Feasibility of skin-friction field measurements in a transonic wind tunnel using a global luminescent oil film

The Global Luminescent Oil-Film Skin-Friction field Estimation method (GLOFSFE) is an image-based method that measures the spatio-temporal development of the thickness of an oil film based on its luminescent intensity; processing the GLOF data enables the attainment of the skin-friction field (Liu et al. 2008; Lee et al. 2018). In the present work, only general information is reported about this measurement technique and the related data analysis. For a detailed explanation, discussion, and validation in a low-speed flow, the reader is referred to earlier work, and in particular to Liu et al. (2008); Lee et al. (2018) and Lee et al. (2020c).

In wind-tunnel experiments, the development of a thin oil film applied to the surface of a wind-tunnel model depends, in general, on the wall shear stress, on the pressure gradient, and on the body force. The governing equation of the oil film, i.e., the thin-oil-film equation (Brown and Naughton 1999), is:where h the oil-film thickness, t is the time, \({\varvec{\uptau }}\) is the skin-friction vector, \(\mu\) is the oil dynamic viscosity, p is the pressure, \(\rho\) is the oil density, and \(\mathbf{a}\) is the body acceleration.

In the case of a luminescent oil film, the distribution of the luminescent intensity emitted from the oil is typically recorded by means of a camera system. When the skin friction exerts its action on the interface between the gas (usually air) and the oil, the oil-film thickness varies according to Eq. (1). The measured luminescent intensity I, emitted from an optically thin oil film, is proportional to the oil-film thickness and to the excitation light intensity (Liu and Sullivan 1998; Husen et al. 2018), i.e.:where \(I_{\mathrm{ref}}\) is the reference intensity value, which essentially corresponds to the excitation light intensity distribution (see Sects. 2.2 and 4.2); and \(h_*\) is the unit oil-film thickness, which is a calibration coefficient. The calibration information is obtained by following Lee et al. (2020c); the calibration procedure will be described in Sect. 4.2.

In the present study, the LLS-based method (Lee et al. 2018, 2020c) was used for the GLOF image analysis. As discussed in detail in those publications, this method enables the extraction of the skin-friction field from a sequence of GLOF images with a sufficient number of oil-film image pairs. In this case, the \({\varvec{\uptau }}\)-field is assumed to be time-independent, and the optimal skin-friction field that satisfies Eq. (1) for the given data is determined. In practice, a finite-difference form of Eq. (1) is considered, which can be rearranged into an overdetermined system of equations for an LLS solution. The estimated skin friction \(\hat{\tau _i}\) in the i-direction (\(i=x,y\), where x and y are the stream-wise and span-wise directions, respectively) is obtained as:where \(\hat{\tilde{\tau _i}}\) is the estimated normalized skin friction in the i-direction (obtained from the GLOF image analysis), \(\tau _*\) the unit skin friction, \(\mu\) the oil dynamic viscosity, \(x_*\) the unit length of the spatial resolution, \(t_*\) the time interval of the image pair, and \(h_*\) the aforementioned unit oil-film thickness. Note that, for simplicity, a single \(x_*\) has been used in Eq. (3), but two different unit lengths can be considered for the spatial resolution in the stream-wise and span-wise directions. For a time sequence of GLOF images, the LLS method provides the estimation of the time-averaged, normalized skin-friction field \({\varvec{\hat{{{\tilde{\tau }}}}}}\). In the numerical procedure, the image pairs and the skin-friction field are vectorized, i.e., matrices are converted into column vectors. In particular, \({\varvec{\hat{{{\tilde{\tau }}}}}}\) is vectorized as a high-dimensional vector \(\underline{{\varvec{\hat{{{\tilde{\tau }}}}}}}\). Moreover, the GLOF images are projected onto a structured grid representing the surface of interest. The residual defined from the physical model (i.e., Eq. (1)) is then expressed as a linear system of equations. In the following equations, the effects of the pressure gradient and of the body force terms on the oil-film thickness development will be assumed to be negligible, as compared to the effect of the skin-friction term. As will be discussed in Subsection 2.3.3, this assumption is not mandatory for the application of the GLOFSFE, but it allows a simplification of the present description. In this case, the LLS method provides \(\underline{{\varvec{\hat{{{\tilde{\tau }}}}}}}\) as (Björck 1996; Lee et al. 2018, 2020c):whereandThe operators, vectors, matrices, and dimensions used in Eqs. (5) and (6) are defined as follows: ‘diag’ indicates the diagonal matrix operator; \({{{\underline{\varvec{r}}}}}_k\) is the vectorized ratioed image (\(I/I_{\mathrm{ref}}\)) pair at the moment k, where the intensity ratio is computed for each pixel prior to the projection of the ratioed images onto the surface grid; \({\varvec{\Delta _x}} \in {\mathfrak {R}^{M \times N}}\) is the spatial difference scheme matrix; \({\varvec{\Delta _t}} \in {\mathfrak {R}^{M \times 2M}}\) is the temporal difference scheme matrix; \({\varvec{M_{{\mathrm{c2f}}}}} \in {\mathfrak {R}^{N \times P}}\) is the cell-to-face interpolation matrix; N is the dimension of the vector \(\underline{{\varvec{\hat{{{\tilde{\tau }}}}}}}\); P is the dimension of the the vector \({{{\underline{\varvec{r}}}}}_k\); and M is the total number of the nodes of the surface grid.

The coefficient of determination \(R^2\), as defined by Lee et al. (2018), can be used to assess the quality of the estimated \(\hat{{{\tilde{\tau }}}}\): in fact, \(R^2=1\) would indicate that all data are perfectly represented by the LLS regression, whereas a small value of \(R^2\) would indicate a poor regression. Indications on the normalized root-mean-square error (NRMSE) in the estimated skin friction are reported in Lee et al. (2018) for various values of \(R^2\), which were obtained with various combinations of image parameters (such as the image noise level) in simulated GLOF images. Values of \(R^2\) above 0.5 indicatively corresponded to NRMSE values below \(\pm 5\%\)

The calibration parameters \(\mu\), \(x_*\), and \(t_*\) in Eq. 3 are given by the used oil, by the spatial resolution of the camera system, and by the time interval between the GLOF images, respectively. The calibration parameter \(h_*\) is determined according to the ratioed image film thickness method (Husen et al. 2018), which is based on the oil-droplet volume method. In practice, the unit thickness \(h_*\) is estimated using measurement results according to the following equation:where \({\overline{i}}\) indicates the ensemble average of an arbitrary quantity i, \(\overline{r_{\mathrm{cal}}}\) is the ensemble-averaged ratioed image of the droplet, S is the corresponding area, and \(n_{\mathrm{cal}}\) is the total pixel number of the area.

