Study of the development of a boundary layer using the remote microphone probe technique

Abstract

The experimental identification of the transition region and the different flow structures inside the boundary layer is still a challenge. For the first time, this research uses the remote microphone probe technique to experimentally evaluate the boundary layer development, the transition process, and the type of flow structures. The remote microphone probe technique is an effective and accurate experimental method to measure wall-pressure fluctuations. The development of the boundary layer was evaluated under natural and forced transition for different inflow velocities and angles of attack. Results of the wall-pressure spectrum, spanwise coherence at different chord positions, and the spanwise correlation length close to the trailing edge are presented. Furthermore, boundary layer and far-field noise measurements at several conditions are also shown. This paper shows the growth of the turbulent structures that contain most of the turbulent energy along the airfoil chord. Further, it is demonstrated that the spanwise correlation length increases with the inflow velocity. Results for the no forced transition cases form a complete database to determine the different transition stages, which were linked with different components of the wall-pressure spectra. The primary and secondary instability mechanisms leading the transition process appear in the wall-pressure spectrum as peaks and a hump, respectively. The two- and three-dimensional nature of the boundary layer structures is also discussed by analyzing the spanwise coherence. Finally, it is shown that when two-dimensional structures reach the airfoil trailing edge, a feedback loop between the acoustic waves at the airfoil trailing edge and a point upstream in the airfoil surface is generated. This feedback loop influences the wall-pressure fluctuations along the entire airfoil chord.

Graphical abstract

1 Introduction

Determining the state of the boundary layer is crucial when studying many aerodynamic and aeroacoustic phenomena. The noise produced by an airfoil highly depends on the state of the boundary layer (Brooks et al. 1989). In actual conditions with full-scale models, such as aircraft wings and wind turbine blades, the transition of the boundary layer usually occurs naturally close to the leading edge due to the high Reynolds number. In wind tunnel experiments with small models and low inflow velocity, the Reynolds number is usually much lower; therefore, the boundary layer hardly completes the transition. To overcome the low Reynolds number effects present in wind tunnel experiments, tripping devices are used to trigger the transition of the boundary layer from laminar to turbulent at a specified position along the chord. The use of tripping devices does not guarantee that the boundary layer completes the transition and will be fully developed at the trailing edge (dos Santos et al. 2022; Botero-Bolivar et al. 2022). Several parameters influence the transition and the development of the boundary layer, i.e., the type of surface roughness, trip location, trip height, and the inflow turbulence (dos Santos et al. 2022; Silvestri et al. 2018; Elsinga and Westerweel 2012; Braslow and Knox 1958; dos Santos et al. 2022). The determination of the state of the boundary layer at the trailing edge is crucial for studying the noise produced by an airfoil. However, the experimental determination of the state of the boundary layer and the transition point is still a complex challenge.

The transition of the boundary layer over an airfoil was studied by Gersten et al. (2017), who determined the classic understanding that the natural transition of the boundary layer occurs in five stages: (1) The initial laminar boundary layer becomes unstable at a point denominated the indifference point. (2) At this position, the boundary layer is dominated by two-dimensional Tollmien–Schlichting (T–S) waves. This is explained by the primary instability theory. (3) The T–S waves are amplified and finally broken down into three-dimensional disturbances, modeled by the secondary instability mechanism. (4) Gamma ($\Gamma$) vortical structures are formed, which are converted into (5) turbulent spots at the critical point. Downstream of this point, the boundary layer is fully turbulent and continues developing as a turbulent boundary layer, which is represented by region (6) in the figure. The primary instability theory models the interaction of a laminar boundary layer with perturbations that are superimposed on the laminar flow, such as the wall roughness or non-uniformities in the outer flow. On the other hand, by using tripping devices, which introduce strong disturbances in the boundary layer, the transition process is shortened. Hence, the primary instabilities are bypassed, and the turbulent spots are directly generated (Zaki et al. 2006). This transition process is called bypass. However, if the trip height is significantly lower than the boundary layer thickness at the trip location, the trip would only generate the primary instabilities, and the boundary layer transition follows the natural transition process (Botero-Bolivar et al. 2022).

The most common experimental technique to map the transition region is infrared thermography which determines the transition region by identifying where the surface temperature abruptly decreases after previous heating of the airfoil surface (Raffel and Merz 2014; Crawford et al. 2013). However, this technique requires a complex setup, and no information about the flow field and structures inside the boundary layer is obtained. The transition point can also be identified using hot-film sensors placed along the chord and finding the minimum coherence among the sensors (Zhang et al. 2002; Obara and Holmes 1985; Rudmin et al. 2013). Richter et al. (2016) used three techniques, i.e., differential infrared thermography, Kulite pressure sensors, and hot-films, to detect the unstable transition point along a blade where the angle of attack was varied in cycles. These techniques exhibited good agreement at low angles of attack, but the agreement was reduced at higher values. Klein et al. (2017) used carbon nanotubes to keep a higher constant temperature and a temperature-sensitive paint to detect the transition point. Hot-wire anemometry has also been used to identify the boundary layer transition process by analyzing the velocity time history close to the wall (Schubauer and Skramstad 1947). Although several experimental techniques can be used to detect the transition point, the previous investigations focused on detecting the transition location as a region on the airfoil, not characterizing the transition structures or studying the boundary layer development. A particular work that includes flow information while studying the transition process was conducted by Chapman et al. (1998), who characterized the transitional flow structures by hot-films densely distributed in the chordwise and spanwise directions. They analyzed the auto- and cross-spectrum of the hot-film measurements and used proper orthogonal decomposition (POD) to characterize the flow structures present during the transition process. They identified the primary instability mechanism responsible for generating two-dimensional structures as peaks in the auto spectrum and related it with the first mode of POD in the spanwise direction. The second instability mechanism was associated with a hump in the auto spectrum and with the second mode of the POD analysis. Baltzer et al. (2010) used POD to identify the flow structures using instantaneous flow fields obtained from direct numerical simulations (DNS).

The wall-pressure spectrum is an integral quantity that gives information about what happens across the entire boundary layer, considering large and small turbulent structures, and laminar structures (Glegg and Devenport 2017). Therefore, analyzing the wall-pressure spectrum is an excellent alternative to investigate the development of the boundary layer and obtain additional information about the flow during the transition process. However, up to the best knowledge of the authors, this method has not been used yet to study the transition process and evolution of the boundary layer. Botero-Bolivar et al. (2022) demonstrated that the state of the boundary layer and the flow structures inside the boundary layer of an airfoil are strongly related to the wall-pressure spectrum. In their work, a single surface microphone was located close to the trailing edge to measure the wall-pressure fluctuations.

There are several methods to measure unsteady wall pressure. The most direct method is to install microphones flushed-mounted in the airfoil surface (Brooks and Hodgson 1981; Sanders et al. 2018; Löfdahl et al. 1996). However, this technique is limited to airfoils of sufficient thickness. Also, the space averaging due to the exposure of the relatively large microphone membrane to the hydrodynamic pressure fluctuations and the saturation of the microphones at low-velocities are disadvantages of this technique (Willmarth 1959; Roger 2017; Corcos 1963). A variation of this technique is to place the microphones recessed under a pinhole (dos Santos et al. 2022; Botero-Bolívar 2022; Mish and Devenport 2006; Sanders et al. 2022; Chun et al. 2004; Meloni et al. 2019). This method reduces the microphone’s spatial averaging since the pinhole’s presence converts the hydrodynamic waves into acoustic waves. However, the installation and saturation at low velocities are still strong limitations. An indirect method to measure the wall-pressure fluctuations is using Tomo-Particle Image Velocimetry  (Ghaemi et al. 2012). Despite the high spatial resolution obtained with this technique, the maximum sampling frequency is relatively low compared with flushed-mounted microphones because of limitations in the high-speed laser and cameras. Another technique to measure the wall-pressure fluctuations directly is the remote microphone probe (RMP).

