3.1. Mean and fluctuating scalar fields

Some basic statistics of the scalar concentration fields are obtained and presented, with comparisons among similar independent studies. Figure 3(a) shows a snapshot of the concentration contours $\phi$ measured by PLIF at ${Re} = 10\,000$ . The profiles of the mean concentration $\langle \phi \rangle$ in the measurement region (i.e. $z/D = 10.7$ ) are displayed in figure 3(b). These profiles are normalised by their respective peak concentrations. The mean profiles are well approximated by fitted Gaussian curves, which are shown as lines in the figure. In our experiment, the mean profile consistently exhibits a single peak. These findings are in agreement with those of Lavertu & Mydlarski (Reference Lavertu and Mydlarski2005) and Germaine, Mydlarski & Cortelezzi (Reference Germaine, Mydlarski and Cortelezzi2014). Similar trends in mean profiles have also been reported by Brethouwer et al. (Reference Brethouwer, Boersma, Pourquié and Nieuwstadt1999), Lavertu & Mydlarski (Reference Lavertu and Mydlarski2005) and Germaine et al. (Reference Germaine, Mydlarski and Cortelezzi2014). The widths of the mean profiles slightly increase with the increase of ${Re}$ . This observation may be attributed to the fact that the plume from the needle injector is stirred and spreads closer to the wall due to higher turbulent fluctuations at the larger Reynolds numbers. In our measurement region ( $10.1\leqslant z/D\leqslant 11.2$ ), the mean profiles show negligible changes over downstream positions, as shown in figure 3(c). Several examples of the spatial concentration profiles are presented in figure 3(d), where significant variations appear. A similar trend is also observed in the temporal concentration profiles shown in figure 3(e).

Figure 3.

(a) A snapshot of the concentration by PLIF at ${Re} = 10\,000$ . Here, $\phi _0$ represents the initial concentration in the needle. (b) Mean concentration profiles normalised by their respective peak values ( $\phi _{{peak}}$ ) at the downstream location $z/D = 10.7$ for Reynolds numbers ${Re} = 10\,000$ (), $Re=15\,000$ () and $Re=20\,000$ (). The symbols indicate the profiles from the PLIF measurements, and the solid lines correspond to the best-fit Gaussian profiles, where approximately every other ten points are shown for clarity. (c) Mean concentration profiles normalised by their respective peak values ( $\phi _{{peak}}$ ) at different downstream locations for ${Re} = 20\,000$ and $z/D = 10.2$ (), $z/D = 10.7$ (), $z/D = 11.2$ (). (d) Space signals of the concentration at the pipe centre at $t = 10D/U_b$ (black), $20D/U_b$ (red), $30D/U_b$ (cyan) for ${Re} = 20\,000$ . (e) Time signals of the concentration at $(y/D,z/D)=(0.2,10.2)$ (black), $(y/D,z/D)=(0.2,10.7)$ (red) and $(y/D,z/D)=(0.2,11.2)$ (cyan) for ${Re} = 15\,000$ . ( f) The PDFs of the scalar fluctuations at radial locations $|y/D| = 0, 0.2, 0.4$ for different Reynolds numbers ( ${Re}=10\,000, 15\,000, 20\,000$ ), where the line patterns correspond to the Reynolds numbers, and the colours indicate different radii. The inset shows the PDFs around their peaks.

The PDFs of the concentration fluctuations $\phi '$ at different radial positions ( $|y/D|$ ) are shown in figure 3( f) at downstream position $z/D =10.7$ for the three Reynolds numbers. All PDFs are spiked and skewed. As ${Re}$ increases or the radial position decreases, the width of the PDFs increases and their peaks decrease, as shown in the inset of figure 3( f). The decreasing tails of the PDFs at $|y/D| = 0$ (blue lines in the figure) exhibit an approximate power-law shape, which is independent of ${Re}$ , as discussed in Villermaux, Innocenti & Duplat (Reference Villermaux, Innocenti and Duplat2001). Similar PDF profiles have been reported in Lavertu & Mydlarski (Reference Lavertu and Mydlarski2005), Lepore & Mydlarski (Reference Lepore and Mydlarski2011) and Germaine et al. (Reference Germaine, Mydlarski and Cortelezzi2014) for thermal plumes. However, other tails depend on the position and ${Re}$ (especially close to the wall, e.g. at $|y/D| = 0.40$ , black lines in figure 3 f), which may be related to the inhomogeneity of the turbulent pipe flow, and needs further investigation. The statistical results of the high-resolution PLIF measurements are similar to those of previous studies, indicating the good quality of our measurements. Therefore, according to the above results, the present streamwise measurement region $10.1 \leqslant z/D \leqslant 11.2$ is used for the subsequent analysis of the fractal dimension of the iso-scalar surfaces.