Measuring the three-dimensional model surface by a two-dimensional camera sensor involves the perspective projection transformation. Photogrammetric techniques can be used in wind-tunnel tests to obtain quantitative flow-visualization image data mapped onto the model surface, as done, for example, in Liu et al. (2000).

The three-dimensional coordinates (x, y, z) in the object space and the corresponding two-dimensional coordinates in the image plane (X, Y) are given by the collinearity equations (Mikhail et al. 2001):where \((X_p,Y_p)\) is the principal point, (dX, dY) is the shift of the image point due to the lens distortion, C is the camera principal distance, \((x_C,y_C,z_C)\) is the camera location in the object space, and \(m_{ij}~(i,j=1,2,3)\) are the elements of the rotational matrix, which are functions of the Euler orientation angle \((\omega ,\phi ,\kappa )\). The orientation angles \((\omega ,\phi ,\kappa )\) are the pitch, yaw, and roll angles, respectively, of the camera in the object coordinate system. The lens distortions (dX, dY) include the radial and tangential distortions.

Similar to OFI measurements, the surface angle to the camera position should be considered for the measurement of the oil-film thickness at each location on the surface (Naughton and Liu 2007). The oil-film thickness normal to the model surface is (Husen et al. 2018; Lee et al. 2020c):where I is the image intensity, \(I_{\mathrm{dark}}\) is the dark current image (which accounts for the influence of residual light and camera electronic noise), and \(I_{\mathrm{exc}}\) is the excitation light distribution image. The angle \(\theta _P\) between the surface and the camera position is calculated as:where \(\mathbf{n}\) is the normal vector to the model surface, and \(\mathbf{v}_{\mathrm{camera}}\) is the vector from the model surface point to the optical center. It should be noted here that, according to Husen et al. (2018), the resolution of the oil-film thickness increases when \(\theta _P\) increases, provided that the signal-to-noise ratio of the image corresponding to the quotient of the emission and excitation intensity distributions is the same.

Since the GLOFSFE is based on assumptions, requirements must be fulfilled for quantitative skin-friction measurements. These can be categorized into fluid dynamic and analytical requirements. In this and in the following sections, the skin-friction vector field will be assumed to be oriented in the stream-wise direction, i.e., \(\tau = \tau _x\), since the freestream examined in the present work was quasi-two-dimensional. It should be, however, emphasized that both components of the skin-friction vector (\(\tau _x\) and \(\tau _y\)) were obtained using the GLOFSFE. In fact, the results presented in Sect. 6 will show the stream-wise component of the skin-friction coefficient \(C_{fx}\).

Figures 1 and 2 show the VA-2 supercritical airfoil model of 1-m chord by 1-m span (Krenkel 2012; Leuckert 2012) installed in the DNW-TWG wind tunnel. DNW-TWG is a closed-circuit, variable density wind tunnel (Binder et al. 1992), in which subsonic, transonic, and supersonic flow conditions can be implemented, depending on which one of the three exchangeable test sections is used. In the present study, the experiments were performed in the adaptive-wall test section, which enables subsonic-to-transonic flow conditions to be covered (freestream Mach numbers from M = 0.3 to 0.9). Within this range, the Mach number can be regulated in 0.005-steps.

The freestream Mach number was determined from the measurements of the flow total pressure and freestream static pressure, according to the isentropic flow equation. The flow total pressure was measured in the DNW-TWG settling chamber, whereas the freestream static pressure was measured using a pressure tap located at the upstream region of the test-section side wall. Both pressures were measured by means of absolute pressure sensors with an accuracy of \(\pm 15\hbox { Pa}\). These pressure sensors as well as all other sensors for the measurement of flow parameters were scanned at 330 Hz for each channel, using a 18-bit data acquisition system. The data were averaged over an integration time of 1 s. During the present investigations, the Mach number was kept constant (within \(\Delta M=\pm 0.003\)) around the set value of 0.72.

The freestream Reynolds number was determined from the freestream Mach number, the flow total pressure, the flow total temperature, and the freestream dynamic viscosity. The flow total temperature was evaluated as the average of the measurements of four resistance temperature detectors (Pt100) installed in the DNW-TWG settling chamber. The accuracy of the flow total temperature measurement was \(\pm 0.2\hbox { K}\). The freestream dynamic viscosity was evaluated via Sutherland’s law for the freestream static temperature, which was determined from the freestream Mach number and the flow total temperature according to the isentropic flow equation. In the current work, the uncertainty in the Reynolds number, based on the model chord length c = 1 m, was within \(\Delta Re_c=\pm 0.0035 \cdot 10^7\)).

As mentioned above, the used DNW-TWG test section has adaptive walls on the upper and lower sides, which allow, in general, interference-free contours to be set. These flexible walls can be deflected via servo motors, with a maximal deflection of the actuators of 120 mm and an accuracy of \(\pm 0.1\) mm (Weiand et al. 2017). The wall adaptation is accomplished by means of a non-iterative Cauchy method based on the pressure distribution measured on the walls and on their deflection (Amecke 1986; Rosemann et al. 1995). The test-section upper and lower walls are equipped with 25 and 23 pressure taps, respectively. The wall pressures were measured by means of electronic pressure scanning modules with an accuracy of \(\pm 69\hbox { Pa}\).

In the present work, x is the chord-wise coordinate, positive from the model leading edge to the model trailing edge; y is the span-wise coordinate, positive from the model port side to the model starboard side; and z is the wall-normal coordinate, positive upward. The model rotated around the rotation center at 42.5% of the chord. The angle of attack AoA of the wind-tunnel model was set by means of a servo drive with an accuracy of \(\pm 0.016^{\circ }\).

The airfoil model was instrumented with pressure taps to measure the surface pressure distribution. The pressure taps were arranged into three chord-wise rows, located along the model centerline (\(y/c=0.5\)) and at the span-wise locations \(y/c=0.32\) and \(y/c=0.68\). The model surface pressures were measured using electronic pressure scanning modules with an accuracy of \(\pm 62\hbox { Pa}\). The wind-tunnel model was also equipped with two thermocouples to measure the model surface temperature (and thus estimate the oil temperature). The accuracy of the temperature measurement by means of the thermocouples was \(\pm 0.3\hbox { K}\). A wake rake was mounted 420 mm downstream from the airfoil trailing edge for the estimation of the airfoil drag coefficient via wake-deficit integration. In this case, the total pressures measured by the probes installed in the wake rake were measured by means of electronic pressure scanning modules with an accuracy of ±41 Pa.