In the RMP technique, the microphone is located outside the model and connected to the airfoil surface through a tube and a pinhole at the airfoil surface (Roger 2017; Berntsen 2014). Therefore, the hydrodynamic waves that are originated at the airfoil surface are propagated through the tube as acoustic waves (Englund and Richards 1984). An anechoic termination is connected to the tube to eliminate the reflection of the sound wave by viscous dissipation (Berntsen 2014). The other extreme of the anechoic termination is sealed to avoid flow passing through. The microphone generally is exposed to the acoustic waves from the lateral of the tube before the anechoic termination. Figure 1 shows a schematic view of a remote microphone probe configuration. In the figure, the subscript 0 represents the airfoil surface. Between the airfoil surface and the microphones, there are two tubes of diameter and length $d_1$, $L_1$, and $d_2$, $L_2$. The anechoic termination is represented by the third tube, with diameter $d_3$, and length $L_3$. The main characteristic of this method is that the microphone is located outside the wind tunnel model, allowing the RMP installation in smaller models, higher spatial resolution, and easier replacement of damaged microphones. Furthermore, the RMP technique allows measuring a higher amplitude of pressure fluctuations than flushed-mounted microphones and microphones under a pinhole. As the tubes attenuate the signal, the level at which the microphone is saturated is higher. The spatial averaging effects due to the sensor area are minimal for the RMP because of the pinhole that connects the microphone to the airfoil surface (Roger 2017; Englund and Richards 1984). Finally, RMP can measure frequencies much higher than tomo-PIV, which usually is limited to about 3 kHz (Ghaemi et al. 2012). The main challenge of the RMP technique is that it depends on a reliable calibration in the entire frequency range to convert back the signal measured by the microphone to the equivalent signal at the airfoil surface. Several research have proved the success of using remote microphone probes to measure wall-pressure fluctuations underneath a turbulent boundary layer (Stalnov et al. 2016; Gruber 2012), underneath an unstable laminar boundary layer (Yakhina et al. 2020), and in the main element of a high-lift airfoil to study the instabilities generated by the slat (Kamliya Jawahar et al. 2021). However, none of the previous research has used the RMP technique to study the boundary layer development and deeply analyze the boundary layer transition process over an airfoil.

Fig. 1

Schematic view of the remote microphone probe

In this research, the RMP technique is used for the first time to provide information about the boundary layer development and natural transition process. RMPs are distributed along the chord and span of a NACA 0008 airfoil, a thin airfoil, which makes it similar to a flat plate, but with a pressure gradient that influences the transition process. Microphones along the chord allow analyzing the evolution of the boundary layer. Microphones distributed along the span provide information about the evolution of the spanwise coherence along the chord and the spanwise correlation length. With this research, we expect to better understand the transition process and development of the boundary layer, particularly obtaining additional information about the flow structures convecting in the boundary layer. Finally, this research also studies the effect of the feedback loop created between the boundary layer instabilities and acoustic waves on the wall-pressure fluctuations along the chord. The analysis of the wall-pressure fluctuations measurements is complemented by hot-wire measurements and far-field noise measurements.

Section 2 presents the methodology followed in this research. Section 3 shows the wall-pressure measurements and the evolution of the boundary layer for the cases of forced and natural transition. The analyses consist of the wall-pressure spectra, spanwise coherence, and spanwise correlation length. Section 4 shows the boundary layer measurements. Section 5 presents the far-field noise measurements, and in Sect. 6 the main conclusions of this research are summarized.

2 Methodology

The experiments were conducted in the open-jet configuration of the closed-circuit Aeroacoustic wind tunnel of the University of Twente (de Santana et al. 2018). The wind tunnel has an open test section of 0.9 m width and 0.7 m height enclosed by an anechoic chamber of 6 m$\times$ 6 m $\times$ 4 m. The empty anechoic chamber has a cut-off frequency of 160 Hz.

The airfoil is mounted vertically between two turntables, see Fig. 2, which allows changing the geometric angle of attack ($\alpha _g$) with a precision of 0.5$^{\circ }$. The geometric angle of attack is defined as the angle between the mean flow direction and the airfoil chord line. The reference system is as follows: x is in the streamwise direction, y is perpendicular to the airfoil wall, and z is located in the spanwise direction. The origin of the coordinate system is at the leading edge at mid-span.

Fig. 2

Experimental setup

2.1 Airfoil model and instrumentation

The airfoil used in this research is a 300-mm chord NACA 0008, which has an 8% maximum thickness at 30% of the airfoil chord. This airfoil was chosen because of its small thickness, similar to a flat plate but with a pressure gradient. Furthermore, this airfoil is characteristic of underwater propellers. The airfoil is instrumented with 82 stainless steel tubes under pinholes at the airfoil surface, which are connected to the RMP. The pinholes are distributed along the chord and the span. In this research, we focus on the RMP located as shown in Fig. 3 left. Twelve microphones are located along the chord on both the pressure and the suction sides (black dots in Fig. 3left), which are used to analyze the development of the boundary layer. At $x/c~=$ 0.1, 0.3, and 0.97, three RMP are located in the spanwise direction at a distance of 2.5 mm and 10 mm from the first microphone (blue dots in Fig. 3left) on the suction side to analyze the spanwise coherence. At $x/c~=$ 0.97, 11 microphones in the spanwise direction generate 45 different distances between pairs of microphones, which are used to calculate the spanwise correlation length.

2.2 Test conditions and tripping device

The inflow velocity was varied from 10 m/s to 45 m/s, which corresponds to a Reynolds number range of 0.2 $\times$ 10⁶ to 0.9 $\times$ 10⁶. The effective angle of attack was varied from 0$^{\circ }$ to 7$^{\circ }$ and then moved to the stall condition. The effective angle of attack was determined as the angle for which the XFOIL computed $C_{\textrm{p}}$ best matched the measured $C_{\textrm{p}}$ distribution. As XFOIL does not model well the flow separation, the stall condition was identified in the experiments by analyzing the $C_{\textrm{p}}$ distribution without comparing it to XFOIL results. The static pressure was measured by pressure scanners NetScanner model 9216. Measurements were done during 30 s with a sampling frequency of 300 Hz.

For the cases of forced transition zig-zag strips of 60$^{\circ }$ top angle and 12 mm width are located at 5% of the airfoil chord, see Fig. 3 right. The trip height (k) was varied with the inflow velocity, the angle of attack, and between the suction and pressure sides to keep a ratio of $k/\delta _k$ between 0.4 and 1, where $\delta _k$ refers to the boundary layer height at the trip location. This ratio ensures the transition of the boundary layer without stimulating the generation of large turbulent structures that affect the development of the boundary layer (dos Santos et al. 2022). $\delta _k$ is obtained from numerical simulations with XFOIL (Drela 1989) discussed in Sect. 2.7. Table 1 and 2 show the tripping device height and the boundary layer thickness for the suction and pressure sides for the inflow velocities and angles of attack presented in this study. Note that at stall condition, the boundary layer thickness is not shown

Table 1 Trip height and boundary layer thickness at the trip location (in parenthesis) in mm for different inflow velocities and angles of attack

Table 2 Trip height and boundary layer thickness at the trip location (in parenthesis) in mm for different inflow velocities and angles of attack

Fig. 3

Left: Distribution of the remote microphone probes on the airfoil surface. Right: Schematic representation of the tripping device. All dimensions are in mm

2.3 Remote microphone probes

The RMP configuration used in this research, based on Fig. 1, consists of a pinhole of 0.3 mm diameter, the tube 1 is a stainless steel tube of inner diameter $d_1 = 1.6$ mm. The length of tube 1 ($L_1$) varies according to the position of the pinhole in the airfoil surface. The anechoic termination is a plastic tube of 1.6 mm inner diameter ($d_3$) and length ($L_3$) of 3 m. Next to tube 1, a Knowles FG 23329-P07 microphone is located in a PMMA tube assembly. This microphone has a flat frequency response from 100 Hz to 10 kHz, according to the manufacturer. The junctions of the stainless steel tube, the anechoic termination, and the microphones with the tube assembly were sealed to avoid leakage. Figure 4 shows a picture of the tube assembly and several components of the RMP technique used in this research.

Fig. 4

Picture of the PMMA tube assembly, stainless steel tubes, microphones, and anechoic termination

2.3.1 Remote microphone probes calibration

The most direct technique to measure wall-pressure fluctuations is a microphone flushed-mounted with the airfoil surface. Therefore, the calibration of the remote microphone probes consists of obtaining a transfer function between the remote microphone probe and an ideal microphone flush-mounted with the airfoil surface. An in-situ calibration of each RMP was conducted after installing the airfoil in the wind tunnel using an in-house designed, 3D printed calibrator, as shown in Fig. 5. The calibrator design is based on the one presented by Roger (2017). It contains a reference microphone GRAS 40HP and an FR8 loudspeaker. The noise source was white noise. Because of the calibrator geometry, specifically the distance between the reference microphone and the airfoil surface, a resonance can be seen in the transfer function between the reference microphone installed in the calibrator and the sensor to be calibrated. To avoid the influence of the calibrator, a two-step calibration procedure was adopted. First, the transfer function was determined between the reference microphone (ref) in the calibrator and a microphone flush-mounted (FM), i.e., a GRAS 40HP, see Eq. 1(a). Next, the transfer function was obtained between the reference microphone in the calibrator and each remote microphone probe (RMP), i.e., each Knowles FG 23329-P07, see Eq. 1(b). The transfer function between the microphone flush-mounted and the remote microphone probe is the ratio of the two transfer functions, see Eq. 2. In Eqs. 1 and 2$\phi _{\textrm{x,x}}$ and $\phi _\mathrm {{x,y}}$ are the auto- and cross-spectrum of the microphone signals represented by the subscripts. The equivalent spectrum of the wall pressure at the airfoil surface is calculated according to Eq. 3.