3.2. Fractal dimension in the inertial–convective range

Studies have demonstrated that the geometry of iso-scalar surfaces in turbulent scalar fields can be characterised in terms of the fractal dimension (Prasad & Sreenivasan Reference Prasad and Sreenivasan1990; Constantin et al. Reference Constantin, Procaccia and Sreenivasan1991; Catrakis & Dimotakis Reference Catrakis and Dimotakis1996; Dasi et al. Reference Dasi, Schuerg and Webster2007; Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020). To analyse the fractal properties of the mixing scalar surfaces, the box-counting method is employed in this study. This method has been widely used in many studies of the fractal geometry of iso-scalar surfaces (e.g. Prasad & Sreenivasan Reference Prasad and Sreenivasan1990; Constantin et al. Reference Constantin, Procaccia and Sreenivasan1991; de Silva et al. Reference de Silva, Philip, Chauhan, Meneveau and Marusic2013; Mistry et al. Reference Mistry, Philip, Dawson and Marusic2016). In the box-counting method, the number of boxes, which the iso-scalar surfaces pass through, scales with the box size $r_b$ as $N(r_b) \sim r_b^{-D_F}$ , where $D_F$ is the fractal dimension (Mandelbrot Reference Mandelbrot1982; Sreenivasan Reference Sreenivasan1991).

Figure 4.

Sample concentration fields $\phi /\phi _0$ at ${Re}=20\,000$ . The colour bar is the same as that in figure 3(a). The concentration contours serving as the background and iso-scalar surfaces at different thresholds are marked as solid cyan lines: (a) $\phi_{th}=1.0\langle \phi \rangle$ , (b) $\phi_{th}=2.0\langle \phi \rangle$ , (c) $\phi_{th}=3.0\langle \phi \rangle$ , (d) $\phi_{th}=4.0\langle \phi \rangle$ . See supplementary movie 1 at https://doi.org/10.1017/jfm.2025.10454.

Figure 5.

Coverage count of boxes $\langle N\rangle /(D/\eta _K)$ required to cover different iso-scalar surfaces versus box size $r_b/\eta _K$ (and $r_b/D$ ) and corresponding fractal dimensions ( $D_F$ ) at different thresholds: (a,d) ${Re}=10\,000$ , (b,e) ${Re}=15\,000$ , (c,f) ${Re}=20\,000$ . The grey dashed line in (d–f) denotes $D_F=1.67$ . Magenta arrows indicate the Taylor microscales (plain arrow) and the integral length scale (hollow arrow) at $|y/D|=0$ . Black arrows indicate correspondingly the same scales at $|y/D| = 0.4$ . (Note that the integral scale at $|y/D| = 0.4$ is underestimated due to the limited streamwise measurement field of view, within which the autocorrelation of the streamwise velocity does not yet reach zero.) Taylor microscales and integral length scales at $0\lt |y/D|\lt 0.4$ lie in between those at $|y/D|=0$ and $|y/D|=0.4$ . These arrow symbols have the same meanings in later figures.

The ensemble average of the concentration field over time, $\langle \phi \rangle$ , is taken as the criterion, and the thresholds $\phi _{{th}}/\langle \phi \rangle = 1.0{-}4.0$ are used to extract the iso-scalar surfaces of the scalar concentration fields. In figures 4(a–d), a sample of the concentration contours of $\phi /\phi _0$ is served as the background, and the iso-scalar surfaces with different thresholds are shown as solid cyan lines. At $\phi _{{th}} = 1.0\langle \phi \rangle$ , the edges of the mixing structures are detected; however, the experimental noise is also captured, as shown close to the borders of figure 4(a). As $\phi _{{th}}$ increases, the regions where the concentration is closer to the initial concentration are enclosed by the isolines, e.g. see the bright yellow regions in figures 4(b–d), where significant mixing does not occur.

