Self-similar geometries within the inertial subrange of scales in boundary layer turbulence

The measurements presented herein were acquired using particle image velocimetry (PIV) in the High Reynolds Number Boundary Layer Wind Tunnel at the University of Melbourne. A brief summary of relevant details is provided here and in table 1. A full account of the experiment is given elsewhere (de Silva et al., 2014).

The experiment was conducted under approximately zero-pressure-gradient conditions with free stream velocity $U_{\infty}=$ 20 m s^-1. The boundary layer thickness $\delta=$ 0.3 m is defined using the convention $\delta_{99}=z(U{=}0.99U_{\infty})$, where $z$ is the wall-normal position. The friction Reynolds number under these conditions is $\Rey_{\tau}=\delta u_{\tau}/\nu=$ 12 300, where $u_{\tau}$ is the friction velocity and $\nu$ is the kinematic viscosity of the air in the wind tunnel. For reference, the Taylor microscale Reynolds number is in the range $\Rey_{\lambda}=450-550$ across the region of interest.

An eight-camera setup spanning a large field of view was employed to capture PIV measurements in the streamwise–wall-normal ($x{-}z$) plane. The multi-camera setup spatially resolved a wide range of scales between the interrogation window size $l^{+}=$ 37 and the overall streamwise extent of the field $L_{x}=2\delta$. Throughout this work, the superscript “$+$” indicates wall normalization using $u_{\tau}$ for velocity and $\nu/u_{\tau}$ for length. This spatial range captures a majority of the turbulent scales – excluding the Kolmogorov microscales and very-large-scale motions – in a high-Reynolds-number setting. Even though the smallest motions are unresolved, the resolution is sufficient to identify the transition from the inertial subrange to the dissipative scales as discussed in §2.3. The temporal resolution of the PIV is coarse enough such that each PIV velocity field is treated as an independent realization, and ensemble statistics are computed across realizations.

Relatively uniform flow regions manifest as peaks in histograms of the local streamwise velocity field $u(x,z)$ (Adrian et al., 2000), where the velocity corresponding to each peak is representative of a distinct UMZ (de Silva et al., 2016). At a given instant in time, the number of peaks in the histogram of $u(x,z)$ indicates the number of UMZs in the local flow field. However, this detection is sensitive to the streamwise extent $\mathcal{L}_{x}$ contributing to each histogram, i.e. how “local” the flow field is. In high-Reynolds-number flows, there may be several smaller UMZs across a wide field $\mathcal{L}_{x}\sim\delta$, where these UMZs collectively smooth the histogram such that the individual peaks are concealed (de Silva et al., 2016). Thus, it is often preferable to employ a narrower field scaled in viscous units $\mathcal{L}_{x}^{+}\sim O(10^{3})$ (de Silva et al., 2016) or a fraction of the outer length scale $\mathcal{L}_{x}/\delta\sim O(0.1)$ (Heisel et al., 2020), depending on the region and statistic of interest.

At the same time, the present analysis relies on the continuity of detected UMZ interfaces across the entire 2$\delta$-wide flow field. A multi-step detection method is employed here to account for these seemingly contradictory requirements. The method combines the two most common approaches in the literature: the histogram-based detection of UMZs (de Silva et al., 2016) and the fuzzy clustering detection of their interfaces (Fan et al., 2019). In the first step, the PIV field is divided into twelve segments of width $\mathcal{L}_{x}^{+}=$ 2 000. The local histogram is computed for each segment to estimate the number of peaks (i.e. UMZs) in the segment. The average number of UMZs across all segments, rounded to the nearest integer, is assumed to be representative of the 2$\delta$-wide field.

In the second step, the position of UMZ interfaces are detected using fuzzy clustering as proposed in Fan et al. (2019), where the inputted number of clusters is based on the number of UMZs determined previously. The clustering algorithm assigns each streamwise velocity data point into a cluster such that the overall velocity variance within clusters is minimized. The velocity of the interfaces, corresponding to the boundaries between clusters, is given by the midpoint between cluster centroids. Here, “boundary” and “centroid” refer to the velocity attributes of the clusters and not their spatial properties. Finally, the position of the UMZ interfaces are determined using isocontours of their velocity. Further details, including an example of the histogram peak detection and a comparison of detected UMZ interfaces with alternate approaches, are provided in Appendix A.