$$\begin{aligned} \textrm{TF}_{\textrm{ref,FM}}= & {} \frac{\phi _{\textrm{ref,ref}}}{\phi _{\textrm{ref,FM}}} \;\; (a); \;\;\;\;\; \textrm{TF}_{\textrm{ref, RMP}}~=~\frac{\phi _{\textrm{ref,ref}}}{\phi _{\textrm{ref,RMP}}}\;\; (b); \end{aligned}$$

(1)

$$\begin{aligned} |\textrm{TF}_{\textrm{FM, rmp}} |^2= & {} \frac{|\textrm{TF}_{\textrm{ref, RMP}} |^2}{|\textrm{TF}_{\textrm{ref, FM}} |^2} \end{aligned}$$

(2)

$$\begin{aligned} \Phi _{\textrm{pp}}= & {} \frac{\phi _{\textrm{rmp, rmp}}}{|\mathrm {TF_{\textrm{FM, rmp}}} |^2} \end{aligned}$$

(3)

Fig. 5

Schematic representation of the calibrator device

The coherence between pairs of microphones for each step of the calibration procedure is analyzed to ensure that both microphones measure a signal that is correlated. Figure 6 left shows the coherence for each calibration step for a RMP connected to the airfoil surface. In the first step, i.e., between the reference microphone and the flushed-mounted (FM) microphone, the coherence is exactly one in the entire frequency range. In the second step, i.e., between the reference microphone and the remote microphone probe (RMP), the coherence is mainly unity up to 5 kHz. Above this frequency, several peaks with reduced coherence start appearing, as can be seen in Fig. 6 left. These peaks are attributed to the contamination of the signal with electronic noise. Because of the strong attenuation in the tubes from the airfoil surface to the distant microphone, electronic noise starts to play a role at high frequencies, contaminating the signal and reducing the coherence between the microphones. Therefore, results are shown for frequencies up to 5 kHz, where the calibration is strongly reliable. Figure 6 right shows the transfer function for each of the steps, calculated according to Eq. 1 (a) and (b); and also the final transfer function as given by Eq 2. In the first step of the calibration, the transfer function is mostly zero in the entire frequency range. The increase at high frequencies is caused by the geometry of the calibrator, as mentioned before. However, this effect is significantly reduced using both steps, and this increase is not observed in the final transfer function. Finally, Fig. 7 shows the spectrum reconstructed using Eq. 3 at each calibration step compared with the spectrum measured by the reference microphone, using white noise as noise source. In both cases, the spectrum is perfectly reconstructed using the transfer function, which validates the calibration methodology. The results shown in Figs. 6 and 7 are for a single remote microphone probe located close to the leading edge of the airfoil. However, similar results are found for all the remote microphone probes.

Fig. 6

Left: coherence between pairs of microphones. Right: transfer function for each calibration step. Remote microphone probe located at $x/c = 0.0035$ and $y/c = 0$

Fig. 7

Autospectrum measured by the reference microphone and the reconstructed spectrum for the calibrated microphone at each calibration step when white noise is generated in the speaker of the calibrator. The remote microphone probe located at $x/c = 0.0035$ and $y/c = 0$

2.3.2 Data acquisition and spectrum determination

The surface-microphone measurements were acquired using four National Instruments PXIe-4499 Sound and Vibration modules installed on a NI PXIe-1073 chassis. The microphones were acquired during 30 s with a sampling frequency of 65536 (2¹⁶) Hz. The current setup allows the acquisition of 64 microphones simultaneously. Therefore, the measurements of the microphones shown in this research were acquired simultaneously.

The wall-pressure spectrum was calculated by adopting the Welch method, using a window size of 2¹³ samples (3.75 s) and a Hanning windowing method with 50% overlap, which resulted in a bin size of 8 Hz. The spectrum is shown in dB, calculated according to Glegg and Devenport (2017), where the reference pressure and velocity used to normalize the power spectral density was $20~\mathrm {\mu }$Pa and 1 m/s and $\Delta f~=~$1 Hz. The wall-pressure spectrum is also normalized using the edge velocity ($U_e$), the friction velocity ($u_\tau$), and the boundary layer thickness ($\delta$) at the position of each surface microphone. These parameters are obtained from XFOIL simulations as explained in Sect. 2.7.

2.4 Boundary layer measurements

The boundary layer was measured using hot-wire anemometry. A single-wire probe (Dantec Dynamics model 55P15) of 5 µm 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. The hot-wire data was acquired with the Dantec StreamLine Pro CTA system and the Dantec StreamWare software in combination with the National Instruments 9215 A/D converter. 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 a reference. The calibration consisted of 32 velocity points distributed logarithmically ranging from 2.5 to 50 m/s.

On average, the velocity was measured at 35 locations across the boundary layer and 5 points in the free stream. The y-locations of these points were spaced logarithmically. The contact of a feeler gauge to the hot-wire prongs determined the probe’s distance to the wall. The gauge accuracy is 0.05 mm. The measurement closest to the airfoil wall is at $y~=~$0.5 mm.

From the measurement of the boundary layer velocity profile, the experimental boundary layer thickness is determined as the distance from the wall where the velocity corresponds to 99% of the edge velocity $U_\textrm{e}$. The experimental boundary layer displacement $\delta ^*$ and momentum thicknesses $\theta$ are determined by performing a trapezoidal numerical integration of the measured boundary layer velocity profile. The measured velocity profile is fitted to the law of the wall using the equation proposed by Coles (1956). The friction velocity $u_\tau$ is determined from this fitting. For the cases of natural transition, the Pohlhausen velocity profile (Thwaites 1949) is used to fit the experimental data. In this case, the fitting calculates the gradient of the external velocity with the downstream position (${\textrm{d}}U/{\textrm{d}}x$).

2.5 Far-field noise measurements

Acoustic measurements were taken using a microphone array equipped with 62 GRAS 40PH microphones distributed following a Vogel spiral distribution located 1.5 m from the airfoil surface. Conventional beamforming was used to localize the far-field noise source. The in-house beamforming algorithm was benchmarked against an array benchmark database (Bahr et al. 2017; Sarradj 2015). The far-field noise spectrum is calculated using the source power integration methodology (Brooks and Humphreys 1999). The region of integration for the noise evaluation was $x/c~\in ~[0.5,~1.5]$ and $y/d \in [-0.3,~0.3]$; where d is the span. The power spectral density (PSD) of the SPI is given at the array center. The uncertainty of the sound power estimated by conventional beamforming in the frequency domain method is around 1 dB (Sarradj et al. 2017).

2.6 Spanwise coherence and spanwise correlation length

The coherence between pairs of microphones is analyzed to obtain information about the two- or three-dimensionality of the boundary layer flow structures. It is compared to the model proposed by Corcos for turbulent boundary layers, shown in Eq. 4, where $b_c$ is the Corcos constant, assumed 1.4 in this work, $U_c$ is the convection velocity, assumed as 0.6U (Stalnov et al. 2016), and $\eta _z$ is the distance between the two points.

$$\begin{aligned} \gamma ^2(f, \eta _z)= \exp {\left( \frac{-4\pi f}{b_c U_c} \eta _z\right) } \end{aligned}$$

(4)

The spanwise correlation length is related to the size of the turbulent structures in the spanwise direction. It is calculated as shown in Eq. 5. In this research, the spanwise correlation length is calculated by integrating numerically the spanwise coherence between every combination of pairs of microphones along the discrete 45 distances generated by the RMP located along the span at $x/c~=$ 0.97.

$$\begin{aligned} l_y=\int _{0}^{\infty }\sqrt{\gamma ^2(f,\eta _z)}{\text {d}}\eta _z \end{aligned}$$

(5)

2.7 XFOIL simulations

XFOIL simulations (Drela 1989) were performed to obtain: I. the surface pressure coefficient ($C_{\textrm{p}}$) distribution over the airfoil needed to determine the effective angle of attack, II. the boundary layer thickness at the trip location to determine the trip height, and III. the friction velocity, the external velocity, and the boundary layer thickness along the chord to normalize the wall-pressure spectrum. The input parameters for XFOIL were the airfoil profile, the Reynolds number based on the airfoil chord and the inflow velocity, the boundary layer transition parameter ($N_\textrm{crit} = 8.68$), the effective angle of attack ($\alpha _e$), and the location of the transition, i.e., 1 for the cases of no forced transition and 0.05 for the cases of forced transition. The no forced transition was assumed to determine the boundary layer thickness at the trip location ($\delta _k$).