Furthermore, the box-counting method is applied to the iso-scalar surfaces at different thresholds. The coverage count $\langle N\rangle$ of boxes, after ensemble averaging over all measured fields (de Silva et al. Reference de Silva, Philip, Chauhan, Meneveau and Marusic2013), is plotted in figures 5(a–c) on log-log axes for three Reynolds numbers, namely 10 000, 15 000 and 20 000. Here, $\langle N\rangle$ decreases as the normalised scale $r_b/\eta _K$ increases, where $\eta _K$ is the Kolmogorov length scale in the pipe centre. (The Kolmogorov length scale in the pipe centre $\eta _K$ is used for the normalisation, in consistency with the Kolmogorov scale used for the later fractal dimensions of one-dimensional space data along the pipe axis in § 3.4. Here, the fractal dimension is obtained over the two-dimensional iso-scalar surfaces; the corresponding Kolmogorov scale for normalisation is not straightforward, given that the Kolmogorov scale has a profile along the pipe radius. Since the fractional statistics are taken mostly for $|y/D|\lt 0.4$ (see figure 4), and the larger thresholds closer to the pipe centre, the average Kolmogorov scale weighted by the mean concentration is $\overline {\eta }_K = [\int _0^{D/2} \langle \phi \rangle \eta _K\,\mathrm{d}y]/[\int _0^{D/2} \langle \phi \rangle\, \mathrm{d}y]\approx 0.93\eta _K$ , and a trivial shift along the horizontal axis is expected in the plots.) With different thresholds ( $0.5 \leqslant \phi _{{th}}/\langle \phi \rangle \leqslant 4$ ), $\langle N\rangle$ decreases as the thresholds increase over a small scale range (approximately $r_b/\eta _K \lt 10$ ), where the scaling gradually approaches $-1$ as $\phi _{{th}}/\langle \phi \rangle$ is increased from $0.5$ to $4$ . Conversely, for the scale range $r_b/\eta _K \gt 10$ , $\langle N\rangle$ does not vary significantly among $\phi _{{th}}/\langle \phi \rangle$ values, visually approaching the scaling $1.67$ .

Figure 6.

(a) Plot of $D_F$ (with the additive law) of one-dimensional $\phi$ along the streamwise direction at $|y/D|=0.3$ . (b) Plot of $D_F$ of one-dimensional $\phi$ with threshold $1.0\langle \phi \rangle$ , along the radial ( $y$ ) direction (averaged over streamwise locations) and along the streamwise direction at five $|y/D|$ values. In (a) and (b), ${Re}= 10\,000$ and the grey dashed line denotes $D_F=1.67$ . See the caption of figure 5 for the meanings of the arrows.

For better examining the scaling, the fractal dimension $D_F$ of the fractal level sets is computed via the equation

(3.1) \begin{equation} D_F = -\frac {\mathrm{d}\!\left[\mathrm{log}\left [\left \langle N\right \rangle /(D/\eta _K)\right ]\right]}{\mathrm{d}\!\left[\mathrm{log}(r_b/\eta _K)\right]}. \end{equation}

Figures 5(d–f) present the fractal dimension for the iso-scalar surfaces of the scalar fields, corresponding to the $\langle N\rangle$ – $r_b$ plots in figures 5(a–c), respectively. The convergence tests of the number of samples are detailed in Appendix B. As illustrated in figures 5(d–f), the fractal dimension for a threshold $\phi _{{th}}$ increases continuously over the small scale range, reaching $D_F\gt 1.67$ , followed by a gradual decrease to approach a nearly constant $1.67\pm 0.05$ , when $r_b/\eta _K$ locates in the inertial–convective range. The uncertainty ( $\pm 0.05$ ) arises from the curve fitting $\langle N\rangle$ – $r_b$ for $r_b/\eta _K\gt 10$ , as suggested in Constantin et al. (Reference Constantin, Procaccia and Sreenivasan1991). The uncertainty of the curve fitting decreases as the fitting region beyond $r_b/\eta _K\gt 10$ narrows. Although the iso-scalar surfaces with the threshold $\phi _{{th}}/\langle \phi \rangle = 1.0$ are contaminated by experimental noise near the wall region (see figure 4 a), this noise has minimal impact on this fractal dimension ( $1.67\pm 0.05$ ). Additionally, to further evaluate the fractal dimension, the one-dimensional $\phi$ at various radial positions $|y/D|$ along the pipe axis is taken to calculate the fractal dimension $D_F$ with the additive law (Mandelbrot Reference Mandelbrot1982). As shown in figure 6(a), the fractal dimension in the inertial–convective range is approximately $1.67$ and is approximately unaffected by the thresholds shown here. Across different radial positions, $D_F$ is approximately $1.67$ within the inertial–convective range; see figure 6(b). Overall, for the three Reynolds numbers considered in the present study, the fractal dimension in the inertial–convective range is approximately $1.67$ , which within the margin of error is in good agreement with the values from experiments with the turbulent jet flow (Constantin et al. Reference Constantin, Procaccia and Sreenivasan1991) and theoretical studies (Mandelbrot Reference Mandelbrot1975; Constantin et al. Reference Constantin, Procaccia and Sreenivasan1991; Grossmann & Lohse Reference Grossmann and Lohse1994).

