Analysis of the remote microphone probe technique for the determination of turbulence quantities

Measuring the statistics of the wall-pressure ﬂuctuations (WPFs) is essential for understanding and evaluating laminar/turbulent boundary layers and ﬂow-induced noise. The remote microphone probe (RMP) technique is one of the most successful experimental approaches to measuring WPFs. Despite its extensive use in academic and industrial research, the scientiﬁc literature currently lacks an evaluation of the RMP inﬂuence on the determination of turbulence quantities, such as the WPF spectrum, the convection velocity, and the turbulence coherence length. To fulﬁll this necessity, this paper analyses the effects of the RMP on the determination of turbulence quantities, including comparisons between the results obtained using RMPs and ﬂush-mounted microphones (FMMs). The RMPs and the FMMs measure the WPFs under a turbulent boundary layer developed on a zero-pressure-gradient ﬂat wall. This study shows that the turbulence quantities determined from RMP measurements are inﬂuenced by the RMP setup, i.e., tube length and inner diameter. The tube length affects the phase between RMPs, which impacts the turbulent structure convection velocity estimation, which can be determined correctly by considering the calibrated phase. Furthermore, the tube inner diameter inﬂuences the coherence between a pair of RMPs, affecting the calculation of the turbulence correlation length. This research also shows that the turbulence quantities determined from the RMP and the FMM measurements differ, mainly in the coherence between pairs of microphones.


Introduction
Hydrodynamic wall-pressure fluctuations (WPFs) on fluid dynamic devices generate flow-induced noise and vibration.These noise sources are present in many engineering applications, such as wind turbines [1], aircraft [2], and ships [3].Noise exposure directly impacts people's well-being and animal life.Thus, stringent regulations have been created to limit the exposure to flowinduced noise [4,5], stimulating research on aeroacoustic noise sources.Measurements of the spatial and temporal characteristics of the WPFs are essential to validate models [6,7] and simulations and better understand the mechanisms of flow-induced noise generation [8][9][10].
The most common method of measuring the WPFs is using microphones mounted flush with the wall [11][12][13][14][15].In this configuration, the microphone is directly exposed to large hydrodynamic pressure fluctuations, which can saturate the microphone for high flow speeds.Furthermore, the finite size of the active sensor area spatially averages the high-frequency component of the measured hydrodynamic pressure fluctuations [16][17][18].Also, the sensor size dictates the minimal distance between two microphones, limiting the smallest turbulent length scales that can be resolved.Additionally, installing flush-mounted microphones (FMMs) is challenging for small-scale models, especially close to a sharp trailing edge where the thickness is small.Microphones can also be mounted in a recessed small cavity under a pinhole [8][9][10]19,20] to diminish spatial averaging effects due to the small cross-section area of the pinhole [17].However, installing microphones under a pinhole is difficult for small-scale models, especially where the model thickness is small.The unsteady pressure field can also be obtained from time-resolved tomographic particle image velocimetry (Tomo-PIV) [21], which has the advantage of a higher spatial resolution.However, this technique results in WPFs only up to % 3 kHz due to the PIV data noise levels [21], which is not sufficient for most flow-induced noise studies.These limitations can be addressed by installing the microphones in a remote configuration, also known as remote microphone probe (RMP).
In the RMP configuration, the microphone is installed outside the scaled model and connected to the wall by a pinhole on the surface and a tube [17,22]; see Fig. 1.This technique can measure higher pressure fluctuation levels compared to FMMs because the RMP tubes attenuate the pressure fluctuation levels, avoiding the microphone saturation [23].Moreover, installing microphones in a remote position also allows the WPF measurement at locations of small thickness, e.g., at the airfoil trailing edge.In addition, it is easier to replace damaged microphones in the RMP technique since they are installed outside the scaled model.Another advantage is that RMPs allow higher spatial resolution than FMMs because the surface area used by each RMP is smaller due to the use of a pinhole and the recessed location of the microphone.Another benefit of the RMP is that pressure transducers can be connected to the end of the anechoic termination to measure the static surface pressure and the WPFs simultaneously.The challenge of this technique is that it depends on a calibration in the entire frequency range.Methods to obtain a reliable calibration have been investigated by Awasthi et al. [24] and van de Wyer et al. [25].
Previous researchers have successfully applied the RMP technique to measure the WPFs generated by unsteadiness in the flow [7,[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].One of the first researches to apply this technique to measure the WPFs under a turbulent boundary layer was done by Kobashi and Ichiho [31].They chose this technique to increase the spatial resolution of the measurements.Perennes and Roger [27] were among the first to measure WPFs with the RMP technique to evaluate flow-induced noise.They measured the WPFs at the trailing edge of a high-lift device.The RMP has been extensively used to evaluate the WPFs close to an airfoil trailing edge to investigate trailing-edge noise [7,28,29,38].Also, the RMP was used to measure the WPFs under a turbulent boundary layer developed on a flat plate [22], downstream of a step [30], and on a butterfly valve subjected to a supersonic flow [23].This technique has also been used to determine the spatial scales of a turbulent boundary layer [39].These studies show that the RMP technique is well suited to measuring the WPFs for different types of unsteady flow and estimating the turbulent scales in a turbulent boundary layer.
The relevance of the RMP for flow-induced noise studies is due to its capability to characterize the turbulent boundary layer developed on a model wall, allowing the modeling of the turbulent flow [7,31,34] and the determination of the correlation between the turbulent flow at the trailing edge and the trailing-edge far-field noise [28,29].In these studies, turbulence quantities are calculated from the RMP time-series data, such as the WPF spectrum, the convection velocity, and the turbulence correlation length.The method to obtain the WPF spectrum on the model surface from the RMP data using the calibration data is well defined in the literature [22,28,29].The convection velocity of the turbulent structures is calculated based on the phase difference between a pair of RMPs.The calibrated RMP phase must be considered to compute the convection velocity because each RMP in the model introduces a specific phase delay due to its tube length.Previous studies have not given information regarding the phase calibration and whether it was applied at all to calculate the convection velocity [7,28,29].Additionally, the influence of the RMP configuration on determining the coherence between different RMPs still needs to be investigated.The coherence is an important parameter because the length scales in the turbulent flow, i.e., correlation length, are determined from the integration of the coherence between a pair of microphones.The correlation length is highly related to the noise level radiated by an airfoil trailing edge [40], making the accurate determination of the coherence important for trailingedge noise.
The main goal of this work is to investigate the influence of the RMP technique on the determination of turbulence quantities relevant to flow-induced noise studies, such as the WPF spectrum, the convection velocity, and the turbulence correlation length.To do so, different RMP configurations, i.e., different tube lengths and diameters, were investigated.The turbulence quantities obtained from RMP measurements are compared with the results from FMM measurements, which is considered the reference value.The measurement results from both RMP and FMM are compared with models, i.e., the Goody model for the WPFs [41] and the Corcos model for the longitudinal and lateral coherence [42].Also, the RMP calibration results are compared with the analytical model developed by Bergh and Tijdeman [43].The paper is organized as follows.First, the basics of the RMP technique are described in Section 2. In Section 3, the analytical and empirical models that are compared to the experimental data are described.In Section 4, the experimental setup and methodology are described.In Section 5, the results of WPFs under a turbulent boundary layer measured by RMPs and FMMs are analyzed and discussed.Finally, in Section 6, the conclusions of this work are stated.