XFOIL yields the boundary layer displacement thickness ($\delta ^*$), the momentum thickness ($\theta$), and the skin friction coefficient ($C_f$) along the chord. The boundary layer thickness along the chord was calculated as, shown in Eq. 6 (Drela and Giles 1987), where H is the shape parameter ($H=\delta ^*/\theta$). The friction velocity $u_\tau$ along the chord was determined from the computed skin friction coefficient ($u_\tau ~=~\sqrt{0.5 U^2 C_f}$).

$$\begin{aligned} \delta ~=~\theta \left( 3.15 + \frac{1.72}{H-1}\right) +\delta ^* \end{aligned}$$

(6)

3 Unsteady wall-pressure measurements

3.1 Boundary layer development with forced transition

Figure 8 shows the measured wall-pressure spectrum at several positions along the chord for different inflow velocities. For all inflow velocities at $x/c~=$ 0.05, the boundary layer is laminar, and the wall-pressure fluctuations are extremely low. Therefore, the signal is contaminated with electronic noise for $f~>$ 1 kHz. Hence, the spectrum above this frequency is neglected. For $U~=~$10 m/s, the wall-pressure spectrum level at $x/c~=$ 0.15 is higher than the spectra measured downstream. This is because, for this velocity at that point, the boundary layer is still in the transition process. The results show that the transition process occurring can be characterized as bypass since no T–S waves are observed, which would be represented by peaks in the wall-pressure spectrum downstream of the tripping device. Turbulent structures are directly formed, characterized by the broadband nature of the wall-pressure spectrum, which develop along the airfoil chord (Yakhina 2017; Sanjose et al. 2019; Nash et al. 1999). The broadband component of the wall-pressure spectrum is caused by the presence of turbulent structures of different sizes ranging from the largest turbulent structures to the Kolmogorov length scale according to the Cascade theory.

Fig. 8

Wall-pressure spectra along the chord for forced transition for different inflow velocities. ${\alpha _e}$ = 0$^{\circ }$

Once the boundary layer develops as a turbulent boundary layer, i.e., at positions $x/c~\ge ~$0.3, the wall-pressure spectrum continuously reduces in the high-frequency range and increases in the low-frequency range. Note that the reduction in the high-frequency range is not noticeable for $U~\ge ~30$ m/s since it occurs for $f~>~$5 kHz. As the boundary layer develops, the boundary layer thickness grows, and therefore, the size of turbulent structures in the direction perpendicular to the wall, which causes an increase in the low-frequency range and shifts the location of the maximum level of the spectrum toward the lower frequencies. On the other hand, the spectrum level in the high-frequency range scales with $\rho u_\tau ^2$ (Glegg and Devenport 2017). Going downstream the airfoil chord, the friction velocity is reduced, also reducing the amplitude of the spectrum in the high-frequency range. Additionally, as the inflow velocity increases, the boundary layer thickness and the turbulence structures are reduced; therefore, the maximum of the wall-pressure spectrum level is shifted toward a higher frequency with the inflow velocity. Analyzing the wall-pressure spectrum at the fixed location $x/c~=~$0.97, the maximum spectrum level occurs at f = 250 Hz, 750 Hz, 1500 Hz, and 2000 Hz, for $U~=~$10 m/s, 20 m/s, 30 m/s, and 45 m/s, respectively, see Fig. 9a, which corresponds to a constant Strouhal number of 0.3, calculated based on the inflow velocity and the boundary layer thickness, as shown in Fig. 9b.

Figure 9b shows that at $x/c=0.97$, when scaled with the outer scales, the wall-pressure spectra for different speeds collapse with less than 1 dB at high frequencies and within 2 dB at low frequencies, within a maximum at $f\delta /U_e$ = 0.3 for 20 m/s $\le ~U~\le$ 45 m/s. For $U~=~$10 m/s and 15 m/s, the boundary layer may still need to develop to scale as a fully turbulent boundary layer. This is also confirmed when analyzing the velocity profile for $U~=~$10 m/s in Fig. 24. At $x/c~=$ 0.88, see Fig. 9c, the wall-pressure spectrum scales with the outer scales, showing the same location of the maximum spectrum level as at $x/c~=$ 0.97 (note that the boundary layer is calculated at each location), except for $U~=~$10 m/s and 15 m/s, which evidences that at this chord position, the boundary layer is not fully developed for these velocities. At $x/c~=$ 0.3, see Fig. 9d, i.e., 75 mm downstream of the trip location, the wall-pressure spectrum scales only for U $\ge$ 30 m/s, whereas for $U~=~$20 m/s and 25 m/s the scaling is not as good as for the downstream positions, evidencing the larger distance needed to complete the transition. This also demonstrates that the bad scaling for $U~=~$10 m/s and 15 m/s is due to the not full development of the boundary layer.

Fig. 9

Wall-pressure spectrum for different inflow velocities and scaled with the outer scales along the chord for forced transition. $\alpha _e$ = 0$^{\circ }$

Figure 10 shows the coherence between a pair of microphones located at a spanwise distance $\eta _z~=~$2.5 mm and 10 mm, for different inflow velocities at $x/c~=$ 0.3 and 0.97. For each velocity, only the range from 0 to 1 is shown. Therefore, the y-axis is shared for adjacent velocity and has two values: 1 for the lower velocity and 0 for the higher velocity. At $x/c~=$ 0.3, the coherence generated by the hydrodynamic pressure fluctuations is zero for all inflow velocities. From Fig. 8, we concluded that at $x/c~=~$0.3 the boundary layer contains only turbulent structures. However, at this chord position, the turbulent structures are so small that do not cause a significant coherence level as such a large distance compared to the size of the turbulent structures, i.e., 2.5 mm. As the acoustic noise is uncorrelated, the low coherence measured by the RMP could be caused by contamination of the acoustic noise produced by the airfoil trailing-edge (Yakhina et al. 2020). However, the RMPs located at $x/c~=~$0.97 that are closer to the noise source measure high coherence levels (max of $\approx$ 0.3) for $\eta _z~=~$2.5 mm, and present a behavior corresponding to a turbulent boundary layer. This is evidence of the development of the boundary layer and the growth of turbulent structures. Furthermore, the coherence is 0 for $\eta _z~=~$10 mm since, at the trailing edge, the turbulent structures are still small compared to such a distance. With this, we can conclude that the coherence measured by the RMP at $x/c~=~$0.3 is caused by the hydrodynamic pressure waves and not by the acoustic waves.

At $x/c~=$ 0.97, the coherence of the microphones at a distance of $\eta _z~=~2.5$ mm is significantly higher than at $x/c~=$ 0.3, presenting a hump characteristic of a turbulent boundary layer, which agree well with Corcos’ model, see Fig. 10b. The peaks observed for $f~>~$0.3 kHz for $U~=~$10 m/s are caused by the contamination of the microphone’s signal with electronic noise. The frequency of the maximum coherence is shifted toward higher frequencies with the increase of the inflow velocity, evidencing a reduction of the size of the turbulent structures in the normal-to-the-wall direction. The maximum coherence occurs at the same frequency as the maximum of the wall-pressure spectrum, which means that the coherence also scales with frequency as $f\delta /U_e$.

Fig. 10

Spanwise coherence at different chord positions for different inflow velocities with forced transition. $\alpha _e$ = 0$^{\circ }$. $b_c~=~1.4$. $\eta _z$ is the spanwise distance between pair of RMPs

Figure 11 left shows the decay of the maximum coherence with the spanwise distance for the different inflow velocities at $x/c~=~$0.97. The coherence is calculated between pairs of microphones distanced by the 45 different combinations created by the 11 remote microphone probes at this chordwise position. The maximum coherence decays exponentially with the distance for all inflow velocities. Furthermore, it slightly increases with the inflow velocity. This is reflected in an increase of the spanwise correlation length with the inflow velocity. The results indicate an increase in the size of the turbulent structures in the spanwise direction with the increase of the velocity, whereas there is a reduction in the size in the normal-to-the-wall direction. Figure 11 right shows the spanwise correlation length scaled with the outer boundary layer parameters. Except for $U~=~$45 m/s in the low-frequency range, the spanwise correlation length collapse in the entire frequency range. The figure also shows the spanwise correlation length calculated with Corcos’ model. As for the spanwise coherence, Corcos’ model works well for frequencies higher than the maximum along the frequency range. Furthermore, Corcos’s model underestimates the spanwise correlation length for higher frequencies.

