from frequency to wave number space using Taylor’s frozen turbulence hypothesis and the convection velocity UC in the direction of flow, Fvðk1Þ ¼ UC 2p F0vðf Þ; (21) where k1 ¼ 2pfUC , Fv is the PSD of the turbulent velocity in wave number space, and F0v is the PSD in frequency space. A value of 0.7 times the mean stream wise velocity measured by the Gill Anemometer at 2.0 m is used for the convection velocity, UC. This choice is supported by the cross spectral and cross correlation data in many studies14–19 and in our correlation measurements outdoors which deter- mined a value of approximately 0.72 for the ratio of the con- vection velocity to the mean wind speed measured away from the surface, independent of the frequency. Figure 4 dis- plays the comparisons of measured and predicted longitudi- nal (a) and vertical (b) turbulence power spectra at different heights. The predictions for different heights are generated from Eq. (7). Table III lists the measurement information, U1, UC and fit parameters C and k for Fig. 4. From Fig. 4(a), it can be seen that the measured and pre- dicted longitudinal turbulence spectra are all very similar except at the 10 cm height. Measurements at all the other heights are close to the prediction in both magnitudes and FIG. 3. The predicted pressure spectra for the three stability conditions: The dotted, dashed, and solid lines are the fits for L¼�10.0 m, L¼�40.0 m, and L¼�1 m, respectively. J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface 627 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23 spectral slopes. The longitudinal turbulence spectrum is not very sensitive to the height, and the measurements and the predictions are close. Figure 4(b) displays the reduction of the measured and predicted vertical turbulence spectra as the ground is approached. The predicted spectral plots all level off at low wave numbers as do the measured spectra. There is a deviation at high wave number for 90 cm height mea- surement. We have not observed this behavior in other data sets. Overall the similarity of the measured and predicted vertical turbulence spectra is satisfying. In summary, there is a reasonable agreement between the predictions using the mirror flow model and measurements of the height dependences of the longitudinal and vertical turbu- lence spectra. These measurements provide support for the use of the models developed here to predict the wind noise measured at and beneath the surface under turbulent wind fields outdoors. B. Investigation of the spectral models for turbulence- shear interaction pressure at and beneath the surface This experiment measures the turbulence power spec- trum, the wind velocity profile, and the pressure fluctuations at the ground surface. The pressure spectra are then pre- dicted from the measured turbulence spectrum and the wind velocity profile fit to the logarithmic profile, and compared to the measured pressure fluctuation spectra. The turbulence data was collected with a Gill Anemometer mounted at 1.0 m height. The wind velocity profile measurements were again taken using six 1-D Anemometers mounted at heights of 10, 50, 90, 110, 150, and 190 cm with the measurement axes aligned with the wind. Pressure measurements were taken with Brüel & Kjær (B&K, Nærum, Denmark) type 4193 1/2 in. microphones powered by a Nexus brand conditioning amplifier. The fre- quency response for the microphone drops off below 0.07 Hz, but the actual low frequency cutoff of the pressure data is 0.1 Hz set by the high pass filter in the Nexus. Figure 5 dis- plays the microphone arrangement for simultaneous flush and foam covered microphone measurements. One microphone was mounted flush in a flat acrylic plate placed flush with the ground surface. The other microphone was mounted in an acrylic plate beneath a 2.54 cm thick sheet of aluminum foam mounted flush in the same surface as the flush micro- phone. All foam used in the measurements here are 40 pores per inch (ppi) aluminum foam with a porosity of about 94%. The piece used in this section is 2.54 cm thick and is approxi- mately 35 cm by 40 cm. The microphone mounted flush with the surface of the plate is mounted upwind from the foam. Figure 6 displays the measurement arrangement. All velocity and pressure data were acquired simultaneously for 900 s in all runs. The pressure data were collected at a sampling rate of 200 Hz. The wind speed data from each 1-D Anemometer are averaged to give the mean wind speed at each height. The measured wind velocity profile is fit to the logarithmic form to determine the values of a and x0 [Eq. (16)]. FIG. 4. Comparisons of measured and predicted longitudinal (a) and verti- cal (b) turbulence power spectra at different heights. The solid lines are the predictions and the symbols represent the measured data for heights of 10 cm (þ), 50 cm (�), 90 cm (*), 110 cm (dot), 150 cm (square), and 190 cm (circle). To make viewing the data easier, each data set and prediction were multiplied by increasing factors of 4 starting with the 50 cm set, i.e., the 50 cm set is multiplied by 4, the 90 cm set by 16 etc. TABLE III. Measurement information, U1, UC, and fit parameters C and k for Fig. 4. Figures Turbulence spectrum U1 (m/s) UC (m/s) C k 4(a) eU11ðk1Þ 2.10 1.47 9.04 20.97 4(b) eU22ðk1Þ 2.47 1.73 12.37 26.25 FIG. 5. (Color online) The flush mounted microphone at the surface and themicrophone mounted beneath the surface before and after the 2.54 cm thick foam covering is placed. 