Remote microphone probe technique
Measurements by RMPs are based on the fact that the WPFs generated by a turbulent flow at the model surface propagate through the tubes as sound waves [26], which are measured by a remote microphone.The tube works as a wave-guide.A sufficiently long tube placed after the microphone acts as an anechoic termination; i.e., this tube eliminates the reflection of the sound waves by viscous dissipation [22].The anechoic termination end is usually sealed to avoid flow due to the pressure difference with the environment.Fig. 1 shows a schematic view of an RMP setup.For an ideal RMP, no resonance occurs, which requires a constant inner diameter and an infinitely long anechoic termination.However, in many applications, it is challenging to maintain the same inner diameter for all the tubes of the RMP.Hence, resonances are present in the WPF measurements.Further, the viscous dissipation introduced by the tubes influences the magnitude and phase of the WPF measured by the RMP.Therefore, a calibration procedure is required to obtain the transfer function between the pressure fluctuations measured by the remote microphone and the actual pressure fluctuations at the surface.
Three regions are identified in Fig. 1: the model surface represented by 0, the tubes that connect the surface to the microphone represented by 1 and 2, and the anechoic termination represented by 3. The two tubes that connect the surface to the microphone have inner diameters d 1 and d 2 and lengths L 1 and L 2 .They can be replaced by one or a sequence of tubes of different diameters and lengths.The number of tubes and their geometric parameters can be tuned to achieve a specific attenuation level.This will be further discussed in Section 3.1.Each design parameter, i.e., the pinhole diameter, the tube inner diameters and lengths, and the anechoic termination inner diameter and length, affects the signal measured by the microphone and needs to be considered during the design phase of the experimental setup.

Analytical and empirical models
This section describes the following models: the analytical formulation for the RMP transfer function developed by Bergh and Tijdeman [43], the empirical expression for the WPF spectrum under a turbulent boundary layer proposed by Goody [41], and the longitudinal and lateral coherence proposed by Corcos [42].These models are compared with the measurement results in Section 5.

Analytical formulation for the RMP transfer function
Bergh and Tijdeman [43] derived an analytical formulation to represent the pressure propagation through thin circular tubes.This formulation can be used during the design of an RMP to target a certain attenuation of the WPFs, ensuring that the RMP does not saturate for a given WPF level.The formulation is based on the Navier-Stokes equations, and the continuity equation, among others.They assumed that the fluid pressure, density, temperature, and velocity had small sinusoidal oscillations without a steady velocity component.Furthermore, the tube radius was assumed to be small compared to its length so that the flow in the tube would be laminar throughout the system.Based on these assumptions, the flow equations were simplified, resulting in a general recursion formula in the frequency domain for a series of connected thin tubes and volumes that relates the pressure on the surface with the pressure at a specific position along the tube length.They compared this theoretical transfer function with experimental results, demonstrating good accuracy.Berntsen [22] compared this analytical transfer function with measurements for different geometric parameters of the RMP, confirming its accuracy.The volume term of the pressure transducer in the Bergh and Tijdeman solution is neglected in this study because, for most RMPs, the microphone diaphragm volume and deflection are negligible.For a series connection of N tubes, the ratio of the pressure (p) at the ending (p j ) and the beginning (p jÀ1 ) of the j th tube of length L j and radius R j is: where j ¼ ½N À 1; N À 2; . . .; 2; 1; TF jÀ1;j is the transfer function between tubes j À 1 and j; J 0 and J 2 are the Bessel functions of first kind of order 0 and 2, f is the frequency, x ¼ 2pf is the angular frequency, c 0 the fluid mean speed of sound, r the fluid specific heat ratio, i the imaginary unit, q the fluid mean density, l the fluid dynamic viscosity, Pr the fluid Prandtl number.The transfer function TF jÀ1;j contains both the attenuation and the phase delay caused by the tubes.For j ¼ N, Eq. 1 yields: With these equations, the pressure ratio between any of the tubes n and m can be computed by: since m P 0; n 6 N, and n > m.Eqs.(1)-( 6) result in the transfer function between the pressure on the surface and the pressure measured by the RMP.