Fig. 11

Left: maximum coherence decay with the spanwise distance at different inflow velocities for forced transition. $x/c~=$ 0.97. ${\alpha _e}$ = 0$^{\circ }$. Right: spanwise correlation length for different inflow velocities calculated as the integration of the coherence of several spanwise microphones for forced transition. $x/c~=$ 0.97. ${\alpha _e}$ = 0$^{\circ }$

3.1.1 Effect of the angle of attack

Figures 12 and 13 show the wall-pressure spectra at various locations along the chord at the suction side for $\alpha _e$ = 3$^{\circ }$ and 7$^{\circ }$ for $U~=~$10 m/s and 45 m/s. The development of the wall-pressure spectrum, going downstream along the airfoil chord, is the same as for the case of $\alpha$ = 0$^{\circ }$ , i.e., an increase of the spectrum level in the low-frequency range and a reduction in the high-frequency range. For $U~=~$10 m/s and $\alpha _e$ = 3$^{\circ }$ (Fig. 12a), the high level of the wall-pressure spectrum at $x/c~=$ 0.15 is a sign of the final stage of the transition process, as was the case also for $\alpha _e$ = 0$^{\circ }$.

Fig. 12

Wall-pressure spectra along the chord at suction side for forced transition for different inflow velocities. ${\alpha _e}$ = 3$^{\circ }$ 

At $\alpha _e$ = 7$^{\circ }$  at the suction side, the transition started upstream of the trip location. This can be concluded from the fact that at $x/c~=$ 0.05 the wall-pressure spectrum is significantly higher than for $\alpha$ = 0$^{\circ }$ at this same location and also higher than the wall-pressure spectra downstream of this position at $\alpha$ = 7$^{\circ }$. This shows that the transition process is in the final stage. According to these results, at $\alpha$ = 7$^{\circ }$ the trip has no significant influence on the boundary layer development. This is clear when comparing the wall-pressure spectrum along the chord for the cases of forced (Fig. 13) and no forced transition (Fig. 22), which are identical at a given chord location. When the angle of attack increases, the pressure gradient increases, promoting an earlier transition of the boundary layer.

Fig. 13

Wall-pressure spectra along the chord at suction side for forced transition for different inflow velocities. ${\alpha _e}$ = 7$^{\circ }$

Figure 14 shows the wall-pressure spectrum at several positions along the chord when the airfoil is at the stall for $U~=~$10 m/s and 45 m/s. For these cases, two additional microphones were analyzed close to the leading edge at $x/c~=$ 0.007 and 0.021. For $U~=~$10 m/s at $x/c~\ge$ 0.05, the wall-pressure spectrum shows a constant decay with the frequency. At $x/c~=~$0.97, the maximum of the spectrum occurs at $f~=~$26 Hz followed by a constant decay of the spectrum level as a function of the frequency following the expression $f^{-0.19}$. This exponent was found by fitting the function $f^a$ to the experimental data, with a coefficient of adjustment of 0.999. Additional decays obtained from the best fit of the experimental data in the low- and high-frequency ranges are shown, with a coefficient of adjustment higher than 0.999 for both cases. The high-frequency range presents a stronger decay of the spectrum level as a function of the frequency, whereas the low-frequency range presents a lower decay. Although the wall-pressure spectra along the chord show relatively similar levels, there is a slight decrease in the level downstream along the airfoil chord. For $U~=~$ 45 m/s along the first half of the chord, there is no clear tendency in variations of the wall-pressure spectra, most probably because of locally intermittent separation of the boundary layer. At $x/c~\ge ~$0.5 the wall-pressure spectrum decays with the frequency as $f^{-0.076}$, and the level is reduced downstream of the airfoil chord, similar to the case of $U~=~$10 m/s. At $x/c~=~$0.97, the maximum spectrum level occurs at $f~=~$126 Hz, which is a higher frequency than the case of $U~=~$10 m/s, due to the reduction of the largest structures because of the increase of the inflow velocity. The reduction of the wall-pressure spectrum downstream along the airfoil chord when the boundary layer is separated is because the detachment of the boundary layer also grows downstream, reducing the effect of the boundary layer in the wall-pressure fluctuations.

Fig. 14

Wall-pressure spectra along the chord for forced transition for different inflow velocities. $\alpha _e$ = stall

Figure 15 shows the influence of the angle of attack on the wall-pressure spectrum close to the trailing edge at $x/c~=$ 0.97, for 10 m/s and 45 m/s inflow velocity. Before the stall, when the angle of attack increases, the low-frequency range increases and the high-frequency range reduces. This behavior is similar to when the spectrum along the chord is discussed. As the angle of attack increases, the boundary layer thickness increases and, therefore, the size of the large turbulent structures inside the boundary layer. Consequently, the low-frequency range of the wall-pressure spectrum is increased, which is associated with larger structures. Contrary, when the angle of attack increases, the friction velocity is reduced (Drela 1989), which causes a reduction in the low-frequency range. Note that the frequency at which the shift of the energy content of the spectrum occurs is the same for the different angles of attack at a given velocity but increases with the inflow velocity because of the reduction in the boundary layer thickness.

When the airfoil is at stall, the boundary layer is detached, and large turbulent structures convect over the airfoil, increasing the wall-pressure fluctuations, but mainly in the low-frequency range. This is illustrated by the case of $U~=~$45 m/s, where a 0.5$^{\circ }$ increase in the geometric angle of attack, from 13$^{\circ }$ to 13.5$^{\circ }$ greatly influences the wall-pressure spectrum because of the flow separation.

Fig. 15

Wall-pressure spectra at $x/c~=$ 0.97 with forced transition for different angles of attack

Figures 16 and 17 show the effect of the angle of attack on the spanwise coherence at $x/c~=$ 0.3 and 0.97 for $U~=~$10 m/s and 45 m/s, respectively. At $x/c~=$ 0.97 the spanwise coherence calculated with Corcos’ theory is also shown. As the $\alpha _e$ increases, the spanwise coherence increases, and its maximum shifts toward lower frequencies due to the growth of the size of the turbulent structures in the normal-to-the-wall direction. When the airfoil has separated flow, higher coherence levels are also observed at $x/c~=$ 0.3, showing the separation starts upstream of this point. Furthermore, as $\alpha _e$ increases, the Corcos model underpredicts the coherence because the geometry of the airfoil starts to play a role and the higher pressure gradient stimulates the growth of the boundary layer thickness. The increase of the coherence with the angle of attack is reflected in an increase of the spanwise correlation length shown in Fig. 18.

The effect of the angle of attack on the spanwise correlation length is less for $U~=~$10 m/s than for $U~=~$45 m/s, as shown in Fig. 18. For $U~=~$10 m/s, the spanwise correlation length is not a monotonic increase with the angle of attack. It is reduced when $\alpha _e$ is increased from 7$^{\circ }$ to stall ($\alpha _g$ = 12.5$^{\circ }$ ). Figure 16 shows that at $\alpha _g$ = 12.5$^{\circ }$  the coherence is reduced going along the chord from $x/c~=$ 0.3 to $x/c~=$ 0.97. This is probably because at $x/c~=$ 0.3 the boundary layer was not completely separated, whereas at $x/c~=$ 0.97 there might be a complete separation, which reduces the spanwise coherence of the wall-pressure fluctuations.

Fig. 16

Spanwise coherence for different angles of attack at different chord positions for forced transition. $U~=~$10 m/s. Re = 0.2 $\times$ 10⁶. $b_c~=~1.4$. $\eta _z$ is the spanwise distance between pair of RMPs

Fig. 17

Spanwise coherence at different chord positions for different angles of attack for forced transition. U = 45 m/s. Re = 0.9 $\times$ 10⁶. $b_c~=~1.4$. $\eta _z$ is the spanwise distance between pair of RMPs

Fig. 18

Spanwise correlation length at $x/c~=$ 0.97 calculated as the integration of the coherence of several spanwise microphones for forced transition

3.2 Boundary layer development with no forced transition

This section discusses the transition process and development of the boundary layer with the no-trip condition. Figure 19 shows the measured wall-pressure spectrum with the RMPs at different positions along the chord for several inflow velocities with natural transition. The inflow velocity was varied from 10 m/s to 45 m/s in steps of 5 m/s to carefully analyze the different stages of the transition of the boundary layer. In some cases, the wall-pressure spectrum is cut off for higher frequencies since there is contamination of electronic noise due to the low levels of pressure fluctuations associated with non-turbulent boundary layers.