628 J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23 Figure 7 displays results of the predicted and measured pressure spectra for measurements on two different days. Table IV lists the measurement information, U1, UC, and fit parameters C, k, a, and x0 for preparing the predictions for Fig. 7. All the measured profile fittings give roughness length values close to that expected for mowed grass (0.01 m). Yu13 presents similar plots for four preliminary measure- ments taken over 2 days and nine measurement sets taken over 3 days. The results from these measurements are well represented by the two sets presented herein. Figures 7(a) and 7(b) represent the two different kinds of results taken during our measurement program. The most distinct difference between Figs. 7(a) and 7(b) is that the measured levels for the flush and foam covered microphones are very similar over a wide wave number range in Fig. 7(a), while the flush microphone levels are roughly 15–20 dB higher in Fig. 7(b). For all measurements presented in the dissertation, 2 days (including 1 day for the preliminary measurements) have results similar to Fig. 7(a); the other 3 days (including 1 day of the preliminary measurements) are similar to Fig. 7(b). There is no apparent difference among the 5 days in setting up the measurement device, and no apparent weather condition difference can be categorized between the two different kinds of results. The measured spectral slope in the inertial range for flush mounted microphones on low level days and for all foam covered microphone measurements are well modeled by the predictions with a value of about �5/3. The high level flush microphone results are not as well behaved, but roughly follow a �5/3 law in the inertial range. The predic- ted spectra for the flush and under foam measurements are very similar in the inertial region and source region. Both increase proportionally to k21 in the source region and then curve down in inertial range after reaching a maximum. At high wave numbers, the foam covered microphones have higher reductions than the flush microphones. The slight dif- ference between the two predictions in each figure in the middle wave number range is due to the small change of coordinate system with respect to the wind velocity profile. The large differences at high wave number are due to the fact that the effective mean source region and the micro- phone are separated by the additional thickness of the foam for the microphone underneath the foam layer. The effective source layer is closer to the surface at high wave number so FIG. 7. Results of the predicted and measured pressure spectra for simulta- neous measurements with a flush mounted microphone at the surface and a microphone beneath the surface with a 2.54 cm thick foam covering on dif- ferent days. The black solid and dashed lines are the measured and predicted spectra for a flush microphone, respectively. The gray solid and dashed lines are the measured and predicted spectra for a microphone beneath the foam covering. TABLE IV. Measurement information, U1, UC and fit parameters C, k, a, and x0 for Fig. 7. Figures Mic. U1 (m/s) UC (m/s) C k a x0 (m) 7(a) Flush, 2.54 cm foam 6.15 4.31 7.33 10.34 1.11 0.008 7(b) Flush, 2.54 cm foam 4.57 3.20 3.33 7.39 0.82 0.007 FIG. 6. (Color online) Measurement setup. The wind in the picture blows from right to left. J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface 629 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23 the small additional separation has a larger effect on the high wave number region of the spectrum. For all our data, the foam covered microphone measure- ments are well predicted both in magnitude and in spectral slope for low and moderate wave number range that meas- urements cover. An examination of all our data found that the measured levels with the foam covered microphone change consistently from figure to figure with the change of the wind speed, while the flush microphone measurement does not follow the wind speed variation systematically. We can find no correlations between the meteorological condi- tions and the presence or absence of the large error in the flush microphone measurements. For example, the wind speed for Fig. 7(b) is lower than for Fig. 7(a), but the flush measurement in Fig. 7(b) has higher levels. The variation of the difference between the flush and under foam measured levels on different days cannot be due to the differences in the atmospheric stability because the atmospheric stability should influence them in the same way. We will examine this contribution