2. Experimental set-up

The experiment was performed in an open-return boundary-layer wind tunnel at Newcastle University. The maximum free-stream speed of the facility is 30 m s⁻¹, with a test section of 0.46 m in width and 0.32 m in height. The flat plate used to develop the turbulent boundary layer is $L_{\textit{plate}} = 3.5$ m long and is equipped with a trailing-edge flap for flow conditioning. A zigzag trip was installed 100 mm downstream of the sharp leading edge across the full width of the flat plate to promote transition to turbulence. The trip has a height of 1.5 mm and a streamwise length of 10 mm (Elsinga & Westerweel Reference Elsinga and Westerweel2012). The free-stream turbulence intensity was found to be approximately 0.3 % after applying a band-pass filter, with the high-pass cutoff frequency defined as $f_{h} = U_{\infty } / (2L_{\textit{plate}})$ (Cattafesta et al. Reference Cattafesta, Bahr and Mathew2010) and the low-pass cutoff set at $f_{l} = 5000$  Hz to remove high-frequency noise from the measurement circuit, as verified through preliminary hot-wire measurements conducted prior to the main experiment. Here, $U_\infty$ indicates the free-stream velocity. A summary of the experimental conditions is given in table 1. Throughout the paper, $x$ , $y$ and $z$ denote the streamwise, wall-normal and spanwise directions, respectively. Lowercase letters indicate instantaneous components. An overbar denotes time-averaged quantities, for example, $\overline {u}$ for time-averaged streamwise velocity. A prime denotes fluctuating components, e.g. $\tau '=\tau -\overline {\tau }$ . The root-mean-square or second moment value is written as $\sigma$ , e.g. $\sigma (\tau )$ , while higher-order statistical moments are given by $S$ for skewness (third moment) and $F$ for kurtosis (fourth moment). The boundary-layer thickness $\delta$ is defined as the wall-normal location where the time-averaged streamwise velocity reaches $0.99U_\infty$ . The slight non-monotonic variation of $\delta$ in table 1 is attributed to the uncertainty associated with the $0.99U_\infty$ threshold definition. The momentum thickness is denoted by $\theta$ . The Reynolds numbers based on the friction velocity and momentum thickness are calculated as $\textit{Re}_\tau = u_\tau \delta /\nu$ and $\textit{Re}_\theta = \theta U_\infty /\nu$ , respectively. For any dimensional length $L$ , the corresponding viscous-scaled quantity is defined as $L^+ = {L u_\tau }/{\nu }$ . The uncertainty of a flow quantity $Q$ is denoted by $\varepsilon _Q$ . To assess the uncertainties of the estimated friction velocity $u_\tau$ and wall-normal position $y$ , a standard least-squares linear regression was applied to the viscous sublayer data ( $y^+ \lt 4.5$ ), following the method of Hutchins & Choi (Reference Hutchins and Choi2002). For other quantities that are functions of independent variables, such as $\textit{Re}_\tau = u_\tau \delta /\nu$ , the corresponding uncertainties were estimated using the error-propagation method of Kline & McClintock (Reference Kline and McClintock1953). The resulting uncertainties of the key flow quantities are summarised in table 2.

Table 1.

A summary of the experimental conditions.

Table 1. Long description

The table presents a summary of experimental conditions for a boundary-layer wind tunnel experiment. It contains 3 rows and 12 columns. The columns are labelled as follows: U infinity in m s^-1, u tau in m s^-1, delta in mm, theta in mm, Re tau, Re theta, l plus R45, w plus R45, l plus R47, w plus R47, D plus LDV, and l plus LDV. The rows provide specific values for each of these parameters. Row 1: 6.11, 0.253, 39.4, 5.40, 664, 2200, 12.6, 3.4, 15.2, 1.7, 0.73, 5.4. Row 2: 8.06, 0.327, 39.7, 5.32, 863, 2860, 16.3, 4.4, 19.6, 2.2, 0.94, 7.0. Row 3: 10.76, 0.422, 36.7, 5.02, 1032, 3600, 21.1, 5.6, 25.3, 2.8, 1.21, 9.0. The table provides detailed measurements and conditions for the experiment, including free-stream velocity, friction velocity, boundary-layer thickness, momentum thickness, Reynolds numbers, and hot-film and LDV probe dimensions in viscous units.

Table 2.

Uncertainty estimates of the experimental parameters.

Table 2. Long description

A table with uncertainty estimates of experimental parameters. The table has three rows and eight columns. The column headers are epsilon_U_infinity in m s^-1, epsilon_u_tau in m s^-1, epsilon_y in micrometres, epsilon_delta in mm, epsilon_theta in mm, epsilon_Re_tau, epsilon_Re_theta, and epsilon_L_plus_equals_1. Row 1: 0.004, 0.002, 2.4, 0.2, 0.02, 6, 24, 0.012. Row 2: 0.008, 0.006, 6.3, 0.6, 0.13, 21, 74, 0.022. Row 3: 0.006, 0.006, 3.9, 0.3, 0.06, 16, 53, 0.017.

Figure 1.

Schematic of the experimental set-up.

Velocity measurements were obtained using an LDV system and were performed approximately 1800 mm downstream of the zigzag trip, where the boundary layer was fully developed. A COHERENT Genesis MX514-1000 SLM laser system with a power output of 300 mW was used as the laser source. Two laser beams with a wavelength of 488 nm (blue) outlined a plane parallel to the wall and were used to measure the streamwise velocity component. The transmitting and receiving optics were mounted on opposite sides of the wind tunnel, as shown in figure 1. The optical system for emitting the laser had a focal length of 480 mm and a beam expander (TSI, Model XPD60) with an expansion ratio of 2.6 was used to achieve a final beam diameter of 2.65 mm. The beam spacing for the laser was 50 mm. The nominal measurement volume had a diameter of $D_{\textit{LDV}} = 0.043$  mm and a length of $l_{\textit{LDV}} = 0.32$  mm, with the corresponding viscous-scaled dimensions summarised in table 1. The LDV signals were processed using a 16-bit Dantec Burst Spectrum Analyser (BSA-F60). The traverse stages provide a positioning resolution of 12.5  $\mathrm{\unicode{x03BC} m}$ on the laser side and 0.625  $\mathrm{\unicode{x03BC} m}$ on the receiver side in the wall-normal direction. The seeding particles were olive oil droplets with a nominal diameter of 1  $\mathrm{\unicode{x03BC} m}$ , yielding a Stokes number less than 0.1 to ensure faithful tracking of the flow (Tropea et al. Reference Tropea2007). The sampling frequency varied from approximately 8 kHz in the free stream to 500 Hz near the wall, and data were acquired for 90 s to achieve statistical convergence.

Wall-shear stress measurements were obtained using two types of hot-film probes, which were tested separately: a Dantec 55R45 flush-mounted sensor ( $l \times w = 0.75 \times 0.2$  mm) and a Dantec 55R47 glue-on sensor ( $l \times w = 0.9 \times 0.1$  mm). To ensure flush mounting, hot-film sensors (models 55R45 and 55R47) were seated in pre-machined recesses. Dial gauge verification showed a maximum surface deviation of $20\ \unicode{x03BC} \text{m}$ , keeping the sensors within $y^+ \lt 0.57$ for the entire Reynolds-number range. Both sensors were operated with a Dantec StreamLine Pro constant-temperature anemometer, set with an overheat ratio of 1.5. The analogue outputs were acquired through a National Instruments system consisting of a BNC-2110 terminal block and a PXIe-6356 high-speed data acquisition card. The sampling frequency was 20 kHz with a recording duration of 90 s for each condition. Tests were performed at three free-stream velocities of 6.1, 8.1 and 10.8 m s⁻¹.

Figure 2.

Viscous-scaled wall-normal profiles of boundary-layer properties: (a) time-averaged streamwise velocity $\overline {u}^+$ , (b) streamwise turbulent intensity $\sigma ({u'}^+)$ .