It has been widely recognized that a transitional and laminar boundary layer cause a wall-pressure spectrum that is composed of peaks in resonance along the entire frequency range (Yakhina 2017; Botero-Bolívar 2022; Sanjose et al. 2019; Nash et al. 1999). However, the origin of the peaks is still a point of discussion. When a laminar or transitional boundary layer containing two-dimensional structures reaches the airfoil trailing edge, the acoustic waves generated at the trailing edge create a feedback loop with a point upstream at the airfoil surface, which amplifies the instabilities of the boundary layer. In this study, we demonstrate for the first time in which condition that feedback loop is originated: For the cases where there are peaks reaching the trailing edge, i.e., U $\le$ 30 m/s, the peaks also appear along the entire airfoil chord, from $x/c~=$ 0.007. The feedback loop changes the wall-pressure fluctuations along the entire airfoil chord since from the forced transition cases, see Fig. 8, we concluded that at $x/c~\le$ 0.05, the boundary layer was undisturbed, and the wall-pressure fluctuations were so low that the microphones reached the electronic noise floor and peaks are observed, whereas for the case of natural transition the spectrum at $x/c~\le$ 0.05 is also composed by peaks that are equivalent to the ones close to the trailing edge. For U$\ge$ 35 m/s, where there are no peaks at the trailing edge, the wall-pressure spectrum at $x/c~\le$ 0.05 with no forced transition matches with the spectrum for the forced transition case, even when peaks caused by the presence of T–S waves occur at some point along the airfoil chord, see $x/c~=$ 0.5. Therefore, the feedback loop is only created when the T–S waves actually reach the airfoil trailing edge and affects the wall-pressure spectrum at the entire airfoil chord, changing the development of the boundary layer and the wall-pressure spectrum. This phenomenon will be widely discussed when analyzing the wall-pressure spectrum, spanwise coherence, boundary layer measurement, and far-field noise spectrum.

For $U~=~$10 m/s, low amplitude peaks start at $f~=~$272 Hz with a 24 Hz delta frequency between each peak. These peaks also appear in the far-field noise spectrum; see Fig. 26. However, they are not found in the streamwise velocity spectrum at any position across the boundary layer; see Fig. 24c. Furthermore, these peaks evolve to a hump at $x/c~=~$0.97. Therefore, those peaks might be related to the feedback loop created when T–S waves reach the trailing edge between the acoustic waves and a point upstream on the airfoil surface. A peak of higher intensity at 434 Hz, followed by its harmonic, is observed along the entire airfoil chord and appears at the same frequency as the ones observed in the wall-pressure spectrum for $U~=~$15 m/s. These peaks are also observed in the velocity spectrum at every position across the boundary layer at $x/c~=~$0.97, and in the far-field noise. Therefore, those tones at the trailing edge are attributed to the T-–S waves originated during the transition process. The exact location of where they originated cannot be determined since the wall-pressure fluctuations are affected by the feedback loop with the acoustic waves. This is demonstrated by the fact that at $x/c~=~$0.3, the peaks do not appear in the velocity spectrum across the boundary layer, and the velocity fluctuations are minimal across the boundary layer, showing the behavior of an undisturbed boundary layer. This analysis will be further discussed in Sect. 4. The level of the spectrum in the entire frequency range is increasing downstream along the airfoil chord. For $U~=~$15 m/s, the main peak also appears at 434 Hz, and the level of the harmonics is slightly increased downstream along the airfoil chord up to $x/c~=$ 0.88, presenting the same behavior as for the case of $U~=~$10 m/s. At $x/c~=$ 0.97, a significant increase in the wall-pressure spectrum level at high frequencies is observed, which indicates that small, turbulent structures started to develop. However, the spectrum is still composed of harmonic peaks overimposed in a broadband component. This wall-pressure spectrum is related to the third transition stage, where there are 3D waves. The presence of 3D waves will be confirmed with the analysis of the spanwise coherence for different spanwise distances, where the peaks have high coherence level constant in the spanwise and low coherence levels for other frequencies. Contrary to the position $x/c~=~$0.3, where a constant coherence level of 1 is found for the several distances, see Fig. 20a, c. At 20 m/s and 25 m/s inflow velocity, the wall-pressure spectrum significantly changes at each downstream position. For $x/c~\le$ 0.73, the wall-pressure spectrum contains the main peaks and several sidelobes, which are superimposed on a hump, contrary to lower velocities, for which the spectrum uniquely contains peaks.

Analyzing the wall-pressure spectrum at $x/c~=$ 0.73 at all inflow velocities, the development of the boundary layer can be seen. Initially, unique T–S waves (for $U~=~$10 m/s and 15 m/s), then the discrete peaks are broken down into peaks with side lobes superimposed on a hump in the same frequency range of the peaks ($U~=~$20 m/s), which indicates the evolution of two-dimensional structures into three-dimensional structures. Subsequently, the hump is amplified, and the peaks are continuously reduced ($U~=~$25 m/s and 30 m/s). At inflow velocities higher than 35 m/s, the spectrum level at the high-frequency range increases significantly compared to the cases of lower inflow velocities, and harmonic humps occur instead of peaks. The harmonic humps present for $U~=~$25 m/s disappear for $U~=~$30 m/s, and finally, the main hump is reduced, leading to the unique broadband spectrum, as seen for the cases of $U~=~$40 m/s and 45 m/s at $x/c~=$ 0.97. The same evolution of the wall-pressure spectrum is observed when analyzing the spectra at $x/c~=$ 0.88. Chapman et al. (1998) explain the harmonic peaks that appear in the wall shear-stress spectrum as evidence of the primary instability mechanism and the hump as the second instability mechanism. Their measurements showed that after the transition, there is a single big structure, causing the hump that we observe in the wall-pressure spectrum. This will be further discussed when analyzing the spanwise coherence. Chapman et al. (1998) demonstrate that skin friction is a footprint of what happens in the boundary layer. Therefore, their conclusions can be used to interpret the results obtained in this research with unsteady wall-pressure measurements. However, measuring the unsteady wall pressure uses more common technologies, and it is more useful for several applications, such as aeroacoustics.

Fig. 19

Wall-pressure spectra along the chord with no forced transition for different inflow velocities. $\alpha _e$ = 0$^{\circ }$ 

For $U~\ge$ 30 m/s the wall-pressure spectrum close to the trailing edge, i.e., $x/c~=$ 0.97, contains a unique broadband component. Therefore, the feedback loop between the acoustics and the unstable hydrodynamic waves is not generated, so the development of the wall-pressure spectrum without the amplification of the boundary layer instabilities caused by the feedback loop can be studied. For x/c $\le$ 0.5, the wall-pressure fluctuations are low because the boundary layer is undisturbed at those positions. This is verified because the wall-pressure spectra at $x/c~=$ 0.05 match with the spectra for the cases of forced transition at the same inflow velocity. In the case primary instabilities are occurring inside the boundary layer, as the feedback loop does not amplify them, they would not be still observed in the wall-pressure spectrum. Another reason could be that the instabilities die in space or time, and there is not enough spatial resolution of the RMP to evaluate the evolution of those instabilities. For $U~=~$35 m/s, 40 m/s, and 45 m/s discrete peaks with a 180 Hz harmonic frequency are present at $x/c~=$ 0.5. The frequency of the first peak increases with the velocity. The first instability mechanism and the generation of T–S waves cause these peaks. At $x/c~=$ 0.73, a prominent hump is observed, which is caused by the secondary instability mechanism. Chapman et al. (1998) concluded that the first instability mechanism leads the transition process and allows the formation of large-scale turbulent structures, as evidenced by the large increase of the wall-pressure spectrum level in the low-frequency range further downstream, whereas the secondary instability mechanism would lead to the formation of the small-scale structures, which increases the high-frequency range of the wall-pressure spectrum. For $U~=~$45 m/s, the boundary layer measurements with and without tripping device are shown in Sect. 4, Fig. 25. The velocity profile scales for both transition conditions and no peaks are observed across the boundary layer. Furthermore, the far-field noise spectrum do not show tones and it matches with the case of forced transition, see Fig. 26. This demonstrates that when at the trailing edge there are no T–S waves, the feedback loop is not created and the wall-pressure fluctuations along the airfoil chord are not changed, even when T–S waves are present at a certain point along the airfoil chord, as for $U~\ge ~$35 m/s.

Figure 20 shows the spanwise coherence between pairs of microphones separated $\eta _z$ = 2.5 mm and 10 mm, at two chord positions for inflow velocities from $U~=~$10 m/s to 45 m/s. At $x/c~=$ 0.3, the coherence evidences the presence of two-dimensional structures, since it is close to one and is not reduced by the spanwise separation of the microphones. This is typical for the T–S waves related to the first stage of a boundary layer transition, which are two-dimensional structures. The coherence is slightly reduced for inflow velocities higher than 30 m/s since the boundary layer is now more developed and the T-–S waves start to convert into three-dimensional waves. Although the coherence shows that at $x/c~=$ 0.3, there are still two-dimensional structures even for velocities U $\ge$ 35 m/s, the wall-pressure spectra no longer show the peaks characteristic of this type of structure for those case (see Fig. 19f, g, h). At higher velocity, due to the absence of two-dimensional structures close to the trailing edge, the feedback loop that amplifies the instabilities of the boundary layer, is no longer formed.

