1 Introduction

In the pursuit of accurate velocity measurements in high Reynolds (Re) number turbulent boundary layer (TBL) flows, spatial averaging effects rising from the use of finite-length sensing probes (most prominently hot-wires), have been a widely acknowledged and discussed problem. As pointed out by Hutchins et al. (2009), hot-wire spatial averaging effects have been the root cause of major disagreements in high-Re streamwise velocity fluctuation profiles and trends in the literature, e.g., with respect to the scaling of the inner-peak (Örlü and Alfredsson 2013) or the existence of an outer peak in the streamwise variance profile (Alfredsson et al. 2011). Due to the importance of this phenomenon, a number of correction schemes for the missing fluctuation energy have been proposed since the seminal work by Hutchins et al. (2009). Miller et al. (2014) assessed the most relevant ones in the literature, including the one by Smits et al. (2011) based on the attached eddy hypothesis, and the one by Segalini et al. (2011) based on the relation of the spanwise Taylor microscale (\(\lambda _g\)) and the two-point streamwise velocity correlation function, finding good results for pipe, channel and zero-pressure gradient (ZPG) TBL flows.

However, most of the aforementioned hot-wire spatial resolution corrections were developed for canonical flows under the premise of universality of the small-scale energy scaled with viscous wall units (Mathis et al. 2011). The one by Segalini et al. (2011), however, might be more generally applicable, although it relies on a correlation between the attenuation and \(\lambda _g\), taking only ZPG TBL data as reference. As discussed in Sanmiguel Vila et al. (2020), this could be a potentially erroneous assumption in non-canonical flows, such as TBLs subjected to adverse pressure gradients (APGs) or flows along rough walls (Gatti et al. 2022). When the flow is decelerated due to the presence of an APG, the overall structure and dynamics of the TBL are greatly affected: internal shear layers appear, with large-scale structures leaving a larger imprint at the wall, and small-scale energy becoming more relevant in the outer region of the TBL as discussed in Sanmiguel Vila et al. (2020) and Pozuelo et al. (2022). The introduction of the pressure gradient, summed with the already-known high Reynolds number effects, leads to an increase in the fluctuations in the outer region of the TBL, which depending on the APG strength, may show a dominant peak in the streamwise component of the Reynolds stresses (Sanmiguel Vila et al. 2020). Nevertheless, the conclusions drawn from spatial attenuation studies in canonical flows have been directly transferred over to the study of APG TBLs. For example, it is common practice to use the same hot-wire length (in viscous units) for flows with different pressure gradients as e.g. done in Mathis et al. (2011) and Harun et al. (2013), thereby seemingly assuming that a comparison between different APG conditions on turbulence measurements may not be biased by spatial resolution effects.

Therefore, despite the plethora of correction techniques available, none of them is directly applicable to APG TBLs, as they were developed with or at least based on data from canonical ZPG TBLs in mind. In this work, we first study the hot-wire resolution effects in APG TBLs and show their differences to the ZPG cases. Furthermore, we assess the applicability and effectiveness of the aforementioned correction methods, as well as propose new pathways in the correction of hot-wire anemometry (HWA) measurements of APG TBLs that can also shed light on ways to include other non-canonical effects.

2 Data Sets

Table 1 Numerical datasets used in the present work
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This section provides a detailed overview of the datasets employed in the present study, which investigates the spatial averaging effects in turbulent boundary layers (TBLs) under varying flow conditions. The datasets, outlined in Table 1, were selected to cover a comprehensive range of TBL scenarios, from Zero Pressure Gradient (ZPG) to non-equilibrium Adverse Pressure Gradient (APG) conditions. This diversity is essential for a robust analysis of spatial averaging effects and their implications in different TBL environments. The datasets include:

  • ZPG (Eitel-Amor et al. 2014): This dataset, coming from well-resolved Large Eddy Simulations (LES) using the pseudo-spectral code Simson (Chevalier et al. 2007), serves as a reference point for understanding TBL behavior in the absence of external pressure forces. It features friction Reynolds numbers (\(Re_\tau\)) around 1000 and 2000, providing a baseline for comparison with other conditions.

