3.1 Flow visualizations

In figure 3, the magnitudes of the normalized instantaneous vorticity $\unicode[STIX]{x1D6FA}/(U_{in}/T_{0})$ , where $\unicode[STIX]{x1D6FA}=(\unicode[STIX]{x1D6FA}_{i}\unicode[STIX]{x1D6FA}_{i})^{1/2}$ with $\unicode[STIX]{x1D6FA}_{i}=\unicode[STIX]{x1D700}_{ijk}\unicode[STIX]{x2202}u_{k}/\unicode[STIX]{x2202}x_{j}$ and $\unicode[STIX]{x1D700}_{ijk}$ being the Levi-Civita symbol, are plotted for all three cases in the $X{-}Y$ plane, which certainly provides insights into the wake-interaction patterns, at least to some extent. Figure 3 shows that perceptible oscillations can be found upstream of the cylinders. A similar spike-like behaviour was also reported in the DNSs of turbulent flow in Trias, Gorobets & Oliva (Reference Trias, Gorobets and Oliva2015), although in their paper a contour plot of the Kolmogorov length scale $\unicode[STIX]{x1D702}$ is presented. Obviously, the Kolmogorov length scale is closely related to the velocity gradients. Thus, the existence of the oscillations is related to the fact that streamlines can be distorted upstream of the cylinders.

Figure 3. Visualizations of the magnitude of instantaneous vorticity $\unicode[STIX]{x1D6FA}/(U_{in}/T_{0})$ in the $X{-}Y$ plane. (a,b) $L_{d}/T_{0}=4$ ; (c,d) $L_{d}/T_{0}=6$ ; (e,f) $L_{d}/T_{0}=8$ . All maps cover only part of the simulation domain and are shown on a linear scale. Note that to visualize the wake-interaction pattern, for each of the cases considered two randomly selected instantaneous flow fields are plotted.

By and large, two different flow patterns can be observed depending on the separation distance between the cylinders. For the case with $L_{d}/T_{0}=4$ , the gap flow is weak and beyond $X/T_{0}=8$ only one large wake can be observed. For the cases with large gap ratios ( $L_{d}/T_{0}=6$ and $8$ ), the gap flow is straight and the two vortex streets persist longer downstream. Thus, depending on the strength of the gap flow, the three simulation cases can be simply divided into two regimes: weak gap flow regime and robust gap flow regime. In the weak gap flow regime, two vortex streets merge into one, whilst in the robust gap flow regime with a large separation between the square cylinders, the two vortex streets do not merge. One could reasonably expect that the two vortex streets in the robust gap flow regime will finally merge in the further downstream region. However, in this study, owing to the limited streamwise length, the combination of the two wakes is not significant, as suggested by figure 2.

3.2 Wake-interaction length scale

In the preceding subsection, two different flow patterns can be identified. In this subsection, we shall demonstrate that the turbulence characteristics in different regimes are quite different based on the wake-interaction length scale. As suggested by Townsend (Reference Townsend1956), the growth of the width of a plane turbulent wake follows $\unicode[STIX]{x1D6FF}\propto \sqrt{T_{0}X}$ . It is tempting to study the wake interactions behind two side-by-side square cylinders by using this scaling law, as has been done by Mazellier & Vassilicos (Reference Mazellier and Vassilicos2010). For a given separation distance $L_{d}$ , the two wakes are expected to meet at a downstream streamwise location $X_{\ast }$ where $\unicode[STIX]{x1D6FF}(X_{\ast })\propto \sqrt{T_{0}X}$ . Consequently, we can now derive the wake-interaction length scale as $X_{\ast }=L_{d}^{2}/T_{0}$ . The original formula for the wake-interaction scale by Mazellier & Vassilicos (Reference Mazellier and Vassilicos2010) has been improved by Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012) to assess the drag coefficient of the wake generators.