further in the conclusions. At high wave number, our predictions model the effects of larger reductions with the foam, but the predictions decrease faster than the measurements. A better prediction at high wave number would require a better understanding of the profile close to the plate surface. The flat plate has a shorter roughness length than the grass which surrounds it, so there is a change in the surface layer profile when the wind blows from the grass to the plate and this may influence the results at high wave number. C. Measurements under different foam thicknesses The measurements from Sec. IV B were repeated with different foam thicknesses as well as with an air gap between the microphone and the foam. Four continuous runs were taken on one day with the same microphone used in each run. In the first three runs, the microphone was mounted directly beneath a 1.27, 2.54, and 5.08 cm thick piece of alu- minum foam, respectively. In the fourth run, the 1.27 cm thick piece of aluminum foam was used with a 3.81 cm air gap to the microphone face. All foams were 40 ppi alumi- num foam with a porosity of approximately 94%. Figure 8 displays the measured and predicted pressure spectra for the four cases. Table V lists the measurement pa- rameters used in preparing Fig. 8. Since the air gap, like the foam, is assumed to have no horizontal average velocity, the same calculation is used for the case with 1.27 cm thick foam and a 3.81 cm thick air gap as with the 5.08 cm thick foam. Figure 8 shows that all the foam covered microphone measurements can be well modeled with our predictions in the measured low and moderate wave number range. At high wave number, the under foam predictions drop off faster than the measurements. In Figs. 8(c) and 8(d), at the very high wave number, the measured plots level off instead of keep rolling down. The 5.08 cm thick foam (or foam plus air gap) reduces the wind noise level sensed by the microphone at that wave number range to lower than the background noise, so the measured level follows the background noise curve rather than decaying as wind noise. Excluding the FIG. 8. Measured and predicted pressure spectra for a microphone directly beneath a 1.27 cm (a), 2.54 cm (b), 5.08 cm (c) thick foam covering and beneath a 1.27 cm thick foam with a 3.81 cm thick air gap (d) in four continuous runs. The solid line is the measurement and the dashed line is the prediction. 630 J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23 background noise effect, it is also found that the larger the distance between the ground surface and the microphone mounting plane, the more accurate the predictions are at high wave number. This is reasonable, because the further the microphone is located below the ground surface, the less sensitive it will be to near surface details of the mean veloc- ity profile. The good prediction in Fig. 8(d) is an indication that our model assumption that there is no mean horizontal velocity in the air gap beneath the foam is reasonable. The foam does not have to extend all the way to the microphone surface to provide additional wind noise reduction. V. CONCLUSIONS This paper provides predictions of the turbulence-shear interaction pressure contributions to wind noise for a flush microphone at the surface and a flush microphone in a sur- face beneath a foam covering. Logarithmic and a multiple exponential wind velocity profiles are used to develop pres- sure spectrum predictions from the measured turbulence spectrum. Investigations with the multiple exponential pro- file model suggest that use of the logarithmic profile for moderately unstable conditions is justified. The height dependences of the longitudinal and vertical turbulence spectra using the mirror flow model by Kraichnan were investigated theoretically and were found to be in rea- sonable agreement with the measurements. The high degree of consistency provides a good support to the reliability and effectiveness of our theory developed by using Kraichnan’s assumptions and approach to model wind noise outdoors. Our theory provides reliable predictions for all the runs at low and middle wave number range of foam covered microphone measurements. At high wave number, our model overestimates the wind noise reduction and underestimates the wind noise level. The refinement of the high wave num- ber prediction would require more attention to evaluation of the velocity field close to the surface. The flush microphone measurements show large varia- tions of the spectral levels relative to the under foam meas- urements on different days and are not well predicted. It is possible that the high level spectra are the results of a differ- ent turbulence-shear interaction from that studied by Kraichnan and this paper. All the indications suggest that the largely varied levels for the flush microphone must originate from the thin boundary layer next to the microphone. The fact that the high level pressure has the same slope as the cal- culated pressure fluctuations implies that the source is an