Self-similarity in over-tripped turbulent boundary-layer flows

Abstract

The scaling universality of structure functions is studied for artificially thickened turbulent boundary-layer flows in over-tripped impacts by using hot-wire measurement datasets. The self-similarity behaviours in the inner and outer regions are examined from the viewpoint of different flow mechanisms. In the inner region, the relative ratios between structure functions for the energy-containing range of scales exhibit universality behaviour, in accordance with Townsend's attached eddy hypothesis. This universality of the energy-containing range of scales extends further away from the wall by increasing the tripping intensity. On the other hand, the impact of the external intermittency on the self-similarity of small-scale turbulence is examined through the intermittent zone in over-tripped conditions. Towards the boundary-layer edge, the structure functions exhibit a growing departure from self-similarity and analytical prediction, and it is demonstrated that the departure is primarily due to external intermittency. Moreover, based on the conditional statistics concentrated in the turbulent regimes, it is revealed that the small scales in the turbulence regime are homogenized in a self-similar behaviour, which is independent of the current tripping conditions.

1. Introduction

1.1. Extended self-similarity

Turbulent flows are characterized by non-Gaussian, intermittent fluctuations over a wide range of different scales (Frisch 1995; Pope 2000). Based on the assumption that the statistical property of the velocity fields is locally isotropic, Kolmogorov's similarity theory predicts that, at sufficiently large Reynolds numbers, the small-scale motions decouple from the large scales and are independent of the boundary or initial conditions, which can be expressed as structure functions

(1.1)\begin{equation}\langle {({\varDelta _r}u)^n}\rangle \propto {r^{{\xi _n}}},\end{equation}

where ${\varDelta _r}u = u(x + r) - u(x)$ is the velocity increment, $u(x + r)$ and $u(x)$ are velocities along the streamwise direction at two points separated by a spatial distance r, $\langle \;\rangle $ represents averaged quantities and ${\xi _n}$ is the scaling exponent. This universality hypothesis indicates that the statistical properties of the velocity fields are self-similar within the inertial range, $\eta \ll r \ll L$, where $\eta $ is the dissipation scale and L is the integral scale of turbulent motions. While the possible existence of universal scaling exponents ${\xi _n}$ is one of the most significant headways of turbulence, a scaling exponent deviation from the Kolmogorov prediction ${\xi _n} = n/3$ has been widely reported, specifically for the higher-order statistics (Anselmet et al. 1984; Frisch 1995; Sreenivasan & Antonia 1997). The ‘anomalous' deviation behaviour from the Kolmogorov scaling has been attributed to the non-Gaussian, strong intermittent fluctuations, which are often known as internal intermittency (Landau & Lifshitz 1963).

Otherwise, extensive investigations have been devoted to the existence of universal scaling laws of $\langle {({\varDelta _r}u)^n}\rangle$ for various kinds of turbulent flows by employing the so-called extended self-similarity (ESS) hypothesis (Benzi et al. 1993a,b). Rather than pursuing the universal scaling exponents ${\xi _n}$ of the nth-order structure function for the inertial subrange (ISR) scales, ESS describes the relative scaling exponent of one given structure function against another as

(1.2)\begin{equation}\langle {({\varDelta _r}u)^n}\rangle \propto {\langle {|{{\Delta_r}u} |^3}\rangle ^{{\xi _n}}},\end{equation}

where the scaling ${\xi _n}$ is computed relative to the third-order structure function of the modulus of the velocity increments. The self-similarity form of the ISR scaling properties in ESS form following (1.2) has been consolidated to hold for various turbulent flows and at high as well as low Reynolds numbers (Grossmann, Lohse & Reeh 1997; Yang et al. 2016b).

In wall turbulence, the dominance of energy-containing range (ECR) scales in the logarithmic region is argued to challenge the scaling universality due to the wall boundedness conditions (Pope 2000). The popularity of the ECR scales, $y < r \ll \delta $ (where y and $\delta $ represent the wall-normal distance and boundary-layer thickness) has been reported in wall-bounded turbulence. Following Townsend's attached eddy hypothesis (Townsend 1976; Meneveau & Marusic 2013; Hu, Yang& Zheng 2019; Marusic & Monty 2019; Wang et al. 2021, 2022), the ECR scales of the normalized even-ordered longitudinal structure functions can be expressed as (Davidson, Nickels & Krogstad 2006; de Silva et al. 2015, 2017)

(1.3)\begin{equation}{\langle {({\varDelta _r}{u_ + })^{2p}}\rangle ^{1/p}} = {D_p}\,\textrm{ln}\left( {\frac{r}{y}} \right) + {E_p},\end{equation}

where ${D_p}$ and ${E_p}$ are constants. The velocity is given in wall units by the subscript $+ $ (${u_ + } = u/{u_\tau }$, where ${u_\tau }$ is the friction velocity). The scaling behaviour by (1.3) in the ECR scales ($\kern 1.8pt y < r \ll \delta $) was confirmed in the logarithmic region from high-Reynolds-number databases, and the scale extent was limited to the given wall-normal distance (Davidson et al. 2006; de Silva et al. 2015). The work of de Silva et al. (2017) further discerned the universality of the scaling for the ECR scales in the ESS framework. They examined the scaling behaviour of the even-order velocity structure functions ${\langle {({\varDelta _r}{u_ + })^{2p}}\rangle ^{1/p}}$ for wall-bounded turbulent flows as the expression

(1.4)\begin{equation}{\langle {({\varDelta _r}{u_ + })^{2p}}\rangle ^{1/p}} = \frac{{{D_p}}}{{{D_m}}}{\langle {({\varDelta _r}{u_ + })^{2m}}\rangle ^{1/m}} + {E_p} - \frac{{{D_p}}}{{{D_m}}}{E_m}.\end{equation}

By means of the ratio of ${D_p}/{D_m}$ as extracted in (1.4), the further reaching universality of the ECR of scales was demonstrated in wall-bounded flows which span a wide range of Reynolds numbers and flow geometries (de Silva et al. 2017; Xia, Brethouwer & Chen 2018; Hu et al. 2019). Specifically, the universality was also observed at $R{e_\tau }\sim \mathrm{{\rm O}(}{10^3})$ ($R{e_\tau } = {u_\tau }\delta /\nu $, $\nu $ is kinematic viscosity), which supports that the scaling for the ECR scales in (1.4) provides more precise measures to examine the structure functions at relatively lower Reynolds numbers.

1.2. Over-tripped turbulent boundary layer

In practical industrial applications, turbulent flows in wind turbines, pipelines, ships and aviation, are in the range of high Reynolds numbers. From the academic aspect, the characteristics of the turbulent boundary-layer (TBL) flows at high Reynolds numbers ($R{e_\tau }$) have always been the focus for researchers (Marusic et al. 2010; Smits, McKeon & Marusic 2011; Marusic et al. 2013; Smits & Marusic 2013; Smits 2020). Admittedly, achieving high-$R{e_\tau }$ TBLs in a canonical form is an infrastructural and economic challenge. Instead, the artificially thickened TBL generated in a given wind tunnel is assumed as one of the cheaper/feasible approaches to mimic the high-$R{e_\tau }$ TBLs.

Since the pioneering work by Klebanoff & Diehl (1951) to artificially generate the TBL flows, numerous studies have been executed in this area by considering the impact of the inflow Reynolds number, tripping device, adaptation length and so on (Erm & Joubert 1991; Fernholz & Finleyt 1996; Castillo & Johansson 2002; Jiménez et al. 2008; Chauhan, Monkewitz & Nagib 2009; Schlatter & Örlü 2010a,b, 2012). For a certain inflow Reynolds number, incorporating some perturbation by the tripping devices at the leading edge of the flat plate is a generally accepted approach to promoting an earlier transition for the development of TBL flows (Erm & Joubert 1991; Fernholz & Finleyt 1996; Jiménez et al. 2008; Chauhan et al. 2009). A small perturbation by the tripping device results in an underdeveloped boundary layer. It is necessary to have a long enough adaptation region for the formation of a fully developed TBL with a high boundary-layer thickness. However, the requirement of a sufficiently long development length is not feasible in a common laboratory environment. On the other hand, for a test section of finite length, a strong perturbation at the leading edge will lead to an over-tripped boundary layer up to surprisingly high Reynolds numbers, by presenting a remarkably increased boundary-layer thickness in the adaptation region (Castillo & Johansson 2002; Schlatter & Örlü 2012; Marusic et al. 2015). Thus, it is more desirable to obtain an artificially thickened higher-$R{e_\tau }$ TBL by an elaborately appropriate trip design (Schlatter & Örlü 2010a, 2012).

A considerable effort has been dedicated to understanding the effects of an exact tripping device with different sizes and shapes on the boundary layer in the adaptation region (Hunt & Fernholz 1975). Rodríguez-López, Bruce & Buxton (2016b) proposed that the adaptive boundary layer could be generated through two mechanisms: wall driven and wake driven, by employing two families of tripping devices which are high aspect ratio uniformly distributed cylinders and low aspect ratio sawtooth fences. For the wall-driven mechanism, the inner structures drive the mixing of the obstacle's wake. On the other hand, the wake-driven mechanism is related to a long adaptation region where the inner structures are reorganized under the influence of highly energetic wake motions. These proposed driven mechanisms have been further confirmed by assessing the geometry parameters of tripping devices, such as aspect ratio, blockage ratio and blockage at the wall (Rodríguez-López, Bruce & Buxton 2017a,b; Buxton, Ewenz Rocher & Rodríguez-López 2018).