  • Flat-plate APGs (Bobke et al. 2017; Pozuelo et al. 2022): These datasets, also well-resolved LES from Simson (Chevalier et al. 2007), offer insights into TBLs under near-equilibrium APG conditions. The range of \(Re_\tau\) in these datasets matches those of the ZPG and wing profile datasets, facilitating comprehensive studies of APG and history effects on TBLs.

  • Wing profile: This new dataset, featuring a profile in the suction side of a NACA 4412 airfoil, addresses TBLs under rapidly changing flow conditions. Obtained through wall-resolved LES, it provides valuable information on the complexities associated with non-equilibrium APG conditions.

Fig. 1
figure 1

Inner-scaled streamwise mean velocity mean (left) and variance (right) profiles of the numerical datasets used in this work. In the APG Pozuelo and ZPG datasets, the high and low \(Re_\tau\) are represented by solid and dashed lines respectively. Black, dashed lines show the linear (\(U^+=y^+\)) and logarithmic (\(U^+=\frac{1}{\kappa } \ln y^+ +B\), with \(\kappa =0.38\) and \(B=4.1\)) velocity profiles

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The latter dataset, new for this study, is a well-resolved LES of the flow around a NACA 4412 airfoil at a chord-based Reynolds number of \(10^6\) and an angle of attack of \(5^\circ\), conducted with the spectral-element method code Nek5000 (Fischer et al. 2008). The implementation of non-conformal meshing and adaptive mesh refinement into Nek5000, as detailed by Tanarro et al. (2020a), has enabled the simulation of a larger domain. This extended domain can capture the broadest turbulent structures around the airfoil’s trailing edge and wake, while maintaining fine resolution in critical areas like the TBL and wake. The profile under analysis is located on the suction side at \(x/c=0.75\), corresponding to a strong, non-equilibrium APG. This profile exhibits a friction Reynolds number nearly matching that of the flat-plate APG TBL by Bobke et al. (2017).

As depicted in Fig. 1, the studied datasets exhibit significant diversity, with matching \(Re_\tau\) cases across different \(\beta\) and flow-history conditions, demonstrating pronounced variations in their mean and variance profiles. These data sets, summarized in Table 1, are exploited to study the effect of probe spatial averaging effects on the streamwise-velocity fluctuations and in conjunction with it the effect on the spanwise Taylor microscale in APG TBLs, which according to Segalini et al. (2011) is associated with the attenuation effect.

3 Correction Schemes

In the present study, two correction schemes for the streamwise variance profile are considered: The one proposed by Smits et al. (2011) and the one by Segalini et al. (2011). The former is based on the universality of the small-scale energy at the wall and the attach-eddy hypothesis, which leads to a compact scheme for canonical wall-bounded flows. The scheme predicts the attenuation of the inner peak (at \(y^+=15\)) based on an empirical relation

$$\begin{aligned} \left. \frac{u^2}{u_m^2}\right| _{y^+=15}=M(L^+) = 0.0091 L^+ - 0.069\, , \end{aligned}$$
(1)

where \(L^+=L/\ell _*\) denotes the viscous-scaled probe length and \(\ell _*\) the viscous length scale. This attenuation is assumed to be universal across different profiles, and from this location (\(y^+=15\)), and following the attached-eddy hypothesis, an exponential decay in the small scales (and hence on the correction needed) is assumed, leading to the following formulation for the correction of missing small-scale energy:

$$\begin{aligned} \frac{u^2}{u^2_m} = M(L^+)\frac{15+\ln (2)}{y^+ + \ln (e^{15-y^+} +1)}\, , \end{aligned}$$
(2)

where \({u^2}\) and \({u^2_m}\) denote the true and measured, i.e. attenuated, streamwise variance, respectively.