Goody WPF spectrum model
Goody [41] proposed an empirical model to calculate U pp beneath a two-dimensional, zero-pressure-gradient turbulent boundary layer: where the scaling factors are given as: where U e is considered as the free-stream velocity U.The inputs for this model were obtained from the boundary layer measurements.

Corcos model for the longitudinal and lateral coherence
The lateral correlation length can be used to evaluate the turbulent structures in the boundary layer in the y-direction.The lateral correlation length is defined as: where n y is the distance between the microphones in the ydirection and c 2 is the coherence between a pair of microphones.
This procedure is applicable as long as n y is smaller than the turbulence correlation length.The experimental determination of this correlation length is challenging because several microphones must be used to perform the integration accurately.Corcos [42] proposed a model to calculate the coherence decay between two points in the y-direction in a turbulent boundary layer as a function of frequency: where U c is the convection velocity and b c is the Corcos constant, which is assumed to be 1.4 [7].Eq. 10(b) is obtained by substituting Eq. 10(a) into Eq.9, yielding a theoretical expression for the calculation of the lateral correlation length.The coherence in the longitudinal direction, which is used to estimate the average size of the turbulent structures that have the largest influence on the WPFs in the streamwise direction, follows the same formulation of the lateral coherence curve shown in Eq. 10(a) but using b c ¼ 2:8 and n x instead of n y , where n x is the distance between the microphones in the streamwise direction.Other expressions for the longitudinal and lateral correlation lengths are available, such as the Efimtsov model [44], which is more accurate than the Corcos model for low frequencies.However, the Corcos model is used in this study because only the coherence formulation is used.Also, the Efimtsov model is comparable with the Corcos model in the mid-and highfrequency ranges [45].

Experimental methodology
This section describes the experimental setup and methodology used to evaluate the influence of the RMP technique on the determination of turbulence quantities.

Wind tunnel facility and flow conditions
The experiments were conducted in the University of Twente AeroAcoustic Wind Tunnel, an open-jet, closed-circuit facility [46].After a contraction of ratio 10:1, the flow enters a closed test section and, subsequently, an open test section.The test section is 0.7 m Â 0.9 m (width x height).The experiments were performed in the open test section.The maximum operating velocity is 60 m/s in the open-jet configuration with a turbulence intensity below 0.08%.More information about the wind tunnel can be found in de Santana et al. [46].The flow temperature was controlled at approximately 20°C.The reference coordinate system considers the streamwise direction axis x, the width direction axis y, and the height direction axis z; see Figs. 2 and 3.
The WPF measurements were conducted under a turbulent boundary layer developed on the bottom wall of the wind tunnel, characterizing a flat plate with a zero pressure gradient.The flat plate case is considered a canonical study case that is adequate to evaluate the RMP concept.The turbulent boundary layer was characterized using hot-wire anemometry on the location where the WPF measurements were performed.The WPF measurements were performed for free-stream velocities U varying from 10 m/s to 50 m/s, with an interval of 10 m/s.The hot-wire measurements were conducted for velocities of 20 and 30 m/s.

Wall-pressure fluctuation measurements
FMMs and RMPs measured the pressure fluctuations under a turbulent boundary layer.Figs. 2 and 3 show a schematic view of the experimental setup.The FMMs are used for reference to compare with the RMPs.Three RMPs (RMP 1, RMP 2, and RMP 3) and three FMMs (FMM 1, FMM 2, and FMM 3) were used during the measurements; see Fig. 3. RMP 1 and RMP 2 and FMM 1 and FMM 2 were placed side by side in the y-direction, corresponding to the wind tunnel width.With this setup, the coherence between the microphones in the lateral direction can be obtained.RMP 3 and FMM 3 were placed in the downstream direction aligned with RMP 2 and FMM 2, respectively; see Fig. 3. Thus, the convection velocity of the turbulent structures and the coherence between the microphones in the streamwise direction can be calculated.
The microphone used for both RMPs and FMMs was the Knowles FG-23329-P07, which is a 2.5 mm diameter omnidirectional electret condenser microphone with a circular sensing area of 0.79 mm.According to the manufacturer, the sensitivity of the microphone is about 45 Pa/mV ± 13 Pa/mV in the flat region of the microphone frequency response (from 100 Hz to 10 kHz).The sensitivity of the microphones used varied from approximately 42 Pa/mV to 48 Pa/mV, comparable to the values observed by Garcia-Sagrado and Hynes [47].The membrane of the microphone was directly exposed to the WPFs, eliminating the influence of the protection grid on the WPF measurements [17,48].Measurements were also performed for a Knowles microphone installed under a pinhole; see Fig. 2.
Microphone data was acquired for 30 s at a sampling frequency of 65,536 Hz by a PXIe-4499 Sound and Vibration module installed in a NI PXIe-1073 chassis.The cut-off frequency of the anti-aliasing filter was 32,178 Hz.