Furthermore, to characterize the scaling in the adaptation region, Marusic et al. (2015) measured the spatial development of high-$R{e_\tau }$ TBL flows from the tripping threaded rods of different diameters. A significant difference was noted in the outer region under the tripping effects, and the influence on the outer region persists for more than 10 m along the streamwise direction. Tang et al. (2024) observed the deviation of the boundary layer from the canonical state, which is tripped by a set of transverse cylindrical rods with incremental diameters at the given inflow Reynolds number. They confirmed that the tripping effects are significant in the outer region by introducing large-scale energetic structures. These observations are consistent with the previous results that the threaded/cylindrical rods featured by the high wall blockage can trigger a long adaptation region by generating a prominent wake in the outer layer (Klebanoff & Diehl 1951; Chauhan et al. 2009; Sanmiguel Vila et al. 2017). These generated energetic wake motions dominate the outer region, which exerts a prominent modification on the external intermittency (Tang et al. 2024). Furthermore, these wake motions transport fluid across the entire wall-normal extent which can disrupt the near-wall regions, and impose a holistic modification on the boundary layer (Marusic et al. 2015; Buxton et al. 2018; Tang et al. 2024). This is very similar to the function of the very-large-scale structures in high-$R{e_\tau }$ TBLs in consideration of the ‘footprints’ and modulation effects by the large scales (Baars, Hutchins & Marusic 2017; Marusic, Baars & Hutchinsc 2017). Importantly, it reveals that the artificially thickened boundary layer has the potential to simulate its canonical counterparts of high-$R{e_\tau }$ TBL flows. In fact, these over-tripped TBL flows widely exist in practical systems, which are stimulated by the effects of roughness, separation, pressure gradients, incoming turbulence, etc.

Notably, the artificially generated TBL flows are beyond canonical flows with the feature of significant changes throughout the boundary layer in all aspects of the mean flow, turbulence energy and scale interactions. However, the existence of self-similarity is less extensively explored under these non-canonical TBL conditions. A sound understanding of the scaling universality behaviour in artificially thickened conditions is essential for generalizing the self-similarity to more general flows. To explore the self-similarity behaviour of the artificially thickened TBL flows, two primary aspects should be considered. Firstly, the equilibrium between the inner layer and the wall is dramatically disrupted by the emergence of large-scale structures generated from over-tripped configurations (Marusic et al. 2015; Buxton et al. 2018; Tang et al. 2024). The inner region undergoes an adaptation process, gradually recovering from the tripping influence. This raises concerns about whether inner-layer self-similarity holds under non-canonical conditions. On the other hand, establishing self-similarity in the outer layer poses a significant challenge due to the dominance of energetic large-scale wake motions and their dependence on the tripping configurations. Moreover, these generated large-scale structures inherently differ from naturally developed very large-scale motions in canonical TBL flows. Consequently, the outer layer is intuitively assumed to lack universality. Therefore, given the above consideration, the current study aims to examine whether self-similar behaviour for the ECR scales is established in the inner region or if the outer-layer self-similarity can be observed in the impact of the external intermittency in the over-tripped conditions.

1.3. Paper outline

The objective of this paper is to study the effect of over-tripped configurations on the self-similarity of the TBL flows in the adaptation region, allowing for meaningful exploration of scaling universality behaviours in artificially thickened TBL flows. The analysis is based on the self-similarity of low-order and higher-order velocity structure functions, by mainly considering two factors: the popularity of the ECR scales in the inner region and the intrinsic scale-sensitive features of intermittency in the outer region. In the remainder of the paper, we introduce in § 2 the experimental dataset of the artificially thickened boundary-layer flows on which the analysis is based. In § 3, we systematically study structure functions at different orders of ECR scales under the tripping influence from the self-similarity perspective. The analysis is carried out in the inner region. On the other hand, the impact of external intermittency on the self-similarity of structure functions is discussed in § 4, and the self-similar behaviours within the ISR are further observed by conditional structure functions. A conclusion is given in § 5.

2. Experimental dataset description

A detailed description of this facility and flow conditions is provided by Tang et al. (2024). Specifications of experimental parameters and tripping configurations are given in table 1. Experiments were conducted in a closed-circuit wind tunnel in Tianjin University, as described in previous studies (Tang et al. 2016; Tang & Jiang 2020). The test section of the tunnel was 2.0 m long, 0.6 m tall and 0.8 m wide. A smooth boundary-layer plate was vertically fastened at the test section. The flat plate had a size of 1.75 m × 0.6 m × 0.015 m (length × width × thickness) with a 4 :1 elliptical leading edge. In the current experiments, it had three different free-stream velocities of ${U_\infty } \approx 5.5$, 9.0 and 13.6 m s⁻¹. Transverse cylindrical rods of different diameters, ${D_c} = 1,\;2,\;3,\;4,\;6,\;8,\;10,\;12,\;14,\;17,\;20\;\textrm{mm}$, were employed as the tripping devices, which were mounted at the position 80 mm downstream of the leading edge of the plate. The current trips have a wide range of $R{e_D} \approx 300{-}17\,000$ ($R{e_D} = {U_\infty }{D_c}/\nu $) which could have a considerable effect on the boundary layer at each free-stream velocity, as suggested in one of the pioneer works in this area by Erm & Joubert (1991). The cylindrical rods were made of ceramic zirconia materials with high hardness and toughness. The ceramic zirconia cylindrical rod was glued onto accurately machined metal inserts which were bolted into a recess and flush with the flat plate wall. The diameter of the tripping rods (${D_c} = 1{-}20$ mm) was assumed to be the only parameter considered at each free-stream velocity. From the basic flow parameters shown in table 1, it can be seen that the boundary-layer flows tripped by the relatively small tripping diameters, such as the cases I-D2, II-D2 and III-D2, are very close to canonical TBLs. For case I-D1, the boundary layer is considered to be under-tripped at the measurement location, and will not be involved in the discussion of this study. Thus, these tripping configurations provide a comprehensive insight into the effect of the tripping rod diameters on the boundary-layer flows from moderate to over-stimulation.

Table 1. Experimental parameters for the profiles of different tripping conditions at different free-stream speeds.

In the experiment, the static wall pressure was measured through four pressure ports in the region $x = 0.78{-}1.58$ m downstream of the leading edge. The coefficient of pressure along the measurement positions was constant to within ${\pm} 0.82\,{\%}$ of the free-stream dynamic head for the moderately tripped cases, which is comparable in quality to related studies (Marusic et al. 2015; Sanmiguel Vila et al. 2017). The pressure deviation was also acceptable in all the over-tripped cases, which should be attributed to the large ratio between the distance from the flat plate to the tunnel ceiling and the maximum displacement thickness (which was below approximately 1.3 %). Hot-wire measurements were carried out at the streamwise location $x = 1.32$ m downstream of the trip. The location in dimensionless form ${x_\theta } = x/\theta $ ($\theta $ is the momentum thickness of boundary layer) is shown in table 1. The streamwise location is located at the adaptive region in the over-tripped conditions, which as expected supplies the data for the examination of the self-similarity in artificially thickened TBL flows. Boundary layer traverses were conducted by a miniature single-sensor boundary-layer probe (TSI-1621A-T1.5). The probe was used with a constant temperature anemometer system of IFA-300 operating at an overheat ratio of 1.7. The tungsten (platinum-coated) hot-wire has a sensitive length of 1.25 mm and a diameter of 4 μm, resulting in a length-to-diameter ratio ($l/d$) of more than 200 (Ligrani & Bradshaw 1987; Hutchins et al. 2009). In terms of spatial resolution, the viscous-scaled wire length (${l^ + }$) is less than ${l^ + } < 48$ for all the measurement cases. Following the suggestions of negligible energy content by Hutchins et al. (2009), the sampling frequency was set up and the corresponding non-dimensional sample interval was $\mathrm{\Delta }{t^ + } < 0.5$ ($\Delta {t^ + } = \Delta tu_\tau ^2/\nu$, where $\Delta t = 1/f$, f is the sampling frequency and ${u_\tau }$ is the friction velocity). Note that the current sampling frequency is higher than the effective frequency that can be resolved by the current hot-wire probe of ${l^ + } \to 48$. The total sampling time at each wall-normal location is given by T, which is normalized in outer variables to give boundary-layer turnover times $T{U_\infty }/\delta $ (in table 1). For converged statistics, these numbers need to be large, since the largest structures in high-$R{e_\tau }$ TBLs can exceed $20\delta $ (Kim & Adrian 1999; Guala, Hommema & Adrian 2006; Hutchins & Marusic 2007b), and we would typically require several hundreds of these events to flow past the hot-wire sensor before we could expect converged statistics. In this study, the total sampling time was set in such a way that the boundary-layer turnover time was in the range of 8500–23 500 for all the measurements. Due to the limit on the amount of memory, the sampling time in each case was changed with the given sampling frequency to make sure that there were more than 8 500 boundary-layer turnover times for all the measurements, which adequately covers the energy contained in the largest scales (Hutchins et al. 2009; Mathis, Hutchins & Marusic 2009) and acquires converged statistics. Calibration was employed by Air Velocity Calibrator Model 1127 of IFA-300 over a velocity range of 0 to 22 m s⁻¹. The hot-wire probe was translated to all the wall-normal positions in the experiments by using a computer-controlled translation stage. The number of wall-normal measurement stations with logarithmical spacing was increased as the boundary-layer thickness increased. In addition, the adjustment of the wall-normal offset of the probe sensor was monitored by a digital microscope-based procedure.