Segalini et al. (2011) follow a different approach, basing the correction on the existing relationship between the spanwise Taylor microscale and the spanwise correlation function of the streamwise velocity fluctuations (\(\rho _{11}\)). The missing energy due to the finite probe length is a function of the two-point spanwise correlation function, whose second derivative at the origin is equal to \(-2/\lambda _g^2\). A Taylor expansion of \(\rho _{11}\), using an ansatz that assures the boundedness of the function, is found by Segalini et al. (2011), which is only dependent on the Taylor microscale. Since the direct and accurate measurement of \(\lambda _g\) would require two spanwise separated hot-wire probes with sufficiently short wire lengths at exactly the same wall-normal distance it is a quantity that is barely measured. The proposed method instead relies solely on two variance profile measurements (which can be performed independently, i.e. they do not need to be performed simultaneously) with different hot-wire lengths, which then – using a minimization algorithm – allows obtaining the correction for the missing energy together with \(\lambda _g\):

$$\begin{aligned} F_2 = \frac{u^2}{u^2_m} = 2 \sum _{j=1}^{4} A_j \frac{ \lambda _g^2}{j^2L^2}\left[ \frac{jL}{\lambda _g}-1 + e^{-\frac{jL}{\lambda _g}}\right] \, , \end{aligned}$$
(3)

where \(A_1=1\), \(A_2=2\), \(A_3=-3\) and \(A_4=1\). Hence, if we have two measurements of the velocity variance \(u_m^2\) made with two wires with different lengths L at the same position and Reynolds number, it is possible to invert the aforementioned problem and determine the desired variance \(u^2\) and \(\lambda _g\).

Thus, at face value, the correction scheme by Segalini et al. (2011) provides some advantages when compared to the one by Smits et al. (2011): it provides not only an estimation of the averaging effect, but also the Taylor microscale, and the scheme is more universal, since it does not rely on the universality of the small-scale energy and does therefore not depend on empirical results specific to ZPGs. However, as critiqued by Miller et al. (2014), it is not without limitations. The method’s accuracy hinges on a minimization process involving two distinct measurement sets, which are not routinely conducted, and this process is susceptible to the amplification of experimental errors.

To address these limitations, we propose the formulation of a universal composite profile for the Taylor microscale in TBLs. This approach draws inspiration from the development of composite profiles for mean velocity profiles in TBLs (see Nickels 2004; Chauhan et al. 2009). The proposed profile delineates three distinct regions within the TBL:

  • An inner layer adjacent to the wall, characterized by dominant viscosity effects and a near-constant Kolmogorov scale, as established in Carlier and Stanislas (2005). In this region, \(\lambda _g\) remains almost constant, with a small linear term as in Blair and Bennett (1987).

  • A logarithmic layer. For the log layer, our methodology diverges from Segalini et al. (2011), who demonstrated that scaling the viscous-scaled Taylor microscale with the square root of the friction Reynolds number results in the convergence of ZPG profiles at various \(Re_\tau\). Our analysis, however, reveals that such convergence does not hold across diverse pressure gradient and history conditions. We achieve a more comprehensive collapse by integrating the shape factor (\(H_{12}=\delta _1/\theta\), with \(\delta _1\) and \(\theta\) representing the displacement and momentum thicknesses, respectively) into the scaling.

  • A wake or outer region, in which a linear relationship between \(\lambda _g\) and \(y/\delta _{99}\) emerges, necessitating a more intricate scaling that incorporates the boundary layer thickness, the shape factor, and the friction Reynolds number.