The examined experimental configuration with a large airfoil model was selected as an optimal condition for the development and validation of measurement techniques, since it enabled investigations at high chord Reynolds numbers with a large measurement surface. However, it was not appropriate for the evaluation of the aerodynamic performance of an airfoil over a broad \(\mathrm{AoA}\)-range. In particular, the adaptation of the test-section walls could not converge at \(\mathrm{AoA}>1.3^{\circ }\), because the local flow velocity in the proximity of the upper and lower walls was outside of the application limits of the wall adaptation algorithm (Amecke 1986; Rosemann et al. 1995; Weiand et al. 2017). Moreover, the wake-rake position was too close to the model trailing edge to measure the airfoil drag correctly, since its distance from the model trailing edge should have been at least 0.7 chord (Barlow et al. 1999). Nevertheless, the wake-rake measurements were useful for the estimation of the drag increase due to the oil-film setup, as will be discussed in Sect. 6.2.

A base coat had to be applied to the wind-tunnel model to prevent contamination of the pressure taps due to the oil. Moreover, in preliminary tests, the original (clean) model surface (see Fig. 1) was shown to provide a low signal-to-noise ratio in the GLOF images. A white matte film (Wrap Film Series 1080-M10, from 3M) was applied as the base coat to the airfoil model, thus covering the central row of pressure taps (see Fig. 2). The base coat, which had a thickness of approximately \(100~\upmu \hbox {m}\) and a width of 320 mm, was applied to the model from 15% of the lower side around the leading edge and over the upper side up to the trailing edge. The pressure distribution on the model surface was monitored and measured by the starboard- and port-side pressure taps when the base coat was applied. Otherwise, the central row of pressure taps, which had a higher spatial resolution, was used (see Fig. 4, Sect. 3.4). In addition to the model pressure distribution, a Schlieren photography system (Krenkel 2012) was used to optically identify shock waves in the chord-wise region at approximately \(29\% \le x/c \le 56\%\).

An overview of the experimental setup in the DNW-TWG test section will be also provided in Fig. 11 (Sect. 5.3), together with the coordinate system and with an example dataset obtained via both optical and conventional measurement techniques.

For the application of the GLOF method (Liu et al. 2008; Husen et al. 2018), a luminescent dye is dissolved into the oil to be applied as a film to the wind-tunnel model surface. When the oil film is illuminated by incident light at the dye-excitation wavelength \(\lambda _{\mathrm{ex}}\), it emits light at \(\lambda _{\mathrm{em}} > \lambda _{\mathrm{ex}}\) (Stokes shift), which can be recorded by a photosensitive device, such as a Charge-Coupled Device (CCD) camera. In the present work, the luminescent oil was made by dissolving a laser dye (Pyrromethene 567, from Exciton) into dimethyl silicone oil (Elbesil Öl B, from L. Böwing) at a weight ratio of approximately 15 ppm, which is near saturation. A bottle containing the mixture was placed in an ultrasonic bath with hot water. Ultrasonic waves were applied, thus quickening the dye dissolution. Luminescent oils with kinematic viscosities (at a temperature of \(25~^{\circ }\hbox {C}\)) of 10,000, 30,000, and 60,000 cSt were prepared for the present investigation, because a high oil viscosity was expected to be needed at the considered transonic flow conditions (see Sect. 4.3).

Figure 3 shows the excitation and emission spectra of Pyrromethene 567 in methanol solution, measured by means of a spectrofluorometer. The excitation spectrum was measured with an emission wavelength of \(\lambda _{\mathrm{em}}=540\) nm, while the emission spectrum was measured with an excitation wavelength of \(\lambda _\mathrm{ex} = 495\) nm. The spectral bandwidths of the excitation and emission wavelengths were set to \(\Delta \lambda _{\mathrm{ex}} = 20\) nm and \(\Delta \lambda _{\mathrm{em}}= 1\) nm, respectively, for both of the spectra. The difference in the spectral bandwidths was due to the difference in the levels of the excitation and emission signals. The larger spectral bandwidth of the excitation wavelength, as compared to that of the emission wavelength, allowed an increase in the signal-to-noise ratio of the measured spectra. An emission peak of 542 nm and excitation peaks of 370 nm and 495 nm can be seen in Fig. 3.

Optical access at DNW-TWG is available through circular windows on the starboard and port sides of the test section. They have a diameter of 100 mm and a vertical distance of 420 mm between the window center and the plane corresponding to the model rotation center. The Schlieren windows are semicircular and have a radius of 162.6 mm, where the center points match the model rotation center.

A UV LED light source (IL-107UV, from HARDsoft), with a peak wavelength of 390 nm, was installed on each side of the test section. These illumination devices were selected to excite the luminescent dye around its first peak (\(\lambda _{\mathrm{ex}} = 370\hbox { nm}\), see Sect. 3.2). Although the magnitude of this peak was lower than that of the second peak at 495 nm (see Fig. 3), the high-power LEDs (which current was set to 15 A) enabled the attainment of sufficient intensity of light emitted by the luminescent oil film, while at the same time maintaining a marked wavelength separation between excitation and emission signals. A 14-bit CCD camera (pco.4000, from PCO) was installed to acquire sequential GLOF images. The camera has a resolution of \(4008\times 2672\) pixels, a pixel-sensor size of 9 \(\upmu\)m, and a maximal frame rate with full image size of five frames per second (fps). A manual focus lens with a focal length of 50 mm (Nikkor 50 mm F/1.2 Ai-S, from Nikon) was installed in front of the camera via a Scheimpflug adapter (Scheimpflug mount, from LaVision). The optical setup was adjusted to focus on the model surface area ranging from 9 to 46% of the chord and from 0.375 to 0.625 of the span. This was the surface region of major interest in the present work: as will be shown in Sect. 6, the main fluid dynamics events on the airfoil upper side (in particular laminar–turbulent transition and shock waves) occurred in this area, which will be named “GLOFSFE-evaluation region” throughout this work.