Figure 2. Long description

The image contains two line graphs labeled as Panel A and Panel B. Panel A: This line graph shows the viscous-scaled wall-normal profiles of the time-averaged streamwise velocity. The x-axis is labeled y+ and ranges from 10^0 to 10^3. The y-axis is labeled u+ and ranges from 0 to 30. The graph includes multiple data series represented by different symbols and colors: blue circles for experimental data with U∞ = 6.1 m/s and Reθ = 2200, orange circles for experimental data with U∞ = 8.1 m/s and Reθ = 2860, and yellow circles for experimental data with U∞ = 10.8 m/s and Reθ = 3600. Additionally, there are lines representing DNS data with Reθ = 2200, DNS data with Reθ = 2400 by Schlatter et al. (2009), and theoretical lines for U+ = y+ and U+ = (1/0.385) ln(y+) + 4.9. An inset within Panel A shows a zoomed-in view of the lower y+ range. Panel B: This line graph shows the viscous-scaled wall-normal profiles of the streamwise turbulent intensity. The x-axis is labeled y+ and ranges from 10^0 to 10^3. The y-axis is labeled σ(u+). The graph includes the same data series and lines as Panel A, showing how the streamwise turbulent intensity varies with y+.

3. Global flow characterisation

Wall-normal profiles of the turbulent boundary layer were obtained from both LDV measurements and DNS performed using the high-order finite-difference solver Xcompact3D (Bartholomew et al. Reference Bartholomew, Deskos, Frantz, Schuch, Lamballais and Laizet2020), which has been extensively validated in previous studies of zero-pressure-gradient turbulent boundary layers (Diaz-Daniel et al. Reference Diaz-Daniel, Laizet and Vassilicos2017; Mahfoze & Laizet Reference Mahfoze and Laizet2021; O’Connor et al. Reference O’Connor, Diessner, Wilson, Whalley, Wynn and Laizet2023, Reference O’Connor, Laizet, Wynn, Edeling and Coveney2024; Wynn et al. Reference Wynn, Parvar, O’Connor, Frantz and Laizet2025). All LDV statistics were computed with transit-time weighting of the seeding particles (Buchhave et al. Reference Buchhave, George and Lumley1979; Tropea Reference Tropea1995). For consistency, the profiles were normalised using the friction velocity $u_\tau$ , determined by a linear fit of the time-averaged streamwise velocity within the viscous sublayer ( $y^+ \lt 4.5$ ), as shown in the inset of figure 2(a). Figure 2(a) demonstrates that the time-averaged velocity profiles measured by LDV and obtained from DNS collapse well with the DNS data of Schlatter et al. (Reference Schlatter, Örlü, Li, Brethouwer, Fransson, Johansson, Alfredsson and Henningson2009) in both the near-wall and logarithmic regions, despite the different Reynolds numbers. The chosen von Kármán constant $\kappa = 0.385$ and additive constant $B = 4.9$ provide a reasonable representation of the classical log law (George Reference George2007). Streamwise turbulence intensity profiles, shown in figure 2(b), collapse below $y^+=10$ and exhibit a peak around $y^+=15$ , consistent with previous observations in high-Reynolds-number boundary layers (Hutchins & Marusic Reference Hutchins and Marusic2007b ; Hutchins et al. Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012). An increase in outer-layer turbulence intensity with Reynolds number reflects the growing influence of large-scale motions (Hutchins & Marusic Reference Hutchins and Marusic2007a ; Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009). Crucially, excellent agreement is observed between LDV measurements at $\textit{Re}_\theta = 2200$ and DNS data at the same Reynolds number, confirming both the accuracy of the experimental facility and the suitability of the DNS database for the subsequent NLR calibration. This collapse of experimental and numerical statistics provides confidence in the combined use of LDV, hot-film measurements and high-fidelity DNS in the present study.

4. Nonlinear regression technique

Although the general framework of the NLR technique was introduced by Agrawal et al. (Reference Agrawal, Whalley, Ng, Dennis and Poole2019), the present study employs a constrained form of NLR to ensure a physically admissible calibration. The inputs for this technique are the first four moments of the wall-shear stress time series. The principle of NLR is to adjust the calibration coefficients such that the first four moments of the calibrated hot-film signal match the true moments as closely as possible through an error minimisation process. In the present study, the wall-shear stress $\tau$ is represented as a third-order polynomial function of the hot-film output voltage $E$ as

(4.1) \begin{equation} \tau = aE^3+bE^2+cE+d,\end{equation}

where $a$ , $b$ , $c$ and $d$ are the calibration coefficients determined through the NLR optimisation. A fourth-order polynomial was also considered, but the calibration results showed negligible differences compared with the third-order case. Other functional forms, including power-law and exponential representations, were investigated. These formulations substantially increased the computational cost of the NLR process and, in some instances, the algorithm failed to converge to a valid calibration curve. The error function is defined as

(4.2) \begin{equation} \epsilon = \left (1 - \frac {\overline {\tau _r}}{\overline {\tau _t}}\right )^2 + \left (1 - \frac {\sigma (\tau _r)}{\sigma (\tau _t)}\right )^2 + \left (1 - \frac {S(\tau _r)}{S(\tau _t)}\right )^2 + \left (1 - \frac {F(\tau _r)}{F(\tau _t)}\right )^2, \end{equation}

where the subscript ‘t’ represents the ‘true’ moments, which are obtained by LDV or DNS in the present study. The subscript ‘r’ represents the moments of the reconstructed wall-shear stress signal measured by the hot-film sensor after the calibration using NLR. To ensure a physically meaningful calibration, a monotonicity constraint was imposed on the polynomial relationship between $\tau$ and $E$ . In other words, the calibration was constrained such that $\tau (E)$ remains strictly monotonic over the entire range of measured voltages. This requirement was enforced by ensuring that $\partial \tau /\partial E = 3aE^2+2bE+c$ remains positive or negative throughout the effective range of $E$ , depending on whether the hot-film circuit produces a positive or negative response, thereby guaranteeing a one-to-one relationship between $\tau$ and $E$ . Without such a constraint, the polynomial may become non-monotonic within the measured voltage range, which would imply that multiple voltage values correspond to the same wall-shear stress, a behaviour that is not physically possible for a correctly functioning hot-film circuit.

Figure 3.

Comparison of results obtained by different calibration techniques: (a) calibration curves of hot film (55R45), with the inset showing the collapse of 50 calibration curves obtained from different initial guesses of the NLR technique; (b) PDFs of wall-shear stress measured at $\textit{Re}_\theta =3600$ .

In the present study the optimal values for the calibration coefficients were calculated using MATLAB’s fmincon function, which is a nonlinear optimisation solver, with initial guesses generated using a random number generator. Since the error function is highly nonlinear, the sensitivity of the calibration to initialisation was examined by generating fifty random sets of initial guesses. The resulting calibration curves collapse almost perfectly within the effective voltage range of the hot film, as shown in the inset of figure 3(a). The average value of the mean absolute percentage error between the instantaneous wall-shear stress signals reconstructed using these fifty calibration curves is 0.13 %. This confirms that the calibration outcome is insensitive to the choice of initialisation and that the constrained NLR procedure consistently converges to an equivalent solution.