Whether the quality of $D_F = 1.67$ is affected by the thresholds is under considerable debate (see e.g. the discussions in Miller & Dimotakis Reference Miller and Dimotakis1991; Lane-Serff Reference Lane-Serff1993; Villermaux & Innocenti Reference Villermaux and Innocenti1999; Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020). To explore this, $D_F$ as a function of the threshold is shown in figure 7, where the thresholds ( $\phi _{{th}}$ ) ranging from $0.01 \langle \phi \rangle$ to $8 \langle \phi \rangle$ are selected. The fractal dimension was obtained from the linear fits of the coverage count in the log-log plots (e.g. figure 5 a) for $r_b/\eta _K \gt 10$ . In figure 7, as the threshold increases, $D_F$ decreases but a clear plateau (approximately 1.67) can be observed for $0.5 \lesssim \phi _{{th}}/\langle \phi \rangle \lesssim 3.5$ . These results are consistent with those of previous experimental studies (Lane-Serff Reference Lane-Serff1993) and DNS studies (Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020). When the threshold is smaller ( $\phi _{{th}}\lt 0.5\langle \phi \rangle$ ), the fractal dimension is larger than the value in the plateau, which can be attributed to the measurement noise in the iso-scalar surfaces. As shown in figure 4(a) and discussed earlier, the measurement noise contaminates the iso-scalar surfaces, resulting in an increase in the coverage count of the boxes, and thus affecting the quality of the power-law scaling. Finally, a high fractal dimension is obtained. For the cases with larger thresholds ( $\phi _{{th}}\gt 4\langle \phi \rangle$ ), the extracted iso-scalar surfaces are within the less mixed regions. As illustrated in figure 4(d), the iso-scalar surfaces are located in regions with high concentration values, leading to smoother surfaces and consequently a smaller $D_F$ . This result makes physical sense. Fluorescent dye with high scalar concentration values is continuously injected into the pipe flow, moving downstream at a finite velocity and being stirred by the fluctuation velocity field (turbulent eddies) of the ambient turbulent flow. This process disperses the dye in all directions, leading to connected and disconnected filaments with sharp borders. When the threshold is relatively low, the borders of the filaments are extracted exactly where mixing occurs, exhibiting fractal characteristics. However, as the threshold becomes large, the scalar concentration within the filament does not mix with the ambient turbulent flow, preventing the development of a wrinkled and fractal surface (Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020).

Figure 7.

The fractal dimension as a function of the threshold ( $\phi _{{th}}/\langle \phi \rangle$ ) for the iso-scalar surfaces in the measurement region. The grey dashed line denotes $D_F=1.67$ , and the two grey dash-dotted lines mark the range of $\phi _{{th}}/\langle \phi \rangle$ , within which $D_F\approx 1.67$ .

Previous studies have employed different criteria to extract iso-scalar surfaces (see e.g. Villermaux & Innocenti Reference Villermaux and Innocenti1999; Dasi et al. Reference Dasi, Schuerg and Webster2007; Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020). To evaluate the impact of these criteria on the fractal dimension, two other criteria, which are commonly used in the aforementioned studies, are used to extract the iso-scalar surfaces, as shown in figures 8(b,c). The first criterion uses $m$ times the local RMS above the local mean ( $\phi _{{th}} = \left \langle \phi \right \rangle + m \phi _{{rms}}$ (e.g. Dasi et al. Reference Dasi, Schuerg and Webster2007; Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020), and the second criterion uses $m$ times the initial concentration ( $\phi _{{th}} = m \phi _0$ ), which is also equivalent to using a constant value as the threshold (e.g. Prasad & Sreenivasan Reference Prasad and Sreenivasan1990; Villermaux & Innocenti Reference Villermaux and Innocenti1999; de Silva et al. Reference de Silva, Philip, Chauhan, Meneveau and Marusic2013; Zhuang et al. Reference Zhuang, Tan, Huang, Liu and Zhang2018; Mohaghar et al. Reference Mohaghar, Dasi and Webster2020). The iso-scalar surfaces determined by the first criterion exhibit a pattern of filaments similar to those obtained using $m$ times the local mean when the threshold values are close; see figures 8(a) and 8(b). When the thresholds are similar, minimal differences are observed between the resulting mixing iso-scalar surfaces. In contrast, the criterion using the initial concentration (constant) retains the mixing surfaces at the centre of the pipe flow; see figure 8(c). The box-counting method is applied to the iso-scalar surfaces using these two criteria to obtain the coverage count of the boxes. As shown in figures 9(a,c), for the range of small scales, $\langle N \rangle$ decreases with the increase of the threshold, and approaching the $-1$ scaling. For larger scales, as $r_b/\eta _K$ increases, the decrease of $\langle N\rangle$ approaches to follow a corresponding scaling. For clarity, the corresponding $D_F$ over large scales ( $r_b/\eta _K \gt 20$ ), the saturated scalings, are presented in figures 9(b,d). A plateau $D_F \approx 1.67$ over a range of thresholds can be observed. These tests clearly demonstrate that although the choice of criteria can influence the details of the extracted iso-scalar surfaces, the general trend of fractal dimension scaling remains consistent, and a constant fractal dimension within a certain range of thresholds can be obtained. This consistency reinforces that the fractal dimension is a reliable descriptor of the complexity of iso-scalar surfaces in turbulent pipe flow of passive scalar mixing, validating both theoretical predictions and experimental observations in turbulent environments (Constantin et al. Reference Constantin, Procaccia and Sreenivasan1991).