av- erage gradient interacting with a turbulence component. We speculate that there are other contributions to the high level pressure, which may be generated by a large average velocity gradient interacting with longitudinal or transverse velocity fluctuations near the surface. The pressure fluctuations generated by a thin layer inco- herent source would decay rapidly with distance. Some evi- dence of rapid decay may be observed in Fig. 8. The measurements all have a slightly higher level than predicted for the 1.27 cm foam but the agreement is better for the 5.08 cm separation between the surface and the microphone. Detailed measurements of the average and fluctuating veloc- ity near the surface would be necessary for a quantitative investigation of this contribution. The research in this paper improves our understanding of the basic characteristics and parameters of the flow that influence the wind noise spectrum generated at and beneath the surface. The success of the model developed for the foam covered microphone lays a good theoretical basis for investigating wind noise generated under porous layers and under trees and bushes. ACKNOWLEDGMENTS This research has been supported by the U.S. Army TACOM-ARDEC at Picatinny Arsenal, NJ. 1R. Raspet, J. Yu, and J. Webster, “Low frequency wind noise contributions in measurement microphones,” J. Acoust. Soc. Am. 123(3), 1260–1269 (2008). 2J. Yu, R. Raspet, J. Webster, and K. Dillion, “Model calculations of wind noise measured in a flat surface under turbulent flow,” in Proceedings of NCAD 2008, Paper no. NCAD2008-73044, 101–110 (2008). 3R. H. Kraichnan, “Pressure fluctuations in turbulent flow over a flat plate,” J. Acoust. Soc. Am. 28(3), 378–390 (1956). 4H. A. Panofsky and J. A. Dutton, Atmospheric Turbulence: Models and Meth- ods for Engineering Applications (Wiley, New York, 1984), pp. 119–143. 5G. K. Batchelor, “Pressure fluctuations in isotropic turbulence,” Proc. Cambridge. Philos. Soc. 47(2), 359–374 (1951). 6W. K. George, P. D. Beuther, and R. E. A. Arndt, “Pressure spectra in tur- bulent free shear flows,” J. Fluid Mech. 148, 155–191 (1984). 7R. L. Panton and J. H. Linebarger, “Wall pressure spectra calculations for equilibrium boundary layers,” J. Fluid Mech. 65(2), 261–287 (1974). 8P. Bradshaw, “ ‘Inactive’ motion and pressure fluctuations in turbulent boundary layers,” J. Fluid Mech. 30(2), 241–258 (1967). 9M. K. Bull, “Wall-pressure fluctuations associated with subsonic turbulent boundary layer flow,” J. Fluid Mech. 28(4), 719–754 (1967). 10W. K. Blake, “Turbulent boundary-layer wall-pressure fluctuations on smooth and rough walls,” J. Fluid Mech. 44(4), 637–660 (1970). 11W. W. Willmarth, “Pressure fluctuations beneath turbulent boundary layers,” Ann. Rev. Fluid Mech. 7, 13–38 (1975). 12A. S. W. Thomas and M. K. Bull, “On the role of wall-pressure fluctua- tions in deterministic motions in the turbulent boundary layer,” J. Fluid Mech. 128, 283–322 (1983). 13J. Yu, “Calculation of wind noise measured at the surface under turbulent wind fields,” Ph.D. dissertation, University of Mississippi, Mississippi, 2009. 14C. Durant and G. Robert, “Experimental study of vibration and acoustic radiation of a pipe induced by fully-developed turbulent air flow,” in The Fourth International Symposium on Fluid-Structure Interactions, Aeroe- lasticity, Flow-Induced Vibration and Noise, 397–402 (1997). 15C. Tropea, A. L. Yarin, and J. F. Foss, Springer Handbook of Experimen- tal Fluid Mechanics (Springer, New York, 2007), p. 762. TABLE V. Measurement information, U1, UC and fit parameters C, k, a, and x0 for Fig. 8. Figures Mic. U1 (m/s) UC (m/s) C k a x0 (m) 8(a) 1.27 cm foam 2.01 1.40 2.83 14.76 0.35 0.006 8(b) 2.54 cm foam 2.44 1.71 6.67 20.75 0.40 0.007 8(c) 5.08 cm foam 3.00 2.10 1.64 8.59 0.57 0.011 8(d) 1.27 cm foamþ 3.81 cm air gap 3.11 2.17 1.73 7.81 0.54 0.007 J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface 631 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23 16J. Kim and F. Hussain, “Propagation velocity of perturbations in turbulent channel flow,” Phys. Fluids 5(3), 695–706 (1993). 17U. Piomelli, J. L. Balint, and J. M. Wallace, “On the validity of Taylor’s hypothesis for wall bounded flows,” Phys. Fluids 1(3), 609–611 (1989). 18L. Ong, “Visualization of turbulent flows with simultaneous velocity and vortic- ity measurements,” Ph.D. Thesis, University of Maryland, College Park, 1992. 19J. M. Wilczak, “Large-scale eddies in the unstably stratified atmospheric surface layer. Part I: Velocity and temperature structure,” J. Atmos. Sci. 41(24), 3537–3550 (1984). 632 J. Acoust. Soc. Am., Vol. 129, No. 2, February 2011 Yu et al.: Wind noise measured at the ground surface Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 130.39.62.90 On: Tue, 02 Sep 2014 10:45:23 s1 cor1 s2 E1 E2 E3 E4 s2A E5 E6 F1 E7 s2B s2B1 E8 E9 E10 E11 s2B2 E12 E13 E14 E15 s2B3 E16 E17 E18 s2B4 s3 E19 E20 F2 T1 T2 s4 s4A E21 F3 s4B F4 T3 F5 F7 T4 F6 s4C F8 s5 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13 B14 B15 T5 B16 B17 B18 B19