The reason for including the shape factor stems from its success in scaling turbulence intensity profiles in APG TBLs (Dróżdż et al. 2015; Vinuesa et al. 2016; Dróżdż et al. 2020). Utilizing the \(\lambda _g\) profiles from the datasets detailed in Sect. 2, we have calibrated the parameters for both the logarithmic and wake regions. This calibration culminates in a piece-wise composite profile, expressed as:

$$\begin{aligned} \lambda _g = {\left\{ \begin{array}{ll} 21 \ell _* + 0.005y^+, &{}\quad y^+<30 \\ (0.7635 \ln (y/\delta _{99}) +3.96)\frac{\ell _*\sqrt{Re_\tau }}{H_{12}}, &{}\quad \ 30<y^+<0.15 Re_\tau \\ (0.35y/\delta _{99} +0.535)\frac{\delta _{99}H_{12}}{\sqrt{Re_\tau }(H_{12}-1)}, &{}\quad \ 0.15<y/\delta _{99}<1 \end{array}\right. } \end{aligned}$$
(4)

To ensure a seamless transition between the different piece-wise components, hyperbolic tangents are employed, mirroring the technique utilized by Smits et al. (2011). It is noteworthy that alternative smoothing functions, such as sigmoid functions, could also serve effectively for this integration. Data covering a wider parameter space than that given in Table 1 would be required to ascertain the validity of the composite profile beyond the range covered with the present data set.

Fig. 2
figure 2

Inner scaled spanwise Taylor microscale profiles as a function of inner-scaled wall-distance for the different datasets (left), and scatter plot of the values of the composite (\(\widehat{\lambda }_g\)) versus the actual (\(\lambda _g\)) Taylor microscale (right). In the left plot, the composite profile is represented by the dashed lines, while the numerical data is represented by the dots

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4 Results

Fig. 3
figure 3

Contours of the pre-multiplied spanwise spectra scaled in viscous units (\(k_z^+ p_{uu}^+\)) in a ZPG (top row), mild, quasi-equilibrium APG (middle row and bottom left) and strong, non-equilibrium APG (bottom right) as a function of the spanwise wavelength (\(\lambda _z^+\)) and wall-normal position (\(y^+\))

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To simulate the effect of spatial resolution in hot-wire anemometry measurements, the approach utilized in Örlü and Schlatter (2011) and Örlü and Schlatter (2013) is used. In short: a box filter (in physical space) with length equal to that of the hot-wire sensor is applied to the velocity signals. This filter is a good surrogate for wires with high length-to-diameter (aspect) ratios, for which end-conduction effects can be neglected, as discussed by Philip et al. (2013). Moreover, the effect of the filtering is simple to simulate, as the convolution in physical space is translated into a simple multiplication of the spectral filter in Fourier space.

We start the analysis by considering the energy content in the streamwise velocity fluctuations of the different datasets. In order to do so, contours of the pre-multiplied spanwise spectra are shown in Fig. 3. One major difference between the ZPG and APG profiles is the emergence of an outer peak, also seen in the fluctuation profiles shown in Fig. 1. While such an outer outer peak is also present for ZPG TBLs at higher \(Re_\tau\) it is apparent for comparably low \(Re_\tau\) TBLs exposed to an APG. It should, however, be noted that the scaling of the outer peak in ZPGs and APGs is different both in wavelength, amplitude and location (Harun et al. 2013; Sanmiguel Vila et al. 2020); with the one for ZPGs actually being located in the overlap region (Alfredsson et al. 2011). Hence it might be more suitable to label it as a second inner-peak. As discussed before, the use of finite-length probes shows itself as a filtering of the small-scale spectral energy across the whole TBL profile. Therefore, the averaging of the streamwise fluctuations experienced at any given wall-normal position is directly related to the amount of small-scale energy present in that region, and not with the total amount of energy. As discussed in Sanmiguel Vila et al. (2020), the presence of small-scale energy in the outer region of the TBL is a major feature of APG TBLs. In the outer region, the pressure gradient and the friction Reynolds number play opposing roles: as the pressure gradient intensity (higher \(\beta\)) increases, more small-scale energy is present in the outer region of the TBL. Contrary, by increasing the friction Reynolds number \(Re_\tau\), the amount of small-scale energy, resulting from the pressure gradient, is diminished, as recently also shown by Deshpande et al. (2023). A special mention should be made to the pressure gradient history (cf. Bobke et al. 2017) leading to the profile: As shown by Vinuesa et al. (2017) and Tanarro et al. (2020b), flow history plays a critical role in the spectral energy content. A TBL subjected to a stronger increasing APG over a shorter development length (in this case the wing case), exhibits not only a stronger outer peak, but also a higher content of small-scale energy in the outer region of the TBL. This is apparent when comparing the pre-multiplied spectral energy contours of the Bobke and wing profiles in Fig. 3.