The wake-interaction length scale $X_{\ast }$ indeed has been successfully applied to fractal-generated turbulence (see, for example, Mazellier & Vassilicos Reference Mazellier and Vassilicos2010; Gomes-Fernandes et al. Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2012; Nagata et al. Reference Nagata, Sakai, Inaba, Suzuki, Terashima and Suzuki2013). In fractal-generated turbulence, the interactions of four wakes are involved, which significantly increase the complexities. Recall that the wake-interaction length scale is based on the interactions of two wakes. To fully investigate the wake interactions with different gap ratios $L_{d}/T_{0}$ and assess whether the evolution of turbulent flow scales with the wake-interaction length scale, various centreline statistics are considered.

Figure 4. Streamwise evolution of the mean velocity $U_{c}$ along the centreline. (a) As a function of $X/T_{0}$ ; (b) as a function of $X/X_{\ast }$ .

In figure 4, we plot the streamwise evolution of the mean centreline velocity $U_{c}/U_{in}$ versus $X/T_{0}$ (a) and $X/X_{\ast }$ (b). Note that in previous studies concerning wake interactions (Zhou et al. Reference Zhou, Wang, Xu and Jin2001; Alam et al. Reference Alam, Zhou and Wang2011; Alam & Zhou Reference Alam and Zhou2013; Alam et al. Reference Alam, Bai and Zhou2016; Zheng & Alam Reference Zheng and Alam2017), the size of the square cylinder, i.e. $T_{0}$ , was extensively used as a scaling parameter for no particular reason. Thus, in this work, in addition to the wake-interaction length scale $X_{\ast }$ , the streamwise distance is also normalized by $T_{0}$ . As one might expect, the value of $U_{c}$ at $X/T_{0}=0$ decreases with the increase of the separation distance $L_{d}/T_{0}$ . However, at $X/T_{0}=26$ , which is the most downstream location considered, $U_{c}/U_{in}$ in the case with $L_{d}/T_{0}=8$ becomes largest. When the streamwise distance $X$ is normalized by $X_{\ast }$ , the two profiles corresponding to the cases with $L_{d}/T_{0}=6$ and 8 collapse onto a single curve at $X/X_{\ast }>0.2$ . Thus, the largest $U_{c}$ found in the case with $L_{d}/T_{0}=8$ at $X/T_{0}=26$ is attributed to its slower spatial evolution with respect to the coordinate distance $X$ . It is not surprising that the wake-interaction length scale cannot be applied to the case with $L_{d}/T_{0}=4$ , since only one large vortex street can be identified in the downstream region, as shown in figure 3. The evolution of the streamwise root mean square (r.m.s.) velocity $U_{rms}$ along the centreline is given in figure 5. It is clear that the profile of the case with $L_{d}/T_{0}=4$ is fundamentally different from the other two profiles. However, due to the limited streamwise length, the decay region is missing for the case with $L_{d}/T_{0}=8$ . Thus, we could not draw a solid conclusion concerning the wake-interaction scale.

Figure 5. Streamwise evolution of the r.m.s. velocity $U_{rms}$ along the centreline. (a) As a function of $X/T_{0}$ ; (b) as a function of $X/X_{\ast }$ .

Figure 6. Streamwise evolution of the skewness of fluctuating streamwise velocity $S_{u}$ along the centreline. (a) As a function of $X/T_{0}$ ; (b) as a function of $X/X_{\ast }$ . The horizontal dashed line indicates $S_{u}=0$ .

Figure 7. Streamwise evolution of the flatness of fluctuating streamwise velocity $F_{u}$ along the centreline. (a) As a function of $X/T_{0}$ ; (b) as a function of $X/X_{\ast }$ . The horizontal dashed line indicates $F_{u}=3$ .