An example flow field with detected UMZ interfaces is shown in figure 1(a). Hereafter, we refer to the UMZ interfaces as velocity isosurfaces owing to their definition. While the present planar measurements limit the detection to isolines on a plane rather than surfaces in a volume, the UMZs and their interfaces have been previously observed in three dimensions (e.g., Chen et al., 2020). The primary limitation of the planar detection employed here is uncertainty in how UMZs connect in the spanwise dimension. For instance, two separate zones of similar momentum may be part of a larger meandering structure (Laskari et al., 2018). Additionally, the detected isosurfaces often delineate isolated UMZ “pockets”, e.g. near the coordinate ($x{\approx}0.4\delta$, $z{\approx}0.65\delta$) in figure 1(a). Three-dimensional measurements are required to determine whether these pockets are connected to a larger UMZ and isosurface in the spanwise direction or are due to small-amplitude velocity variability within a UMZ. To avoid potentially spurious detections in the latter case, small pockets with an enclosed area less than 10$\lambda_{T}^{2}$ are removed, where the Taylor microscale $\lambda_{T}$ is the relevant scaling parameter for the interfaces (Eisma et al., 2015; de Silva et al., 2017; Heisel et al., 2021). The chosen minimum area ensures the interface thickness encompasses no more than half the area of the pockets. While the pocket near ($x{\approx}0.4\delta$, $z{\approx}0.65\delta$) exceeds the minimum area and is therefore retained, the nearby pockets identified as white-colored regions are smaller than the threshold and are not included. Appendix B evaluates how the selected minimum area of pockets affects later results, specifically the fractal dimension of the isosurfaces.

A particular point of interest is the imprint of the detected UMZs and their interfaces on a one-dimensional (1D) velocity signal and its corresponding statistics. Figure 1(b) shows an example 1D transect of $u(x,z{=}0.1\delta)$ from panel 1(a), where the vertical lines indicate crossings of the isosurfaces. The intermittency of the crossings is immediately apparent. As expected, crossings occur at the largest “jumps” in velocity near $x\approx\delta$ and $x\approx 1.7\delta$. Previous studies have shown that regions of high shear generally align with UMZ interfaces (de Silva et al., 2017; Gul et al., 2020; Chen et al., 2021). However, due to the continuity of the isosurfaces, the interfaces also extend into regions where the shear magnitude is lower. This attribute of the isosurfaces is apparent near $x\approx 0.4\delta$ and $x\approx 0.7\delta$ in 1(b) where the clusters of crossings do not align with large velocity changes. Lastly, the filtering of UMZ pockets is evident near $x\approx 0.25\delta$ and $x\approx 0.75\delta$ where a crossing is excluded because the corresponding spatial region in panel 1(a) is too small.

The methods used to estimate the approximate limits of the inertial subrange are discussed here due to variability in conventions across and within disciplines. As discussed in the introduction, the limits are specifically for statistics of the streamwise velocity component. Later figures indicate these subrange limits for the purpose of placing trends and transitions within the spectrum of turbulent scales. The scaling arguments employed here are preferred over directly detecting the –5/3 region in the energy spectrum due to uncertainties in both the spectrum estimate and detection procedure. The estimated limits are supported by later statistics including figure 5(b).

The transition from the inertial subrange to the dissipative scales is approximated based on the Kolmogorov lengthscale $\eta=(\nu^{3}/\epsilon)^{1/4}$, where $\epsilon$ is the average rate of turbulent energy dissipation. The dissipation was estimated as $\epsilon\approx 15\nu\langle(\partial u/\partial x)^{2}\rangle$ assuming local isotropy. Studies have shown the dissipative range to start near the angular wavenumber $k_{x}=0.1\eta^{-1}$(e.g., Saddoughi & Veeravalli, 1994). The equivalent length in physical space, i.e. $2\pi/k_{x}\approx 63\eta$, is used here as the lower limit of the inertial subrange.