Fig. 4
figure 4

Streamwise velocity fluctuations in a ZPG (top row), mild, quasi-equilibrium APG (middle row and bottom left) and strong, non-equilibrium APG (bottom right). Full black lines are used for the original fluctuation profile, while dashed and dotted lines represent attenuated profiles corresponding to 20 and 50 viscous units length, respectively. Profiles reconstructed using the correction by Smits et al. (2011) are shown in blue (with line-style depending on the probe-length from which the profile was reconstructed), whereas profiles reconstructed using the correction by Segalini et al. (2011) (which uses both attenuated profiles) are shown in red

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Fig. 5
figure 5

Scatter plot of the inner scaled Taylor microscale (\(\lambda _g\)) predicted by Segalini et al.’s method (using \(L^+=20\) and 50) against the actual one

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Fig. 6
figure 6

Streamwise velocity fluctuation attenuation contours (\(F_2=uu_m/uu\)) for ZPG (top row), mild APG (middle row, bottom left), and strong APG (bottom right) are shown relative to wall-normal position (\(y^+\)) and probe length (\(L^+\)). Full contours represent actual attenuation; dashed, dotted, and dash-dotted lines represent predicted attenuation by Smits et al. (2011), Segalini et al. (2011) (using \(\lambda _g\) from timeseries), and Segalini et al. (2011) (using \(\lambda _g\) from composite profile in Sect. 3), respectively. Horizontal lines mark \(y^+=12\) and \(y/\delta _{99}=0.15\)

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In order to illustrate these points, the influence of using different hot-wire lengths (20 and 50 viscous units) for measuring TBL profiles under different pressure gradient and Reynolds numbers is demonstrated in Fig. 4. The top row of this figure shows the well-known effects of insufficient spatial resolution in ZPG TBLs. Notably, the small-scale dominated inner layer is subject to significant attenuation, while the outer layer’s amplitude remains unaffected by the increased probe size. Conversely, in APG TBLs, as visualized in the lower panels of Fig. 4, both the inner and outer layers exhibit attenuation, which intensifies with longer probes (\(L^+ = 50\)) and is more pronounced at lower \(Re_\tau\) and higher \(\beta\) values. This effect is most prominent in the wing profile dataset, indicative of strong non-equilibrium APG conditions. It can hence be anticipated that correction methods based on the assumption of small-scale universality limited to the inner layer (such as the one by Smits et al. 2011) will not be able to correctly account for the missing small-scale energy in the outer layer.

To evaluate existing correction schemes by Smits et al. (2011) and Segalini et al. (2011), these were applied to variance profiles simulating measurements with the indicated probe lengths, and the results are shown in Fig. 4 as well. In this respect it should be noted that the correction scheme by Smits et al. (2011) has been proposed and validated against ZPG TBLs flows and as such the application of their method to APG TBL flows, albeit done in the literature, has not been promoted by the authors. While the smaller probe predominantly affects the inner region for all datasets, the longer probe impacts both the near-wall and outer peaks of the variance profile. The results for APG TBLs related to longer probe length reveal a crucial limitation in the correction scheme proposed by Smits et al. (and other similar correction schemes): although it effectively corrects the inner peak, it inadequately addresses the outer region, in which, as shown in Fig. 3, some small-scale energy is present in the case of strong APGs. This limitation is rooted in the design and underlying fundamental assumption of these schemes, specifically the attached eddy hypothesis, which implies a \(y^{-1}\) factor in the correction, underscoring the need for a more comprehensive approach to correct spatial averaging effects in non-canonical TBLs, e.g., under the influence of APGs.