Figures 6 and 7 show the evolution of the centreline skewness $S_{u}$ and flatness $F_{u}$ , where $S_{u}=\langle u^{\prime 3}\rangle /\langle u^{\prime 2}\rangle ^{3/2}$ and $F_{u}=\langle {u^{\prime }}^{4}\rangle /\langle u^{\prime 2}\rangle ^{2}$ . Extreme events with negative $S_{u}$ and large $F_{u}$ can only be found in the cases with larger gap ratios ( $L_{d}/T_{0}=6$ and 8). In contrast, the profiles of $S_{u}$ and $F_{u}$ for the case with $L_{d}/T_{0}=4$ are more or less flat. The most intense events occur at $X/X_{\ast }\simeq 0.2$ and with downstream distance, $S_{u}$ and $F_{u}$ gradually acquire the corresponding values of a Gaussian distribution. Note that the location with the largest negative value of $S_{u}$ , i.e. $X/X_{\ast }\simeq 0.2$ , is close to the corresponding location in the turbulence generated by a single square grid (Zhou et al. Reference Zhou, Nagata, Sakai, Suzuki, Ito, Terashima and Hayase2014a ; Laizet et al. Reference Laizet, Nedić and Vassilicos2015b ). It is worth mentioning that in a previous study of the turbulence generated by a single square grid (Zhou et al. Reference Zhou, Nagata, Sakai, Suzuki, Ito, Terashima and Hayase2014a ), the corresponding gap ratio was 9.1, which is close to the cases with $L_{d}/T_{0}=8$ . It is therefore clear from figure 6 that only for the cases with larger $L_{d}$ , can the extreme events be found and the distribution of the velocity fluctuations in the upstream region is highly non-Gaussian. As shown in figures 6 and 7, for the case with larger gap ratios ( $L_{d}/T_{0}=6$ and 8) the use of the wake-interaction length scale $X_{\ast }$ does return a better description of the streamwise evolution of $S_{u}$ and $F_{u}$ , albeit that the two profiles do not completely collapse and the profiles for the case with $L_{d}/T_{0}=8$ are not well converged.

In this subsection, we test the wake-interaction length scale $X_{\ast }$ and reveal that for the case with $L_{d}/T_{0}=4$ , the streamwise evolution of various centreline statistics does not scale with $X_{\ast }$ . Recall that the derivation of $X_{\ast }$ is based on the interactions of two independent wakes. And it is demonstrated that for the case with $L_{d}/T_{0}=4$ the two wakes merge into one in the downstream region (i.e. $X/T_{0}>8$ ). As a result, it is not surprising that the wake-interaction length scale cannot be applied. However, for the case with larger separation distance (i.e. $L_{d}/T_{0}=6$ and 8), some of the turbulence characteristics more or less scale with $X_{\ast }$ , as happens in the case of the grid-generated turbulence, rather than simply $X/T_{0}$ . Note that for a typical low-blockage space-filling grid, the corresponding $L_{d}/T_{0}$ is much larger. For example, as shown in Nagata et al. (Reference Nagata, Sakai, Inaba, Suzuki, Terashima and Suzuki2013) the ratios of the largest mesh size to the thickness of the largest bars considered are 12.4, 13.8 and 14.0, respectively. Thus, the separation distances considered (i.e. $L_{d}/T_{0}=4,6$ and 8) may not well represent the two wake problem when the wakes are far enough from each other.

3.3 Extreme events

As mentioned in the introduction, extreme events can be closely related to the appearance of the $-5/3$ spectra. In this subsection, the characteristics along with the formation mechanism of the extreme events shall be discussed. It can be seen that extreme events are found in the cases with relatively larger gap ratios, namely, $L_{d}/T_{0}=6$ and 8. For the case with the largest gap ratio $L_{d}/T_{0}=8$ , however, the dual-wake flow still undergoes transition at the furthest downstream location at which the numerical data are available and the corresponding skewness $S_{u}$ takes a large negative value. From now on we therefore concentrate attention on the case with $L_{d}/T_{0}=6$ , which allows us to perform a complete investigation of the extreme events.

Figure 8. PDF distributions of (a) $u^{\prime }$ and (b) $\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}t$ at $X/T_{0}=6$ and 26. The corresponding Gaussian fits at $X/T_{0}=26$ (represented by red open circles) are also plotted for comparison.