The separation of the excitation and emission lights was accomplished also by the installation of appropriate optical filters in front of the LEDs and of the camera. Sharp-edged band-pass filters for the wavelength range of \(385\pm 35\hbox { nm}\) were placed in front of the LEDs to block light at lower and higher wavelengths. A 550-nm long-pass filter was installed in front of the camera lens to block the excitation light, while at the same time allowing the light emitted by the oil film to be captured. In fact, the cut-on wavelength of 550 nm indicates 50% of the peak transmittance, and the transmittance curve of the used optical filter is relatively smooth. Thus, the emission intensity was sufficiently high for the GLOF measurement, even though the cut-on wavelength was close to the emission peak.

The flow parameters were a freestream Mach number of 0.72, a flow total pressure of 80 kPa, a flow total temperature of approximately 310 K, and a freestream Reynolds number based on the chord length \(c=1\hbox { m}\) of \(Re_c=10^{7}\). The angle of attack (\(\mathrm{AoA}\)) was varied as -0.4, 0.0, 0.4, 0.8, 1.2, 1.5, 1.8, and \(2.0^{\circ }\), thus enabling the study of various boundary-layer stability situations on the model upper side. This was, in general, the surface at lower pressure, i.e., with more negative values of the pressure coefficient \(C_p\). The pressure distributions measured on the clean configuration surfaces (i.e., without the base coat) are shown in Fig. 4, where curves with different colors correspond to different angles of attack. It should be noted here that the pressure distributions on both model surfaces (upper and lower sides) are presented in this figure, but the focus of the present work was on the model upper surface. Therefore, only the upper side pressure distributions will be considered in the following sections, when the discussion will refer to Fig. 4.

The pressure distributions at \(\mathrm{AoA} > 1.3^{\circ }\) are not expected to be representative for the VA-2 airfoil at free-flight conditions, since the adaptation procedure of the test-section walls did not converge in these cases (see Sect. 3.1). Therefore, pre-defined wall contours were implemented at \(\mathrm{AoA} > 1.3^{\circ }\). The pressure distributions on the model upper surface showed, for a significant portion of the chord length, weakly accelerated flow at \(\mathrm{AoA}=-0.4^{\circ }\), and a quasi-zero pressure gradient at \(\mathrm{AoA}=0.0^{\circ }\); the pressure minimum was enhanced with increasing \(\mathrm{AoA}\) up to \(0.8^{\circ }\), whereas shock waves occurred at larger \(\mathrm{AoA}\), with a downstream shift of the shock location with increasing \(\mathrm{AoA}\).

The evaluation of the requirements on the conditions for the application of the GLOFSFE is presented in this Subsection by following the structure of Sect. 2.3.

A calibration process is necessary for establishing the relationship between the variables measurable via GLOF and the desired variables, i.e., to move from spatio-temporal image data to the oil-film dynamics. The calibration parameters were determined according to a calibration procedure, which was discussed in detail in Lee et al. (2018, 2020c). A set of calibration images (shown in Fig. 7) were acquired with the wind tunnel at rest, as summarized below.

The GLOF measurements were conducted as follows. Before wind-tunnel operation, the luminescent oil was applied to the upstream area of the GLOFSFE-evaluation region, as shown in Fig. 9a. The oil was not applied further upstream of the GLOFSFE-evaluation region, since the leading-edge area would have been particularly sensitive to surface roughness (see Sect. 4.1.1). An oil with a high kinematic viscosity of 60,000 cSt (at a temperature of \(25~^{\circ }\hbox {C}\)) was selected. This choice was dictated by the necessity to avoid that the oil would drop during the wind-tunnel preparation time (for example, during the safety check and the wind-tunnel pressure adjustment) and that the oil would separate from the model trailing edge during the runs, thus damaging the wake-rake probes or contaminating the wind tunnel. After the wind-tunnel operation was started, the oil film spread during the flow acceleration. The oil-film thickness was visually monitored during the spreading process. The image acquisition was initiated when the oil covered the GLOFSFE-evaluation region and, at the same time, the image intensity was below saturation. After acquisition of a set of 241 sequential images, the angle of attack was varied, and the image recording was repeated as soon as stable flow conditions were attained again. The series of measurements was then terminated before the oil film reached the model trailing edge, thus avoiding damages of the wake-rake probes or contamination of the wind tunnel. A final oil-film distribution is shown in Fig. 9b.

Each wind-tunnel operation was called a series. Four series with various orders of \(\mathrm{AoA}\) were conducted (S1 to S4), as described below and summarized in Table 1. The different series were run to investigate the most appropriate starting and running conditions for GLOF measurements in DNW-TWG, and enabled an evaluation of the repeatability of the GLOFSFE estimations, as discussed in Sect. 6.1.

The camera parameters, i.e., the intrinsic parameters (focal length, scale factor, principal point, and radial and tangential distortions) and the extrinsic parameters (camera location and orientation angles), were obtained via camera calibration. The camera calibration parameters were determined by pairs of three-dimensional space coordinates of the model surface and two-dimensional image coordinates of the grid images (see Sect. 4.2). These coordinate pairs were provided by the grid images at the various \(\mathrm{AoA}\)s, where the image coordinates were obtained by reading the grid image points. The corresponding model surface coordinates were converted to the spatial coordinates, which were determined by the model geometry, the model installation position, and the \(\mathrm{AoA}\). The camera calibration was carried out using a MATLAB code by Heikkila (2000).

After subtraction of the averaged dark current images, all images were projected onto the three-dimensional model surface. The target model surface coordinates were set to the GLOFSFE-evaluation region, i.e., the area ranging from \(x/c = 9\%\) to 46% and from \(y/c = 0.375\) to 0.625 (see Sect. 3.3); the resulting pixel density was 5600 pixels per meter. The corresponding image coordinates were obtained from Eq. (8), and the image value was calculated by bicubic interpolation. The projected calibration images and a representative oil-film image are shown in Fig. 10.

The time-sequential oil-film thickness distributions in the model surface domain were analyzed using the GLOFSFE code (Lee and Liu 2018), written in MATLAB. All 241 luminescent oil images acquired for each data point were paired into 240 image pairs, and then, an average skin-friction field was obtained for each data point, as described in Sect. 2.1. The oil-film thickness h was evaluated according to Eq. (9), where \(h_*\) was obtained from the oil-droplet images (see Sect. 4.2) via Eq. (7), and the distribution of the angle \(\theta _P\) was calculated from Eq. (10); here, the normal vector \(\mathbf{n}\) was obtained from the model geometry data, and \(\mathbf{v}_{\mathrm{camera}}\) was calculated from the optical center point of the camera. The range of \(\theta _P\) was from \(57^{\circ }\) to \(68^{\circ }\); therefore, the oil-thickness resolution was approximately double, as compared to that for \(\theta _P~=~0^{\circ }\) (Husen et al. 2018).