The Taylor microscale $\lambda_{T}^{2}=\langle{u^{\prime}}^{2}\rangle/\langle(\partial u/\partial x)^{2}\rangle$ has also been used as the upper limit of the dissipative range of scales for a variety of flow applications (see, e.g., Cava et al., 2012; Debue et al., 2018; Bandyopadhyay et al., 2020). Further, experimental evidence suggests the intense shear layers and largest vortex cores are proportional to $\lambda_{T}$ (Eisma et al., 2015; de Silva et al., 2017; Heisel et al., 2021). Both $\lambda_{T}$ and $\epsilon$ were estimated here using hotwire anemometry measurements under the same flow conditions. The ratio of the parameters is $\lambda_{T}/\eta\approx$ 40 to 45 within the region of interest studied herein. Later results include a limited number of data points between the two possible limits $\lambda_{T}\approx 40\eta$ and $63\eta$, such that there is no meaningful distinction between the two scaling choices with respect to conclusions drawn herein.

The transition from the inertial subrange to the production range is approximated using a dissipation-based lengthscale $L_{\epsilon}=u_{\tau}^{3}/\epsilon$ (Davidson & Krogstad, 2014). The friction velocity $u_{\tau}$ was measured directly from the wall shear stress as $u_{\tau}=(\nu\partial U/\partial z)^{1/2}$ using separate high-magnification PIV in the viscous sublayer. The measured value was corroborated using a drag balance facility and the Clauser chart method (de Silva et al., 2014). In canonical logarithmic regions with production and dissipation in equilibrium, the length scale simplifies to $L_{\epsilon}=\kappa z$ and the transition occurs approximately at $z\approx L_{\epsilon}/\kappa$ (de Silva et al., 2015), where $\kappa$ is the von Kármán constant. The limit $L_{\epsilon}/\kappa$ is therefore employed here, noting that the $L_{\epsilon}$ basis is preferred over $z$ due to its general extension to roughness layers and other non-equilibrium conditions (Davidson & Krogstad, 2014; Chamecki et al., 2017; Ghannam et al., 2018).

With respect to the limits discussed above, the PIV interrogation window size $l/\eta=14{-}18$ is within the dissipative scales and the streamwise field extent $L_{x}/L_{\epsilon}=20{-}55$ exceeds the inertial subrange. These ranges are based on the $\eta$ and $L_{\epsilon}$ values observed within the logarithmic region of the boundary layer. The values confirm that the measurement spatial range captures the full extent of the inertial subrange within the region of interest.

A direct method for assessing potential fractal geometries is the box counting technique (for applications in turbulence see, e.g., Sreenivasan & Meneveau, 1986; Moisy & Jiménez, 2004; de Silva et al., 2013). In this method, the domain is partitioned into boxes of size $b$, and the number of boxes $N_{b}$ required to enclose the shape is counted. The process is repeated for a series of box sizes by varying $b$. In principle, the box can have as many dimensions as the domain, e.g. a cube box for a flow volume. A square box in the two-dimensional PIV field is employed here. Figure 2(a) shows an example streamwise velocity isosurface and the boxes of size $b$ required to capture the shape.

The statistics for $N_{b}$ resulting from the box counting routine are shown as colored symbols in figure 2(b) for the detected UMZ interfaces. Due to the large range in $z$ spanned by each convoluted isosurface, the statistics are grouped based on the velocity $u_{i}$ of the isosurfaces. The wall-normal position associated with each isosurface is then taken as the position $z(U{=}u_{i})$ where the mean velocity matches $u_{i}$. The groups in figure 2(b) correspond to four $z/\delta$ positions within the logarithmic region. The purpose of associating the velocity groups with a position $z$ is to estimate the inertial subrange limits using local values of $\eta(z)$ and $L_{\epsilon}(z)$. These limits – whose definitions are given in §2.3 – are shown as vertical dashed lines in figure 2(b). In figure 2 and later figures, the normalizations relevant to both limits of the inertial subrange are shown for completeness.