5. Nonlinear regression using moments obtained by LDV

In this section the LDV measurements within the viscous sublayer are used to construct an instantaneous wall-shear stress signal, from which the statistical moments serve as inputs to the NLR procedure. Here, the wall-shear stress measured by LDV is calculated as $\tau = \mu ({u}/{y}) \big |_{y^+ \lt 5}$ . For context, a standard hot-film calibration is also presented to illustrate its limitations: the amplitude of the instantaneous wall-shear stress is underestimated due to heat conduction into the substrate (Eckelmann Reference Eckelmann1974; Alfredsson et al. Reference Alfredsson, Johansson, Haritonidis and Eckelmann1988; Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009; Whalley et al. Reference Whalley, Dennis, Graham and Poole2019). In the standard calibration approach, the time-averaged wall-shear stress is inferred from LDV measurements of the mean streamwise velocity gradient within the viscous sublayer at several free-stream velocities, and these values are plotted against the corresponding time-averaged hot-film voltage, after which the polynomial relationship given in (4.1) is used to establish the mapping between $E$ and  $\tau$ . Since the results obtained using the 55R47 hot-film sensor showed no noticeable differences compared with those from the 55R45 sensor, only one sensor is presented for each result in the following sections for clarity. Figure 3(a) shows the calibration curves obtained by these two different calibration techniques. The yellow solid line represents the result of a standard calibration, with yellow asterisks marking the calibration points. The red solid line represents the NLR calibration result obtained using wall-shear stress moments derived from LDV measurements at $y^+ = 3.6$ with $U_\infty = 10.8$  m s⁻¹. Compared with the yellow standard calibration curve, the red NLR calibration curve shows that the resulting wall-shear stress is more sensitive to changes in hot-film voltage. Probability density functions (PDFs) of the instantaneous wall-shear stress time series measured by LDV and by the hot film calibrated using the two techniques were also compared, as shown in figure 3(b). The black solid line and red circles represent the results measured by LDV at $y^+ = 3.6$ and measured by the hot film calibrated using the NLR technique, respectively. These two PDFs collapse, which indicates that the NLR technique performs as expected, because the similarity in the PDF distribution suggests that the moments of the two time series are nearly the same. Compared with the two PDFs, a greater concentration in the range of $0.18 \leq \tau \leq 0.3$ is observed in the PDF of the hot-film probe with a standard calibration. This is consistent with the instantaneous wall-shear stress time series shown in figure 4(a), and reaffirms previous findings that standard calibration procedures lead to damping of wall-shear stress fluctuations. In contrast, the NLR calibration technique recovers the sensitivity of the hot film to the true wall-shear stress fluctuations.

Figure 4.

Comparison of instantaneous wall-shear stress measured by different techniques at $\textit{Re}_\theta =3600$ : (a) a 30 s time series, (b) the first 0.5 s time series. (Hot film: 55R45.)

Figure 4. Long description

Two line graphs compare instantaneous wall-shear stress measured by different techniques over different time series. Panel A: A line graph displays a 30-second time series of wall-shear stress (τ) measured by different techniques. The x-axis represents time (t) in seconds, ranging from 0 to 30 seconds. The y-axis represents wall-shear stress (τ) in Pascals (Pa), ranging from 0 to 1.0 Pa. Three data series are shown: τ measured by LDV at y+ = 3.6 (black line), τ measured by hot film with NLR (moments obtained by LDV) (orange line), and τ measured by hot film with standard calibration (yellow line). The graph shows fluctuations in wall-shear stress over time for each measurement technique. Panel B: A line graph displays the first 0.5 seconds of the time series of wall-shear stress (τ) measured by different techniques. The x-axis represents time (t) in seconds, ranging from 0 to 0.5 seconds. The y-axis represents wall-shear stress (τ) in Pascals (Pa), ranging from 0 to 0.8 Pa. The same three data series are shown as in Panel A. The graph shows detailed fluctuations in wall-shear stress over a shorter time period, highlighting the differences in measurements by the different techniques.

As shown in figure 3(a), one may ask why these two calibration curves look completely different and how they should be interpreted. The standard calibration curve is constructed from measurements taken at several free-stream velocities and, therefore, represents an averaged response of the hot film over a wide range of operating conditions. In contrast, the NLR calibration curve is obtained using the statistical moments of the wall-shear stress at a single flow velocity, with the mean wall-shear stress explicitly prescribed as one of the input constraints. As a result, although the two calibration curves differ markedly in shape, both recover the correct mean wall-shear stress at the corresponding operating condition, while the NLR calibration additionally adjusts the sensitivity of the hot film to accurately capture the fluctuating component of the wall-shear stress. For the fluctuating component of the wall-shear stress, however, the two approaches behave very differently. The standard calibration becomes unreliable because the instantaneous voltage response of the hot film is weakened by heat conduction into the substrate, which reduces its sensitivity to rapid variations in wall-shear stress. The NLR approach overcomes this limitation by effectively recovering the gain that maps the voltage fluctuations back to the true wall-shear stress fluctuations. As demonstrated later in figure 6(a), even high-frequency contents of the wall-shear stress can be retrieved once the appropriate NLR calibration curve is applied. These considerations imply that, for any given free-stream velocity, the mean wall-shear stress can be obtained using the standard calibration procedure, whereas an accurate reconstruction of the instantaneous fluctuations requires a NLR calibration curve obtained at that same operating condition.

Figure 5.

Comparison of instantaneous wall-shear stress obtained at different wall-normal locations from DNS at $\textit{Re}_\theta = 2200$ .

To verify the calibration result for the NLR technique, a comparison was made between the instantaneous wall-shear stress measurements obtained by LDV and the hot film, as shown in figure 4. Here, the LDV probe was positioned directly above the location of the hot-film probe at $y^+$ = 3.6 and the two data sets were acquired simultaneously. A 30 s time series for comparing the overall results is shown in figure 4(a). Compared with the wall-shear stress measured by LDV (the black solid line), the measurement by a flush-mounted hot film (the yellow solid line) using a standard calibration technique shows a significantly smaller fluctuation magnitude. However, the measurement by the same hot film with NLR calibration (the red solid line) displays a fluctuation magnitude comparable to that of the LDV results. A 0.5 s time series for examining the details is shown in figure 4(b). Again, the wall-shear stress measured by the hot film using a standard calibration technique fluctuates with a reduced magnitude compared with the LDV results, while the measurement using NLR calibration recovers in magnitude and fits well with the LDV results with minor differences. These differences could be attributed to the slightly different measurement volumes of the LDV and hot-film probe and their different wall-normal measurement locations (i.e. the LDV probe is located at $y^+$ = 3.6 whereas the hot-film probe is located at $y^+ = 0$ ). It should be noted that in the present study, the LDV did not measure wall-shear stress directly but instead inferred it from the wall-normal gradient of the streamwise velocity from within the viscous sublayer using Newton’s law of viscosity, i.e. $\tau = \mu ( {{\rm d}u}/{{\rm d}y})$ , for $y^+\lt 5$ . As shown in figure 5, the wall-shear stress signals computed from DNS velocity data using the same approach at different wall-normal locations are plotted together. Here, the wall-shear stress corresponding to $y^+=0$ was calculated from the instantaneous velocity closest to the wall using $\tau = \lim \nolimits _{y^+ \to 0} \mu ({{\rm d}u}/{{\rm d}y})$ . Although the time series from different locations exhibit similar trends, slight discrepancies remain, suggesting that even within the viscous sublayer region, the inferred wall-shear stress depends on the precise wall-normal position of the measurement. The amplitude of these wall-shear stress signals decreases slightly with increasing wall-normal distance, indicating that the statistical moments at different wall-normal locations differ marginally. As a result, the wall-shear stress measured by the hot film, when calibrated using the moments obtained from LDV at $y^+=3.6$ , can be expected to contain a small systematic deviation.

Figure 6.

Comparison of power spectra of wall-shear stress measured by different techniques. (a) Dimensional spectra from experiments at $\textit{Re}_\theta = 3600$ . (b) Inner-variable normalised spectra from experiments at $\textit{Re}_\theta = 3600$ compared with DNS data at $\textit{Re}_\theta = 2200$ . (Hot film: 55R45.)

Figure 6. Long description

Two line graphs compare power spectra of wall-shear stress measured by different techniques. Panel A: A line graph displays dimensional spectra from experiments. The x-axis represents frequency (f) in Hertz (Hz) on a logarithmic scale ranging from 10^1 to 10^3. The y-axis represents the power spectral density (φ_ττ(f)) in Pascals squared per Hertz (Pa^2 Hz^-1) on a logarithmic scale ranging from 10^-12 to 10^-4. The graph includes multiple lines representing different measurement techniques: black for LDV at y+ = 3.6, blue for hot film (NLR, slotting technique), orange for hot film (NLR, Welch PSD), yellow for hot film (standard calibration, Welch PSD), and purple for DNS data. Panel B: Another line graph shows inner-variable normalised spectra from experiments compared with DNS data. The x-axis represents normalised frequency (f+) on a logarithmic scale ranging from 10^-3 to 10^-1. The y-axis represents normalised power spectral density (φ_ττ+(f+)) on a logarithmic scale ranging from 10^-6 to 10^2. The same color scheme is used to differentiate the measurement techniques. The graphs illustrate how different methods of measuring wall-shear stress compare in terms of their power spectra, highlighting variations and similarities across techniques.