RMP setup
Each RMP consisted of two tubes (tubes 1 and 2) connected by a tube assembly where the microphone was installed; see Fig. 2. Tube 1 was connected to the surface by a pinhole of 0.5 mm.Devenport et al. [49] showed that a 0.5 mm pinhole is sufficient to resolve boundary layer pressure fluctuations up to a frequency of 20 kHz.Tube 2 is the anechoic termination, which had a tube length of 3 m and inner diameter of 1.6 mm and was sealed.Different tube lengths and diameters were used to evaluate the RMP setup.Table 1 shows the RMP configurations tested.Each configuration consists of three RMPs: RMP 1, RMP 2, and RMP 3. The length L 1 and inner diameter d 1 of tube 1 varied among the configurations.For configurations 1, 3, 4, and 5, tube 1 of RMP 1, RMP 2, and RMP 3 had the same length.However, for configuration 2, tube 1 of RMP 1 and RMP 3 had a length of 0.45 m, whereas tube 1 of RMP 2 had a length of 0.1 m.This configuration was tested to determine the effect of different tube lengths on the convection velocity and the lateral coherence calculation, which is a common situation encountered in airfoil models.The length L 2 and inner diameter d 2 of the anechoic termination, i.e., tube 2, was the same for all configurations and RMPs.

RMP calibration
The objective of RMP calibration is to obtain the transfer function between the signal measured by the RMP and the WPFs at the surface.In-situ calibration must be performed to obtain the characteristic transfer function of each RMP because of mounting effects and manufacturing errors [28].
In this study, an in-house calibrator device inspired by the design in Ref. [17] was used.The calibrator consists of a speaker to generate the excitation, a reference microphone, and a hole or cavity where the RMP pinhole is placed; see Fig. 4. The calibrator was manufactured using selective laser sintering (SLS) 3D printing.The calibrator dimensions are given in this paper's supplementary material.A Visaton FR 8 speaker is placed at the larger opening of the convergent section, sealed by an o-ring to reduce acoustic leakage.The speaker was connected to a white noise source.An o-ring seals the contact between the calibrator and the surface.The reference microphone (GRAS 40PH) is flush-mounted in the inner tube of inner diameter d cal ¼ 7 mm.The distance between the reference microphone and the surface is L cal ¼ 7:5 mm.Due to the distance between the pinhole and the reference microphone L cal , a peak is observed in the transfer function between the reference microphone and the microphone to be calibrated.This is discussed later on in this section.In this study, this peak is referred to as the calibrator peak.
A two-step calibration method was used; i.e., the final transfer function is obtained from two intermediate transfer functions, each one determined in a different calibration step.First, the unsteady pressure generated by the speaker is measured by the reference microphone and the RMP (see Fig. 4a), resulting in the transfer function between these two microphones.The second step consists of measuring the unsteady pressure generated by the speaker with the reference microphone and an FMM (see Fig. 4b), resulting in a transfer function between these two microphones.This method has also been applied in Refs.[17,28].The advantage of the two-step calibration procedure is the minimization of the calibrator-peak amplitude in the final transfer function.However, this effect is not entirely eliminated because the acoustic field within the calibrator will not be the same for both calibration steps because the RMP line participates in the acoustic system, altering the calibrator peak [25].The calibration procedure for each RMP was repeated at least two times to verify the repeatability of the transfer function obtained.The calculation of the experimental transfer function is discussed below.The calibration following this procedure was performed for the RMPs and the Knowles microphone under the pinhole.
Two transfer functions must be obtained for the two-step calibration: 1. between the RMP and the calibrator reference microphone; 2. between an FMM at the surface and the calibrator reference microphone.The pressure measured by the RMP in the frequency domain P m ðf Þ is related to the pressure measured by the calibrator reference microphone in the frequency domain Similarly, the pressure measured by the FMM at the surface in the frequency domain P 0 ðf Þ is related to the pressure measured by the reference microphone in the frequency domain P ref ðf Þ by the transfer function TF ref;0 ðf Þ: where P m ; TF ref;m ; P ref ; P 0 , and TF ref;0 are complex variables.The transfer functions are defined as: where U m;m is the auto-spectral density of the pressure time-series measured by the RMP, U 0;0 is the auto-spectral density of the wall Fig. 5a shows the experimentally obtained transfer functions for the first and second calibration steps (TF ref;m j Exp and TF ref;0 j Exp , respectively), the experimental final transfer function (TF 0;m j Exp ), and the final analytical transfer function (TF 0;m j Analytical ).The calibrator peak is observed at 11.5 kHz.This peak is observed in both calibration steps.The influence of the calibrator peak on the final transfer function (TF 0;m j Exp ) reduces because a two-step calibration was performed; see Eq. 15.However, the calibrator peak is not eliminated because the acoustic field within the calibrator is not the same for both calibration steps because the RMP line participates in the acoustic system, altering the calibrator peak.
To verify if the measured pressure by the pair of microphones involved in a calibration step was correlated, the coherence (c 2 ) between these microphones was evaluated.Fig. 5b shows that the coherence is approximately 1 up to approximately 10 kHz for different RMPs, indicating that the signal measured by the reference microphone and the RMPs were strongly correlated up to this frequency.Therefore, the calibration and the transfer functions obtained are reliable up to 10 kHz.
The accuracy of the analytical formulation to predict the RMP transfer function is now analyzed.Fig. 6 shows the analytical and experimental final transfer functions for different RMP configurations.Configurations 1 and 3 have the same inner diameter (d 1 =1.5 mm) and different tube lengths (L 1 =0.45 m and 0.1 m, respectively).Configurations 3 and 4 have the same tube length (L 1 =0.1 m) and different inner diameters (d 1 =1.5 mm and 0.5 mm, respectively).There is a good agreement between the experimental and analytical transfer functions up 1 kHz for all the cases; see Fig. 6a.However, the analytical transfer function underpredicts the attenuation caused by the RMP tubes for higher frequencies.This discrepancy is most probably due to installation effects, confirming that an in situ calibration is essential to obtain accurate calibration results [17,28].Longer tubes (with the same inner diameter) and smaller diameter tubes (with the same tube length) increase the attenuation.Note that the tube diameter considerably influences the attenuation for low frequencies, whereas the tube length significantly affects the attenuation in the highfrequency range.Fig. 6b shows that the tube length significantly influences the transfer function phase, whereas the tube inner diameter has little effect.Section 5.2 discusses this effect in the convection velocity determination.Furthermore, the analytical formulation predicts the transfer function phase well in the entire frequency range.
During the measurements of the WPFs caused by the turbulent boundary layer, the cross-spectral density of the WPFs at the surface U 0;0 j mic i;mic j between microphone i and microphone j is reconstructed using the transfer function TF 0;m for each microphone in combination with the RMP measurement of the unsteady pressure: U 0;0 j mic i;mic j ¼ U m;m j mic i;mic j TF 0;m j mic i TF Ã 0;m j mic j ð16Þ where U m;m j mic i;mic j is the cross-spectral density of the unsteady pressure calculated between the remote microphone i and remote microphone j when exposed to the turbulent boundary layer.TF 0;m j mic i and TF 0;m j mic j correspond to the final transfer function of microphone i and j, respectively, and TF Ã 0;m is the complex conjugate of TF 0;m .If i ¼ j; U 0;0 j mic i;mic j reduces to the auto-spectral density U 0;0 j mic i and U m;m j mic i;mic j reduces to the auto-spectral density U m;m j mic i .Note that U 0;0 j mic i;mic j accounts for the level attenuation and phase due to the RMP setup.