For statistically self-similar geometries, the resulting relation between $b$ and $N_{b}$ follows a power law

A more rigorous test of the power law relation is achieved by estimating the local (in scale) fractal dimension as $D_{2}(b)=-d\log{(N_{b})}/d\log{(b)}$, which is shown in figure 2(c). The dimension for each position is approximately constant within the estimated inertial subrange of scales near the value $D_{2}\approx 1.2$ represented by horizontal lines in figure 2(c). This value matches previous estimates of $D_{2}$ using the same PIV experiment and a different method to detect UMZ interfaces (de Silva et al., 2017). The results are also consistent with fits to each curve in the inertial subrange which yield values for $D_{2}$ between 1.2 and 1.22. The range of scales exhibiting the constant $D_{2}\approx 1.2$ in figure 2(c) is specifically confined to the inertial subrange, which grows with increasing distance from the wall and spans a full decade for $z/\delta=0.2$ ($\bigtriangledown$). Previous studies of UMZ interfaces employed narrower streamwise segments $\mathcal{L}_{x}=2\,000\nu/u_{\tau}\approx 0.2\delta$ such that the upper limit of the inertial subrange was not evident in the box-counting results (de Silva et al., 2017). While there is evidence for a constant fractal dimension for scalar concentration isosurfaces in simulations of isotropic turbulence (Iyer et al., 2020), the monofractal behavior confined to the inertial subrange in figure 2 is new for velocity isosurfaces within the boundary layer.

Several tests were conducted to assess the robust nature of the results in figure 2. The first test identified velocity isosurfaces between the UMZ interfaces. For a UMZ whose bounding interface isosurfaces are defined by the velocities $u_{i}$ and $u_{i+1}$, an isosurface of the velocity $0.5(u_{i}+u_{i+1})$ was assumed to represent facets internal to the UMZ. The dashed lines in figure 2(a) correspond to these internal isosurfaces, and the grey filled symbols in figure 2(b,c) show the corresponding box counting results. The internal isosurfaces have the same self-similar behavior as the UMZ interfaces, where the fractal dimension $D_{2}\approx 1.2$ is constant within the inertial subrange. The second test analyzed isosurfaces of the fixed velocity $U(z{=}0.1\delta)$ which is independent of the UMZ detection. Again, the same result for $D_{2}$ (not shown here) was observed. The final test repeated the analysis using measurements under varying flow conditions (de Silva et al., 2014) including for rough surfaces (Squire et al., 2016), and the same quantitative results were observed. The findings of these tests suggest the constant fractal dimension within the inertial subrange is a general feature of velocity isosurfaces internal to the boundary layer, and is not a unique property of UMZ interfaces dependent on the present flow conditions.

As discussed elsewhere (de Silva et al., 2017), the observed fractal dimension $D_{2}\approx 1.2$ is lower than the approximate value 1.33 reported for the turbulent-nonturbulent interface (TNTI) (Sreenivasan & Meneveau, 1986; de Silva et al., 2013; Chauhan et al., 2014; Borrell & Jiménez, 2016). One-dimensional box counts presented in appendix C suggest the lower $D_{2}$ value estimated here may be due to anisotropy in the shape of the largest geometric features along the velocity isosurfaces. The expected one-dimensional fractal estimate $D_{1}\approx$ 1/3 is achieved by excluding local regions where the largest features yield a trivial box-counting result. As seen in appendix B, the dimension $D_{2}$ is also sensitive to the treatment of “pockets” described in §2.2. Filtering of pockets introduces a subjective distinction between small-scale features and isosurfaces that should be counted towards statistical self-similarity. Based on appendices B and C, the differing values for $D_{2}$ may be due to limitations and subtle differences in methodology. Accordingly, the critical result here for $D_{2}$ is its constancy within the inertial subrange of scales rather than its precise numerical value.

As introduced previously in figure 1(b), the convoluted isosurfaces appear as “crossings” along 1D streamwise transects of the flow. These transect crossings provide low-order information on the position of velocity changes along $x$. The information is considered low order because the complexity of the continuous velocity signal is reduced to discrete instances where the signal crosses the isosurface velocity $u_{i}$.

The crossings of detected UMZ interfaces are closely related to so-called zero crossings (Liepmann, 1949; Liepmann et al., 1951; Badri Narayanan et al., 1977; Sreenivasan et al., 1983; Kailasnath & Sreenivasan, 1993; Poggi & Katul, 2009). In a zero crossing analysis, point measurements such as hotwire anemometry are used to locate where the velocity signal crosses the mean velocity, i.e. where the fluctuating velocity $u^{\prime}=u-U$ is zero. In the context of the present analysis, zero crossings are restricted to a single streamwise velocity isosurface $u_{i}=U(z)$ for a transect measured at $z$. The crossings presented here are generalized to allow for multiple isosurfaces that can differ from the local mean velocity.