The power spectral density (PSD) of the wall-shear stress signals is presented in figure 6(a). The black line represents the LDV result, which was calculated using the variable window slotting technique proposed by Tummers & Passchier (Reference Tummers and Passchier1996) to account for the irregular sampling of the LDV data. The red and blue lines were obtained from the hot-film signals calibrated using the NLR technique. The red line was calculated using the conventional PSD approach based on Welch’s method (Welch Reference Welch1967), while the blue line was calculated using the slotting technique after downsampling the hot-film data to match the irregular LDV sampling. Comparison of the two curves shows good agreement up to around 1 kHz, indicating that the slotting technique provides a reliable spectral estimate within its resolvable frequency range. The reduced spectral content beyond 1 kHz in the blue curve is a consequence of the limited sampling frequency of the LDV near the wall. The yellow line shows the PSD of the wall-shear stress signal measured by the hot film using standard calibration. Although the overall trend is similar, its magnitude is more than an order of magnitude lower than that obtained by LDV. In contrast, the NLR-calibrated hot-film and LDV results agree closely across the entire overlapping frequency range, confirming that the NLR calibration successfully recovers the amplitude of the wall-shear stress signal and provides a reliable measurement of its temporal behaviour.

Figure 6(b) presents the PSDs of the wall-shear stress signals normalised using inner variables to facilitate comparison with DNS results. The black, red and yellow lines correspond to the LDV, NLR-calibrated hot-film and standard-calibrated hot-film results at $\textit{Re}_\theta =3600$ , respectively, while the purple line represents DNS data at $\textit{Re}_\theta =2200$ . Although the Reynolds numbers are not identical, no significant Reynolds-number dependence of the normalised PSD was observed within the range $2200 \lt \textit{Re}_\theta \lt 3600$ , as shown later in figure 30(a). The standard-calibrated hot film continues to show a lower energy level compared with the LDV and DNS results, which is unsurprising due to the probe’s intrinsic inability to correctly capture true wall-shear stress fluctuations with a standard calibration process. In contrast, the NLR-calibrated hot film agrees closely with the LDV and DNS spectra across the resolvable frequency range. Furthermore, the NLR-calibrated hot film can capture spectral energy beyond 1 kHz, i.e. for approximately $f^+ \gt 0.1$ , where the LDV signal becomes limited, although the fluctuation energy in this frequency range is relatively small compared with the overall spectrum (see figure 30 b), and its distribution follows the same trend as that observed in the DNS results.

6. Nonlinear regression using estimated moments

Since the first four moments are the only inputs to the NLR technique, the accuracy of the calibration depends on how accurately these moments represent the true wall-shear stress. The first moment of the wall-shear stress is the mean, which is obtained by a linear fit of the near-wall streamwise velocity profiles as shown in figure 2. Therefore, it remains constant when measured at different wall-normal locations in the linear region. For this reason, we only discuss how the second to fourth moments of the wall-shear stress change with varying measurement locations within the viscous sublayer.

Figure 7(a) shows the profiles of the second moment of the viscous-scaled wall-shear stress time series, i.e. $\tau ^+ = \tau / \overline {\tau }$ (Alfredsson et al. Reference Alfredsson, Johansson, Haritonidis and Eckelmann1988), obtained by LDV and DNS at the same Reynolds number. The two profiles collapse, showing excellent agreement between the experiments and simulations. In both cases, the second moment linearly increases as the wall is approached until $y^+ \approx 2$ . Closer to the wall, the DNS results show that this moment approximately levels off to a plateau, as indicated by the black dotted line. Experimentally, it is usually very difficult to measure the instantaneous velocity in the viscous sublayer, which makes the determination of the second- or higher-order moments of the wall-shear stress particularly challenging. In the present study, a careful optical set-up with a beam expander was employed to reduce the measurement volume, enabling measurements down to $y^+ = 2.5$ (equivalent to $y = 89\,\unicode{x03BC}$ m at $\textit{Re}_\theta = 3600$ ). However, no clear plateau was observed in the experimental data. Therefore, to estimate the true second moment of the wall-shear stress, the data obtained within $3 \lt y^+ \lt 5$ were used, and a linear fit was applied to extrapolate the values to $y^+ = 2.1$ , taken here as the onset of the plateau region. An example using DNS data at $\textit{Re}_\theta = 2200$ is shown by the black dashed line in figure 7(a), indicating that the estimated value at $y^+=2.1$ coincides with that at the wall ( $y^+=0$ ). Figure 7(b) presents the dimensional second-moment profiles of the wall-shear stress time series measured by LDV at different Reynolds numbers. Blue, red and yellow circles represent results at $U_\infty = 6.1$  m s⁻¹, $U_\infty = 8.1$  m s⁻¹ and $U_\infty = 10.8$  m s⁻¹, respectively, showing that a higher free-stream velocity leads to larger second-moment values. The corresponding blue, red and yellow stars indicate the estimated second moments obtained by the linear-fit extrapolation to $y^+=2.1$ as introduced above.

Figure 7.

(a) Distribution of the viscous-scaled second moment of the wall-shear stress obtained by DNS and LDV at the same Reynolds number. (b) Dimensional second moment of the wall-shear stress measured by LDV at different wall-normal locations at different Reynolds numbers.

Figure 8.

(a) Distribution of the third moment of the wall-shear stress obtained by DNS and LDV at the same Reynolds number. (b) Distribution of the third moment of the wall-shear stress obtained by LDV at different Reynolds numbers.

Figure 9.

(a) Distribution of the fourth moment of the wall-shear stress obtained by DNS and LDV at the same Reynolds number. (b) Distribution of the fourth moment of the wall-shear stress obtained by LDV at different Reynolds numbers.

Figure 10.

Comparison of instantaneous wall-shear stress measured by different techniques at $\textit{Re}_\theta = 3600$ . (Hot film: 55R45.)

Figure 11.

The PDFs of the inner-scale normalised wall-shear stress measured by different techniques at $\textit{Re}_\theta =2200$ . (Hot film: 55R47.)

Figures 8(a) and 9(a) show the third- and fourth-order moments of the wall-shear stress obtained from DNS and LDV as functions of the wall-normal location at matching Reynolds numbers. For both moments, a similar trend to that of the second moment is observed: the values increase approximately linearly towards the wall and then approach a plateau. To estimate the true third- and fourth-order moments of the wall-shear stress, the same linear-fit technique used for the second moment was applied, fitting the profiles within $3 \lt y^+ \lt 5$ , as indicated by the black dashed lines in the figures. The extrapolated values from these fits correspond to $y^+ = 2.7$ and $y^+ = 2.0$ for the third and fourth moments, respectively. Interestingly, although the profiles measured by LDV exhibit a slightly gentler slope compared with the DNS results as $y^+$ decreases, the extrapolated values agree closely with those obtained from DNS. Figures 8(b) and 9(b) show the extrapolated third- and fourth-order moments of wall-shear stress measured by LDV at different Reynolds numbers. The results indicate that these higher-order moments show negligible sensitivity to Reynolds number within the range tested.

A comparison of the instantaneous wall-shear stress time series measured by a hot-film probe calibrated using moments obtained from the LDV signal at $y^+ = 3.6$ and using the estimated moments was conducted, as shown in figure 10. Clearly, both signals fit well with the LDV results, indicating that such variations in the moments have only a minor influence on the recovered wall-shear stress. As shown in the zoomed view in figure 10, the wall-shear stress measured by the hot-film using estimated moments exhibits a larger fluctuation magnitude than that obtained using moments directly from LDV. This behaviour is expected since a higher value of the second moment was applied during the NLR calibration.

For direct comparison with the DNS results, figure 11 presents the PDFs of the viscous-scaled wall-shear stress measured at $\textit{Re}_\theta = 2200$ . The experimental and numerical profiles show good collapse. The inset highlights that the PDFs measured by the hot film calibrated using estimated moments are closer to those of the wall-shear stress signal obtained from DNS than those calibrated using LDV moments. A slight discrepancy remains between the DNS and LDV results measured at $y^+=3$ , which can be attributed to differences in the measurement volumes. The LDV probe volume corresponds to 0.73 and 5.4 viscous units in the streamwise and spanwise directions, respectively, whereas the DNS data were averaged over a grid size of 12.0 and 8.1 viscous units, respectively.