A better reconstruction of the original (non-averaged) data is obtained with the correction scheme proposed by Segalini et al. (2011). In fact, both the inner and outer layer lie on top of the original data after the correction is applied. As described in Sect. 3, the method relies on an approximation of the velocity-fluctuation correlation function using the spanwise Taylor microscale (valid in the limit of small probe lengths). The method estimates with a good degree of accuracy both the corrected variance of the streamwise velocity fluctuations as well as its spanwise Taylor microscale (shown in Fig. 5). The out performance of the method proposed by Segalini et al. (2011) over the one by Smits et al. (2011) is clearly reflected in Fig. 6, in which the contours of the fluctuation attenuation predicted by both methods for different probe lengths at different wall-normal positions are overlayed on top of the actual attenuation. As discussed previously, the assumption of the attached eddy hypothesis leads to a decrease proportional to \(y^{-1}\) in the correction scheme by Smits et al. (2011), a faster rate than the actual one, specially for APGs. This leads to an under-correction of the attenuation starting in the overlap layer of the TBL. In cases in which the small-scale energy is confined to the inner layer (i.e. ZPGs or mild APGs at high \(Re_\tau\)), this departure from the actual attenuation is not as drastic as long as the probe lengths are small. However, for high \(\beta\) and lower \(Re_\tau\) cases, larger deviations from the actual attenuation are observed starting from the overlap layer at moderate probe lengths. On the other hand, the attenuation predicted by the correction scheme by Segalini et al. (2011) is almost on top of the actual attenuation for all the considered cases across the whole TBL, deviating from the actual attenuation only when the probe lengths are large as a result of the Taylor expansion needed to arrive at the relationship between \(\rho _{11}\) and \(\lambda _g\). Nevertheless, this method has two major drawbacks: it requires two measurements (although not simultaneous) using different wire lengths, and, as noted by Miller et al. (2014) the optimization process used is prone to amplifying experimental noise.

In order to circumvent the issues presented by both the Smits et al. (2011) and Segalini et al. (2011) correction schemes, both approaches should be combined. In Sect. 3, we proposed the development of a correction scheme based on a single expression, as in the Smits et al. (2011) correction, which should be capable of providing a meaningful correction for the streamwise velocity fluctuations in the outer layer of non-canonical flows (i.e., APGs). The proposed methodology utilizes the relationship between the spanwise Taylor microscale and the streamwise velocity two-point correlation function, following the approach of Segalini et al. (2011). As shown in Fig. 2, a good agreement was found between the composite profile proposed and the Taylor microscale for the different datasets by decomposing the profile into three regions: an inner, log and outer region. These three separate regions are also present in the attenuation contours shown in Fig. 6. Near the wall, the attenuation is almost constant up to \(y^+\approx 12\) across all datasets, supporting the claims by Smits et al. (2011), about the importance of the Kolmogorov scale as the driving factor for attenuation (which is constant in the near-wall region). Using Segalini et al. ’s scheme (either with the actual \(\lambda _g\) or the one from the composite profile), also results in a constant attenuation in this region, due to the constant value that \(\lambda _g\) takes in the viscous sublayer. After a non-linear overlap layer, a logarithmic region emerges in the \(F_2\) map. Although Smits et al. ’s scheme predicts an exponential decay in this region (and extending all the way to the free-stream), the decay is seen to be lower than the \(y^{-1}\) rate in the actual data, and better captured by the scheme proposed by Segalini et al. . Furthermore, the decay does not persist uniformly towards the free-stream, displaying a discernible shift in the decay factor in the outer region, more accurately captured by the Taylor microscale-based model.