Inner-normalized mean streamwise velocity profiles, ${\langle U\rangle ^ + }$ versus ${y^ + }$, are shown in figure 1 for all the tripping conditions at different free-stream velocities. Considering the strong impact of the tripping configurations on the mean flow fields, the friction velocity, ${u_\tau }$, is estimated from the raw mean velocity by fitting a composite profile by Rodríguez-López, Bruce & Buxton (2015). As shown, the mean velocity profiles for different tripping diameters collapse in the near-wall region. For comparison, the inner velocity profile of Musker (1979) is superimposed as a black line, in which the von Kármán constant $\kappa = 0.41$ and the constant $B = 4.86$ (Nagib & Chauhan 2008; Marusic et al. 2013; Segalini, Örlü & Alfredsson 2013). The agreement also suggests that the near-wall region is more prone to adapting quickly to a canonical TBL under the tripping impact, which is similar to the previous investigations (Schlatter & Örlü 2012; Rodríguez-López, Bruce & Buxton 2016a; Rodríguez-López et al. 2017b). On the contrary, a significant difference is observed in the outer layer, showing a progressively repressive wake region with increasing ${D_c}$. Similar to the TBLs stimulated by other kinds of tripping conditions (Marusic et al. 2015; Dogan, Hanson & Ganapathisubramani 2016), the current over-tripped effects could result in a suppression of the wake region by introducing the generated wake flows, which leads to lower velocity in the modified region, as shown in figure 1.

Figure 1. Inner-normalized mean velocity profiles for the different ${D_c}$ at different free-stream speeds: (a) case I, (b) case II and (c) case III. The solid black lines show the Musker profile (Musker 1979) with constants $\kappa = 0.41$ and $B = 4.86$.

Prior to calculating the higher-order moments of the streamwise structure function, it is necessary to inspect the statistical convergence of the higher moments, which can be verified by examining the pre-multiplied probability density function (p.d.f.) of velocity increments, ${(\Delta {u_ + })^n}P(\Delta {u_ + })$, following the approach used in previous works (Meneveau & Marusic 2013; Yang, Marusic & Meneveau 2016a; de Silva et al. 2017). Figure 2 plots the results for case III-D20 at two representative wall-normal heights in the inner and outer regions. For the inner region, the reference location is at ${y^ + } \approx 80$, and the other reference location is at $y/\delta \approx 0.87$, corresponding to an intermittency parameter of $\gamma (y) \approx 0.55$ ($\gamma $ represents the proportion of time that the hot-wire probe records turbulent fluids). It can be noted that an acceptable convergence degree of the current dataset is up to $n = 8$. The results show acceptable ‘closure’ of the pre-multiplied p.d.f. with the smooth tails in the sense that the structure function, which is the area under the curve, is well captured. However, convergence at the order of $n = 10$ is moderate, therefore, for the current analysis results at the order of $n > 8$ should be considered with due caution. Similar convergence results are obtained for the other data used in the current study.

Figure 2. Pre-multiplied p.d.f. of ${(\Delta {u_ + })^n}P(\Delta {u_ + })$ at two representative wall-normal heights in the inner and outer region for case III-D20: (a) at ${y^ + } \approx 80$ and (b) at $y/\delta \approx 0.87$, with the spatial distance $r \approx \delta $. Curves are multiplied by an arbitrary factor ${K_n}$ to get the normalized maximum for all orders.

3. Self-similarity in the inner region of over-tripped boundary layers

3.1. Structure functions in the inner region

To explore the self-similarity of structure functions in the inner region under the impact of the over-tripped conditions, the even-order structure functions up to the tenth order are plotted in figure 3. The structure functions are normalized by the different characteristic scales, which are the Kolmogorov scales (Kolmogorov velocity ${u_K} = {(\nu \langle \varepsilon \rangle )^{1/4}}$ and Kolmogorov length $\eta = {({\nu ^3}/\langle \varepsilon \rangle )^{1/4}}$, where $\langle \varepsilon \rangle = 15\nu {(\partial u/\partial x)^2}$ is the estimate of the mean dissipation rate based on the local-homogeneity assumption) and the Taylor scales (the root mean square of the velocity fluctuations ${u_\lambda } = {\langle {u^2}\rangle ^{1/2}}$ and Taylor length scale $\lambda = {(15\nu u_\lambda ^2/\langle \varepsilon \rangle )^{1/2}}$). In figure 3, the line colour for each order structure function switching from dark to light represents an increase of the rod diameter from ${D_c} = 1$ to 20 mm in case III. In the plot, the higher-order structure functions are presented since they hold the information about internal intermittency (Landau & Lifshitz 1963), which represents a stochastic behaviour of very intense fluctuations with a higher frequency of occurrence than that predicted by a Gaussian distribution and occurs predominantly at the small scales. In the current study, with increasing tripping rod diameter, the large-scale structures generated by the tripping configurations are enhanced, which provides a modification effect on the small scales in the inner region (Mathis et al. 2009; Baars et al. 2017; Marusic et al. 2017; Tang et al. 2024). Thus, under the tripping influence in the over-tripped conditions, whether the similarity of the ISR could be established arouses our interest.

Figure 3. Distribution of even-order structure functions up to the tenth order at ${y^ + } \approx 80$ for various tripping conditions with different rod diameters in case III. The structure functions are normalized by (a) the Kolmogorov scales and (b) the Taylor scales. The line colour from light to dark corresponds to increasing tripping rod diameters from ${D_c} = 1$ to 20 mm in case III.

Figure 3 shows the even-order structure functions at the wall-normal location of ${y^ + } \approx 80$. In figure 3(a), for the second order, there is reasonable support for self-similarity for various tripping configurations, since an adequate collapse of structure functions can be observed almost over the entire r space. At the smallest scales, the collapse is perfect, which suggests the structure functions obey the classical Kolmogorov scaling $\langle {({\varDelta _r}u)^2}\rangle /u_K^2 \propto \; {(r/\eta )^2}$. At the largest scales, the self-similarity still holds with acceptable accuracy despite the tripping influence and finite-Reynolds-number effects. However, the situation is different for the higher-order structure functions. The self-similarity of structure functions is not strictly valid. Specifically, the eighth- and tenth-order structure functions normalized by the Kolmogorov scales (${u_K}$ and $\eta $) reveal a non-collapsing and clearly non-self-similar arrangement over the entire range of scales. In addition, as regards the performance of the Kolmogorov similarity at the smallest scales, it requires higher-resolution datasets for observation. Similarly, the structure functions do not satisfy the self-similarity over the entire scale range after normalization with the Taylor scales (figure 3b). As shown, except for the collapse at the intermediate scales, the Taylor scales lead to a certain discrepancy in the dissipative range and at the large scales on increasing the order. This finding confirms the general understanding of turbulence that the structure functions do not feature universality over the entire range of scales, specifically at low to moderate Reynolds numbers (Pearson & Antonia 2001).

As indicated in the previous investigation (Landau & Lifshitz 1963), the non-universality of the higher-order structure functions is highly sensitive to different effects, such as internal intermittency and finite-Reynolds-number effects. Especially, in the current study, the generated large-scale wake structures by the over-tripped rods have amplitude and frequency modification effects on the small scales in the inner region, as presented in Appendix A. Both the amplitude and frequency modulation coefficients are increased with the tripping diameter, which means that generated large-scale structures could alter the internal intermittency behaviour based on modulation effects and result in the non-universality of the higher-order statistics, as shown in figure 3.

Figure 4 plots the comparison of the p.d.f.s of the velocity gradients $\partial u/\partial x$ at the wall-normal height ${y^ + } \approx 80$ for various tripping cases III-D1–D20. It shows that the p.d.f.s are non-Gaussian and have stretched exponential tails, which is very similar to the p.d.f.s in many other kinds of turbulent flows, such as isotropic turbulence (Gotoh, Fukayama & Nakano 2002) and turbulent jet flows (Gauding et al. 2021). On increasing the tripping rod diameter, the p.d.f.s have almost a consistent distribution. Careful observation shows that the tails of the p.d.f.s become slightly stretched in the over-tripped cases, which should be related to the footprint of large-scale wake structures generated by the tripping condition with increasing rod diameter (the evidence of the footprint effect of large scales in the near-wall region is exhibited in Appendix B).

Figure 4. Comparison of p.d.f.s of velocity gradients at ${y^ + } \approx 80$ for various tripping conditions with different rod diameters in cases III-D1–D20. The black dashed line indicates a normal distribution. The curves are normalized by the standard deviation ${\langle {(\partial u/\partial x)^2}\rangle ^{1/2}}$.