7. Nonlinear regression using moments obtained based on Reynolds number

Figure 12 shows the relationship between the second moment of the wall-shear stress and the Reynolds number $\textit{Re}_\theta$ . For both the DNS and experimental results, together with data from previous studies (Komminaho & Skote Reference Komminaho and Skote2002; Simens et al. Reference Simens, Jiménez, Hoyas and Mizuno2009; Schlatter & Örlü Reference Schlatter and Örlü2010a ), the viscous-scaled second moment increases with increasing Reynolds number. The present experimental and numerical results collapse well with the empirical relationship given by

(7.1) \begin{equation} \sigma (\tau ^+) = 0.300+0.015\ln (\textit{Re}_{\theta }), \end{equation}

which is derived from

(7.2) \begin{equation} \sigma (\tau ^+) = 0.298+0.018\ln (\textit{Re}_{\tau }) \end{equation}

with $\textit{Re}_\tau = 1.13\times \textit{Re}_{\theta }^{0.843}$ , originally proposed by Schlatter & Örlü (Reference Schlatter and Örlü2010a ) for turbulent boundary layers. These relationships enable the second moment to be estimated directly from $\textit{Re}_\theta$ or $\textit{Re}_\tau$ , which are more readily obtained than direct measurements of wall-shear stress fluctuations in practice.

Figures 13 and 14 present the third and fourth moments of the wall-shear stress as functions of Reynolds number. Both moments show an increasing trend with Reynolds number similar to that observed for the second moment. Based on this behaviour, empirical relationships of the same functional form are proposed for the third and fourth moments, as given by

(7.3) \begin{equation} S(\tau ) = 0.502+0.067\ln (\textit{Re}_{\theta }) \end{equation}

and

(7.4) \begin{equation} F(\tau ) = 1.536+0.426\ln (\textit{Re}_{\theta }), \end{equation}

where $S$ and $F$ represent the third and fourth moments of wall-shear stress, respectively. The parameters of these fits were determined using the present DNS and experimental data together with data from previous studies (Schlatter & Örlü Reference Schlatter and Örlü2010a ; Diaz-Daniel et al. Reference Diaz-Daniel, Laizet and Vassilicos2017), giving a total of 30 data points.

Figure 12.

Distribution of the inner-scale normalised second moment of wall-shear stress obtained by DNS and LDV at different Reynolds numbers.

Figure 13.

Distribution of the third moment of wall-shear stress obtained by DNS and LDV at different Reynolds numbers.

Figure 14.

Distribution of the fourth moment of wall-shear stress obtained by DNS and LDV at different Reynolds numbers.

With these empirical equations, the statistical moments required for the NLR process can be obtained without the need for velocity measurements within the viscous sublayer. For instance, the first moment (the mean wall-shear stress) can be readily estimated from the flow using methods such as the Clauser chart (Clauser Reference Clauser1956) or the Musker profile (Musker Reference Musker1979; Kendall & Koochesfahani Reference Kendall and Koochesfahani2008). The Reynolds numbers used in these equations can be determined through various approaches, including evaluating the full boundary-layer mean velocity profile to compute the momentum thickness for $\textit{Re}_\theta$ or to compute the boundary-layer thickness for $\textit{Re}_\tau$ . An example calibration curve is shown in figure 15 for $U_\infty = 10.8$  m s⁻¹, corresponding to $\textit{Re}_\theta = 3600$ , where the first moment was obtained from the same flow using LDV data and the higher moments were estimated from the empirical equations. For comparison, the NLR calibration curves based on LDV-derived moments (§ 5) and those using extrapolated moments (§ 6) are also shown. The three NLR calibration curves exhibit only minor differences, demonstrating that a flush-mounted hot-film probe can be calibrated for absolute instantaneous measurements of wall-shear stress using the proposed empirical relationships, with only the Reynolds number of the flow as input.

Figure 15.

A comparison of calibration curves obtained using different moments. (Hot film: 55R45, $\textit{Re}_\theta = 3600$ .)

The wall-shear stress signal measured by the hot film calibrated with only the first moment and Reynolds number obtained from the flow is shown in figure 16. The black solid line represents the LDV measurement at $y^+=3.6$ . The red line corresponds to the flush-mounted hot-film probe calibrated using LDV moments (§ 5), while the purple line corresponds to the hot film calibrated with moments obtained from the empirical equations. As a result, the purple line is almost indistinguishable in the figure because it almost completely overlaps with the red line. The corresponding PDFs of the wall-shear stress time series obtained by the two calibration approaches are shown in figure 17, where the curves collapse onto each other, which demonstrates the reliability of the calibration when only the first moment and Reynolds number are used.

Figure 16.

Comparison of instantaneous wall-shear stress measured by different techniques at $\textit{Re}_\theta = 3600$ . (Hot film: 55R45.)

Figure 17.

The PDFs of the wall-shear stress measured using different techniques at $\textit{Re}_\theta = 3600$ . (Hot film: 55R45.)

Figure 18.

Power spectral densities of wall-shear stress signals obtained from hot-film data calibrated using different sets of statistical moments. (Hot film: 55R45, $\textit{Re}_\theta = 3600$ .)

Figure 18 presents the PSDs of the wall-shear stress signals obtained from hot-film data calibrated using three different sets of moments. The red line represents the result calibrated with moments measured by LDV, the blue line corresponds to the calibration using the estimated moments introduced in the previous section,and the purple line corresponds to the calibration using the moments estimated from the empirical Reynolds-number-based relations discussed in the present section. All three spectra show very similar distributions. The blue line lies slightly above the red line, consistent with the observation in § 6 that the wall-shear stress measured by the hot film using estimated moments exhibits a slightly larger fluctuation magnitude than that obtained using LDV moments. This behaviour is expected since a higher value of the second moment was applied during the NLR calibration. The purple line almost completely overlaps with the red line, consistent with the comparison of PDFs shown in figure 17, providing further evidence of the accuracy of the calibration when only the first moment and Reynolds number are used. These results demonstrate that all three approaches for obtaining the moments lead to nearly identical wall-shear stress spectra, indicating that in practical applications, any method can be adopted without significant differences in outcome.

8. Case study I: detection and characterisation of hibernating turbulence in turbulent boundary-layer flows

Hibernating turbulence has been identified as a dynamically distinct state within near-wall turbulence, characterised by significantly reduced wall-shear stress and attenuated near-wall activity (Xi & Graham Reference Xi and Graham2012; Kushwaha et al. Reference Kushwaha, Park and Graham2017; Whalley et al. Reference Whalley, Park, Kushwaha, Dennis, Graham and Poole2017, Reference Whalley, Dennis, Graham and Poole2019). To date, experimental evidence of these low-drag events has been primarily restricted to transitional or low-Reynolds-number internal liquid flows. In such cases, the high thermal conductivity of the working fluid relative to the sensor substrate facilitates accurate hot-film measurements. However, the direct observation and characterisation of hibernating turbulence in external aerodynamic flows (such as turbulent boundary layers in wind tunnels) have largely remained elusive (Ryu et al. Reference Ryu, Davis, Park, Zhang and Yoo2021). This is due to the significant heat transfer into the substrate when operating in air, which severely attenuates the instantaneous signal. Case study I addresses this limitation by leveraging the NLR calibration technique to recover high-fidelity wall-shear stress measurements. This approach enables the detection and characterisation of hibernating events in a turbulent boundary layer at a Reynolds number an order of magnitude higher than the previously reported channel flow investigations.