It should be mentioned that similar experiments of passive scalar mixing in an open-channel turbulent boundary layer show a low fractal dimension; see Dasi et al. (Reference Dasi, Schuerg and Webster2007). Their experiments showed less dense concentration fields with fewer iso-scalar flow structures, possibly due to the lower turbulence intensity from the background turbulence, leading to a limited number of filament structures observed in one image for the box-counting method. In contrast, in our experiments, the initially homogeneous fluid with dye from the injection system is stretched and swirled by the turbulence in pipe flow. As a result, the core with a high concentration is partially preserved, while clear fluid is entrained into the core, and the dye fluid is folded to generate filaments. The number of filaments observed in our experiments is greater than that observed in Dasi et al. (Reference Dasi, Schuerg and Webster2007), which statistically increases the coverage count for the box-counting method. This demonstrates the presence of multiscale structures of mixing iso-scalar surfaces deformed by turbulent eddies.

Figure 8.

An instantaneous sample of iso-scalar surfaces with different criteria when ${Re}=20\,000$ : (a) $\phi _{{th}}=4.0\langle \phi \rangle \in [0.59, 5.02]$ , (b) $\phi _{{th}}=\langle \phi \rangle + 2\phi _{{rms}} \in [0.31, 4.88]$ , (c) $\phi _{{th}}=0.04\phi _0 = 4$ . The unit of $\phi _{{th}}$ is $\unicode{x03BC}$ g l^–1.

Figure 9.

Box-counting results for iso-scalar surfaces are shown for two threshold criteria when ${Re}=20\,000$ : (a,b) $m$ times the local RMS above the local mean, specifically, $\phi _{{th}} = \langle \phi \rangle + m \phi _{{rms}}$ ; (c,d) $m$ times the initial concentration, namely $\phi _{{th}} = m \phi _0$ . (a,c) The number of boxes required to cover iso-scalar surfaces with different thresholds versus box size $r_b/\eta _K$ . The fractal dimension $D_F = 1.67$ at large scales is represented by the dashed line, whereas $D_F = 1$ at small scales is represented by the dash-dotted line. (b,d) Fractal dimension $D_F$ as a function of the threshold, where the grey dashed line indicates $D_F=1.67$ . See the caption of figure 5 for the meanings of the arrows.