An important property of the crossings is the streamwise distance $\delta x_{i}$ between consecutive isosurface crossings. For reference, the zero crossings literature refers to the distance between crossings as the interpulse period (Sreenivasan & Bershadskii, 2006; Cava et al., 2012) or the persistence (Perlekar et al., 2011; Chamecki, 2013; Chowdhuri et al., 2020). Figure 3(a) illustrates the definition of $\delta x_{i}$ using isosurfaces of detected UMZ interfaces that cross a 1D signal at $z=0.1\delta$. Because multiple UMZ interfaces are present within the boundary layer, the segments of length $\delta x_{i}$ can be bounded by either the same velocity isosurface or different isosurfaces. Separate statistics are presented for these two scenarios.

Figure 3(b,c) shows probability distributions of $\delta x_{i}$ for transects along four wall-normal positions within the log region. The distributions are presented as number densities $N_{d}$, where the total area under the distribution represents the number of measured $\delta x_{i}$ events. The number density allows for a direct comparison between the two crossing scenarios described above: for a given value of $\delta x_{i}$, the scenario with larger $N_{d}(\delta x_{i})$ is relatively more frequent and thus contributes more to overall statistics. The distribution for all values of $\delta x_{i}$, shown as a solid line figure 3(b,c), is simply the sum of the individual $N_{d}$ distributions.

The distribution shape for $\delta x_{i}$ between crossings of the same isosurface is in close agreement with previous observations for the zero crossing interpulse period. Intermediate distances are well approximated by a power law (Bershadskii et al., 2004), and the shape transitions to an exponential tail across longer distances (Sreenivasan et al., 1983; Cava et al., 2012; Chamecki, 2013). The exponential trend is more apparent from the log-linear plots presented in figure 3(c). Previous studies have described the shortest interpulse periods using a log-normal distribution (Badri Narayanan et al., 1977; Sreenivasan & Bershadskii, 2006), but the present PIV measurements do not resolve the trends at the dissipative scales where the log-normal distribution is expected.

Within the inertial subrange of scales, the power law exponent for $N_{d}(\delta x_{i})$ between crossings of the same isosurface is approximately –1.5 regardless of position $z$. The slope again agrees with results for zero crossings (Bershadskii et al., 2004; Cava et al., 2012). Based on the amplitude of $N_{d}$ in figure 3(b), $\delta x_{i}$ values corresponding to the inertial subrange are predominately due to crossings of the same isosurface. Even though $\delta x_{i}$ values between different isosurfaces are not self-similar in this region, the behavior along a single isosurface has leading-order importance such that the overall curve still approximates a power law. However, the overall power law exponent (–1.3) is somewhat flattened by the contributions of $\delta x_{i}$ between crossings of different isosurfaces. The trends in figures 2 and 3(b) demonstrate that the signature of self-similarity in the inertial subrange is reflected by the geometry of individual streamwise velocity isosurfaces. The behavior across consecutive isosurfaces, which is related to the amplitude of velocity variations, affects the power law exponent but does not contribute directly to the self-similarity.

Across longer $\delta x_{i}$ crossing intervals, the $N_{d}$ trends transition and the statistics between different isosurfaces gain leading-order importance within the production range of scales. The exact transition point where the $N_{d}$ curves intersect is sensitive to the detection of UMZs, and the result is discussed here qualitatively. The transition corresponds to a shift from a power law distribution to an exponential tail seen in figure 3(c). The tail slope varies moderately between wall-normal positions, suggesting a normalization parameter other than $L_{\epsilon}$ is required to describe variations in the tail slope. The results agree with previous works that observed the exponential tail to deviate from the integral scales (Chamecki, 2013). This trend is not the focus of the present analysis, and is not explored further here.