The estimated skin friction \({\hat{\tau }}\) was calculated from Eq. (3), where \(\hat{{{\tilde{\tau }}}}\) was obtained from the GLOF images [see Eqs. (4)–(6)] using the aforementioned GLOFSFE-evaluation code. The areas where the given data were ill-posed were eliminated from the evaluation process. The ill-posed nodes were identified at the locations where the determinant of the system (Lee et al. 2020c) was lower than a threshold, which was four times the double-precision floating-point relative accuracy. It should be emphasized here that ill-posed nodes were therefore detected also when the estimated wall shear stress approached zero, i.e., also at separation and reattachment locations as well as in their vicinity. This aspect will be discussed in Sect. 6.

Figure 11 shows the GLOFSFE results (estimated oil-film and skin-friction distributions) for a representative data point at \(M=0.72\), \(Re_c=10^7\), and \(\mathrm{AoA}=1.8^{\circ }\) (S1-07, see Sect. 6). In this figure, the pressure distribution on the model surface, the wake deficit, the camera position from the camera calibration, the locations of the Schlieren windows, and the Schlieren image are also shown.

The oil-film thickness in wall units \(h^+\) was evaluated from the estimated oil-film thickness h and skin-friction \({\hat{\tau }}\) (Lee et al. 2020c):where \(\rho _{\mathrm{air}}\) is the air density, and \(\nu _{\mathrm{air}}\) is the air kinematic viscosity. The air properties \(\rho _{\mathrm{air}}\) and \(\nu _{\mathrm{air}}\) were computed from the measured surface pressure distribution and flow total quantities, according to the isentropic flow equations.

A thorough evaluation of the GLOF measurement uncertainties was presented in Lee et al. (2020c). Grounding on that work and on the skin-friction estimation presented in Eqs. (3) and (7), the following uncertainty sources are reviewed for the present study: oil dynamic viscosity, unit length, unit time, calibration oil-droplet volume, calibration image, and normalized skin friction. In any case, an estimation of the overall uncertainty \(\Delta C_{fx}\) in the skin-friction coefficient determined via GLOFSFE will be provided in Sect. 6.1.

The estimations obtained from the GLOF image analysis of all data points examined in the present work are summarized in Figs. 12, 13, 14 and 15. In general, the variables will be presented without the accent \(\hat{}\), since it will be specified that the related quantities are estimated via GLOFSFE. The estimated, average oil-film thickness distributions \({\bar{h}}\) and the coefficients of determination \(R^2\) (Lee et al. 2018, 2020c) are shown in Figs. 12 and 13, respectively. The chord-wise skin-friction distributions \(C_{fx}\) estimated via GLOFSFE are presented in Fig. 14. In these three figures, the gray-masked areas indicate the regions where the GLOF data were unavailable or where the nodes were identified as ill-posed (see Sect. 5.3). Figure 15 shows \(C_{fx}\) profiles, obtained from Fig. 14 as averages along the span-wise regions where the \(C_{fx}\) and \({\bar{h}}\) distributions were essentially two-dimensional.

One of the major challenges of GLOF measurements at the examined test conditions becomes apparent when Figs. 12 and 13 are analyzed. The areas with high \(R^2\) values (i.e., where the data are well represented by the LLS regression) correspond to the regions where the oil film was thick. In these areas, however, the flow was turbulent, and the oil-film thickness was larger than the thickness of the viscous sublayer (see Sects. 2.3.1 and 4.1.1), leading to an increase in the skin friction in Fig. 14, as compared to the clean surface. The increased skin friction was therefore due to a physical effect (the interaction between the oil film and the boundary layer), and not to errors in the GLOF image analysis or in the numerical processing. As an extreme case, the maximal value of the estimated \(C_f\) in the turbulent-flow regions was larger than the reference value \(C_{f,\mathrm{T}}\) (see Sect. 4.1.1) by a factor of 36. In contrast, areas with low \(R^2\) values (i.e., where the LLS regression was poor) correspond to the regions where the oil film was thin. As discussed in Lee et al. (2020c), the thinner oil film leads to increased uncertainties due to the image noise and the image processing, and thus to a reduction in the coefficient of determination and in the confidence of the skin-friction estimation. This affected the estimations obtained in the laminar-flow regions, where the oil film was thin. Nevertheless, the oil-film thickness in these regions was below the critical limit of roughness discussed in Sects. 2.3.1 and 4.1.1, thus allowing a quantitative estimation of the laminar skin-friction coefficient and of the location of transition onset, which will be presented below as well as in Sect. 6.3.

A dedicated discussion is considered necessary for the data points with the expected presence of laminar separation bubbles. In fact, at the larger angles of attack (\(\mathrm{AoA}\ge 0.8^{\circ }\)), the laminar boundary layer was expected to undergo separation as it encountered a marked adverse pressure gradient, particularly pronounced at \(\mathrm{AoA}\ge 1.2^{\circ }\) (see Fig. 4). The strong amplification of disturbances within a separated boundary layer was expected to lead to transition in the separated shear layer, with the resulting turbulent motion eventually inducing the reattachment to the surface and the formation of a laminar separation bubble (see Miozzi et al. (2019), among others). The critical lines (i.e., the separation and reattachment lines), as well as the areas in their vicinity, are regions where the wall shear stress is zero or nearly zero. As mentioned in Sect. 5.3, the nodes in regions of vanishing wall shear stress are detected as ill-posed in the GLOFSFE code. The gray-masked stripes, essentially oriented in the span-wise direction, visible in Figs. 13 and 14 at approximately \(x/c<30\%\) (data series S1 and S2), very likely originated from critical lines, and most probably from the reattachment lines. The stream-wise position and extent of the gray-masked stripes was probably determined by both physical factors and time history of the oil-film. This latter influence will be discussed below, focusing on the presence of multiple gray-masked stripes in various data points of series S1 and S2. With regard to the time-averaged representation of a laminar separation bubble, the reverse-flow region downstream of the separation location is known as a region of low, negative skin friction, which is also named “dead-air” or “dead-water” region (see Costantini et al. (2019) and Miozzi et al. (2019), among others). Also the low value of the negative skin friction in the dead-air region could not be resolved via GLOFSFE at the present experimental conditions, thus possibly contributing to the stream-wise extent of the gray-masked stripes. Between the dead-air region and the reattachment location, a region of more intense reverse flow is expected within the laminar separation bubble, with larger negative wall shear stress culminating in a negative skin-friction peak upstream of reattachment. The large skin friction in the separated flow region upstream of reattachment (as well as in the turbulent region downstream of the reattachment location) is expected to be very efficient in removing oil from the reattachment location area. This was very likely the reason for the formation of stripes of very low oil-film thickness, which can be seen in Fig. 12. Their chord-wise locations approximately matched those of the gray-masked stripes in Figs. 13 and 14, thus indicating that the very low GLOF signal was most probably the cause for the unattainability of skin-friction estimations in these regions. Moreover, laminar separation bubbles are also expected to exhibit significant unsteadiness, especially in the region close to the reattachment location. This also probably contributed to the difficulty of detecting the negative skin friction values within the laminar separation bubbles, and especially upstream of the reattachment location.