3.2. Relative relations of structure functions for the ECR scales

In the spirit of the ESS hypothesis, the relation of the velocity structure functions as expressed in (1.4) for the ECR scales is examined to further explore universality in this part. Following the previous work from de Silva et al. (2017), figure 5 plots the relation of ${\langle {({\varDelta _r}{u_ + })^{2p}}\rangle ^{1/p}}$ versus $\langle {({\varDelta _r}{u_ + })^2}\rangle$ at ${y^ + } \approx 80$ for all the cases III-D1–D20. All the results are computed at a fixed wall-normal location of ${y^ + } \approx 80$, similar to that processed in figures 3 and 4. In the plot, the tenth-order $(2p = 10)$ structure function is also involved to highlight any subtle differences in the ESS framework. In figure 5, the distributions of ${\langle {({\varDelta _r}{u_ + })^{2p}}\rangle ^{1/p}}$ versus $\langle {({\varDelta _r}{u_ + })^2}\rangle$ for almost all the scales are shown by the coloured lines, from which the ESS scaling cannot be extracted directly, as shown by de Silva et al. (2017). However, for the ECR-scale range of $y < r < \delta $, as marked by the symbols, the results obviously reveal a convincingly extended range of scaling behaviour in the ESS-inspired framework. The results show a good collapse of the higher-order moments up to $2p = 10$ and provide direct support for the investigation of de Silva et al. (2017). Furthermore, the current study extends the application of the ECR-scale similarity to the over-stimulated boundary layers, which are in an adaptation region to recover from the leading-edge tripping effects and return to the canonical state. The good agreement for each higher order $2p = 4,\;6,\;8,\;10$ in figure 5 suggests that the scaling behaviour of ECR scales is independent of the current tripping influence.

Figure 5. The ESS plot for higher-order moments for the same databases shown in figures 3 and 4. The results are computed at the wall-normal location of ${y^ + } \approx 80$; ${\langle {({\varDelta _r}{u_ + })^{2p}}\rangle ^{1/p}}$ versus $\langle {({\varDelta _r}{u_ + })^2}\rangle $: (a) $p = 2$, (b) $p = 3$, (c) $p = 4$, (d) $p = 5$, as shown in the coloured lines for almost the entire scale range of r. The data in the ECR-scale range of $y < r < \delta $ are shown by the symbols. The solid black lines (——) correspond to a fit following (1.4) in the ECR-scale range at case III-D20.

The extended range of scaling behaviour in figure 5 provides an opportunity to have a reliable estimation of the scaling coefficients (slopes ${D_p}/{D_1}$, $p = 2,\;3,\;4,\;5$). To further quantify the relation, the ratios of ${D_p}/{D_1}$ are computed based on a linear fit in the ECR-scale range $y < r < \delta $. The work of de Silva et al. (2017) utilized this procedure to exhibit a good description of the ECR-scale similarity at $R{e_\tau } \approx 900$, which indicates that ECR scaling coefficients can be confidently estimated from databases at low Reynolds numbers when no clear logarithmic region exists. The current study confirms the scaling universality of ECR scales based on the datasets in the range of $R{e_\tau } \approx 500{-}3500$, especially under the tripping influence. Figure 6 shows the estimates of the higher-order slope ratios ${D_p}/{D_1}$ over a range of ECR scales for various tripping rods with different diameters. We can note that the coefficient ratio ${D_p}/{D_1}$ holds a consistent distribution for all the tripping cases. For comparison, the results in the inner region of TBL flow ($\kern 1.8pt {y^ + } \approx 150$, $R{e_\tau } \approx 1600$) by Sillero, Jiménez & Moser (2013) are also shown in the plot. As shown, ${D_p}/{D_1}$ in the current study is in accordance with the result by Sillero et al. (2013). The consistent scaling behaviour confirms that the relation of the structure functions in the ECR scales is independent of the different tripping conditions in the current study.

Figure 6. Distribution of ratios ${D_p}/{D_1}$ versus p, from the relation in figure 5. The symbol of the blue circle represents the results of TBL DNS datasets ($\kern 1.8pt {y^ + } \approx 150$, $R{e_\tau } \approx 1600$) from Sillero et al. (2013).

Furthermore, to explore the performance of the structure functions for the ECR scales at a broader wall-normal extent, figure 7 plots the distribution of ratios ${{D_p}/{D_1}}{(p = 2,3,4,5)}$ across the entire boundary layer for all the tripping cases. Similar to figure 6, the coefficient ratio calculated from the direct numerical simulation (DNS) TBL datasets by Sillero et al. (2013) ($\kern 1.8pt {y^ + } \approx 150$, $R{e_\tau } \approx 1600$) is employed for comparison (black lines). As shown, a good collapse of the ratio is noted in the inner region for the various tripping conditions, which indicates that the scaling universality of the ECR scales is independent of the tripping rod diameter. Meanwhile, the values of ${D_p}/{D_1}$ in the inner region appear unchanged. This constant of ${D_p}/{D_1}$ is consistent with the suggestions by de Silva et al. (2017) in the log region of high-Reynolds-number TBLs. In addition, based on the ESS-inspired framework, the universality of the scaling of the ECR scales extends beyond the logarithmic region of the boundary layer. From the insets in figure 7, it can be seen that the universality extends almost up to $y/\delta \approx 0.5$ for all the tripping cases. Since the boundary-layer thickness is increased with the tripping diameter, it can be deduced that a further reaching universality is performed by increasing the tripping rod diameter.

Figure 7. Distribution of ratios ${D_p}/{D_1}$ along the wall-normal heights of $\kern 1.8pt {y^ + }$ ($\kern 0.06em y/\delta $ in the insets) for all the tripping cases III-D1–D20; (a) $p = 2$, (b) $p = 3$, (c) $p = 4$ and (d) $p = 5$. The coefficient ratio value calculated from the TBL DNS datasets (${z^ + } \approx 150$, $R{e_\tau } \approx 1600$) from Sillero et al. (2013) is also plotted in black lines for comparison. The results of ratios ${D_p}/{D_1}$ for cases I and II are shown in Appendix C.

From these findings, it can be concluded that, even though the tripping impacts are likely to be embodied in structure functions at different orders, as shown in figures 3 and 4, we are able to exhibit the similarity of ECR scales by accurately quantifying the universal scaling of the ratios between two structure functions (${D_p}/{D_1}$) over a much larger wall-normal extent. Analogously, recent work on high-order moments (Yang et al. 2016a,b; Xia et al. 2018) also indicated that the extent of logarithmic scaling as a function of y increases when examining ratios between moment functions in ESS form. The observation of the ESS region could be attributed to the argument that the bulk flow or viscous effects have a similar action on the structure function for the ECR scales (Yang et al. 2016a,b), which can be further extended under the over-tripped conditions.

Additionally, the relative relation of structure functions in (1.4) is challenged in the outer part around the edge of the boundary layer. This region is dominated by the turbulent/non-turbulent intermittency, rather than the ECR scales following Townsend's attached hypothesis. Hence, it is reasonable to note an obvious discrepancy in the examination of the scaling for ECR scales in the ESS framework, as shown in figure 7. In the following, the similarity behaviour in the outer region will be examined by considering the external intermittency.

4. Self-similarity in the outer region with external intermittency

In this section, we examine whether structure functions in the outer region admit self-similarity under the influence of the tripping condition and discuss the effects of external intermittency.

4.1. The external intermittency

The external intermittency is described as the flow alternating between turbulent and substantially irrotational non-turbulent motions in the outer-layer region (Corrsin & Kistler 1955; Townsend 1976). The turbulent region and non-turbulent region are separated by the turbulent/non-turbulent interface (TNTI), which is assumed to be a sharp and highly contorted superlayer (Bisset, Hunt & Rogers 2002; Chauhan et al. 2014; da Silva et al. 2014; Philip et al. 2014). The mean intermittency distribution is argued to be independent of Reynolds number (Fiedler & Head 1966). In the artificially thickened boundary-layer flows, the generated wake structures from the leading-edge trips can provide a remarkable alteration to the external intermittency in the adaptive region (Rodríguez-López et al. 2016a; Buxton et al. 2018; Tang et al. 2024), by inquiring into the statistics of the outer-layer region, such as mean velocity, variance, energy spectra and high-order statistics.

To observe the external intermittency behaviour, it is necessary to detect the TNTI, which is a continuous ongoing research activity in turbulent flows. Several detecting methods have also been proposed for the experimental datasets, by considering that the intrinsic background turbulence in free stream. For particle image velocimetry measurements, the TNTI was detected by the technique based on turbulent kinetic energy (Chauhan et al. 2014), local homogeneity (Reuther & Kähler 2018), fuzzy clustering of the streamwise velocity field (Fan et al. 2019; Younes et al. 2021), track of the Lagrangian particle trajectories (Long, Wu & Wang 2021) and so on. For the current one-dimensional flow data by hot-wire measurement, the detection of intermittency relies upon identifying whether the probe is measuring turbulent or non-turbulent fluids. Therefore, a turbulent kinematic energy criterion proposed by Chauhan et al. (2014) is utilized to detect the TNTI in this study. This procedure has been extensively used in recent works (de Silva et al. 2013; Chauhan et al. 2014; Kwon, Hutchins & Monty 2016; Saxton-Fox & McKeon 2017; Buxton et al. 2018; Chen et al. 2023), and only a brief description will be given here.