The results in figure 3 were reproduced using randomized data sets to ensure the $N_{d}$ distributions are not an artifact of the methodology. Synthetic velocity fields generated using random Gaussian noise do not contain any large-scale organization or UMZ signature (de Silva et al., 2016), and all observed trends in $N_{d}$ are lost. In a separate test, the streamwise velocity fluctuations in each PIV field were randomized in their phase (Theiler et al., 1992). Phase randomization of the two-dimensional Fourier transform preserves the original two-dimensional energy spectrum and correlations of the velocity. The phase-randomized data produces the same overall shape of $N_{d}$ due to the relation between the crossing distributions and the spectrum (Bershadskii et al., 2004; Poggi & Katul, 2009; Heisel, 2022). However, there are fewer crossings between different isosurfaces and its distribution is no longer exponential within the inertial subrange. The result indicates that the observed transition in statistics between the inertial and production ranges is related to aspects of the flow structure that are distorted during phase randomization.

The figure 3 trends can be used to interpret the transition from the inertial subrange to the production range of scales in the context of coherent structures. The $\delta x_{i}$ values between different isosurfaces represent the distances between adjacent UMZ edges. The distance across UMZs, and more generally the overall geometry of the large-scale velocity regions, becomes the limiting factor within the production range. The confinement of the self-similar isosurfaces by the large-scale organization of the flow is consistent with the sharp decline of $N_{d}$ in figure 3(b) and the deviation from a constant fractal dimension in figure 2(c). In this sense, the statistically relevant property of the flow geometry is the self-similarity of isosurfaces in the inertial subrange, and the size of the coherent velocity regions in the production range. This interpretation of the production range is essentially the same as the existing viewpoint (Perry & Chong, 1982) and is supported by direct observations of UMZ sizes (Heisel et al., 2018, 2020) and distances between coherent velocity regions (Dong et al., 2017).

The remainder of the analysis quantifies how the self-similar geometries influence scale-dependent statistics such as the structure function. The second-order longitudinal structure function of the streamwise velocity is defined as

To assess the contribution of spatial features such as the detected streamwise velocity isosurfaces to the structure function statistics, the ensemble averaging operator in equation (2) can be replaced by a conditional averaging operator based on an additional parameter related to the spatial features. The parameter $\delta x_{o}=x_{i}-x_{o}$ describes the distance from a given reference point $x_{o}$ to the nearest crossing $x_{i}$ of a detected streamwise velocity isosurface. Figure 4(a) shows the definition of $\delta x_{o}$ for an example isosurface. Rather than ensemble averaging across all $x_{o}$, the conditional averaging operator only considers instances where $\delta x_{o}$ is a specified value. The result is a structure function dependent on both $r_{x}$ and the distance $\delta x_{o}$ to the nearest isosurface:

The traditional structure function using equation (2) is discussed here prior to introducing the conditional results. Figure 4(b) shows $\langle\Delta u^{2}\rangle$ for the wall-normal position $z=0.1\delta$. A power law slope of 2 describes the smallest $r_{x}$ increments, but is not assessed here due to spatial resolution limitations. This power law exponent is predicted from the Kármán–Howarth equation when viscous diffusion dominates over inertial mechanisms (Pope, 2000). In the inertial subrange, Kolmogorov’s second similarity hypothesis predicts $\langle\Delta u^{2}\rangle\sim r_{x}^{2/3}$, which is well-supported by measurements and simulations (Pope, 2000). However, the 2/3 signature in figure 4(b) is not strictly constant within the inertial subrange. The changing slope within the inertial subrange is a known limitation of structure function statistics, and is in part due to the inclusion of other small- and large-scale effects in the cumulative velocity increment $\Delta u$ (see, e.g., Davidson & Pearson, 2005).

Conditional structure functions $\langle\Delta u^{2}\rangle|_{\delta x_{o}}$ for four values of $\delta x_{o}$ are shown in figure 4(c). The four values were chosen to represent a range within and beyond the inertial subrange. By fixing the position of the nearest detected isosurface $\delta x_{o}$, and considering the isosurfaces are a low-order representation of velocity changes as discussed previously, it is expected that the resulting conditional average will yield a large velocity increment near $r_{x}\approx\delta x_{o}$. Large “jumps” in $\langle\Delta u^{2}\rangle|_{\delta x_{o}}$ are indeed observed at $r_{x}\approx\delta x_{o}$. Further, each conditional curve converges to the overall result $\langle\Delta u^{2}\rangle$ across long distances where the imposed $\delta x_{o}$ condition is no longer relevant. The thickness of the jumps (in terms of $r_{x}$) is in part due to viscous effects that act to smooth the velocity gradients. Accordingly, the jump thickness for $\delta x_{o}\approx 30\eta$ spans most of the dissipative range, but appears increasingly sharp for the other cases along the logarithmic abscissa.