Fig. 20

Spanwise coherence for different inflow velocities at different chord positions with no forced transition. $b_c~=~1.4$. ${\alpha _e}$ = 0$^{\circ }$. $\eta _z$ is the spanwise distance between pair of RMPs

At $x/c~=$ 0.97, the spanwise coherence provides valuable information about the type and evolution of the flow structures convecting in the boundary layer. At 10 m/s, the coherence is only slightly lower than one caused by the increase of the distance between the RMP, which indicates the presence of the two-dimensional structures, also confirmed by the discrete peaks as a unique component of the wall-pressure spectrum at this position. For $U~=~$15 m/s, the coherence is significantly reduced with the spanwise distance, indicating that the two-dimensional structures are no longer present. However, some peaks are observed in the spanwise coherence at both distances at the same frequency as the peaks of the wall-pressure spectrum, related to the presence of three-dimensional waves occurring in the third stage of the boundary layer transition. For $U~=~$20 m/s, there are peaks in the low-frequency range, corresponding to those of the wall-pressure spectrum. The peaks are more evident for $\eta _z$ = 10 mm because of the reduction of the coherence with the distance for the turbulent structures that are already developing. For frequencies higher than 1 kHz, the coherence is almost zero because the turbulent structures responsible for the broadband component at high-frequency in the wall-pressure spectrum are still too small to cause high coherence levels. For inflow velocities higher than 25 m/s, there is a pronounced hump in a specific frequency range in accordance with the hump observed in the wall-pressure spectra, which is caused by the prominent structure generated after the transition by the secondary instability theory. This hump is reduced as the inflow velocity is increased since the turbulent structures are being developed. At 45 m/s, the coherence is significantly low and shows no hump.

3.2.1 Effect of the angle of attack

Figure 21 shows the wall-pressure spectra at various locations along the chord on both suction and pressure sides for $U~=~$10 m/s. On the suction side, the secondary instability mechanism is evidenced by the hump that starts to occur at $x/c~=$ 0.5. However, the first instability mechanism is not easily identified. Small peaks are exhibited along the chord superimposed on the hump. On the pressure side, small peaks are observed along the entire airfoil chord, similar to what was observed for $\alpha _e$ = 0$^{\circ }$ , and the secondary instability mechanism is not identified.

Fig. 21

Wall-pressure spectra along the chord with no forced transition for different inflow velocities. ${\alpha _e}$ = 3$^{\circ }$. $U~=~$10 m/s. Re = 0.2 $\times$ 10⁶

At $\alpha _e$ = 7$^{\circ }$  and $U~=~$10 m/s, see Fig. 22a, at $x/c~=$ 0.05, the wall-pressure spectrum shows a hump at 2.5 kHz, which is most probably related to the secondary instability mechanism. For x/c < 0.05, the wall-pressure spectrum is very low, showing that no turbulent structures causing wall-pressure fluctuations are present in the boundary layer. Further downstream, the wall-pressure spectra are composed of a single broadband component, showing that there are only turbulent structures convecting inside the boundary layer. For the case of 45 m/s inflow velocity, see Fig. 22b, at $x/c~=$ 0.021, the wall-pressure spectrum is significantly higher than further downstream, with small peaks harmonic of 320 Hz. This is similar to the wall-pressure spectrum at $x/c~=$ 0.97 for $U~=~$15 m/s at $\alpha _e$ = 0$^{\circ }$ , see Fig. 19b. At x/c $\le$ 0.021 the wall-pressure spectrum is very low, suggesting no wall-pressure fluctuations, i.e., a laminar boundary layer. At x/c $\ge$ 0.021 the wall-pressure spectra are composed of a single broadband component, suggesting a developing turbulent boundary layer. At both velocities, as the pressure gradient is high, the transition region is shorter than the distance between pairs of microphones located chordwise. Hence, the different stages of the transition are not observed. Furthermore, for U =45 m/s, the transition occurs before $x/c~=$ 0.05, as was discussed in Sect. 3.1.1, and the trip height is much lower than the boundary layer at the trip location so that there is no influence of the trip on the increase of the boundary layer thickness. This is concluded from comparisons of the wall-pressure spectra along the chord at x/c $\ge$ 0.05 of both cases, forced (Fig. 13) and no forced transition (Fig. 22), which are identical.

Fig. 22

Wall-pressure spectra along the chord at the suction side for no forced transition for different inflow velocities. ${\alpha _e}$ = 7$^{\circ }$

Figure 23 shows the wall-pressure spectrum at different angles of attack for 10 m/s and 45 m/s close to the trailing edge. For angles of attack higher than 5$^{\circ }$, the wall-pressure spectrum is similar to the spectrum observed for forced transition, showing that for those angles there is a fully turbulent boundary layer and there is no influence of the tripping device.

Fig. 23

Wall-pressure spectra at $x/c~=$ 0.97 at suction side with no forced transition for different angles of attack

4 Boundary-layer measurements

This section shows the boundary layer measurements at two positions along the chord for forced and no forced transition to provide insights about the state of the boundary layer for the several conditions linked to what was observed in the wall-pressure spectra. Figure 24 shows the boundary layer measurements at $x/c~=$ 0.3 and 0.97 for $U~=~$10 m/s for the cases of natural and forced transition. The figure shows the main velocity profile, velocity fluctuations, and velocity spectrum at several locations across the boundary layer: the measurement closest to the airfoil wall and at 0.25, 0.5, and 0.75 times the boundary layer thickness for each case. For the case of natural transition at both chord locations, the Polthausen velocity models the main velocity profile; see Fig. 24a. At $x/c~=~$0.97, the shape factor is 2.37, which is close to the Blasius shape factor for the laminar boundary layer (2.56). Contrary, at $x/c~=~$0.3, the shape factor is 1.22, which no corresponds to a laminar boundary layer. However, the boundary layer thickness at this location is too small (1.3 mm), and therefore, the measurement uncertainties are larger. For the cases of forced transition, the velocity profiles are modeled by the logarithmic law of the wall. The shape factor (H) for these cases are 1.19 and 1.46 at $x/c~=~$0.3 and 0.97, respectively. The shape factors are slightly outside of the range expected for a turbulent boundary layer (1.3$-$1.4), which demonstrates that the boundary layer is not fully developed. This was also the conclusion when analyzing the scaling of the wall-pressure spectra for $U~=~$10 m/s, in Fig. 9b–d. This demonstrates that analyzing the wall-pressure spectrum gives an accurate indication of the development of the boundary layer.

Fig. 24

Boundary layer measurements and fitted velocity profile for different positions along the chord and transition conditions. ${\alpha _e}$ = 0$^{\circ }$. $U~=~$10 m/s. Re = 0.2 $\times$ 10⁶. dU/dx is obtained from the fitting of the Polthausen velocity profile, and $u_\tau$ is obtained from the fitting of the logarithmic law of the wall with Cole’s wake function. NT is natural transition, and FT is forced transition

Figure 24b shows that the velocity fluctuations for the case of natural transition at $x/c~=$ 0.3 are minimal, and they are similar to ones in the free-stream, showing that the boundary layer is laminar and most probably before the indifference point. At $x/c~=$ 0.97, the velocity fluctuations slightly increase and show a maximum at $y/\delta$ = 0.4. The velocity spectrum for the case of natural transition at $x/c~=$ 0.97 shows the presence of T–S waves across the entire boundary layer, see Fig. 24c. The peaks caused by the T–S waves are at the same frequencies of the ones observed in the wall-pressure spectrum, see Fig. 19a. Above 1 kHz, the hot-wire measurements do not show the presence of any other peak, and the velocity spectrum is flat. This demonstrates that the T–S waves are only present for lower frequencies, as observed in the wall-pressure spectrum. At $x/c~=$ 0.3 for the natural transition, the velocity spectrum is flat as a function of the frequency, which is expected since the velocity fluctuations are minimal. No tones caused by the presence of T–S waves are observed across the boundary layer. This demonstrates that the peaks observed at the wall-pressure spectrum along the airfoil chord are caused by the feedback loop created by the T–S waves reaching the trailing edge and the acoustic waves generated at the trailing-edge discontinuity and not by the real presence of the T–S waves along the airfoil chord.