The intermittency interface detector function is introduced to evaluate the external intermittency, which is defined as

(4.1)\begin{equation}\varphi (i) = \frac{{100}}{{U_\infty ^2}}\frac{1}{3}\sum\limits_{j ={-} 1}^1 {{{({u_{i + j}} - {U_\infty })}^2}} ,\end{equation}

where the index i is an arbitrary instant in the temporal domain and the summation over index j indicates a mean over three consecutive measurements in a time series. The turbulent (non-turbulent) fluids can be detected as those for which $\varphi (i)$ is higher (lower) than a given threshold ${\varphi _{th}}$. A binary representation of the flow is obtained using the threshold, and the binary representation is considered as

(4.2)\begin{equation}\psi (i) = H(\varphi (i) - {\varphi _{th}}) = \left\{ {\begin{array}{@{}cc@{}} {1,}&{\varphi (i) \ge {\varphi_{th}}}\\ {0,}&{\varphi (i) < {\varphi_{th}}} \end{array}} \right., \end{equation}

where H is the Heaviside function and $\psi (i)$ is the so-called intermittency function. Then, as proposed by Klebanoff (1955), an intermittency parameter $\gamma (y)$ at the wall-normal location of y is defined as

(4.3)\begin{equation}\gamma (y) = \frac{1}{N}\sum\limits_{i = 1}^N {\psi (i,y)} ,\end{equation}

which quantifies the proportion of time that the flow is turbulent. Close to the wall, the flow is expected to be fully turbulent, $\gamma (y) = 1$, the non-turbulent fluids gradually have more occurrence far away from the wall, and $\gamma (y) = 0$ in the free stream. It is well known that the intermittency profile of $\gamma (y)$ in the TBL can be described by the error function (Corrsin & Kistler 1955; Klebanoff 1955; Fiedler & Head 1966; Hedleyt & Keffer 1974)

(4.4)\begin{equation}\gamma (y) = \frac{1}{{{\sigma _I}\sqrt {2{\rm \pi} } }}\int_y^\infty {\textrm{exp}\left( { - \frac{{{{(y - {Y_I})}^2}}}{{2\sigma_I^2}}} \right)\textrm{d}y} ,\end{equation}

here, ${Y_I}$ is the wall-normal location of the mean interface where $\gamma ({Y_I}) = 0.5$ and ${\sigma _I}$ is the standard deviation of the instantaneous interface position ${y_I}$ relative to the mean position ${Y_I}$. Here, ${Y_I}$ and ${\sigma _I}$ are the estimated parameters by fitting the measurement intermittency profile to the function of (4.4). From the above procedure, it can be seen that the intermittency is dependent on the given threshold value ${\varphi _{th}}$, which is related to the values of ${Y_I}$ and ${\sigma _I}$. Thus, by fitting the error function in (4.4), a threshold of 0.07–0.08 is chosen to calculate the intermittency profile $\gamma (y)$ for all the cases in the current study. The wall-normal intermittency profiles $\gamma (y)$ in the outer scaling $y/\delta $ are shown in figure 8. As shown, $\gamma (y)$ exhibits a normal distribution for all the tripping conditions in the current study, by presenting good fittings to the error function in (4.4). Considering that the current tripping conditions make a significant alteration to the flow fields in the wake region, this could challenge the fit precision based on the wake function in determining the ‘true’ boundary-layer thickness (the wall distance where $\bar{U} = {U_\infty }$ exactly) (Chauhan et al. 2009). Thus, the conventional boundary-layer thickness ($\kern 1.8pt y = \delta $ where $\bar{U}/{U_\infty } = 0.99$) is used in this study, which is smaller than the ‘true’ boundary-layer thickness (Chauhan et al. 2009). As shown in figure 8, $\gamma \approx 0.2{-}0.4$ (increase with ${D_c}$) at $y/\delta = 1$ in all cases, which is consistent with the previous investigations (Kovasznay, Kibens & Blackwelder 1970; Hedleyt & Keffer 1974; Chen & Blackwelder 1978; Chauhan et al. 2014). The standard deviation ${\sigma _\gamma }$ of the interface location is presented as a function of the tripping rod diameter in figure 9. The solid line indicates the range of ${\sigma _\gamma } \approx 0.15{-}0.18$, which is obtained from the data of canonical TBL flows (Corrsin & Kistler 1955; Hedleyt & Keffer 1974). It shows that ${\sigma _\gamma }$ has a good agreement with the results in the moderately tripped cases. Then, increasing ${D_c}$ alters the intermittent region by introducing strong perturbations. The intermittency region rapidly extends further away from the wall by holding a wider proportion in the boundary layer, as shown in figure 8, thus, an increased ${\sigma _\gamma }$ is noted in figure 9.

Figure 8. Intermittency parameter profile $\gamma (y)$ as a function of $y/\delta $ in outer scaling for different tripping rod diameters with different free-stream velocities; (a) case I, (b) case II, (c) case III. Coloured lines represent the result by fitting (y) to the error function in (4.4).

Figure 9. Standard deviation of the interface location ${\sigma _\gamma }$ as a function of tripping rod diameter ${D_c}$ for the cases I, II and III. Solid line represents the range of ${\sigma _\gamma } \approx 0.15{-}0.18$.

With this definition of the intermittency parameter $\gamma $, any conventionally averaged quantity $\langle \phi \rangle $ can be decomposed as

(4.5)\begin{equation}\langle \phi \rangle = \gamma {\langle \phi \rangle _T} + (1 - \gamma ){\langle \phi \rangle _N},\end{equation}

where ${\langle \phi \rangle _T}$ and ${\langle \phi \rangle _N}$ indicate the conditional averaging results by only considering either turbulent or non-turbulent regimes (Mellado, Wang & Peters 2009; Gauding et al. 2021). The conditional averages are defined with the binary indicator function, $\psi $, as

(4.6)\begin{equation}{\langle \phi \rangle _T} = \langle \psi \phi \rangle , \end{equation}

and

(4.7)\begin{equation}{\langle \phi \rangle _N} = \langle (1 - \psi )\phi \rangle .\end{equation}

4.2. Intermittency structure function

Referring to the external intermittency, when we calculate the statistical moments of the velocity increments of two points separated by a spatial distance r, different conditions should be considered: (I) both points are located within the turbulent regime, (II) both points are within the non-turbulent regime and (III) one point in the turbulent regime and the other one in the non-turbulent regime. For locally homogeneous turbulence, the corresponding probabilities for the different conditions can be obtained as ${\gamma _{TT}} = \gamma - {\textstyle{1 \over 2}}{\varTheta _I}$, ${\gamma _{NN}} = 1 - \gamma - {\textstyle{1 \over 2}}{\varTheta _I}$ and ${\gamma _{TN}} = {\varTheta _I}$, where ${\varTheta _I} = \langle {(\psi (x + r) - \psi (x))^2}\rangle $ is known as the intermittency structure function (Kuznetsov, Praskovsky & Sabelnikov 1992). It is clear that ${\gamma _{TT}} + {\gamma _{NN}} + {\gamma _{TN}} = 1$. From the probabilities in different conditions, it can be deduced that the ratio

(4.8)\begin{equation}\frac{{{\gamma _{TT}}}}{\gamma } = 1 - \frac{{{\varTheta _I}}}{\gamma },\end{equation}

which represents the conditional probability that one point of the velocity increment is in the turbulent regime considering that the other point is also in the turbulent regime. From the above equations, it can be deduced that ${\gamma _{TT}}/\gamma \to 1$ (${\gamma _{TT}} \to \gamma $) as ${\varTheta _I} \to 0$ if $r \to 0$. Figure 10 shows ${\gamma _{TT}}/\gamma $ against spatial distance $r/\eta $ at different wall-normal positions the intermittency of which is in the range of $\gamma = 0.1{-}0.9$ for case III-D20 as a representative case. It is expected that in the inner region with the higher value $\gamma \to 0.9$, ${\gamma _{TT}}/\gamma $ is close to unity with $r \to 0$ and has a relatively slowly decreasing trend with increasing separation distance r. Further away from the wall, ${\gamma _{TT}}/\gamma $ decreases remarkably at larger r. A similar distribution was reported in the turbulent jet flow by Gauding et al. (2021). Due to the external intermittency of the alternation between turbulent and non-turbulent fluid and the decreased proportion of turbulent excursion, it is reasonable to have lower ${\gamma _{TT}}/\gamma $ toward the edge of the boundary layer ($\gamma \to 0.1$) at the larger-separation distance.

Figure 10. Conditional probability ${\gamma _{TT}}/\gamma $ against spatial distance, $r/\eta $, at different wall-normal positions with the intermittency parameter in the range of $\gamma = 0.1{-}0.9$ for case III-D20.

To further explore the feature of the intermittency function $\psi $, one-dimensional random telegraphic signals ${\psi _{ts}}$ are introduced, which have randomly distributed values of ${\psi _{ts}} = 1$ in turbulent regimes and ${\psi _{ts}} = 0$ in non-turbulent regimes, with the corresponding probabilities of $P({\psi _{ts}} = 1) = {\gamma _{ts}}$ and $P({\psi _{ts}} = 0) = 1 - {\gamma _{ts}}$. The second-order moment of the increment ${\varTheta _{I,ts}}(r)$ can be expressed by the autocorrelation function of ${\psi _{ts}}$, as

(4.9)\begin{equation}{\varTheta _{I,ts}}(r) = \langle {({\psi _{ts}}(x + r) - {\psi _{ts}}(x))^2}\rangle = 2({R_{{\psi _{ts}}}}(0) - {R_{{\psi _{ts}}}}(r)) = 2({\gamma _{ts}} - {R_{{\psi _{ts}}}}),\end{equation}

where ${R_{{\psi _{ts}}}}(r)$ represents the analytical expression of the autocorrelation of ${\psi _{ts}}$, which is expressed as (Machlup 1954; Fitzhugh 1983; Thiesset et al. 2020)

(4.10)\begin{equation}{R_{{\psi _{ts}}}}(r) = \langle {\psi _{ts}}(x + r){\psi _{ts}}(x)\rangle = {\gamma _{ts}}({\gamma _{ts}} + (1 - {\gamma _{ts}})\,{\textrm{e}^{ - r/{\mathcal{L}_{ts}}}}).\end{equation}

In (4.10), ${\mathcal{L}_{ts}}$ is a characteristic length scale, ${\mathcal{L}_{ts}} = 2{\gamma _{ts}}(1 - {\gamma _{ts}}){[{\lim _{r \to 0}}({\varTheta _{I,ts}}(r)/r)]^{ - 1}}$, that relates to the probability of ${\psi _{ts}}$ transiting from a value of 1 to 0 and vice versa (Machlup 1954; Fitzhugh 1983; Thiesset et al. 2020).