The behavior of the jumps in $\langle\Delta u^{2}\rangle|_{\delta x_{o}}$, in comparison to the approximate power law within the inertial subrange, demonstrates the features underlying the ensemble-averaged structure function. The power law relation is not apparent from any given instantaneous velocity increment $\Delta u(x_{o},r_{x})=u(x_{o}+r_{x})-u(x_{o})$. Rather, it is the stochastic result of variability in the positions of the isosurfaces across numerous instances, where the position is parameterized as $\delta x_{o}$ in figure 4. The distributions in figure 3 already showed that the variability is statistically self-similar in the inertial subrange, a direct result of the fractal dimension in figure 2. The structure function power law may therefore result from a composite of “jumps” across self-similar velocity isosurfaces. This possibility is explored in the following section using a simplified velocity signal.

The crossing positions in §3.2 (figure 3) encompass the geometric properties of the isosurfaces evaluated in §3.1 (figure 2). Here, the measured 1D velocity signal is reduced to these crossing properties. By removing aspects of the signal unrelated to the isosurface crossings, the simplified signal is used to demonstrates how the self-similar signature of the multi-dimensional isosurface geometries translates to scale-dependent statistics of the longitudinal signal, specifically the structure function.

In a zero crossing analysis, the simplification is accomplished using the telegraph approximation (TA) (e.g., Bershadskii et al., 2004). The TA signal is assigned a value of 1 or 0 depending on the sign of the fluctuating velocity: $u_{TA}=1$ where $u^{\prime}>0$ and $u_{TA}=0$ where $u^{\prime}<0$. The basic principles of the TA signal are generalized here to construct a simplified signal $u_{\pm}$ based on the detected streamwise velocity isosurfaces. Hereafter, $u_{\pm}$ is referred to as the “iso-crossing” signal.

The 1D iso-crossing signal is assigned an initial value $u_{\pm}(x{=}0)=0$. At each isosurface crossing position $x_{i}$, $u_{\pm}$ is increased by 1 if $\partial u/\partial x>0$ and is decreased by 1 if $\partial u/\partial x<0$ based on the gradient at the crossing. The iso-crossing signal therefore contains information on the position and sign of the velocity change for each crossing. The signal is equivalent to $u_{TA}$ for mean velocity crossings $u_{i}=U$, and additionally allows for multiple isosurfaces of any arbitrary velocity $u_{i}$.

Figure 5(a) shows an example iso-crossing signal in comparison with the measured velocity $u$. To negate the effect of the velocity magnitude in the visual comparison, the $u$ signal is normalized to achieve the same standard deviation as the $u_{\pm}$ signal, and the values are shifted to achieve $u(x{=}0)=0$. The presence of multiple isosurfaces with different velocities $u_{i}$ leads to $u_{\pm}$ values beyond 0 and 1. The four observed values of $u_{\pm}$ ranging from –1 to 2 indicate that the example 1D signal spans four UMZs.

Structure function statistics for $u$ and $u_{\pm}$ are presented in figure 5(b,c). To facilitate comparison of the trends, the iso-crossing structure functions are increased by a constant factor such that the curves are shifted closer to the measured results. The shift does not affect the shape or slope of the curves in figure 5(b). The shift does increase the slope of the linear region for the $u_{\pm}$ curves in figure 5(c), but it does not change the fact that an approximately linear region exists.

The primary difference between the structure functions is the excess variability for the iso-crossing signal at small $r_{x}$ distances corresponding to the dissipative scales. The larger values are due to the removal of viscous smoothing effects in favor of discontinuous jumps in $u_{\pm}$. Otherwise, despite the limited information in the iso-crossing signal as seen in figure 5(a), the resulting structure function captures several key features of $\langle\Delta u^{2}\rangle$. For figure 5(b) in the inertial subrange, a power law is observed with a cleaner signature for $u_{\pm}$ than $u$. The fitted power law exponent is approximately 0.55 for the iso-crossing signal, which is smaller than the value close to 2/3 observed for $u$. The difference in power law exponent is further discussed later in this section.