Figure 25a shows the velocity profile for forced and natural transition for $U~=~$30 and 45 m/s at $x/c~=$ 0.97. Note that for the natural transition case for $U~=30$ m/s, three small peaks are observed in the wall-pressure spectrum at $x/c~=$ 0.97, whereas for $U~=~$45 m/s, no peaks are observed at that position, see Fig. 19e, h. For the cases of natural transition, both fittings, the Polthausen velocity and the logarithmic law of the wall are shown. Note that for the cases of natural transition, the Polthausen velocity profile does not model the experimental velocity profile, indicating that the boundary layer is already in a transitional state. Furthermore, for $U~=~$45 m/s, the mean velocity for forced and natural transition collapse, see Fig. 25a. A small difference in the calculated friction velocity is found, i.e., 4.1% difference. The boundary layer thickness is 0.0054 m and 0.0075 m and the shape factor is 1.32 and 1.31 for the natural and forced transition cases, respectively. According to the shape factor the boundary layer is turbulent in both cases. Thus, the differences in the friction velocity and boundary layer thickness are due to the different transition locations and processes for each case. For the case of forced transition, the boundary layer has a bypass transition process forced at $x/c~=~$0.5; whereas for the case of no forced transition, the boundary layer has a natural transition process where the primary instabilities are present at $x/c~=~$0.5, the secondary instability is observed at $x/c~=~$73, and finally a turbulent boundary layer is observed at $x/c~=~$0.97.

Figure 25d compares the velocity spectrum at several positions across the boundary layer for the cases of natural and forced transition for $U~=~$45 m/s. For the position closest to the wall, the case of forced transition presents a higher level in the low-frequency range since this case has larger turbulent structures due to the earlier transition. This increase is reflected in the increase of the velocity fluctuations close to the airfoil wall observed in Fig. 25b. At y/$\delta$ = 0.25 and 0.5, the velocity spectrum is similar for both cases. Finally, at y/$\delta$ = 0.75, the case of forced transition presents a lower spectrum level since there is a stronger influence of the free stream for this case. Note that for the same y/$\delta$ ratio, for the case of force transition, the measurement is further away from the airfoil wall.

For $U~=~$30 m/s the mean velocity and the velocity fluctuations across the boundary layer do not collapse for the cases of forced and no forced transition, showing that for the case of no forced transition, the boundary layer is not fully turbulent yet, which is also concluded from the small peaks in the wall-pressure spectrum, see Fig. 19e. Figure 25c compares the velocity spectrum for several positions across the boundary layer for the cases of forced and no forced transition. Close to the airfoil wall, the spectrum for the case of no forced transition has a higher level in the high-frequency range and a lower level in the low-frequency range than the case of forced transition. This trend is caused because, in the case of no forced transition, the turbulence is still governed by the smaller structures present in the inner part of the boundary layer that are developing. Therefore, the velocity spectrum level is higher in the higher frequency range, and also the velocity fluctuations are significantly higher closer to the airfoil wall and reduced in the outer part of the boundary layer compared to the forced transition case, see Fig. 25b.

Fig. 25

Boundary layer measurements and fitted velocity profile for different transition conditions and inflow velocity. ${\alpha _e}$ = 0$^{\circ }$. Measurements at x/c=0.97. dU/dx is obtained from the fitting of the Polthausen velocity profile and $u_\tau$ is obtained from the fitting of the logarithmic law of the wall with the Cole’s wake function. NT is natural transition and FT is forced transition

5 Far-field noise measurements

Figure 26 shows the trailing-edge far-field noise spectrum for several inflow velocities with forced and no forced transition, which generate several boundary layer states that have been discussed in the manuscript. The figure also shows the background noise of the wind tunnel for each velocity. For $U~=~$10 m/s with natural transition, the far-field noise spectrum contains tones in the low-frequency range corresponding to the ones observed in the wall-pressure spectra along the chord for $x/c~\le$ 0.88. However, these peaks do not appear in the velocity spectrum across the boundary layer at $x/c~=~$0.97. The tone at 434 Hz observed in the far-field noise corresponds to the peak observed in the velocity spectrum and in the wall-pressure spectrum, see Figs. 24c and 19a. For $U~=~$25 m/s, the far-field noise spectrum shows tones overimposed in a hump at $\approx$ 1 kHz, corresponding to the peaks observed in the wall-pressure spectrum along the entire airfoil chord, see Fig. 19d. For $U~=~$45 m/s, no tones in the far-field noise spectrum are observed, and the case of forced transition matches the case of no forced transition. The hump observed at high frequency is due to blunt trailing-edge noise, which occurs when the boundary layer thickness is thick compared to the airfoil trailing-edge thickness. For the natural transition case, the blunt trailing-edge noise occurs in a higher frequency due to the smaller boundary layer thickness, as discussed in Sec. 4 Brooks and Hodgson (1981); however, this phenomenon is out of the scope of this work. For $U~=~$45 m/s T–S waves are not reaching the trailing edge, and therefore, the feedback loop is not created, despite T–S waves appearing at some point along the airfoil chord, see Fig. 19h. Contrary, for $U~=~$25 m/s and 10 m/s, where T–S reach the trailing edge, the feedback loop is created, and tones are observed in the far-field noise spectrum at the same frequencies as the wall-pressure spectrum. This also demonstrates that the feedback loop is only created when the T–S waves reach the trailing edge, which changes the development of the boundary layer and the wall-pressure fluctuations along the airfoil chord.

Fig. 26

Airfoil trailing-edge noise generated by several boundary layer states at the airfoil trailing edge and wind tunnel background noise (BG). NT is natural transition, and FT is forced transition

6 Conclusions

This research presents a detailed analysis of the boundary layer development along a NACA 0008 airfoil in relation to different inflow velocities and angles of attack, using wall-pressure fluctuations measured by remote microphone probe technique. Both forced transition by tripping at 5% of the airfoil chord and no forced transition was investigated. The analyses were based on the wall-pressure spectra, the spanwise coherence, and the spanwise correlation length. Furthermore, boundary layer measurements at several positions along the chord, for forced and no forced transition, complemented the analysis to better understand the stated of the boundary layer. The results demonstrated that the evolution of the boundary layer, both qualitatively and quantitatively, can be very well determined, with very good spatial resolution by analyzing the wall-pressure fluctuations caused by the different flow structures inside the boundary layer.

The results show that when two-dimensional structures, known as T–S waves, reach the trailing edge, a feedback loop occurs between the acoustic waves generated at the trailing edge and a point upstream of the airfoil chord. This feedback loop amplifies the disturbances inside the boundary layer, which cause the boundary layer to be disturbed more upstream when compared with a case of no feedback loop. This effect generates peaks in the wall-pressure spectrum even close to the leading edge where the boundary layer originally is undisturbed. This was also confirmed with far-field noise measurements. When the feedback loop is created, the far-field noise spectrum is composed of tones at the same frequencies as the ones appearing in the wall-pressure spectrum. Contrary, when the feedback loop is not created, the far-field noise spectrum is composed of a single broadband component.

The results for forced transition showed that downstream of the transition point, the frequency of the maximum level of the wall-pressure spectrum is shifted toward a lower frequency as going downstream along the airfoil chord due to the growth of the turbulent structures that contain most of the energy. The behavior is similar when increasing the angle of attack, and the opposite behavior is seen when increasing the inflow velocity since the turbulent structures are reduced with the increase of the Reynolds number. The wall-pressure spectrum caused by a fully-developed turbulent boundary layer scales with the edge velocity and the boundary layer thickness, showing a constant Strouhal number of the maximum spectrum level for all the conditions.

The cases of no forced transition clearly showed many aspects of the boundary layer transition process. The wall-pressure spectrum clearly helps to identify the stage of the transition and the flow structures inside the boundary layer. Initially, the undisturbed boundary layer (1) is characterized by very low wall-pressure fluctuations. T–S waves appear at the indifference point (2) because of the primary instability mechanism, which are two-dimensional structures characterized in the wall-pressure spectrum by discrete peaks in the entire frequency range. The two-dimensional waves evolve into three-dimensional waves (3), led by the secondary instability mechanism. This stage is identified by peaks with their sidelobes over imposed on a hump. At stages (4) and (5), turbulent structures are formed that are characterized by a broadband component in the wall-pressure spectrum which starts at the high-frequency range. These stages are also characterized by discrete peaks caused by the presence of some remaining 2D structures that disappear from the stage (4) to (5). At the end of the stage (5), the wall-pressure spectrum is composed of a unique broadband component of a significantly higher level than a fully-developed turbulent boundary layer. At stage (6), a fully turbulent boundary layer has been developed. During the boundary layer development, the maximum spectral level is shifted toward lower frequencies because of the growth of the turbulent structures, similar to the case of forced transition.

This research presents a complete dataset where for the first time, the remote microphone probe technique is extensively used to analyze the boundary layer state, transition, and development. The detail with which all aspects of the transition could be determined from the measurements opens several future investigations.