Figure 9. Time traces of (a) $u^{\prime }/U_{rms}$ and $v^{\prime }/V_{rms}$ and (b) time traces of $\unicode[STIX]{x1D6FA}$ and $u^{\prime }/U_{rms}$ at $X/T_{0}=6$ and $Y/T_{0}=0$ . The origin of time for all lines is the same. Two different time frames (TF1 and TF2) with the appearance of extreme events can be identified.

Figure 8 shows the PDFs of the streamwise velocity fluctuations $u^{\prime }$ and their time derivative $\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}t$ at $X/T_{0}=6$ and 26. Note that at $X/T_{0}=6$ the most intense events occur, whereas the turbulent flow is almost Gaussian distributed at $X/T_{0}=26$ (i.e. $S_{u}\simeq 0$ and $F_{u}\simeq 3$ ). The distributions of $u^{\prime }$ and $\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}t$ at $X/T_{0}=6$ have significant fat tails compared with those at $X/T_{0}=26$ , suggesting the probability of the rare events (i.e. intense value of $u^{\prime }$ and $\unicode[STIX]{x2202}u^{\prime }/\unicode[STIX]{x2202}t$ ) at $X/T_{0}=6$ is considerably larger.

To provide deeper insight into the extreme events, figure 9(a) presents the time traces of the centreline velocity fluctuation $u^{\prime }/U_{rms}$ and $v^{\prime }/V_{rms}$ at $X/T_{0}=6$ , where $v^{\prime }$ and $V_{rms}$ are the vertical velocity fluctuations and the corresponding r.m.s. velocity, respectively. The variations of $u^{\prime }$ and $v^{\prime }$ are relatively smooth and exhibit quasi-periodic (regular) oscillations except for the TF1 and TF2. Figure 9(b) shows the time evolution of the magnitude of vorticity $\unicode[STIX]{x1D6FA}$ and $u^{\prime }$ at $X/T_{0}=6$ , which suggests that non-turbulent regions (i.e. quiescent regions with negligibly small magnitude of vorticity) are dominant. The occurrences of the extreme events are only found in the turbulent region (i.e. fluctuating region with high vorticity). Note that in the case of the dual-wake flow, the most intense events occur at a location where the flow is not always in the turbulent region, as in the turbulence generated by a single square grid (see figures 5 and 8 in Zhou et al. (Reference Zhou, Nagata, Sakai, Suzuki, Ito, Terashima and Hayase2014a ); figures 9 and 10 in Laizet et al. (Reference Laizet, Nedić and Vassilicos2015b )). Another finding from figure 9 concerning the occurrence of the extreme events is that $u^{\prime }$ and $v^{\prime }$ are closely correlated (i.e. in phase within TF1 and anti-phased within TF2). More quantitatively, the joint PDFs of $u^{\prime }$ and $v^{\prime }$ at $X/T_{0}=6$ are shown in figure 10. The joint PDFs at a further downstream location $X/T_{0}=26$ are also presented for comparison. Figure 10(a) suggests that the flow particles with either high positive or high negative $v^{\prime }$ tend to possess large negative $u^{\prime }$ . This observation is therefore in accord with the assumption that the large magnitude of $v^{\prime }$ is more likely to be associated with a large negative $u^{\prime }$ . The contour lines corresponding to rare events are approximately symmetrical with respect to the horizontal line, i.e. $v^{\prime }/V_{rms}=0$ , confirming that $u^{\prime }$ and $v^{\prime }$ can be either in phase or out of phase with each other. At $X/T_{0}=26$ , however, all contour lines acquire a ‘pancake’ shape instead, suggesting that $u^{\prime }$ and $v^{\prime }$ are essentially statistically uncorrelated.

Figure 10. Joint PDFs of $u^{\prime }/U_{rms}$ and $v^{\prime }/V_{rms}$ . (a) At $X/T_{0}=6$ ; (b) at $X/T_{0}=26$ . A total of six contour levels (i.e. 0.1, 0.05, $0.01$ , $0.005$ , $0.001$ and $0.0005$ ) are shown.