Experiments were conducted in the same wind-tunnel facility as previously described at a free-stream velocity of $U_\infty = 6.0$  m s⁻¹, corresponding to a friction Reynolds number of $\textit{Re}_{\tau } = 690$ and a momentum-thickness Reynolds number of $\textit{Re}_{\theta } = 1930$ . To investigate the spatio-temporal characteristics of low-drag states, instantaneous wall-shear stress was measured using the NLR-calibrated hot-film sensor, while the instantaneous streamwise velocity was simultaneously acquired via LDV positioned directly above the hot film. The LDV probe was traversed to multiple wall-normal locations spanning the near-wall and logarithmic regions ( $y^+ = 3.5$ to $150$ ). Given the stochastic and infrequent nature of hibernating turbulence, data were acquired for a duration of two hours at each wall-normal position. This extensive sampling period was essential to capture a sufficient number of hibernating events to ensure the statistical convergence of the conditional samples and ensemble averages. Hibernating events were identified from the reconstructed wall-shear stress time series using a dual-criterion threshold. An event was defined as a period during which the instantaneous wall-shear stress dropped at least 10 % below the mean value ( $\tau _w \leq 0.9\overline {\tau }_w$ ) and remained below this threshold for a continuous duration exceeding $T^+ ={Tu_\tau ^2}/{\nu }= 200$ .

Figure 19(a) shows the conditionally sampled and ensemble-averaged wall-shear stress signals corresponding to hibernating turbulence events with durations of $T^+ \gt 200$ . Here, $\langle \boldsymbol{\cdot }\rangle _{{cond}}$ represents conditional ensemble averages obtained by averaging over all detected events. Each event is aligned such that $t^+ = 0$ corresponds to the instant at which the instantaneous wall-shear stress first drops 10 % below the mean wall-shear stress. Individual events are represented by thin grey lines, while the ensemble average is shown by the thick black line. From $t^+ = 0$ , a rapid decrease in both the mean level and fluctuation intensity of the wall-shear stress is observed, followed by a plateau region corresponding to the hibernating state, located approximately within $10 \leq t^+ \leq 200$ . A substantial number of events persist for significantly longer durations than $T^+ = 200$ . Examples of such long-duration events are highlighted in figure 19(a), with the red and blue lines indicating individual events with $T^+ \gt 300$ and $T^+ \gt 400$ , respectively. Figure 19(b) presents the results for the subset of events with longer durations selected from figure 19(a). Individual events are again shown as thin grey lines, with the ensemble average represented by the thick black line. Compared with the $T^+ \gt 200$ case, the $T^+ \gt 400$ events exhibit a more sustained low-stress plateau, yet the mean ensemble-averaged wall-shear stress is nearly identical in both cases.

Figure 19.

Wall-shear stress signals conditioned on hibernating turbulence events: (a) events with $T^+ \gt 200$ , (b) events with $T^+ \gt 400$ . Individual events are shown as thin grey lines and the ensemble average is represented by the thick black line. The red and blue lines indicate individual events with $T^+ \gt 300$ and $T^+ \gt 400$ , respectively. The ensemble averages are based on 6554 and 624 events for (a) and (b), respectively.

Figure 19. Long description

Two line graphs depict wall-shear stress signals conditioned on hibernating turbulence events. Panel A: This line graph shows individual events as thin grey lines and the ensemble average as a thick black line. The red and blue lines indicate individual events with specific conditions. The x-axis is labeled t+ and ranges from -100 to 500. The y-axis is labeled τ/τ, τ/τh and ranges from 0 to 3. Panel B: This line graph also shows individual events as thin grey lines and the ensemble average as a thick black line. The x-axis is labeled t+ and ranges from -100 to 500. The y-axis is labeled τ/τ, τ/τh and ranges from 0 to 3. The ensemble averages are based on 6554 and 624 events for Panel A and Panel B, respectively.

Figure 20.

Streamwise velocity conditioned on hibernating turbulence events: (a) streamwise velocity signals measured at $y^+ = 15$ , with individual events shown as thin grey lines and the ensemble average as the thick black line; the red and blue lines indicate conditionally sampled instantaneous streamwise velocity during hibernating events with durations of $T^+ \gt 300$ and $T^+ \gt 400$ , respectively. (b) Ensemble-averaged velocity signals at different wall-normal locations.

Figure 20(a) shows the time series of the streamwise velocity measured by LDV at $y^+ = 15$ , conditioned on the detected hibernating events. The velocity is normalised using the usual time-averaged friction velocity. Individual events are represented by thin grey lines, while the ensemble average is shown by the thick black line. A distinct difference is observed between the instantaneous evolution of the streamwise velocity and that of the wall-shear stress. While the detection of hibernating events is based on the wall-shear stress signal remaining below a prescribed threshold for a given duration, the corresponding streamwise velocity exhibits substantial fluctuations around the ensemble average, as illustrated by the highlighted blue and red lines in figure 20(a). Figure 20(b) presents the ensemble-averaged streamwise velocity signals at multiple wall-normal locations, again normalised using the usual time-averaged friction velocity. The black dotted lines indicate the time interval corresponding to the plateau region of the detected hibernating events in the wall-shear stress signal, which is estimated from the cross-correlation between the LDV velocity signal and the wall-shear stress signal. As the wall-normal distance increases, the corresponding low-velocity plateau in the LDV signal progressively shifts to earlier times. This behaviour is consistent with the inclination of turbulence structures in the turbulent boundary layer. At the same time, the coherence between the wall-shear stress and the streamwise velocity progressively weakens with increasing wall-normal distance, indicating a reduced coupling between near-wall dynamics and outer-layer flow structures, until ultimately there is a loss of communication with the wall.

Figure 21.

Wall-normal profiles of boundary-layer properties: (a) streamwise velocity profiles, (b) streamwise turbulence intensity profiles.

Figure 21. Long description

The image contains two line graphs labeled as Panel A and Panel B. Panel A shows streamwise velocity profiles, while Panel B displays streamwise turbulence intensity profiles. Panel A: The line graph in Panel A plots the dimensionless streamwise velocity (u+) against the dimensionless wall-normal distance (y+). The x-axis is labeled y+ and ranges from 10^0 to 10^3. The y-axis is labeled u+ and ranges from 0 to 30. The graph includes multiple data series: DNS with Reθ = 2000 (solid black line), experimental data with U∞ = 6.08 m/s and Reθ = 1930 (yellow circles), a linear profile (dotted black line), von Kármán log law (dashed black line), a shift from von Kármán log law (dashed red line), Virk MDR log law (dotted red line), and hibernating events (T+ > 200) normalised using τ_h (blue stars). Panel B: The line graph in Panel B plots the dimensionless streamwise turbulence intensity (σ(u+)) against the dimensionless wall-normal distance (y+). The x-axis is labeled y+ and ranges from 10^0 to 10^3. The y-axis is labeled σ(u+) and ranges from 0 to 3. The graph includes multiple data series: DNS with Reθ = 2000 (solid black line), experimental data with U∞ = 6.08 m/s and Reθ = 1930 (yellow circles), hibernating events (T+ > 200) normalised using τ_h (blue stars), and hibernating events (T+ > 200) normalised using τ (green stars). The graphs show wall-normal profiles of boundary-layer properties, including the mean streamwise velocity and streamwise turbulence intensity, comparing DNS, experimental results and conditionally sampled hibernating-event data.

Figure 21(a,b) shows the mean and fluctuating streamwise velocity profiles, respectively. The black solid line represents the DNS data, while the yellow circles denote the usual time-averaged experimental results. The blue symbols correspond to the conditionally sampled and ensemble-averaged streamwise velocity profiles during hibernating turbulence events with durations of $T^+ \gt 200$ . Here, both the streamwise velocity and the wall-shear stress used for normalisation are averaged over the lower than average hibernating interval, i.e. over the range indicated by the black dotted lines in figure 20(b). The red dotted line indicates the maximum drag reduction (MDR) asymptote, $U^+ = 11.7 \ln y^+ - 17$ (Virk Reference Virk1971), while the red dashed line represents a shifted von Kármán log law. In the viscous sublayer, the blue symbols still follow the linear profile, indicating that the near-wall mean shear remains governed by viscous scaling. At intermediate wall-normal locations, the conditionally sampled and ensemble-averaged streamwise velocity profile approaches the MDR asymptote, which is usually associated with the highly drag-reduced flows of polymeric solutions rather than the Newtonian fluid flows shown here. This behaviour is consistent with previous observations of hibernating turbulence in channel flows from both experiments (Whalley et al. Reference Whalley, Park, Kushwaha, Dennis, Graham and Poole2017, Reference Whalley, Dennis, Graham and Poole2019) and DNS (Kushwaha et al. Reference Kushwaha, Park and Graham2017). Notably, these new boundary-layer observations are at a friction Reynolds number approximately one order of magnitude higher than the previous channel flow investigations. This suggests that the hibernating state exhibits a universal near-wall structure that is largely independent of flow configuration and persists to fully developed wall-bounded turbulent flows. Further away from the wall, the blue symbols deviate from the MDR asymptote to follow the shifted von Kármán log law, suggesting that the outer part of the boundary layer retains a logarithmic structure, before the final point peeling away at the start of the wake region for the lower-Reynolds-number conditionally sampled boundary layer. Here, the shifted von Kármán log law is a consequence of scaling with the lower than average wall-shear stress. The streamwise turbulence intensity profiles are shown in figure 21(b), normalised with either the usual time-averaged wall-shear stress (green symbols) or the lower than average wall-shear stress (blue symbols) found during the hibernating turbulence plateau. Both forms of scaling show a significant reduction in streamwise turbulence intensity throughout the boundary layer, more so when scaling with the usual time-averaged wall-shear stress.