From the expression of (4.9) and (4.10), an analytical intermittency structure function of the random telegraphic signal can be expressed as

(4.11)\begin{equation}{\varTheta _{I,ts}}(r) = 2{\gamma _{ts}}(1 - {\gamma _{ts}})(1 - {\textrm{e}^{ - r/{\mathcal{L}_{ts}}}}).\end{equation}

From (4.11), it can be seen that, at large separations, ${\varTheta _{I,ts}}(r \to \infty ) = 2{\gamma _{ts}}(1 - {\gamma _{ts}})$, and for the small-separation limit it can be deduced that

(4.12)\begin{equation}\mathop {\lim }\limits_{r \to 0} {\varTheta _{I,ts}}(r) = 2{\gamma _{ts}}(1 - {\gamma _{ts}})\frac{r}{{{\mathcal{L}_{ts}}}} + {\rm O}({r^2}),\end{equation}

which indicates that ${\varTheta _{I,ts}}(r)$ is proportional to r when the separation distance r is small compared with ${\mathcal{L}_{ts}}$.

Then, we can compare the analytical intermittency structure function of the random telegraphic signal, ${\varTheta _{I,ts}}$, with the intermittency structure function ${\varTheta _I}$ of the wall turbulence signals for case III-20 in the current study. In figure 11, the intermittency structure function ${\varTheta _I}$ is presented for the different wall-normal positions, which is normalized by the large-scale limit $2\gamma (1 - \gamma )$. The separation distance r is normalized by the characteristic length scale $\mathcal{L}$. The plot shows a good collapse between ${\varTheta _{I,ts}}/2{\gamma _{ts}} (1 - {\gamma _{ts}})$ and ${\varTheta _I}/2\gamma (1 - \gamma )$ at the scales of the small- and large-scale limits. A clear discrepancy is noted in the intermediate-scale range, which becomes more obvious as increasing wall-normal heights (decreasing intermittency parameter $\gamma $). This discrepancy was also reported in the intermittent range scales of turbulent jet flows (Gauding et al. 2021) and liquid–gas turbulence (Thiesset et al. 2020). In fact, the TNTI is widely argued to be a self-similar fractal which can be quantified by the fractal dimension (Sreenivasan & Meneveau 1986). The self-similar fractal behaviour has been confirmed in different classes of flows, such as boundary layer, jet flow, plane wake and mixing layer (Sreenivasan & Meneveau 1986; Meneveau & Sreenivasan 1991). Furthermore, Tang et al. (2024) confirmed that, under the impact of the tripping conditions in the current study, the boundary-layer flows remain self-similar fractals with identical fractal dimensions. Thus, the discrepancy in the intermediate-scale range in figure 11 is due to the fact that the morphology of the TNTI is not fully random but instead characterized by the self-similar fractal behaviour of turbulence.

Figure 11. Comparison of the intermittency structure function ${\varTheta _I}$ (coloured lines) at different wall-normal positions ($\gamma = 0.1{-}0.9$) with the analytical intermittency structure function of the random telegraphic signal (${\varTheta _{I,ts}}$, blue dashed lines). The black dashed lines indicate the analytical small- and large-scale limits. Data for case III-D20.

As mentioned earlier, the characteristic length scale $\mathcal{L}$ relates to the transiting probability of $\psi $, which is equivalent to the jump frequency transiting from a value of 1 to 0 and vice versa. The jump frequency is expressed as

(4.13)\begin{equation}{f_\psi } = \left\langle {\left|{\frac{{\partial \psi }}{{\partial x}}} \right|} \right\rangle = \frac{{{n_I}}}{{{\mathcal{L}_0}}},\end{equation}

where ${f_\psi }$ represents the number of turbulent/non-turbulent (non-turbulent/turbulent) transitions ${n_I}$ per length in the streamwise direction. Following the previous investigations (Debye, Anderson & Brumberger 1957; Thiesset et al. 2020; Gauding et al. 2021), it can be deduced that ${\lim _{r \to 0}}({\varTheta _I}/r) = \langle |\partial \psi /\partial x|\rangle$ from the small-separation limit in (4.12), which means that the small-separation limit of the second-order structure function is equal to the jump frequency. Figure 12(a) shows the jump frequency ${f_\psi }$ as a function of the wall-normal heights for all the cases III-D1–D20. In the plot, ${f_\psi }$ has a clear maximum for each tripping configuration, which means that the alteration between turbulent and non-turbulent happens most frequently. On increasing the tripping rod diameter, the maximum ${f_\psi }$ moves outwards with a gradually decreasing trend, meanwhile, the wall-normal span of ${f_\psi }$ has a broader extent, which implies a growing wall-normal extent of the TNTI in the over-tripped conditions. This is consistent with the results of the intermittency parameters in figure 8. It is shown that, with increasing tripping rod diameter, the intermittency region rapidly extends further away from the wall by holding a wider proportion of the boundary layer (as shown in the inset of figure 12a). Specifically, the intermittency parameter exhibits a normal distribution with an increasing standard deviation ${\sigma _\gamma }$. Here, ${\sigma _\gamma }$ is suggested to be a measure of the width of the intermittent zone. Thus, ${\sigma _\gamma }$ is employed for the normalization in figure 12(b), and ${f_\psi }$ exhibits a self-preserving shape with intermittent factor in the relationship of ${f_\psi }{\sigma _\gamma } \propto \gamma $. This finding is consistent with the growth of the characteristic length scale of TNTI with increasing tripping diameter in the framework of self-similar behaviour.

Figure 12. (a) Jump frequency ${f_\psi }$ as a function of the wall-normal height ${y^ + }$ ($\kern 1.8pt y/\delta $ in the inset), (b) self-preservation of ${f_\psi }$ after normalization with the standard deviation of intermittent parameter ${\sigma _\gamma }$, as a function of $\gamma $. The data are for the various tripping conditions with different rod diameters (cases III-D1–D20). The results of cases I and II are shown in Appendix D.

4.3. Structure functions and the effects of external intermittency

The current tripping configurations provide a predominant influence on the intermittency in the wake region, such as the jump frequency, which attracts our interest as to whether the self-similarity of the structure functions can be observed in the intermittent zone. Figure 13 shows the normalized even-order structure functions up to the tenth order at the different wall-normal heights for the case III-D20. As shown in figure 13(a,b), the structure functions are normalized by the Kolmogorov scales (${u_K}$ and $\eta $) and the Taylor scales (${u_\lambda }$ and $\lambda $), respectively, similar to figure 3. It seems that the second order shows the self-similarity as regards the wall-normal heights in the range of $\gamma = 0.1{-}0.9$, by showing an acceptable collapse in either form of the normalizations. The remarkable collapse is noted at the small scales; the distribution can be explained by the validation of classical Kolmogorov scaling. At the large scales, a certain discrepancy is noted in the Kolmogorov-scale normalizations, which should be attributed to the external intermittency in the over-tripped conditions. It is evident that the higher-order structure functions (fourth to tenth order) exhibit a clearly non-collapsing and non-self-similar arrangement over the entire range of scales through the intermittent zone. This result is expected in higher-order statistics, which are very sensitive to various factors, especially external intermittency, and probably promoted under the influence of the tripping conditions. Additionally, by comparing with the results in the inner region, as shown in figure 3(a), it is indicated that the spatial resolution effect of the hot-wire probe on the estimation of the Kolmogorov length scale should be neglected from the reasons causing the non-collapsing distribution in figure 13(a).

Figure 13. Distribution of even-order structure functions up to the tenth order at different wall-normal heights in the range of $\gamma = 0.1{-}0.9$ for case III-D20. The structure functions are normalized by (a) the Kolmogorov scales and (b) the Taylor scales, as similar in figure 3. The line colour from dark to light means that the wall-normal position moves outwards, corresponding to an intermittency parameter from $\gamma = 0.9$ to 0.1.

To reveal the cause behind the lack of self-similarity of higher-order structure functions, we present the characteristics of intermittency. Following Batchelor & Townsend (1949), the turbulent signals are intermittent when the fine structure of the turbulence tends to be locally concentrated, intermittent in nature and randomly scattered through the fluid in a spotty way. This spatially spotty pattern becomes more prominent with increasing order of the velocity derivative. Hence, the higher-order moment of velocity derivative is a suitable tool to probe turbulent intermittency, and the flatness is defined as

(4.14)\begin{equation}F = \frac{{{{\left\langle {\left( {\dfrac{{\partial u}}{{\partial x}}} \right)}^4 \right\rangle }}}}{{{{\left\langle {\left( {\dfrac{{\partial u}}{{\partial x}}} \right)}^2\right\rangle }}^2}},\end{equation}

which is widely accepted to represent intermittency at the fine scales. Note that, for the velocity increment ${\varDelta _r}u = u(x + r) - u(x)$, when $r \to 0$, ${\varDelta _r}u$ at small scales is essentially equivalent to the derivative.