However, it is pertinent to note that in ZPG or near-equilibrium APGs with low \(\beta\) values at high \(Re_\tau\), the impact of small-scale energy in the TBL’s outer regions is minimal, implying that spatial resolution effects become significant only with considerably long probes, as discussed by Deshpande et al. (2023). Yet, under specific conditions such as rapidly evolving, non-equilibrium, strong APGs (high \(\beta\)), particularly at lower \(Re_\tau\), the use of moderately sized hot-wire probes can lead to noticeable attenuation in the measured velocity variance. However, due to the absence of high-resolution numerical data for these conditions at high \(Re_\tau\), it remains uncertain whether attenuation in the outer region is also a concern in these scenarios.

5 Conclusions and Outlook

In this work, we analyzed the effect of hot-wire length on the attenuation of the streamwise velocity fluctuation profiles in canonical and non-canonical TBLs. We focused our attention on three kinds of profiles: ZPGs, near-equilibrium APGs, and non-equilibrium APGs. For the latter a new high-fidelity numerical simulation of the flow around a NACA 4412 wing profile was carried out thereby extending previous simulations (Hosseini et al. 2016; Vinuesa et al. 2018; Tanarro et al. 2020a, b). Two correction methods based on different approaches are analyzed: On the one hand, Smits et al. (2011) propose a correction based on the attached eddy hypothesis and empirical fittings to ZPG datasets, and features a single expression; and, on the other hand, Segalini et al. (2011) uses the relationship between the spanwise Taylor microscale (\(\lambda _g\)) and the two-point correlation function (\(\rho _{11}\)), which is more general but requires two sets of measurements with different hot-wire lengths. Furthermore, we propose a universal composite profile for \(\lambda _g\), based on a three-layer approach (viscous/inner layer, logarithmic region, and outer/linear region), which is able to model \(\lambda _g\) for the studied datasets. This composite profile can be used in conjunction with the relationship between the attenuation and \(\lambda _g\) proposed by Segalini et al. (2011), but it can also serve as a self-standing correlation extending existing correlations in the literature for \(\lambda _g\) as e.g. the one by Blair and Bennett (1987), which is valid for ZPG TBL flows.

We show that the correction by Smits et al. (2011) designed for ZPG flows can not be applied for non-canonical flows since it fails to capture the presence of small-scale energy in lower \(Re_\tau\) and high \(\beta\) cases, leading to an under-correction of the outer-peak characteristic of APGs. We therefore issue a note of caution to limit its use only to canonical TBLs, i.e., either to ZPGs or in near-equilibrium APGs with low \(\beta\) and high \(Re_\tau\). On the contrary, the correction scheme proposed by Segalini et al. (2011) is shown to perform well in both canonical and non-canonical flows. Nevertheless, in its original form it has one major drawback: it necessitates of two independent fluctuation profiles measured with different hot-wire lengths, and involves an optimization process which could result in the amplification of experimental uncertainties. By looking at the contours of fluctuation attenuation, \(F_2(y^+,L^+)\), we show that the composite profile developed for the Taylor microscale is able to follow the attenuation predicted using Segalini et al. ’s method with the actual \(\lambda _g\). The use of this composite profile avoids the need for two independent measurements which, as claimed by Miller et al. (2014), could lead to the amplification of experimental uncertainties.

The proposed method can hence be used to compensate for spatial resolution effects in various APG TBL flows as demonstrated here for the data listed in Table 1, but also literature data covering the same parameter space (Monty et al. 2011 not shown here) to \(Re_\tau =3000\) and \(\beta <3\) which previous methods could not. This is, in particular, important in APG TBL studies in which authors have previously assumed that spatial resolution effects are limited to the inner layer, thereby excluding \(L^+\) as a possible reason for observed differences in the outer layer. Hence, even for flow conditions that are not covered by the flow cases considered here, the proposed method might serve as an indicative tool to assess how far from the wall—for a specific wire length—attenuation might prevail.