As can be seen in Figs. 12, 13, 14 and 15, the repeatability of the results was limited. This was due to the following issues:As discussed in the beginning of this subsection, the GLOFSFE estimations in the turbulent-flow regions can be considered only as qualitative. They can still be meaningful for the examination of the topology of the skin-friction distribution, in a manner similar to that of conventional oil-flow visualizations: the identification of turbulent wedges in Fig. 14, as presented above, is an example of such qualitative analysis. However, the turbulent skin-friction estimations of the current study cannot be analyzed in quantitative terms. This includes the challenge in clearly and reliably identifying the location corresponding to the end of the transition region, as defined in Sect. 4.1.1: the location where \(C_f\) reached \(C_{f,\mathrm{T}}=3 \cdot 10^{-3}\). An alternative to this definition may have been the location where \(C_f\) reached its maximum, after the transition process had started. Because of the inaccurate skin-friction estimations downstream of the location of transition onset, however, a maximum in the skin-friction distribution could not be reliably identified. This can be seen in Fig. 15, where a distinct peak in the skin-friction coefficient downstream of the location of transition onset was not detectable for various data points, as due to the issues discussed above.

In contrast, a quantitative analysis of the skin-friction estimations in the laminar-flow regions was feasible for some of the examined data points, since the oil-film thickness was below the critical limit of surface roughness. At the same time, the uncertainty in the skin-friction values estimated in these areas was relatively high, as indicated by the low value of the coefficient of determination \(R^2\). As a first step of this analysis, the distributions of the average oil-film thickness and of the skin-friction coefficient were surveyed to identify the data points with reliable estimations in the laminar-flow regions. By consideration of the issues discussed above, the following data points were discarded:The chord-wise profiles of the skin-friction coefficient estimated for the remaining data points were examined. As described for Fig. 15, these are the averages of the \(C_{fx}\) distributions in the laminar regions where the flow was essentially two-dimensional; accordingly, the areas affected by turbulent wedges were excluded. The \(C_{fx}\) profiles were analyzed to extract the standard deviation (or the difference, when only two data points were available) of the estimated skin-friction coefficients in evaluable parts of the laminar-flow regions, i.e., where \(C_{fx}\) profiles were obtained for different data points over the same portion of the chord. Representative uncertainties \(\Delta C_{fx}\) for the estimations of the skin-friction coefficient were then determined as stream-wise averages of the extracted distributions of the \(C_{fx}\) standard deviation (or difference) over these portions of the chord. The obtained, representative uncertainties ranged from approximately \(\Delta C_{fx}=3 \cdot 10^{-5}\) to \(1.5 \cdot 10^{-4}\). These values correspond to approximately \(\pm 8\%\) to \(\pm 38\%\) of the reference laminar skin-friction coefficient \(C_{f,\mathrm{L}}\). As already mentioned in Sect. 5.4, a full uncertainty analysis, such that conducted in Lee et al. (2020c), could not be carried out in the present work. This was due not only to the missing information on the offset error in the oil-film thickness estimation (see above), but also to the variation of the local flow conditions along the stream-wise direction over the examined airfoil surface. Furthermore, the oil-film thickness distributions for the evaluable data points were different even for the same angle of attack (see Fig. 12). As a conservative estimation, the maximal value \(\Delta C_{fx}=1.5 \cdot 10^{-4}\) from the aforementioned analysis based on the \(C_{fx}\) profiles was taken as a representative uncertainty of the skin-friction coefficient distributions in the laminar-flow regions. This value appears reasonable as a conservative estimation, since it would also cover the contributions of all uncertainties in the GLOFSFE parameters described in Sect. 5.4.

The location of transition onset \(x_{\mathrm{T,onset}}/c\) could be also determined quantitatively for the evaluable data points identified in the aforementioned analysis. Additionally, also the data points S3-03 to S3-04 and S4-01 to S4-02 could be considered for the evaluation of \(x_{\mathrm{T,onset}}/c\). In contrast to the previous analysis, for this scope, the short stream-wise extent of the measurable laminar-flow region was sufficient to detect the increase in \(C_{fx}\) related to the initiation of the transition process. As introduced in Sect. 4.1.1, transition onset was detected at the location where \(C_f\) became larger than \(C_{f,\mathrm{L}}=4 \cdot 10^{-4}\). In Fig. 15, it can be seen that the locations of transition onset detected for the evaluable data points were in reasonable agreement, with a maximal deviation of \(\Delta x_{\mathrm{T,onset}}/c = 5\%\) at \(\mathrm{AoA}=0.8^\circ\). In most of the other cases, the deviation was \(\Delta x_{\mathrm{T,onset}}/c = 3\%\) or less.

The GLOFSFE estimations in the laminar-flow regions and the detected locations of transition onset will be further discussed in Sect. 6.3, where a comparison with the transition locations measured via TSP in recent experiments (Costantini et al. 2020) will be also presented.

As a final consideration in this subsection, it should be remarked that the CFL number was sufficiently low even in the turbulent-flow regions. As a representative example, the CFL number (calculated from Eq. (13)) in the turbulent-flow region was approximately 2.7 at \(x/c=30\%\) for the data point S2-02. The coefficient of determination at this location was \(R^2=0.53\), so that the numerical process appeared to be valid even when the CFL number exceeded unity. The maximally allowable CFL number \({ CFL }_{\mathrm{max}}\) could be larger than one, because the oil-film distribution has a continuous length.