For the purpose of testing the original Kolmogorov similarity hypothesis, the evolution of high-order statistics of the velocity derivatives with $R{e_\lambda }$ has been widely examined. It was indicated that the statistic of F is affected by two factors, which are the finite-Reynolds-number effect and the flow conditions (Antonia et al. 2015, 2017; Djenidi et al. 2017; Meldi, Djenidi & Antonia 2018; Tang et al. 2018). Figure 14(a) shows the distribution of the flatness factor F against $R{e_\lambda }$ for all the data with the intermittency parameter in the range of $\gamma = 0.1{-}0.9$ (for cases III-D1–D20). In the plot, with increasing tripping diameter ${D_c}$, both F and $R{e_\lambda }$ exhibit a broader extent. Each of the tripping conditions indicates that the flatness factor of the velocity derivative F is increased on reducing $R{e_\lambda }$ (also $\gamma $). From the numerous investigations, the $R{e_\lambda }$ dependence of F in a fitting power law has been sought empirically in different kinds of flows, and the fitting power law $F\sim (1.14 \mp 0.19)Re_\lambda ^{0.34 \pm 0.03}$ was derived from an abundance of turbulent data (Antonia, Satyaprakash, & Hussain 1982; Jiménez et al. 1993; Wang et al. 1996; Sreenivasan & Antonia 1997; Gotoh et al. 2002; Ishihara et al. 2007; Ishihara, Gotoh & Kaneda 2009). The fitting power law is drawn in figure 14(a) for comparison. It can be seen that, at a high intermittent factor ($\gamma \to 0.9$), F shows a good collapse with the power law of $R{e_\lambda }$. The collapse means that the high-order statistics of fine-scale structures in the wall-normal region with the higher $\gamma $ are almost independent of the impact of external intermittency. The results are supported by Djenidi et al. (2017) that a constant flatness factor can be noted in the region $0.3 \le y/\delta \le 0.6$ (corresponding to high $\gamma $), in which both F and $R{e_\lambda }$ are independent of the distance to the wall. On decreasing the intermittent factor (further away from the wall), F has an obvious deviation from the fitting power law of $R{e_\lambda }$ by showing significantly greater values in figure 14(a), which could be attributed to the effect of the flow conditions of external intermittency. Similarly, figure 14(b) shows the dependence of F on the external intermittency parameter $\gamma $. As shown, F increases from an inner value close to 6 ($\gamma \to 0.9$) to a value around 50 ($\gamma \to 0.1$) close to the edge of the boundary layer. Considering that the velocity derivative can emphasize the fine-scale components but not cut out the low-frequency (large-scale) parts of the signal (Kuo & Corrsin 1971), the dependence of F confirms that the fine scales do not decouple from the large scales under the tripping effects and further supports that the external intermittency has a dominant impact on the fine scales. In fact, the remarkable change of F was also discussed by Gauding et al. (2021) in turbulent jet flows. They linked the change of F to the local kinematics of TNTI by considering the non-equilibrium in the energy cascade and the direction of the inter-scale transport (Watanabe, da Silva & Nagata 2019, 2020), which is probably further promoted in the current over-tripped conditions.

Figure 14. Flatness factor of the velocity gradients F as a function of (a) Reynolds number $R{e_\lambda }$ (light to dark red colour represents an increase of intermittent parameter from $\gamma = 0.1$ to 0.9, the black line indicates the power law $F = (1.14 \mp 0.19)Re_\lambda ^{0.34 \pm 0.03}$) and (b) the intermittency parameter $\gamma $ (the line colour indicates the tripping rod diameter, ${D_c}$).

To further understand the impact of external intermittency, the conditional flatness factor of the velocity derivatives is defined as (following the expression of (4.6))

(4.15)\begin{equation}{F_T} = \frac{{\left\langle {\psi {{\left( {\dfrac{{\partial u}}{{\partial x}}} \right)}^4}} \right\rangle }}{{{{\left\langle {\psi {{\left( {\dfrac{{\partial u}}{{\partial x}}} \right)}^2}} \right\rangle }^2}}}.\end{equation}

The binary indicator function of intermittency $\psi $ is introduced to consider only the turbulent portion of the outer-layer flow. Figure 15(a,b) shows the distribution of conditional flatness factor ${F_T}$ against $R{e_\lambda }$ and $\gamma $, respectively. In figure 15(a), ${F_T}$ and $R{e_\lambda }$ show a good agreement with the power law ${F_T} = (1.14 \mp 0.19)Re_\lambda ^{0.34 \pm 0.03}$ for all the data at different wall-normal heights through the intermittent region. It is clear that the effect of the external intermittency, which leads to the obvious deviation (significantly high values of F in figure 14a), is excluded by examining the conditional flatness factor ${F_T}$, which is further confirmed by the distribution of the p.d.f.s of the velocity derivatives in figure 16. The power-law dependence of ${F_T}$ on $R{e_\lambda }$ in figure 15(a) implies that, in the turbulent regimes of the outer layer, the large-scale fluctuations play a role in determining the turbulence intermittency, the level of which increases with $R{e_\lambda }$ (Batchelor & Townsend 1949).

Figure 15. Conditional flatness factor of the velocity gradients ${F_T}$ as a function of (a) Reynolds number $R{e_\lambda }$ and (b) the intermittency parameter $\gamma $. Note that the presentation is completely consistent with that in figure 14 for comparison.

Figure 16. Conventional (a) and conditional (b) p.d.f.s of the velocity derivatives at different wall-normal positions as indicated in the legend of intermittent factor $\gamma $. The black dashed line indicates a normal distribution. The curves are normalized by the standard deviation $\sigma = {\langle {(\partial u/\partial x)^2}\rangle ^{1/2}}$ and the conditional standard deviation ${\sigma _T} = {\langle {(\psi \,\partial u/\partial x)^2}\rangle ^{1/2}}$, respectively. The density functions are estimated over abscissa intervals of bin width $0.15\sigma $ ($0.15{\sigma _T}$). Data for case III-D20.

In figure 15(b), ${F_T}$ is almost constant and less sensitive to the variations of $\gamma $. Obviously, these findings are independent of the tripping conditions, which means that the strong turbulent transport of the wake flow could lead, in a homogenized statistical sense, to the turbulent regions. The homogenization of turbulent regions was speculated by Corrsin & Kistler (1955) and later confirmed by free shear flows (Mellado et al. 2009; Gauding et al. 2021). The current results expand this feature into the wall-bounded shear turbulent flows, especially in the over-tripped conditions. In addition, the analogy suggests that the external intermittency has a comparable role in the TNTI in the outer region for both the free and wall-bounded shear flows.

To reveal the mechanism behind the fine-scale coupling related to the external intermittency, we study the p.d.f.s of the velocity derivatives at different wall-normal heights through the intermittent zone. In fact, the flatness factor of the velocity derivatives can be obtained in the integral expression as

(4.16)\begin{equation}F = \int_{ - \infty }^\infty {{{({u_{x,\sigma }})}^4}\sigma P({u_{x,\sigma }})\,\textrm{d}{u_{x,\sigma }}} ,\end{equation}

where ${u_{x,\sigma }} = (\partial u/\partial x)/\sigma $, $P({u_{x,\sigma }})$ is the p.d.f. of the normalized velocity derivative and $\sigma = {\langle {(\partial u/\partial x)^2}\rangle ^{1/2}}$ is the standard deviation. Figure 16(a) shows the p.d.f.s of the velocity derivative $(\partial u/\partial x)/\sigma $ at the different wall-normal heights through the intermittent zone. Since the fourth moment is heavily dependent on the large values of ${u_{x,\sigma }}$, the flatness factor F in figure 14 is a measure of the relative extent of the skirts of the probability density curves. Clearly, it can be seen in figure 16(a) that the p.d.f.s depart from normality further away from the wall towards the boundary-layer edge. The departure is enhanced by showing that the tails of $P({u_{x,\sigma }})$ become increasingly stretched, which is a characteristic footprint of the external intermittency. In particular, the far tails represent large velocity gradients that partly stem from the thin interfacial layer between turbulent and non-turbulent fluids (Elsinga & da Silva 2019). At the same time, a distinct peak emerges around ${u_{x,\sigma }} = 0$. This peak originates from non-turbulent regions outside of the turbulent envelope where the velocity gradients are close to zero. The combination of these behaviours leads to the flatness factor F increasing as it moves toward the boundary-layer edge, as shown in figure 14.

It is now of interest to compare the results with the conditional p.d.f.s that account only for the turbulent portion of the flow. The conditional p.d.f. is defined as $P({u_{x,{\sigma _T}}}|\psi = 1)$ in the condition of $\psi = 1$. A similar procedure was executed by Gauding et al. (2021) to observe the self-similarity in the shear-layer region of turbulent jet flows. The parameter $P({u_{x,{\sigma _T}}}|\psi = 1)$ in figure 16(b) exhibits self-similarity. The departure from normality has the same shape, which has a higher probability than the normal curve in the neighbourhood of zero and a lower probability at the intermediate values. This kind of distribution was consolidated as a typical feature of turbulent signals with internal intermittency in the inner region of TBL flow (Kuo & Corrsin 1971), similar to figure 4. From this comparison between figure 16(a,b), we can conclude that external intermittency is the relevant mechanism that destroys self-similarity through the intermittent zone.