Spectrum and coherence determination
The auto and cross-spectral densities and the coherence were calculated using the Welch method.The data was averaged using blocks of 8192 (0.125 s) samples and windowed by a Hanning windowing function with 50% overlap, resulting in a frequency resolution of 8 Hz.The spectral level for the pressure measurements is shown in decibel, calculated as in Section 8.4 of Glegg and Devenport [50].The reference pressure and frequency used were p r ¼ 20lPa and Df ref ¼ 1 Hz.

Boundary layer measurements
The turbulent boundary layer at the surface of the wind tunnel wall was measured using hot-wire anemometry.A single-wire probe (Dantec Dynamics model 55P15) of 5 lm diameter and 1.25 mm wire length was used to measure the streamwise velocity.The probe was mounted in a Dantec Dynamics 55H22 probe support installed on a symmetric airfoil.This airfoil was fixed in a 3D traverse system, allowing the probe translation with a resolution of 6.5 lm.The hot-wire data was acquired with the Dantec   The data was recorded for 20 s with a sampling frequency of 65,536 Hz and an anti-aliasing filter with a cut-off frequency of 30 kHz.The hot-wire calibration was performed in situ in the closed test section with a Prandtl tube as reference.The velocity measurements had a maximum system uncertainty of 5% with a 95% confidence interval.This uncertainty was computed following the guidelines provided by Dantec Dynamics [51], which considers the calibration equipment, calibration linearization, A/D board resolution, probe positioning, and temperature variations.The boundary layer was measured at the location of the FMM 1.On average, the velocity was measured at 75 locations across the boundary layer and 7 points in the free stream.The z-locations of these points were spaced logarithmically.The probe's distance to the wall was determined by the contact of a feeler gauge to the hot-wire prongs.The gauge accuracy is 0.05 mm, and the probe's distance to the wall was 0.8 mm.
From the measurement of the boundary layer velocity profile, the experimental boundary layer thickness d is determined as the distance from the wall where the velocity corresponds to 99% of the free-stream velocity U.The experimental boundary layer displacement d Ã and momentum thicknesses h are determined by performing a trapezoidal numerical integration of the measured boundary layer velocity profile.The shape factor is determined as H ¼ d Ã =h.The measured velocity profile is fitted to the logarithmic law of the wall: where u is the streamwise velocity, u s is the friction velocity, z is the normal-to-the-surface direction, m is the fluid kinematic viscosity, and the constants j and B are considered j ¼ 0:38 and B ¼ 5.The friction velocity u s is determined from this fitting with a coefficient of determination R 2 of approximately 0.99.The wall-shear stress is calculated from the friction velocity as s w ¼ qu 2 s .These boundary layer parameters are used as input to the Goody model [41] for the WPF spectrum, and are shown in Table 2.This table also shows the coefficient of determination R 2 of the fitting between the measured velocity profile and the law of the wall.Fig. 7 shows the velocity and turbulence intensity profiles determined from the hot-wire measurements.Also, the logarithmic law of the wall fitted to the measured velocity profile is included.The axes are normalized with z þ and u þ for the velocity profile and with the boundary layer thickness d and the freestream velocity U for the turbulence intensity profile.The turbulence intensity profile is determined from the root-mean-square of the streamwise velocity fluctuations u rms .A good agreement between the measured velocity profile and the law of the wall is observed.This is reflected in the coefficient of determination value, which is almost 1; see Table 2. Also, the velocity profiles overlap well for the different free-stream velocities.The velocity profiles deviate slightly for z=d > 0:85, and the turbulence intensity profiles deviate slightly for z=d > 0:6.