The aerodynamic coefficients estimated for all examined data points are presented in Fig. 16. For the reasons discussed in subsection 3.1, the values of the aerodynamic coefficients should be regarded only as qualitative. With the base coat applied, the spatial resolution of the measured pressure distribution, which served for the estimation of the lift coefficient \(C_L\) and of the pitching moment coefficient \(C_M\), was even further reduced, as compared to the clean configuration. In fact, the pressure taps on the lower side of the leading-edge area were covered by the base coat and were thus unavailable; additionally, \(C_L\) and \(C_M\) had to be calculated from the starboard-side pressure distribution (at y/c = 0.68), where the number of pressure taps was less than that of the central pressure tap row. Nevertheless, the estimated values of \(C_L\), \(C_M\), and of the drag coefficient \(C_D\) could be considered for the evaluation of relative changes in the aerodynamic coefficients due to the oil-film setup. This included not only the global influence of the oil film, but also the effect of the base coat, which was surveyed by comparing the aerodynamic coefficients estimated for the clean configuration (i.e., without the base coat) with those of the base-coat only cases (i.e., without the oil film).

The estimated \(C_L\) and \(C_M\) were in agreement for all the examined cases despite the presence of the base coat and of the oil film (deviations in the lift and pitching moment coefficients within \(\Delta C_L = \pm 2.5\%\) and \(\Delta C_M = \pm 1\%\) from the clean configuration values), except for the case at \(\mathrm{AoA}=1.8^\circ\), at which the maximal \(\Delta C_L\) and \(\Delta C_M\) were approximately \(\pm 4\%\) of the clean configuration values. This case was found to be particularly sensitive to even very small variations (within the parameter accuracy presented in Sect. 3.1) in the Mach number, in the flow total temperature, and in the wind-tunnel wall contour; for this reason, it resulted difficult to reproduce the shock-wave position, as confirmed by the Schlieren images.

A global effect of the oil-film setup can be seen in the variation of the drag coefficient. Comparing the drag coefficient estimated for the clean configuration with that of the base-coat only cases, the presence of the base coat was found to already lead to a significant increase in \(C_D\) for most of the examined data points. This was very likely due to the generation of turbulent wedges from the span-wise steps at the base-coat edges (see Sect. 6.1), but also the change in airfoil contour due to the application of the base coat may have contributed to the \(C_D\) increase.

In the presence of the oil film, the larger additional increases in the drag coefficient were found for the data points with large turbulent wedges (e.g., S2-03, S2-04, S4-05, and S4-06, see Figs.  14 and 16). Moreover, the trend observed in Fig. 16 is that data points with thinner oil film corresponded to smaller additional increases in \(C_D\). This can be seen, for example, by a comparison of S1-01 with S1-15, S1-02 with S1-14, etc. In fact, the first data points of series S1 were also affected by the oil ‘bump’, which led to premature transition (see Sect. 6.1). When the oil film is thin, the additional increase in the drag coefficient is essentially negligible, as shown, e.g., by the comparison of the base-coat only cases with the data points S1-07 to S1-15. The general trend of larger additional \(C_D\) increases with larger oil-film thickness supports the discussion of the oil-film influence on the estimated skin-friction distributions, which was presented in the previous subsection. However, it should be emphasized that the base coat already led to significant increases in the drag coefficient: in particular, at \(0.8^{\circ } \le \mathrm{AoA} \le 1.5^{\circ }\) the effect of the base coat on the wake deficit was even larger than that due to the oil film.

The evaluable data points from the analysis presented in Sect. 6.1 were further surveyed to select, for each angle of attack, a single data point with a higher confidence level in the laminar-flow regions. For this reason, the data points S1-09 to S1-15 were discarded, since the coefficient of determination \(R^2\) was smaller than that of the data points S1-04 to S1-07 and S3-05 to S3-07. Moreover, the data points S3-05 and S3-06 were preferred to the corresponding data points from series S4 (S4-06 and S4-05, respectively), because these latter data points were affected by a large turbulent wedge around \(y/c=0.51\).

The \(C_{fx}\) distributions of the selected data points are shown in Fig. 17. The profiles of \(C_{fx}\) and \({\bar{h}}\) along with the reference values \(C_{f,\mathrm{L}}\), \(C_{f,\mathrm{T}}\), and the estimated viscous sublayer thickness \(\delta _{\mathrm{v}}= 5 \mu _{\mathrm{air}} / \sqrt{\tau \rho _{\mathrm{air}}}\) (see Eq. (21)) are shown in Fig. 17a. The corresponding \(C_{fx}\) distributions from Fig. 14 are shown in Fig. 17b. In Fig. 17a, it can be seen that noisy distributions of \(C_{fx}\) and \({\bar{h}}\) corresponded to the low h regions, approximately where \({\overline{h}}^+ < 3\). Interestingly, it appears that the quality of the GLOFSFE estimations in the laminar-flow regions improved as \({\overline{h}}^+\) approached \({{{\overline{h}}}}^+ \sim 5\), in a manner similar to the observations of Lee et al. (2020c) for a turbulent flat-plate boundary layer. However, larger values of \({\overline{h}}^+\) (approximately \({{{\overline{h}}}}^+ > 7\)) in the laminar-flow region close to the leading edge led to overestimations of the skin-friction coefficient, as will be shown below. These overestimations were observed for the data points S1-04 to S1-08, where the oil film was thick in the GLOFSFE-evaluation region close to the leading edge because of the oil ‘bump’ affecting these data points (see Fig. 12 and the related discussion in Sect. 6.1).

To obtain reference skin-friction values in the laminar-flow region, compressible boundary-layer calculations were carried out using the laminar boundary-layer solver COCO (Schrauf 1998), which is a well established code for boundary-layer stability analysis and laminar wing design (see Streit et al. (2015) and Schrauf and von Geyr (2020), among others). The surface pressure distributions, measured by means of the pressure taps, served as inputs for the boundary-layer computations, together with the freestream Mach number, the chord Reynolds number, and the freestream static temperature. The model surface was reasonably assumed to be an adiabatic wall. The stream-wise distributions of the laminar friction coefficient computed via COCO for the examined angles of attack are also shown in Fig. 17a (\(C_{f,\mathrm{num}}\)). Moreover, zoomed-in plots of the comparisons between the profiles of \(C_{fx}\) estimated via GLOFSFE and those computed using COCO are presented in Fig. 18. In this latter figure, only the chord-wise region up to \(x/c=40\%\) and the \(C_{fx}\) values between \(10^{-4}\) and \(10^{-3}\) are shown in the horizontal and vertical axes, respectively. The reference value for the laminar skin-friction coefficient \(C_{f,\mathrm{L}}\) with the representative uncertainty estimated in Sect. 6.1 are also presented in Fig. 18. It should be emphasized here that the numerical computations, performed with a laminar boundary-layer solver, could not take into account the transition process; they therefore provide skin-friction data that are meaningful for comparison only in the region upstream of the location corresponding to the transition onset, as observed in the experiments. Moreover, COCO predicted laminar separation to occur in the examined region for \(0.8^{\circ } \le \mathrm{AoA} \le 1.8^{\circ }\); for this reason, no numerical data are available downstream of \(x_{\mathrm{S}}/c \sim 12\%\), 16%, 18%, and 31% for the angles of attack \(\mathrm{AoA} = 0.8^{\circ }\), \(1.2^{\circ }\), \(1.5^{\circ }\), and \(1.8^{\circ }\), respectively. In Figs. 17a and 18, it can be seen that the value assumed in Sect. 4.1.1 for the reference laminar skin-friction coefficient \(C_{f,\mathrm{L}}=4 \cdot 10^{-4}\) was, in general, representative for the computed laminar \(C_f\). This was observed for a significant chord-wise portion of the investigated airfoil region in the majority of the examined cases, except for the cases at \(\mathrm{AoA} = 0.8^{\circ }\), \(1.2^{\circ }\), and \(1.5^{\circ }\), where laminar separation was predicted close to the leading-edge area (i.e., at \(x/c < 20\%\)).