Finally, we revisit the self-similarity of the velocity structure functions. Motivated by the previous discussion, a conditional structure function is defined as

(4.17)\begin{equation}{\langle {({\varDelta _r}u)^n}\rangle _{TT}} = \langle \psi (x + r)\psi (x){(u(x + r) - u(x))^n}\rangle, \end{equation}

for which both ending points are restricted to the turbulent regime of the boundary layer. In the previous investigation, Sabelnikov et al. (2019) defined structure functions in a similar fashion to distinguish in non-premixed flames between burnt and unburnt regions.

Figure 17 presents the normalized conditional structure functions ${\langle {({\varDelta _r}u)^n}\rangle _{TT}}$ up to the tenth order at different wall-normal heights through the intermittent zone. The structure functions are normalized by conditional Kolmogorov scales (${u_{K,T}} = {(\nu \langle \psi \epsilon \rangle )^{1/4}}$ and ${\eta _T} = {({\nu ^3}/\langle \psi \epsilon \rangle )^{1/4}}$) and conditional Taylor scales (${u_{\lambda ,T}} = {\langle \psi {u^2}\rangle ^{1/2}}$ and ${\lambda _T} = {(15\nu u_{\lambda ,T}^2/\langle \psi \varepsilon \rangle )^{1/2}}$) that account only for the turbulent portion of the flow. Compared with the conventional structure functions $\langle {({\varDelta _r}u)^n}\rangle $ (in figure 13), the conditional structure functions ${\langle {({\varDelta _r}u)^n}\rangle _{TT}}$ reveal a significantly improved collapse up to the tenth order, as shown in figure 17(a,b). Especially, on increasing the order, the self-similar arrangement can still be noted. In this sense, ${\langle {({\varDelta _r}u)^n}\rangle _{TT}}$ is assumed to have a reasonably good universality. More importantly, the observation signifies that turbulent regions homogenize across the intermittent zone even under the most over-tripped condition.

Figure 17. Normalized conditional structure functions ${\langle {({\varDelta _r}u)^n}\rangle _{TT}}$ up to the tenth order at different wall-normal heights, where both end points are located inside the turbulent regime. The structure functions are normalized by the conditional (a) Kolmogorov scales and (b) Taylor scales. The line colour has the same meaning as that in figure 13. Data for case III-D20.

Considering the collapse of the conditional high-order structure functions, we now examine the scaling exponents of structure functions and external intermittency in the following. Other than the turbulent signal in the near-wall region, the turbulent signals in the intermittent zone are dominated by external intermittency, the behaviour of which is enhanced by the tripping configurations. Thus, the self-preservation/similarity behaviour of the conditional higher-order structure functions will not be examined for the ECR scales but rather examined in a relative scaling form by the ESS hypothesis as

(4.18)\begin{equation}\frac{{{\xi _p}}}{{{\xi _2}}} = \frac{{\textrm{d}\,\textrm{log}\,\langle {{\textrm{(}{\varDelta _r}{u_T})}^p}\rangle }}{{\textrm{d}\,\textrm{log}\,\langle {{\textrm{(}{\varDelta _r}{u_T})}^2}\rangle }},\end{equation}

which is calculated by the gradient of the pth-order conditional structure function against the second order. The ESS approach in (4.18) provides an extended wide scaling range relative to the restricted subrange, which offers an opportunity for better estimates of the relative ISR scaling exponents (Benzi et al. 1993a,b). Figure 18 shows the scaling exponents ${\xi _p}/{\xi _2}$ ($\kern 0.06em p = 4,\;6,\;8,\;10$) at different wall-normal heights. It shows ${\xi _p}/{\xi _2}$ as a function of the external intermittency parameter $\gamma $. Clearly, the relative scaling exponents remain constant with $\gamma $ and collapse for all the tripping cases. This result indicates that the small scales in turbulent regimes are homogenized in a self-similar behaviour, which is independent of the wall-normal height and the influence of the current tripping wake flows.

Figure 18. Dependence of the relative scaling exponents ${\xi _p}/{\xi _2}$ on the intermittency parameter $\gamma $ for all the tripping conditions (cases III-D1–D20) and for (a) $p = 4$, (b) $p = 6$, (c) $p = 8$, (d) $p = 10$.

Furthermore, the p-model, ${\xi _p} = 1 - \textrm{lo}{\textrm{g}_2}({0.7^{p/3}} + {0.3^{p/3}})$, developed by Meneveau & Sreenivasan (1991), is employed for comparison. The $p$-model is based on a multi-fractal approach and is known to be able to predict the ISR scaling exponents of homogeneous isotropic turbulence with very good accuracy (Sreenivasan & Antonia 1997). Figure 19 plots the average relative scaling exponents (as shown in figure 18) through the intermittent zone $\gamma = 0.1{-}0.9$ for cases I, II and III. The plot shows an acceptable agreement with the prediction from the p-model. Hence, the relative scaling exponents confirm the self-similarity of higher-order structure functions in the turbulent regime even under the influence of the over-tripped conditions. In addition, the average relative scaling exponents in figure 19 involve the cases at all Reynolds numbers in this study, the agreement indicates that the self-similarity of higher-order structure functions is independent of the current Reynolds numbers.

Figure 19. The average of relative scaling exponents ${\xi _p}/{\xi _2}$ up to the tenth order in the intermittent zone ${\gamma = 0.1{-}0.9}$ for cases I, II and III. The standard deviation is denoted by the error bar. For comparison, the $p$-model, ${\xi _p} = 1 - \textrm{lo}{\textrm{g}_2}({0.7^{p/3}} + {0.3^{p/3}})$, by Meneveau & Sreenivasan (1991) (solid line) is shown.

5. Conclusions

In artificially thickened TBL flows from over-tripped conditions, the generated large-scale structures not only alter the intermittency behaviour in the outer region but also provide a modification to the small scales in the inner region. Thus, the effect of the internal and external intermittencies on the self-similarity of TBL in the adaptive region was examined in this study.

In the inner region, by utilizing the ESS hypothesis, the relative scaling of the velocity structure functions was presented with a further reaching universality for the ECR scales. The scaling for the ratio between the velocity structure functions was based on the log-law scaling for the ECR scales in the logarithmic region of high-Reynolds-number TBLs (Davidson et al. 2006; de Silva et al. 2015, 2017), which is in accordance with Townsend's attached eddy hypothesis. The current study consolidated that the quantitative measures of the scaling behaviour for the ECR scales can be effectively estimated from databases at low and moderate Reynolds numbers ($R{e_\tau } \approx 500{-}3500$). Moreover, in the current over-tripped conditions, the boundary layer is thickened with increasing tripping rod diameter, and the constant exponent scaling extends further away from the wall. It can be concluded that the scaling universality for the ECR scales in ESS form is independent of the tripping conditions, which is enhanced with a further reaching universality in the over-tripped conditions.

Then, we studied the self-similarity of structure functions at different wall-normal positions through the intermittent zone. In the intermittent zone, the phenomenon of external intermittency originates from the TNTI, which separates the turbulent and non-turbulent regimes from the hot-wire signals. The morphology of the TNTI was described by the intermittency structure function at different scales. It was indicated that the over-tripped conditions provide a significant modification to the external intermittency. Even though the non-Gaussianity and external intermittency are enhanced under the over-tripped impacts, an acceptable self-similarity was noted for the second-order velocity structure functions through the outer region. However, it was expected that the external intermittency exhibits a remarkable influence on the high-order structure functions. The breakdown of self-similarity was mainly contributed to by the combination of the appearance of higher velocity derivatives and the alternation of turbulent and non-turbulent fluids, which was supported by corresponding statistics such as the flatness factor of velocity derivatives and the jump frequency of turbulence–non-turbulence (vice versa). In fact, the effect of external intermittency on fine-scale turbulence was amplified by increasing the tripping rod diameter. It was manifested that the fine scales are not decoupled with the large scales which are related to the external intermittency at all the tripping conditions.

By defining the turbulent statistics conditioned on fully turbulent regimes, the collapse was noted in the conditional flatness factor and the p.d.f.s of the velocity derivatives, which are nearly independent of the intermittency parameter $\gamma $. Then, the self-similarity behaviour of the conditional higher-order structure functions was further examined. The relative scaling exponents obtained by the ESS hypothesis remain constant against the intermittency parameter across the intermittent zone. An agreement of the conditional scaling exponents with the $p$-model was further examined. It was revealed that the fine scales in the turbulent regime in the artificially thickened TBLs are homogenized in a self-similar behaviour, which is independent of the tripping conditions and also the Reynolds number in the current study. In addition, it should be noted that the above conclusions of the scaling universality of the structure functions are also suitable for the canonical cases, as moderately tripped TBLs with relatively small tripping diameters were involved in the current study.

In practice, the TBL flows over wind turbines, ships, vehicles and aviation systems, experience the effects of leading-edge configurations or upstream flow conditions (such as, roughness, surface curvature, separation, blowing, suction, etc.). Thus, the practical TBL flows are not only at the condition of high Reynolds number, but also beyond canonical flows with the feature of significant changes in all aspects of the mean flow and the turbulence. The current study presents evidence of the self-similarity of non-canonical TBL flows at relatively high $R{e_\tau }$ under leading-edge tripping impacts. With regards to this, the observation in this study provides a potential avenue to assess/predict more general and practical TBL flow behaviour.