9. Case study II: reconstruction of wall-shear stress from near-wall hot-wire measurements

To further demonstrate the capability of the NLR-calibration technique in extracting additional physical information from existing measurements, a second case study is presented in which historical hot-wire data are revisited. While hot-wire anemometry is traditionally used to measure velocity fluctuations, the measured signals inherently contain information related to wall-shear stress when the sensor is located within the viscous sublayer. However, such information is not directly accessible using conventional calibration approaches, as hot-wire measurements close to the wall are significantly affected by wall heat conduction, which leads to calibration inaccuracies (Hutchins & Choi Reference Hutchins and Choi2002). In this case study, the NLR technique is applied to reconstruct wall-shear stress from hot-wire measurements obtained within the viscous sublayer of a zero-pressure-gradient turbulent boundary layer. This enables the transformation of the hot-wire voltage signal into wall-shear stress, thereby providing new physical insight into near-wall turbulence dynamics that cannot be obtained from the original interpretation of the data. This example highlights the potential of the NLR approach as a post-processing tool for reinterpreting historical datasets and extending their scientific value beyond their original scope.

The data examined here were acquired using a multi-channel hot-wire probe system, consisting of a spanwise hot-wire rake and a single I-type hot-wire probe located beneath the centre of the rake, as shown in figure 22(a). The measurements were originally conducted to investigate the interaction between large-scale structures in the logarithmic region and bursting events in the near-wall region (Chen et al. Reference Chen, Iwano, Sakai and Ito2021). The experimental set-up is identical to that reported in Chen et al. (Reference Chen, Iwano, Sakai and Ito2021), although the specific flow conditions differ. For completeness, a brief summary of the experimental conditions is provided here. The mean streamwise velocity profile measured by the single I-type hot-wire probe is shown in figure 22(b). The free-stream velocity is $U_\infty = 10.84$  m s⁻¹ and the Reynolds number based on the momentum thickness is $\textit{Re}_\theta = 2863$ . The friction velocity and the wall-normal position of the hot wire were estimated by fitting the measured mean streamwise velocity profile to DNS data at a comparable Reynolds number over the range $6 \leq y^+ \leq 180$ , as highlighted by green symbols in figure 22(b). Within this region, the mean streamwise velocity profile is largely insensitive to the Reynolds number, as shown in figure 2(a), and the hot-wire data are not affected by conduction to the wall. The friction velocity is $u_\tau = 0.430$  m s⁻¹ and the boundary-layer thickness is $\delta = 33.8$  mm. The hot wires have a sensing length of $1$  mm and a diameter of $5\,\unicode{x03BC}$ m. The wall-normal distance between the single I-type probe and the rake is fixed at $5.1$  mm.

Figure 22.

(a) Multi-channel hot-wire probe, consisting of a spanwise hot-wire rake and a single I-type hot-wire probe located beneath the centre of the rake. (b) Mean streamwise velocity profile measured by the single I-type hot-wire probe.

The dataset used for the NLR reconstruction corresponds to the lowest wall-normal measurement location of the single I-type probe, indicated by the red marker in figure 22(b). At this position, the nominal wall-normal location of the probe is approximately $y^+ = 1.2$ , while the rake is located at $y^+ = 148$ , within the logarithmic region. For the NLR implementation, the first moment is determined from the friction velocity, while the second to fourth moments are obtained from the empirical relations based on $\textit{Re}_\theta$ shown in (7.1), (7.3) and (7.4). Figure 23 shows a comparison of the wall-shear stress signals obtained using the conventional velocity calibration and the NLR-based calibration, respectively. It can be observed that both the mean and fluctuation magnitude of the wall-shear stress estimated using the velocity calibration are significantly larger than those obtained using the NLR approach. This discrepancy arises from the limitations of the conventional method when applied in the viscous sublayer. In the velocity-based approach, the wall-shear stress is estimated as $\tau = \mu ({u}/{y})$ . However, near the wall, the measured streamwise velocity is overestimated due to conduction to the wall, introducing errors in the numerator. At the same time, the nominal wall-normal position of the single I-type hot-wire probe at the lowest measurement location may not be accurate. Once the probe prongs come into contact with the wall, the wire no longer follows the traverse motion reliably, leading to additional uncertainty in the wall-normal position and consequently in the denominator. The combined effect renders the wall-shear stress obtained using the velocity calibration highly unreliable in the viscous sublayer. In contrast, the NLR approach is not affected by the uncertainty in the wall-normal location of the sensor, provided that the probe remains within the viscous sublayer. This is because the NLR method directly establishes a relationship between the measured voltage signal and the wall-shear stress, without requiring an intermediate conversion to velocity. As a result, it provides a more robust and physically consistent estimate of wall-shear stress in the near-wall region.

To verify the accuracy of the wall-shear stress reconstructed using the NLR technique, the viscous-scaled PSD of the signals is computed and compared with the hot film and DNS results, as shown in figure 24(a). The PSD obtained from the hot-wire measurements using NLR collapses well with that obtained from the hot film using NLR, and both are in good agreement with the DNS data, despite the difference in Reynolds number, probe type and facility. This consistency indicates that both the mean and the fluctuations of the wall-shear stress are successfully recovered from the hot-wire data using the NLR approach. For comparison, the PSD of wall-shear stress obtained from the hot wire using the conventional velocity calibration is also included, which deviates significantly from both the NLR-based results and the DNS data. This discrepancy arises because both the mean and the fluctuations of the wall-shear stress are inaccurately estimated when using the standard velocity calibration approach.

Figure 23.

Comparison of wall-shear stress signals obtained from hot-wire measurements using velocity calibration and NLR.

Figure 24.

Viscous-scaled PSD of wall-shear stress: (a) comparison of PSD obtained using different measurement and calibration techniques, (b) comparison of PSD obtained by applying NLR to a white-noise signal and hot-wire signals measured at different wall-normal locations.

Figure 24. Long description

Two line graphs depict the viscous-scaled power spectral density (PSD) of wall-shear stress. Panel A compares PSD obtained using different measurement and calibration techniques. The x-axis represents the frequency in viscous units (f+) on a logarithmic scale ranging from 10^-3 to 10^-1. The y-axis represents the PSD of wall-shear stress (φ_ττ(f+)) on a logarithmic scale ranging from 10^-6 to 10^2. The graph includes multiple lines representing different data sources: τ measured by hot wire based on velocity calibration at Re_θ = 2863, τ measured by hot wire using NLR at Re_θ = 2863, τ measured by hot film using NLR at Re_θ = 3600, and τ from DNS data at Re_θ = 2200. Panel B compares PSD obtained by applying NLR to a white-noise signal and hot-wire signals measured at different wall-normal locations. The x-axis and y-axis are the same as in Panel A. The graph includes multiple lines representing different wall-normal locations (y+) and a white-noise signal. The legend indicates the specific y+ values and the application of NLR to hot-film and hot-wire data. Annotations indicate increasing y+ values in Panel B.