Turbulence quantities under a turbulent boundary layer
This section discusses the measurements of the turbulent boundary layer developed over a flat wall.RMPs with different tube lengths and inner diameters were investigated, and the effects of these geometric parameters on the wall-pressure spectrum, convection velocity, and coherence between a pair of microphones were analyzed.For comparison, results for the FMMs are also included.The nomenclature used in the sub-indices in the following sections is: ''0" refers to the surface, ''m" refers to the RMP or pinhole microphone, and ''ref" refers to the reference microphone in the calibrator.

Wall-pressure fluctuation spectrum
The WPF spectrum obtained from the RMP measurements was corrected by the RMP transfer function determined from the calibration, called the reconstructed spectrum.The reconstructed RMP WPF spectrum was compared with the WPF spectrum obtained from the FMM measurements.Fig. 8a shows the WPF spectrum measured by the RMPs (U m;m j RMP ), and the reconstructed spectrum (U 0;0 j RMP ), which is the WPF spectrum on the surface.Even though the different RMPs shown in Fig. 8a have the same tube lengths and diameter, the measured spectrum U m;m j RMP differs among them.This is attributed to installation effects [17,28] and  possible differences in the frequency response of the microphones.
However, the reconstructed spectral curves U 0;0 j RMP for the different RMPs overlap well.This clearly shows that the experimental transfer functions capture the WPF attenuation well, including the installation effects and possible differences in the frequency response of the microphones.The RMP reconstructed WPF spectrum agrees with the spectral curve obtained from the FMM measurements within 5 dB up to 10 kHz.For frequencies higher than 10 kHz, the calibrator peak observed in the transfer function contaminates the reconstructed spectrum.This confirms that the maximum usable frequency with the calibrator used and the two-step calibration procedure is 10 kHz.A maximum discrepancy of % 5 dB between U 0;0 j RMP and U 0;0 j FMM up to 2 kHz is observed.This discrepancy is due to the pinhole used on the RMP setup; this will be discussed in the following paragraph.Fig. 8b shows the WPF spectrum for RMP 1 for different RMP configurations.The spectra measured by the different RMP configurations U m;m j RMP are considerably different because of the different tube lengths and diameters used for each configuration.However, the reconstructed spectra U 0;0 j RMP match well for the different RMP configurations because an appropriate calibration is performed, as shown in Refs.[22,23].
The WPF spectral level measured by the RMP is lower than that measured by the FMM for low frequencies.The level difference at 104 Hz is 3.5 dB for 20 m/s and 5.4 dB for 30 m/s; see Fig. 9a.This figure also shows the WPF spectrum reconstructed from the measurement using the Knowles microphone under a pinhole, i.e., U 0;0 j PH .This spectrum U 0;0 j PH is obtained by applying the experimental transfer function (TF 0;m in Fig. 9b), following the same procedure as for the RMP.The spectral level of the microphone under the pinhole U pp j PH is similar to the level of the RMP for low frequencies.We do not expect the low-frequency discrepancy between the FMM and RMP to be due to the background noise because the microphone under a pinhole experiences a lower attenuation level than the RMP for low frequencies, but the WPF spectrum for the microphone under a pinhole still presents the same tendency as the RMP, leading us to believe that this discrepancy is due to the pinhole effect on the measurements.The pinhole influence on the results can be due to its effect on the acoustic field during the two-step calibration or on the hydrodynamic field during the measurements of the WPFs under the turbulent boundary layer.To verify these hypotheses, the transfer functions obtained from the two-step calibration of the microphone under the pinhole are analyzed.Fig. 9b shows that the transfer function between the microphone under the pinhole and the reference microphone (TF ref;m ) is the same as the transfer function between the FMM and the reference microphone (TF ref;0 ) for frequencies lower than 1 kHz.This indicates that the influence of the pinhole on the transfer function is limited to high frequencies (f > 1 kHz).Thus, the level difference in the WPF spectrum for low frequencies is attributed to the pinhole influence on the hydrodynamic field.This interpretation is supported by the spectral level difference between the RMP and the FMM because the difference increases with the flow speed.If the level difference was due to the calibration, the difference should be constant with flow speed.Fig. 10 shows the WPF spectrum measured by the FMM 1 and the RMP 1 compared with the Goody model.The inputs for this model were obtained from the boundary layer measurements, which are shown in Table 2.The Goody model overlaps well with the FMM results for all velocities up to 6 kHz for 20 m/s and 9 kHz for 30 m/s.This results in a Strouhal number St ¼ f d=U of approximately 43.Past this frequency, the measurements show a level attenuation, probably due to the spatial averaging caused by the finite size of the transducer.Awasthi et al. [24] have also reported an attenuation of the spectral level for St > 40 due to spatial averaging effects.For high frequencies, the spectrum for both RMP and FMM follows x À5 up to 6 kHz for 20 m/s and 9 kHz for 30 m/s, i.e., St % 43.Several studies report different values for the lowfrequency slope of the WPF spectrum, usually ranging from À0.7 to À0.8 [52][53][54]; however, there is no universally agreed value.
The FMM spectrum in the low-frequency range follows x À0:7 , which is the same as predicted by Goody [41] and observed in other studies [24,53].Due to the pinhole effect on the hydrodynamic field, the WPF spectral level measured by the RMP follows a different scaling x À0:6 .This is a point of attention because WPF spectrum models are usually developed based on the scalings observed from experimental results.In the current work, the two techniques result in different scalings for low frequencies.Depending on the technique used, the scaling changes, influencing the accuracy of a developed WPF spectrum model.A limitation of the FMM is that it can be saturated for high flow speeds because the microphone membrane is directly exposed to the WPFs.To verify the saturation condition of a microphone measuring a stochastic phenomenon, e.g., a turbulent boundary layer [50], the probability density function (PDF) of the time-history signal is analyzed.This PDF should have a Gaussian shape because of the stochastic behavior of turbulence [55].Fig. 11 shows the PDF curves for the RMP in different configurations and the FMM.At 30 m/s, the PDF for all cases behaves as a Gaussian curve.There is a considerable difference between the PDF shapes for the RMPs and the FMM due to the different pressure levels measured by each microphone.At 50 m/s, the FMM is saturated since the curve is mainly flat and clipped at the extremes, whereas the curves for the RMPs follow a Gaussian shape.Hence, the RMPs are better suited than the FMM for high turbulence levels, e.g., high Reynolds and Mach numbers and for applications exposed to a turbulent inflow.The flow conditions to which the microphone will be submitted during the tests should be considered during the RMP design.In this way, the RMP can be designed to obtain sufficient attenuation to avoid microphone saturation and, at the same time, sufficiently low attenuation to guarantee that the signal has a high enough amplitude that the microphone can measure.A first estimate of the attenuation for an RMP can be obtained using the analytical formulation developed by Bergh and Tijdeman [43].This formulation predicts the attenuation reasonably well up to 1 kHz, which is the frequency range where the highest levels of WPFs are encountered.