The fractal dimension $D_F\approx 1.35$ characterises the turbulent/non–turbulent interfaces in turbulent jet flows, where the ambient flow remains quiescent, and entrainment occurs at the interfaces (Prasad & Sreenivasan Reference Prasad and Sreenivasan1990; Mistry et al. Reference Mistry, Philip, Dawson and Marusic2016, Reference Mistry, Dawson and Kerstein2018). The fractal dimension agrees with the theoretical predictions of Prasad & Sreenivasan (Reference Prasad and Sreenivasan1990) and Constantin et al. (Reference Constantin, Procaccia and Sreenivasan1991). Recent investigations have demonstrated that the fractal dimension of turbulent/non-turbulent interfaces is influenced by the applied threshold at low turbulence intensities, yielding values at approximately 2.1 (below the typical 2.35) in three-dimensional measurements (e.g. Er, Laval & Vassilicos Reference Er, Laval and Vassilicos2023; Zhang, Watanabe & Nagata Reference Zhang, Watanabe and Nagata2023). Similarly, other studies have also shown that the fractal dimension of interfaces between a turbulent ambient flow and a turbulent jet increases with the turbulence intensity of the ambient flow (e.g. Kankanwadi & Buxton Reference Kankanwadi and Buxton2020; Kohan & Gaskin Reference Kohan and Gaskin2022; Chen & Buxton Reference Chen and Buxton2023). Thus the presence of background turbulence can notably affect the fractal dimension of the mixing surfaces because the turbulent eddies can stir the mixing parcel and generate a mixing and wrinkling surface. In the present study, the flow emerging from the injection needle exhibits jet-like characteristics, yet the surrounding flow is a fully developed turbulent pipe flow. In the far field of the injection needle (approximately $220d$ downstream), active and well-developed mixing can be seen in the current measurement. There are no obvious entrainment interfaces between the ambient flow (pipe flow) and jet flow (from the needle), as mixing structures still exist in the region close to the wall, even with a high threshold; see figure 4(d). The present experimental conditions match better with the condition that the theories of fractal dimension consider for the iso-scalar surface in the passive scalar mixing (Constantin et al. Reference Constantin, Procaccia and Sreenivasan1991). Therefore, the fractal dimension $D_F \approx 1.35$ for the turbulent/non–turbulent interfaces is not observed in this study. Instead, the iso-scalar surfaces of the mixing structures in this study tend towards fractal dimension $1.67$ in the turbulence (Constantin et al. Reference Constantin, Procaccia and Sreenivasan1991).

3.3. Critical crossover length scale

Figure 10.

(a) An illustration of extracting the crossover length scale from the $\langle N \rangle{-}r_b$ plot ( ${Re}=20\,000$ ). (b) The ratio $r^*/\eta _K$ from different studies of the passive scalar mixing. (c) The crossover length scale determined using the second-order structure function of velocity increments. (d) The ratio $r^*/\eta _K$ obtained from the second-order velocity structure function (open symbols) and velocity spectra (half-filled symbols). The ratio of $r^{**}$ to $r^*$ versus ${Re}_\lambda$ is shown in the inset of (b). Here, ${Re}_\lambda$ in Constantin et al. (Reference Constantin, Procaccia and Sreenivasan1991) is evaluated as $Re_\lambda \approx \sqrt {2{Re}}$ (Pope Reference Pope2000). In the legends, ATBL refers to the atmospheric turbulent boundary layer, TBL denotes the turbulent boundary layer, HIT denotes homogeneous isotropic turbulence, and TSF denotes turbulent swirling flow. See the caption of figure 5 for the meanings of the arrows.

In the theory proposed by Constantin et al. (Reference Constantin, Procaccia and Sreenivasan1991), a crossover length scale, denoted as $r^*$ , represents the transition between smooth surfaces at small scales and fractal surfaces at larger scales. Constantin et al. (Reference Constantin, Procaccia and Sreenivasan1991) predicted that the ratio $r^*/\eta _K$ is independent of the Reynolds number, suggesting a universal value. Grossmann & Lohse (Reference Grossmann and Lohse1993) supported this prediction and estimated the ratio to be approximately $10$ , the crossover length scale of the two distinct regimes of the scaling in the second-order structure function of velocity increments.

This crossover length scale is examined in this study. The method for estimating $r^*$ is illustrated in figure 10(a). The log-log plot of the coverage count $\langle N\rangle$ of boxes of size $r_b/\eta _K$ is divided into two parts: the left part (for $-0.3\lt \mathrm{log_{10}}(r_b/\eta _K) \lt 0.5$ ) and the right part (for $\mathrm{log_{10}}(r_b/\eta _K)\gt 1.7$ ). The left and right parts are linearly fitted, with the latter having slope $-1.67$ . Thresholds ranging from $2.0 \langle \phi \rangle$ to $4.0\langle \phi \rangle$ in 0.5 increments are used. The average and standard deviation of the crossover length scales across different thresholds provide the value and error bar, respectively. Our results, shown in figure 10(b), indicate that the critical crossover length scale is approximately 10 times the Kolmogorov scale. The results from studies of Constantin et al. (Reference Constantin, Procaccia and Sreenivasan1991) and Iyer et al. (Reference Iyer, Schumacher, Sreenivasan and Yeung2020) were extracted using the same method, and are shown in figure 10(b) with different colour symbols. The $r^*/\eta _K$ value in this study exhibits a slight increase with the increase of ${Re}$ , similar to the value of Constantin et al. (Reference Constantin, Procaccia and Sreenivasan1991) in a jet with a slightly larger Reynolds number, and smaller than the values of Iyer et al. (Reference Iyer, Schumacher, Sreenivasan and Yeung2020) in a homogeneous isotropic turbulence, which has much larger ${Re}_\lambda$ .