Since the PSD of wall-shear stress obtained from the hot-wire and hot-film measurements using the NLR approach collapse well with each other, it may be questioned whether the NLR procedure artificially enforces this agreement. To address this concern, the NLR method was also applied to a white noise signal and hot-wire voltage signals measured at different wall-normal locations. The resulting PSDs of these reconstructed signals are shown in figure 24(b). Here, only the signals measured within the viscous sublayer, i.e. those measured at $y^+ = 1.2$ and $y^+ = 4.1$ , exhibit close agreement with the hot-film-based result across the entire frequency range. With increasing wall-normal distance, the reconstructed signals exhibit progressively higher energy at high frequencies, leading to increasing deviation from the hot-film result. This behaviour reflects the fact that the signals measured away from the wall are no longer representative of wall-shear stress dynamics. These results demonstrate that the NLR approach does not artificially force the reconstructed signals to follow a predefined spectral trend. Instead, it preserves the intrinsic characteristics of the input signal. Therefore, if the measured signal does not originate from wall-shear stress related dynamics, the application of NLR will not yield a PSD consistent with that of the true wall-shear stress.

In the original study of Chen et al. (Reference Chen, Iwano, Sakai and Ito2021), the well-known variable-interval time-averaging (VITA) method (Blackwelder & Kaplan Reference Blackwelder and Kaplan1976) was applied to investigate the relationship between VITA events in the logarithmic region and those in the buffer region. The former were associated with the passage of hairpin vortices (Adrian et al. Reference Adrian, Meinhart and Tomkins2000), while the latter correspond to the well-known bursting phenomena (Kim et al. Reference Kim, Kline and Reynolds1971; Lee et al. Reference Lee, Eckelman and Hanratty1974; Blackwelder & Eckelmann Reference Blackwelder and Eckelmann1979), which are recognised as a primary source of turbulence production (Lu & Willmarth Reference Lu and Willmarth1973). It was found that the presence of hairpin vortices in the logarithmic region slightly enhances bursting activity near the wall. However, the extent to which hairpin vortices in the logarithmic region quantitatively influence the wall-shear stress remains unclear. To address this question, the same VITA technique as used in Chen et al. (Reference Chen, Iwano, Sakai and Ito2021) was applied to the streamwise velocity fluctuations measured by each probe of the hot-wire rake. An example of the fluctuation velocity signal measured by the central probe in the rake and the corresponding wall-shear stress reconstructed from the single I-type probe are shown in figures 25(a) and 25(b), respectively. The red dotted lines indicate the time instants at which VITA events are detected in the logarithmic region. The blue- and red-shaded windows have a width of 6.78 ms, corresponding to a streamwise distance of approximately $1.5\delta$ under Taylor’s hypothesis, where the local mean streamwise velocity in the logarithmic region is taken as the convection velocity. It can be observed that the detected VITA events are associated with sudden accelerations in the instantaneous streamwise velocity signal, which are characteristic of sweep and ejection events linked to hairpin vortices. The black dotted lines in figure 25(b) denote the estimated arrival time of these events at the wall, accounting for the inclination of turbulent structures. This time delay is determined from the streamwise shift between the stagnation points identified in the velocity field in the logarithmic region and the wall-shear stress field, as shown in figures 27(a) and 27(b). As shown in figure 25(b), in most cases the wall-shear stress decreases during the ejection phase and approaches a local minimum around the black dotted lines. Subsequently, during the sweep phase, a significant increase in wall-shear stress is observed.

Figure 25.

Time series of velocity fluctuation and wall-shear stress: (a) velocity fluctuation measured by the central probe in the hot-wire rake at $y^+ = 148$ , (b) wall-shear stress reconstructed from the single I-type probe using NLR. Red dotted lines indicate the instants of VITA events detected in the logarithmic region, while the black dotted lines denote the corresponding arrival time at the wall.

To further quantify the influence of these VITA events on the instantaneous wall-shear stress, figure 26 shows the PDFs of the maximum wall-shear stress evaluated separately within time windows of 6.78 ms before and after each VITA event reaches the wall. These time windows correspond to the ejection (blue-shaded) and sweep (red-shaded) regions before and after the black dotted lines in figure 25(b), respectively. It is observed that the PDF of the maximum wall-shear stress during the ejection phase peaks at approximately $1.45\,\bar {\tau }$ and decays rapidly as $\tau _{\textit{max}}/\bar {\tau }$ increases beyond 3.2. In contrast, the PDF during the sweep phase peaks at a higher value of approximately $1.8\,\bar {\tau }$ and exhibits a longer tail, extending to $\tau _{\textit{max}}/\bar {\tau } = 4$ . These results indicate that VITA events in the logarithmic region can induce extremely high wall-shear stress events in the viscous sublayer, with sweep phases producing stronger extreme responses than ejection phases.

Figure 26.

The PDFs of the maximum wall-shear stress within time windows associated with VITA events detected in the logarithmic region. The PDFs are constructed from 742 individual VITA events.

To investigate the spatial footprint of hairpin vortices detected in the logarithmic region on the wall-shear stress, the ensemble-averaged velocity distribution in the logarithmic region and the corresponding wall-shear stress fluctuation distribution, conditioned on the occurrence of VITA events, are computed and shown in figures 27(a) and 27(b), respectively. The streamwise direction is reconstructed using Taylor’s hypothesis. Since VITA events are identified independently by each probe in the rake, the spanwise distributions of both velocity and wall-shear stress are reconstructed by aligning the signals according to the relative probe positions associated with each detected event. This follows a conditional reconstruction approach similar to that of Hutchins et al. (Reference Hutchins, Monty, Ganapathisubramani, Ng and Marusic2011), in which a two-dimensional streamwise velocity field was reconstructed using a single hot-wire probe in combination with a spanwise array of wall-mounted hot-film sensors. As shown in figure 27(a), a symmetric Y-shaped streamwise velocity distribution is observed, which is consistent with that observed in Chen et al. (Reference Chen, Iwano, Sakai and Ito2021). Specifically, the low-momentum region flanked by high-speed fluid corresponds to the necks of the hairpin vortex, while the location of the detected VITA event ( $x/\delta = 0$ ) can be interpreted as the stagnation region downstream of the vortex head (Adrian et al. Reference Adrian, Meinhart and Tomkins2000). A similar Y-shaped pattern is observed in the wall-shear stress fluctuation field in figure 27(b). However, due to the inclination of the turbulent structures, the corresponding stagnation point in the wall-shear stress distribution is shifted downstream by approximately $0.73\delta$ relative to that in the logarithmic region. Although both ejection and sweep events associated with hairpin vortices contribute to a positive Reynolds shear stress production, their effects on wall-shear stress are fundamentally different (Lu & Willmarth Reference Lu and Willmarth1973). Ejection events ( $u' \lt 0$ , $v' \gt 0$ ) tend to reduce the wall-shear stress, whereas sweep events ( $u' \gt 0$ , $v' \lt 0$ ) lead to an increase in wall-shear stress when averaged over many events, as illustrated in the schematic in figure 27(c). In contrast to the streamwise velocity field, where regions of high and low speed are sharply defined, the corresponding wall-shear stress distribution exhibits a more gradual variation. This is because the ensemble averaging is conditioned on VITA events detected in the logarithmic region, and the corresponding wall-shear stress signals are not perfectly aligned in time across different events. As a result, while instantaneous wall-shear stress signals (see red-shaded regions in figure 25 b) may exhibit sharp peaks, sometimes reaching up to four times the mean value before rapidly decaying, the ensemble-averaged response shows a more moderate variation as shown in figure 27(b). Additionally, the influence of the logarithmic-region events extends over a substantial streamwise distance, with both low and high wall-shear stress regions (associated with ejection and sweep events, respectively) spanning more than $3\delta$ in the streamwise direction. With reference to the data shown in figure 27(b), a fluctuation of approximately 0.022 Pa corresponds to a 10 % variation relative to the mean wall-shear stress, highlighting the contribution of hairpin-vortical flow structures to skin-friction drag from an ensemble-averaged perspective.

Figure 27.

Ensemble-averaged distributions conditioned on VITA events detected in the logarithmic region: (a) streamwise velocity fluctuation, (b) wall-shear stress (WSS) fluctuation, (c) schematic illustration of the wall-shear stress distribution associated with a hairpin vortex.