For figure 5(c) in the production range, the structure functions nearest the wall are approximately linear. The linear trend is consistent with the predicted logarithmic dependence $\langle\Delta u^{2}\rangle\sim\ln(r_{x})$ that is analogous to the $k_{x}^{-1}$ signature in the energy spectrum (Davidson et al., 2006; Davidson & Krogstad, 2006). Further, the linear slope for the measured $u$ signal closely matches previous observations of 2.4 (de Silva et al., 2015). However, the linear slope for the iso-crossing structure function is not representative due to the manual shift discussed above.

In addition to capturing the inertial subrange power law and production range logarithmic trends, the iso-crossing structure function also identifies where the transition occurs between the two scaling regions. The result is entirely consistent with the crossing distribution in figure 3. The $N_{d}$ distributions exhibit an inertial subrange power law owing to the self-similar geometry of the isosurfaces and a transition to an exponential tail at scales where the overall geometry of the large-scale velocity regions gains leading-order importance. Figure 5 demonstrates how these attributes of the isosurface geometry and their crossing properties propagate to structure function statistics. The same trends in the transition to the production range are also apparent from the spectrum of a TA signal (Huang et al., 2021), likely due to the TA signal exhibiting the same exponential cutoff in figure 3 imposed by the larger-scale geometries.

The primary attribute absent from the iso-crossing signal is the variability in velocity amplitude at positions between the crossings. This absence is responsible for the difference in power law exponent between the structure functions for $u$ and $u_{\pm}$ seen in figure 5(b). A more representative iso-crossing signal can be achieved by simply incorporating additional isosurfaces at velocity increments between the detected UMZ interfaces. Figure 6 uses results at $z=0.1\delta$ to demonstrate how $u_{\pm}$ and its structure function change with the inclusion of additional isosurfaces. Unlike for previous results, the smallest isolated isosurface pockets are not excluded from the signal in figure 6. While these pockets obfuscate the signature of fractal self-similarity as discussed in §3.1, the pockets must be included for a full representation of the velocity signal.

The number of isosurface level sets is parameterized by the resulting velocity difference $\Delta u_{i}$ between adjacent isosurfaces. The iso-crossing signal can be considered as a transformation from spatial coordinates to a velocity grid. Rather than discretizing the signal at fixed positions or times, the signal is mapped to fixed velocities whose resolution is given by $\Delta u_{i}$. As the number of level sets increases, the iso-crossing signal captures the velocity variability in finer increments of velocity resolution and more closely resembles $u$. The result for $\Delta u_{i}/\sigma_{u}=0.5$, shown for $x/\delta=$ 0.8 to 1.2 in figure 6(a), features 8 isosurfaces across the range of $u$. Yet the structure function resulting from this $u_{\pm}$ (ensemble-averaged across all PIV fields) shown in figure 6(b) has a power law exponent 0.62, within 10% of 2/3. An even larger improvement occurs within the dissipative range, where the sharp velocity jumps are distributed across additional steps spanning a wider distance as $\Delta u_{i}$ decreases.

The additional level sets in figure 6 correspond to internal isosurfaces within the detected UMZs. Figure 2 showed that the internal isosurfaces exhibit the same self-similarity as the UMZ interfaces. As a result, every streamwise velocity isosurface $u_{i}$ represented by the $u_{\pm}$ signals in figure 6 will have self-similar crossing distributions analogous to the power law in figure 3. The velocity amplitude variability captured by the finer $\Delta u_{i}$ increments is therefore expected to also exhibit self-similar behavior, specifically in terms of the distances between positions where the velocity reaches the same amplitude. In this sense, the self-similar signature of individual isosurfaces extends to velocity amplitude upon considering the collective contribution of many isosurfaces to a 1D transect of the flow geometry. While an infinite number of isosurface level sets is required to fully represent the continuous velocity signal, trends across the turbulence spectrum can be reasonably recovered with a limited account of isosurface properties. Most notably, the original iso-crossing signal in figure 5 with a minimal number of isosurfaces is able to reproduce a majority of the scale-dependent behavior in the inertial and production regions.