Convection velocity
The determination of the average velocity at which the turbulent eddies with the largest influence on the WPFs are convected inside a turbulent boundary layer, i.e., the convection velocity U c , is of great interest for aeroacoustic studies.The turbulent eddies convected in the streamwise direction cause a phase delay in the unsteady pressure measured by the microphones aligned in the downstream direction.The convection velocity is determined from the phase gradient spectrum (d/ i;j =df ) between microphones i and j in the streamwise direction: where g x is the streamwise distance between the microphones, which is g x ¼ 8 mm for this study.The phase is calculated from the cross-power spectral density (CPSD) between microphones i and j.
The tubes used in the RMP cause a phase delay in the WPFs measured by the remote microphone; see Fig. 6b.Thus, if the RMPs aligned in the streamwise direction have different tube lengths, these RMPs will have a phase delay between themselves due to: 1. the turbulent eddies being convected, and 2. the different tube lengths.Therefore, the calculation of the convection velocity based on the phase gradient is affected by the RMP configuration.As an example, the convection velocity is calculated for the FMM 2 and  FMM 3 and between the RMP 2 and RMP 3 for configuration 2, which has different tube lengths for RMP 2 and RMP 3. The convection velocity is determined from the phase gradient of the cross-PSD in the range from 1 kHz to 3.5 kHz.The convection velocity is 0:18 U for the RMP for the case where RMP-induced phase delay is not considered in the calculation; i.e., the cross-PSD is calculated directly from the measured RMP time series.For the FMM, the convection velocity is 0:59 U. Considering the RMP-induced phase delay, i.e., using Eq.16 to calculated the cross-PSD, the convection velocity is 0:57 U, which is much closer to the FMM.Thus, these results show that it is crucial to account for the RMP phase delay in the convection velocity calculation.

Longitudinal and lateral coherence
Fig. 12 shows the lateral and longitudinal coherence of a pair of microphones separated by 8 mm for different RMP configurations; refer to Table 1.This figure also shows the coherence obtained from the FMMs and the coherence for the Corcos model.For the lateral direction, the coherence for the RMPs increases for low frequencies compared to the FMMs.The maximum coherence in the lateral direction measured with the RMP is shifted to low frequencies, but it follows the decay proposed by Corcos.This better agreement is helpful in experimental practices because the Corcos constant is usually calculated by fitting the analytical expression to the experimental results.Contrarily, the RMP coherence in the longitudinal direction decays with frequency following a different tendency compared to the FMM and the Corcos model.The difference in the coherence values between the RMP and the FMM might be associated with the pinhole effect on the hydrodynamic field, which is mainly observed for frequencies lower than 1 kHz.The difference in coherence values between the RMP and the FMM also indicates that different correlation length values are obtained depending on the technique used to measure the WPFs because the correlation length is calculated from the integration of the coherence; see Eq. 9.
For both lateral and longitudinal directions, the coherence for the RMPs with the largest tube diameter (configurations 1, 2, and 3) is higher than that for the RMP with the smaller tube diameter (configuration 4) for frequencies lower than 1 kHz.The lower coherence values for RMP in configuration 4 compared to the other configurations are probably due to the higher attenuation of the WPF levels for this configuration compared with the other configurations.This higher attenuation is due to the tube diameter.Fig. 6a shows that the attenuation is more than 10 dB at 100 Hz for configuration 4 (tube diameter of 0.5 mm) compared to almost no attenuation for configurations 1 and 3 (tube diameter of 1.5 mm) at the same frequency.Due to the higher attenuation for low frequencies, the RMP in configuration 4 is more sensitive to other pressure fluctuation sources (for example, background noise) than the other configurations, resulting in a drop in the coherence; i.e., the signal-to-noise ratio is worst for configuration 4. Since the smaller tube diameter is more sensitive to the other sources of pressure fluctuations, extra measurements were performed to verify whether the background noise effects can be observed in the coherence spectrum.These extra tests consisted of measuring the WPFs under the turbulent boundary layer developed on the wind tunnel wall with RMPs spaced in the y-direction by g y = 8, 21, 57, 152, and 304 mm.Fig. 13 shows the coherence values for these measurements.The coherence decreases as the  distance increases, reaching a value of almost zero in the entire frequency range for g y P 57 mm.These results show that the frequency range where lower coherence values are observed for an RMP with a smaller tube diameter (100 6 f 6 500 Hz) is not due to the direct measurement of the background noise since no coherence is observed when only background noise is present (g y P 57 mm).The tube length does not influence the coherence of a pair of microphones in any direction, as observed for configurations 1 and 3. Specifically, configuration 2, in which the tube lengths of the RMP 1 and RMP 2 are different, demonstrates that the use of different tube lengths does not influence the coherence between the microphones.Conversely, the tube diameter affects the coherence in the low-frequency range, i.e., the coherence reduces as the tube diameter reduces; see configuration 4.