The formation of the extreme events can be explained by using a Lagrangian description. One can imagine that the local fluid particles along the centreline with large magnitude of $v^{\prime }$ have a better chance of being from a location with a larger vertical distance from the centreline. In the near field of a square cylinder, the turbulent wake consists of vortex tube clusters, which are usually characterized by a large magnitude of vorticity, and the velocity deficit is still significant. These slower vortex tube clusters serve as obstacles and can easily slow down the local turbulent flow, resulting in the extreme events with negative $S_{u}$ . Relying on the direction of the vertical movement, $u^{\prime }$ and $v^{\prime }$ can be either in phase or out of phase with each other along the centreline as shown in figures 9 and 10.

4.1 One-dimensional energy spectra

Before investigating the relationship between the extreme events and the energy spectra, we need to obtain the shape of the one-dimensional energy spectra first. The one-dimensional energy spectra of the streamwise velocity fluctuation $u^{\prime }$ at $X/T_{0}=6$ and 26 along the centreline for the case with $L_{d}/T_{0}=6$ are given in figure 11. Note that at the upstream location (i.e. $X/T_{0}=6$ ) the turbulent flows are highly non-Gaussian distributed and highly intermittent, whereas at the far downstream location (i.e. $X/T_{0}=26$ ) the flow field becomes approximately Gaussian distributed and approximately fully turbulent. The spectra at the two chosen locations display pronounced peaks at the normalized frequency $St=fT_{0}/U_{in}=0.142$ corresponding to the vortex shedding frequency (more specifically, the regular and slow oscillation of $u^{\prime }$ at $X/T_{0}=6$ as shown in figure 9). The dominant frequency $fT_{0}/U_{in}=0.142$ is approximately half the value found in the flow behind a single square cylinder. For example, as presented in Portela, Papadakis & Vassilicos (Reference Portela, Papadakis and Vassilicos2017) when the inlet Reynolds number is $Re_{T_{0}}=3900$ , the corresponding $fT_{0}/U_{in}$ is 0.267. In a previous experiment of dual-wake flows with a slightly smaller gap ratio $L_{d}/T_{0}=5$ and much larger inlet Reynolds number $Re_{T_{0}}=47\,000$ , the returned $St$ is 0.128, which is close to our simulation result (Alam & Zhou Reference Alam and Zhou2013).

Figure 11. (a) Energy spectra of $u^{\prime }$ at $X/T_{0}=6$ and 26 for the case with $L_{d}/T_{0}=6$ . The vertical dashed line indicates $St=fT_{0}/U_{in}=0.142$ ; (b) the corresponding compensated energy spectra $E_{uu}(fT_{0}/U_{in})^{5/3}$ . The horizontal (i.e. frequency) axis is linear.

Compared with $X/T_{0}=26$ , at $X/T_{0}=6$ the vortex shedding contains more energy; that is the peak value of $E_{uu}$ is much larger at $X/T_{0}=6$ . However, except for the narrow frequency range corresponding to vortex shedding, the energy spectrum at $X/T_{0}=6$ is always much smaller than that at $X/T_{0}=26$ . This is due to the smooth evolution of $u^{\prime }$ as shown in figure 9, albeit the extreme events bearing large fluctuating energy (see figure 9). Note that even the velocity fluctuations at $X/T_{0}$ = 6 and 26 appear to have more or less identical strengths (magnitudes of $U_{rms}$ ) (see figure 5), their intrinsic properties are actually quite different as already shown in figures 8 and 11.

The energy spectrum at $X/T_{0}=6$ exhibits a well-defined $-5/3$ power law with a range for more than a decade. With downstream distance, however, the range of the $-5/3$ power law is eroded, albeit the corresponding r.m.s. velocity is even slightly larger (see figure 5) and the local Reynolds numbers based on the Taylor microscale remains approximately unchanged, that is, $Re_{\unicode[STIX]{x1D706}}\simeq 100$ (not shown herein). Another interesting finding is that for both locations, the shedding frequency $St=0.142$ defines the beginning of the $-5/3$ power-law energy spectra.