As introduced in Sect. 1, reference transition locations have become available via recent experiments (Costantini et al. 2020), in which the VA-2 airfoil model was equipped with a TSP and investigated for the same test cases as those examined in the present work. The measured surface temperature distributions were analyzed along five evaluation sections, which were located at different span-wise positions. For each evaluation section, transition was detected at the location corresponding to the maximal temperature gradient in the stream-wise direction, according to the procedure described in Costantini (2016). The transition locations \(x_{\mathrm{T}}/c\) and the related uncertainties \(\Delta x_{\mathrm{T}}/c\) reported for the TSP data from Costantini et al. (2020) are the averages and the standard deviations of the transition locations detected at the five evaluation sections. Because of the difference in the definition of \(x_{\mathrm{T,onset}}/c\) and \(x_{\mathrm{T}}/c\), the transition location from the TSP data analysis is expected to be always more downstream than the location of transition onset estimated via GLOFSFE. Nevertheless, the distance between these two locations may be small when transition is induced by a strong adverse pressure gradient (see Costantini (2016)). A comparison of the locations of transition onset estimated via GLOFSFE with the transition locations measured via TSP is presented in Fig. 19. Note here that the reported values of \(\Delta x_{\mathrm{T,onset}}/c\) correspond to the maximal deviation of the locations of transition onset estimated via GLOFSFE for the evaluable data points at a certain \(\mathrm{AoA}\) (see Sect. 6.1), i.e., it is an uncertainty of a different nature than that reported for the TSP results.

With consideration of the representative measurement uncertainty, the comparisons in Figs. 17a and 18 show, in portions of the chord for some of the examined data points, reasonable agreement of GLOFSFE estimations and numerical results. Moreover, the comparison of the transition locations presented in Fig. 19, considering also the reported experimental deviations and uncertainties, shows significant agreement of the GLOFSFE estimations with the TSP results for most of the investigated angles of attack.Figure 20 shows the estimations of the transition regions and of the shock-wave positions for the selected data points with varying angle of attack. As discussed in Sect. 6.1, the reported values of \(x_\mathrm{T,onset}/c\) (blue diamonds) are quantitative estimations obtained using the GLOFSFE, whereas the locations of transition end (blue squares) can be considered only as qualitative. The shock-wave positions were estimated from the measured surface pressure distributions at the center positions of the adverse pressure gradient interval, and the uncertainties (presented as green bars) were evaluated from the tap interval width. Note here that the starboard-side pressure taps were used in this case, since the central row of pressure taps was covered by the base coat (see Sect. 3.1). Therefore, the resolution of the pressure distribution was reduced, as compared to Fig. 4. The shock-wave positions could be confirmed by the Schlieren images in the cases at \(\mathrm{AoA}=1.8^{\circ }\) (see Fig. 11) and \(\mathrm{AoA}=2.0^{\circ }\), since in these cases, the shock wave occurred within the available Schlieren windows (see Fig. 2 and Sect. 3.3). The reported shock-wave positions (magenta crosses) were determined by taking the location corresponding to the middle of the dark strip in the time-averaged Schlieren image, while their uncertainties (magenta bars) were evaluated from the variation of the dark strip location in the time series of the images. Moreover, the red crosses in Fig. 20 indicate the locations where the estimated oil-film thickness \({\bar{h}}\) matched the estimated viscous sublayer thickness \(\delta _{\mathrm{v}}\).

As can be seen in Fig. 20, the estimated transition region was found to move upstream when the angle of attack was increased from \(\mathrm{AoA} = -0.4^{\circ }\) to \(0.8^{\circ }\). This evolution was induced by the more pronounced pressure minimum at \(x/c \sim 8\%\) and by the following stronger adverse pressure gradient (see Fig. 4). As discussed above, at \(\mathrm{AoA}=0.8^{\circ }\), transition occurred very likely within a laminar separation bubble, induced by the markedly adverse pressure gradient observed in this case. At larger angles of attack, shock waves were seen in the airfoil pressure distribution, and confirmed by the Schlieren images in the cases of \(\mathrm{AoA}=1.8^{\circ }\) and \(2.0^{\circ }\). The downstream movement of the transition region with increasing angle of attack for \(\mathrm{AoA} \ge 1.2^{\circ }\) was related to the change in the pressure distribution at these transonic flow conditions. At \(\mathrm{AoA}=1.2^{\circ }\) and \(1.5^{\circ }\), the shock-wave location matched that of the transition region: in these cases, boundary-layer transition was induced by the shock wave. In contrast, at \(\mathrm{AoA} = 1.8^{\circ }\) and \(2.0^{\circ }\), transition was found to start at a location upstream of the shock-wave position. As discussed above, the adverse pressure gradient in the region at approximately \(30\% \le x/c \le 37\%\) appeared to induce boundary-layer transition at the two largest angles of attack. The evolution of the transition region with varying angle of attack estimated via GLOFSFE was generally in agreement with the change in the transition location as a function of \(\mathrm{AoA}\) measured via TSP (Costantini et al. 2020)—see Fig. 19.

Finally, it is interesting to note in Fig. 20 that \(x(h=\delta _{\mathrm{v}}) \le x(C_f=C_{f,L})\) held in most of the cases, clearly indicating that the oil-film thickness in the turbulent-flow regions was larger than the critical limit of roughness.