Conclusions
This study discussed the influence of the RMP technique on the determination of turbulence quantities relevant to flow-induced noise studies, such as the WPF spectrum, the convection velocity, and the turbulence coherence.The main contribution of this study is to show for the first time that: 1. turbulence quantities determined from RMP measurements are influenced by the RMP setup, i.e., tube length and diameter, and 2. turbulence quantities determined from the RMP and the FMM measurements are different.
The analytical formulation developed by Bergh and Tijdeman [43] predicts the attenuation for the RMP reasonably well up to 1 kHz, which is the frequency range where the highest levels of WPFs are encountered.Thus, this model is helpful during the design of an RMP because a certain attenuation can be targeted, ensuring that the RMP does not saturate for a given WPF level.The calibration procedure and calibrator used in this study result in reliable measurements up to 10 kHz.
The WPF spectrum measured by the FMM matches well with the Goody model up to St ¼ 40.The spectrum determined from the RMP and the FMM measurements differs in the lowfrequency range, which is attributed to the pinhole effects on the hydrodynamic field.The RMP spectral level follows x À0:6 whereas the FMM spectral level follows x À0:7 for low frequencies.This is an important observation because WPF spectrum models are usually developed based on spectral scaling from experimental results.Thus, this shows that depending on the experimental technique used, a different scaling can be observed, influencing the accuracy of the WPF spectrum models developed based on these measurements.For high frequencies, the spectral level for the RMP and the FMM follows the same decay (x À5 ).
The RMP tube length and diameter influence the transfer function and the measured spectrum, but these effects are considered when an appropriate calibration is performed.Also, an in situ calibration is paramount to obtaining reliable results because it accounts for installation effects.The RMP tube length also affects the phase between different RMPs, directly impacting the turbulent structure convection velocity calculation.Correct estimations of the convection velocity are obtained with the RMP when the RMP-induced phase delay is considered in the calculation.The coherence between the different RMPs exposed to a turbulent boundary layer differs from the FMM results.Also, the coherence is sensitive to the signal-to-noise ratio when the RMP setup results in a high attenuation level for low frequencies.As a result, the calculation of the turbulence correlation length is affected by the RMP setup.
The observations made in this manuscript are for the experimental setup used.However, we expect that the RMP setup still influences the convection velocity calculation and coherence between microphones for other more complex setups since these effects are already noticeable for a straightforward setup.Further investigations are needed to evaluate the influence of other aspects of the RMP setup, such as the number of bends in the tube, on the turbulence quantity determination.

Fig. 3 .
Fig. 3. Top view of the experimental setup.

2 Fig. 4 .
Fig.4.Ideal two-step calibration procedure with a calibrator.The schematic view is not to scale.
(a) Experimental and analytical transfer functions for RMP 1 (b) Coherence between the reference microphone in the calibrator and the RMPs

Fig. 5 .
Fig. 5. Transfer functions (a) and coherence (b) for an in situ two-step calibration for RMPs in configuration 1.

F
.L. dos Santos, L. Botero-Bolívar, C.H. Venner et al.Applied Acoustics 208 (2023) 109387 StreamLine Pro CTA system and the Dantec StreamWare software in combination with the National Instruments 9215 A/D converter.

Fig. 6 .
Fig. 6.Comparison of the analytical and experimental transfer function level and phase for different RMP configurations.

Fig. 9 .
Fig. 9. Left: PSD of the WPFs measured by RMP 1 (conf.1), FMM 1, and the microphone under the pinhole for two velocities.Right: Experimental transfer functions for the two-step calibration of the microphone under the pinhole.

Fig. 10 .
Fig. 10.PSD of the WPFs measured by RMP 1 in configuration 1 and FMM 1 and the Goody model for two free-stream velocities.p r ¼ 20lPa and Df ref ¼ 1 Hz.

Fig. 11 .Fig. 12 .
Fig. 11.Probability density function of the time signal for the FMM and RMPs for two free-stream velocities.

Fig. 13 .
Fig. 13.Coherence between a pair of RMPs in the lateral direction spaced g y U ¼ 30 m/s.

Table 1
Experimental conditions for the remote microphone probes.

Table 2
Boundary layer parameters measured.