Similar observations have already been reported in various kinds of turbulent flows. For instance, in the near field of grid-generated turbulence (single square grid (Laizet et al. Reference Laizet, Nedić and Vassilicos2015b ; Paul et al. Reference Paul, Papadakis and Vassilicos2017), regular grid (Isaza, Salazar & Warhaft Reference Isaza, Salazar and Warhaft2014), fractal grid (Laizet, Vassilicos & Cambon Reference Laizet, Vassilicos and Cambon2013; Hearst & Lavoie Reference Hearst and Lavoie2014)), the power spectrum exhibits a well-defined power law with an exponent close to $-5/3$ and the range of the $-5/3$ scaling narrows with downstream distance. Zhou et al. (Reference Zhou, Nagata, Sakai and Watanabe2018) demonstrated in a dual-plane jet flow, the range of the $-5/3$ power law also decreases further downstream. Even downstream of a single square prism, a similar observation can also be found (Portela et al. Reference Portela, Papadakis and Vassilicos2017).

4.2 Filtered fluctuating velocity fields

To take the observation in the preceding subsection (i.e. well-defined $-5/3$ spectra can be found at both streamwise locations considered) one step further, the relationship between the energy spectra and the extreme events will be discussed here. At $X/T_{0}=6$ , the flow field is not fully turbulent; there are alternations between laminar large-scale oscillations and strong turbulent fluctuations (e.g. TF1 and TF2). To shed light on the physical mechanism responsible for the wider range of the $-5/3$ power law at $X/T_{0}=6$ in the high-frequency range, we first attempt to subtract the signal in the low-frequency range. The velocity fluctuating signals are passed through a highpass filter with the low cutoff frequency $fT_{0}/U_{in}$ being 2.0. Thus, the filter data are expected to contain signals with frequency $fT_{0}/U_{in}$ larger than 2.0, where the $-5/3$ power law is absent. Throughout this paper, the velocity data are filtered using eighth-order zero-lag (i.e. bidirectional) Butterworth filters (e.g. highpass, bandpass and lowpass filters). And the scripts ${<}$ , ${\sim}$ and ${>}$ stand for statistics associated with high-, intermediate- and low-frequency components, respectively. For instance, the velocity fluctuation can be decomposed as $u^{\prime }=u_{{>}}^{\prime }+u_{{\sim}}^{\prime }+u_{{<}}^{\prime }$ . The use of high-order filters allows us to efficiently remove the unwanted components. Considering the fact that filters chosen are bidirectional, the filtered signals can be expected to be delay compensated. A further validation of the involved filtering processes is given in supplementary appendix D.

Figure 12. (a) Energy spectrum of the filtered velocity signal $u_{{<}}^{\prime }/U_{in}$ subject to a highpass filter with a cutoff frequency of 2.0. Energy spectrum of the original velocity signal (i.e. entire signal) $u^{\prime }$ is also shown for comparison. The vertical dashed line indicates $fT_{0}/U_{in}=2$ . (b) Time traces of the filtered signals $u_{{<}}^{\prime }/U_{in}$ . The time evolution of the original velocity signals $u^{\prime }$ shown in figure 9 is also plotted as a comparison.

Figure 12(a) shows the spectrum of filtered velocity fluctuations $u_{{<}}^{\prime }$ corresponding to the high-frequency range $fT_{0}/U_{in}>2.0$ . For comparison, the spectrum of the entire fluctuating signal $u^{\prime }$ already shown in figure 11 is also included. The profile corresponding to $u_{{<}}^{\prime }$ in the high-frequency range remains virtually intact and unaffected. In contrast, the fluctuating signals within the low-frequency range are efficiently subtracted. We now apply the chosen highpass filter to the time traces of $u^{\prime }$ shown in figure 9. As can be seen in figure 12(b), except for the fluctuating signals in the TF1 and TF2, the filtered signals are in a relatively quiescent state with negligible magnitude. This observation suggests that the high-frequency signals contribute to the extreme events.