Moreover, according to Grossmann & Lohse (Reference Grossmann and Lohse1993), the ratio $r^*/\eta _K$ marks the transition from the viscous range to the inertial range, where fluid motion appears fractal. Grossmann & Lohse (Reference Grossmann and Lohse1993) observed this ratio in the second-order structure function of velocity increments, namely, $S_2(r) = \langle [u^\prime (r+z)-u^\prime (z)]^2\rangle = \langle [\Delta u^\prime (r)]^2 \rangle$ , where $\Delta u^\prime (r)$ denotes the velocity difference over a distance $r$ along the $z$ direction. In the present study, the longitudinal second-order structure functions are plotted in figure 10(c). From the two known scalings, i.e. $S_2(r)\sim r^2$ and $S_2(r)\sim r^{2/3}$ (Pope Reference Pope2000), their intersection gives a crossover length scale $r^{**}/\eta _K$ , shown in figure 10(d) as black symbols. The $r^{**}/\eta _K$ values of other experimental studies from their second-order velocity structure function (van Atta & Chen Reference van Atta and Chen1970; Antonia, Satyaprakash & Hussain Reference Antonia, Satyaprakash and Hussain1980; Mestayer Reference Mestayer1982; Anselmet et al. Reference Anselmet, Gagne, Hopfinger and Antonia1984) and the velocity spectra (Debue et al. Reference Debue, Kuzzay, Saw, Daviaud, Dubrulle, Canet, Rossetto and Wschebor2018) are shown in figure 10(d), in addition to the theoretical predictions (Effinger & Grossmann Reference Effinger and Grossmann1987; Zhang et al. Reference Zhang, Bodenschatz, Xu and Xi2021) and experimental collections (Pope Reference Pope2000). Our results are consistent with their results, indicating that the crossover length scale is approximately 10 times the Kolmogorov scale.

The inset of figure 10(b) shows that the ratio of $r^*/\eta _K$ (from the scalar fractal surfaces) and $r^{**}/\eta _K$ (from the velocity structure function) is approximately $1$ , in good agreement with the theoretical prediction of Grossmann & Lohse (Reference Grossmann and Lohse1993). This comparison demonstrates that the geometric complexity of iso-scalar surfaces is intricately linked to the dynamics of turbulent mixing.

3.4. Fractal dimension in the viscous–convective range

Given by the geometric measure theory, the scaling behaviour of the Hausdorff volume $H$ of a passive scalar graph over a three-dimensional ball $B_r$ with radius $r$ is given by $H(g(B_r)) \sim r^{D_g}$ , where $g$ is the graph over the sphere. The scaling dimension of the graph, $D_g$ , is connected to the fractal dimension by $D_F = D_g -1$ (Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020); see details in Grossmann & Lohse (Reference Grossmann and Lohse1994), Eckhardt & Schumacher (Reference Eckhardt and Schumacher1999) and Iyer et al. (Reference Iyer, Schumacher, Sreenivasan and Yeung2020). Theoretical predictions of the fractal dimension $D_F$ (Grossmann & Lohse Reference Grossmann and Lohse1994) at $Sc=1$ and $2000$ are shown in figure 11; meanwhile, the $D_F$ of the DNS with $Sc = 1$ from Iyer et al. (Reference Iyer, Schumacher, Sreenivasan and Yeung2020) is extracted from their figure and shown as symbol lines.

Figure 11.

The fractal dimension $D_F$ versus $r_b/\eta _K$ (and $r_b/\eta _B$ ). The data are obtained from two-dimensional PLIF images for ${Re}=10\,000$ (squares) with threshold $0.5\langle \phi \rangle$ , and the data points are skipped for clarify. The fractal dimensions predicted by the geometric measure theory model ( $D_g$ ) (Grossmann & Lohse Reference Grossmann and Lohse1994) and DNS (Iyer et al. Reference Iyer, Schumacher, Sreenivasan and Yeung2020) are shown using a relationship $D_F = D_g - 1$ with the additive law. The inset shows $D_F$ with thresholds $0.5\langle \phi \rangle$ – $3\langle \phi \rangle$ . See the caption of figure 5 for the meanings of the arrows.