We further divide the intermediate range (i.e. $0.2<fT_{0}/U_{in}<2.0$ ), where the $-5/3$ spectrum can be found, into two subranges ( $u_{{\sim}}^{\prime }=u_{{\sim},l}^{\prime }+u_{{\sim},h}^{\prime }$ ) to explore the erosion of the $-5/3$ scaling. Figures 13(a) and 13(b) show the energy spectrum and the velocity signal in the high-frequency zone of the $-5/3$ scaling range (i.e. $0.8<fT_{0}/U_{in}<2.0$ ). The fact that the non-zero fluctuations are mainly found in TF1 and TF2 indicates an intimate relationship between intermittent events and the extended $-5/3$ power law. In other words, we could safely draw the conclusion that the occurrences of the extreme events (TF1 and TF2) contribute to the formation of the $-5/3$ power law in the high-frequency range (i.e. $fT_{0}/U_{in}\gtrsim 0.8$ in figure 12 b). The range over which the $-5/3$ power law can be found therefore is determined by two turbulence behaviours, one being the large-scale vortex shedding and the other being the occurrence of the extreme events. With increasing downstream distance, the strength of the extreme events becomes weaker (see figure 6 a), which results in the deterioration of the range of the $-5/3$ power law.

Figure 13. (a) Energy spectrum of the filtered velocity signal $u_{{\sim},h}^{\prime }/U_{in}$ subject to a band-pass filter with a lower cutoff frequency of 0.8 and a higher cutoff frequency of 2.0. Energy spectrum of the original velocity signal (i.e. entire signal) $u^{\prime }$ is also shown for comparison. The vertical dashed lines indicate $fT_{0}/U_{in}=0.8$ and 2.0. (b) Time traces of the filtered signal $u_{{\sim},h}^{\prime }/U_{in}$ . The time evolution of the original velocity signal $u^{\prime }$ shown in figure 9 is also plotted as a comparison.

Figure 14. (a) Energy spectrum of the filtered velocity signal $u_{{\sim},l}^{\prime }/U_{in}$ subject to a band-pass filter with a lower cutoff frequency of 0.2 and a higher cutoff frequency of 0.8. Energy spectrum of the original velocity signal (i.e. entire signal) $u^{\prime }$ is also shown for comparison. The vertical dashed lines indicate $fT_{0}/U_{in}=0.2$ and 0.8. (b) Time traces of the filtered signal $u_{{\sim},l}^{\prime }/U_{in}$ . The time evolution of the original velocity signal $u^{\prime }$ shown in figure 9 is also plotted as a comparison.

It must, however, be pointed out that there is a large part of the $-5/3$ power law which is not in the highpass filtered field, that is, the intermediate-frequency range $0.2\lesssim t/(T_{0}/U_{in})\lesssim 0.8$ . To investigate what is responsible for this part of the spectrum, we use a bandpass filter to extract the velocity signals within this range. And the higher and lower cutoff frequencies are set to $fT_{0}/U_{in}=0.8$ and 0.2, respectively (see figures 14 a and 14 b). The time-traces of the filtered signals $u_{{\sim}}^{\prime }$ demonstrate that the strength of the filtered velocity $u_{{\sim}}^{\prime }$ is weak and as expected the oscillation behaviour possesses smaller periods. In contrast to the formation of the $-5/3$ scaling in the high-frequency range where the turbulent motions are dominant, in the intermediate-frequency range both laminar and turbulent motions contribute to the appearance of the $-5/3$ scaling.

Figure 15. (a) Energy spectrum of the filtered velocity signal $u_{{>}}^{\prime }/U_{in}$ subject to a lowpass filter with a lower cutoff frequency of 0.2. Energy spectrum of the original velocity signal (i.e. entire signal) $u^{\prime }$ is also shown for comparison. The vertical dashed line indicates $fT_{0}/U_{in}=0.2$ . (b) Time traces of the filtered signal $u_{{>}}^{\prime }/U_{in}$ . The time evolution of the original velocity signal $u^{\prime }$ shown in figure 9 is also plotted as a comparison.