Since $D_F$ is independent of the Reynolds numbers in this study (see figure 5), $D_F$ at ${Re}=10\,000$ is shown in figure 11, where the spatial and temporal resolutions are most close to resolving the small-scale motions. When the box size $r_b$ is very small, the iso-scalar field exhibits smoothness, i.e. $D_F\approx 1$ . As $r_b$ increases, $D_F$ increases gradually to reach a maximum $D_F\approx 1.85$ at $r_b/\eta _K \approx 7$ , where the iso-scalar surfaces are wrinkled. For further larger length scales, the passive scalar field manifests the fractality $D_F \approx 1.67$ ; see figure 11. Albeit that the fractal dimension indicates ‘wrinkled’ iso-scalar surfaces at some small scales, as predicted for $Sc\gtrsim 1$ from the theory of Grossmann & Lohse (Reference Grossmann and Lohse1994), this curve is close to the prediction for $Sc=1$ , much smaller than the prediction for $Sc=2000$ (Grossmann & Lohse Reference Grossmann and Lohse1994); see the black solid line in figure 11. The $D_F$ value is obtained from the two-dimensional scalar field. The averaged resolution of the scalar measurements is approximately $35\eta _B$ in space and approximately 0.016 $\tau _K$ in time, where $\tau _K$ is the Kolmogorov time scale. The measurements insufficiently resolve the flow motions in the viscous–convective range; correspondingly, the averaging effect of the camera pixel gives smoothed scalar signals, leading to smaller $D_F$ . Based on this analysis, the one-dimensional $\phi$ at a single pixel ( $y/D=0$ ) along the pipe axis is taken to obtain the fractal dimension $D_F$ with the additive law (Mandelbrot Reference Mandelbrot1982). Along this spatial line, the resolution is approximately $5\eta _B$ within the measurement plane, and approximately $\eta _B$ in the depth direction of the laser sheet, close to the Batchelor scale. This $D_F$ is shown as the in figure 12(a), which agrees with the theoretical curve of Grossmann & Lohse (Reference Grossmann and Lohse1994) for $Sc=2000$ . With a larger threshold, i.e. $0.2\langle \phi \rangle$ , $D_F$ still shows $D_F\approx 2$ for $1\lesssim r_b/\eta _K \lesssim 5$ , but gets smaller for $r_b/\eta _K \lt 1$ , as shown in figure 12(b). The agreement of the measured $D_F$ from the one-dimensional space $\phi (y/D=0)$ with the theoretical curve of Grossmann & Lohse (Reference Grossmann and Lohse1994) remains for the criteria of $\phi _{{rms}}$ and $\phi _{0}$ , as shown in figure 12(c).

Figure 12.

(a) The fractal dimension $D_F$ (with the additive law) of one-dimensional spatial PLIF results at $y/D = 0$ versus $r_b/\eta _K$ (and $r_b/\eta _B$ ) with threshold $0.05\langle \phi \rangle$ . The lines denote the prediction of Grossmann & Lohse (Reference Grossmann and Lohse1994). (b) The $D_F$ values from the same spatial sequence with different thresholds of $\langle \phi \rangle$ . (c) The $D_F$ values from the same spatial sequence with different criteria. The data points are skipped for clarity.

To further evidence the influence of the resolution on the fractal dimension, two test cases were considered for one-dimensional data from $\phi (y/D=0)$ at ${Re}=10\,000$ . In the first case, the data are sub-sampled with skipping size $\varDelta _s \approx 0.2\eta _K$ . In the second case, the data are filtered with a box filter of size $\varDelta \approx 0.2\eta _K$ , followed by sub-sampling the filtered sequence with skipping $\varDelta _s=\varDelta$ points. This operation mimics the PLIF measurements by a camera sensor with a coarser spatial resolution, given that a pixel of a camera sensor works like a box for collecting the light. In figure 12(a), the $D_F$ value of the sub-sampled sequence (denoted by the pink hollow squares) remains consistent with that of the raw sequence, which indicates that the constant $D_F$ persists in accordance with the definition of the fractal (box-counting) dimension. The $D_F$ value from the sequence with $\varDelta \approx 0.2\eta _K$ and $\varDelta _s\approx 0.2\eta _K$ is smaller than that from the raw sequence in the viscous–convective range, but is unchanged in the inertial–convective range. With a larger filter size ( $\varDelta \approx 0.3\eta _K$ and $\varDelta _s\approx 0.3\eta _K$ ), $D_F$ in the viscous–convective range decreases further, and a monotonic increase of $D_F$ with $r_b/\eta _K$ can be observed. This demonstrates that the fractal dimension at the small scales is particularly sensitive to the measurement resolution and measurement quality. This is one possible technical cause for the fractal dimension $D_F\approx 2$ in the viscous–convective range being overlooked in previous experiments.