Let us finally consider the behaviour in the low-frequency range (i.e. $fT_{0}/U_{in}\lesssim 0.2$ ), which stems from large-scale vortex shedding. A lowpass filter with the cutoff frequency of $fT_{0}/U_{in}=0.2$ is used. Expect for TF1 and TF2, as shown in figure 15(b) the large-scale periodic oscillations of $u_{{>}}^{\prime }$ corresponding to the energy containing range hold most of the energy.

Figure 16 shows the PDFs of the normalized velocity components $u_{{>}}^{\prime }/U_{rms}^{{>}}$ , $u_{{\sim}}^{\prime }/U_{rms}^{{\sim}}$ and $u_{{<}}^{\prime }/U_{rms}^{{<}}$ at $X/T_{0}=6$ . The probability distribution of $u^{\prime }/U_{rms}$ is shown for comparison. Except for the profile corresponding to $u_{{<}}^{\prime }$ , all other probability distributions are negatively skewed and the profiles of $u_{{\sim}}^{\prime }$ and $u_{{<}}^{\prime }$ exhibit pronounced fatter tails. The statistical characteristics of $u^{\prime }$ , $u_{{>}}^{\prime }$ , $u_{{\sim}}^{\prime }$ and $u_{{<}}^{\prime }$ (e.g. r.m.s., skewness and flatness) at $X/T_{0}=6$ and 26 are listed in table 2. For instance, the skewness of $u_{{>}}^{\prime }$ is obtained as $S(u_{{>}}^{\prime })=\langle u_{{>}}^{\prime 3}\rangle /\langle u_{{>}}^{\prime 2}\rangle ^{3/2}$ . At the downstream location $X/T_{0}=26$ , the definitions of the components $u_{{>}}^{\prime }$ , $u_{{\sim}}^{\prime }$ and $u_{{<}}^{\prime }$ are the same as those at $X/T_{0}=6$ . The negative values of $S(u_{{>}}^{\prime })$ and $S(u_{{\sim}}^{\prime })$ at $X/T_{0}=6$ imply that the probability distributions within different frequency ranges are negatively skewed, whereas $S(u_{{<}}^{\prime })\simeq 0$ suggests a symmetrical shape. And the velocity signals in the intermediate- and high-frequency ranges are significantly intermittent for $F(u_{{\sim}}^{\prime })=33$ and $F(u_{{<}}^{\prime })=79$ . These findings are in accord with the corresponding probability distributions shown in figure 16. With downstream distance at $X/T_{0}=26$ , the magnitudes of the flatness are close to that of a Gaussian distribution.

The intermittent features found at $X/T_{0}=6$ in the intermediate- and low-frequency ranges are distinctly different from the so-called ‘dissipation-range intermittency’, which is believed to be only associated with scales that are comparable to or smaller than the Kolmogorov dissipation scale (see, for example, Frisch (Reference Frisch1995) and references therein). It has been suggested that the concept of ‘dissipation-range intermittency’ does not necessarily violate the self-similarity in the inertial subrange, which is not the case for our study.

Figure 16. Probability distributions of $u_{{>}}^{\prime }/U_{rms}^{{>}}$ , $u_{{\sim}}^{\prime }/U_{rms}^{{\sim}}$ and $u_{{<}}^{\prime }/U_{rms}^{{<}}$ at $X/T_{0}=6$ . The profile of $P(u^{\prime }/U_{rms})$ is also shown for comparison.

Table 2. Strength (i.e. r.m.s.), skewness and flatness of $u^{\prime }$ , $u_{{>}}^{\prime }$ , $u_{{\sim}}^{\prime }$ and $u_{{<}}^{\prime }$ at $X/T_{0}=6$ and 26.

We close this subsection with a short remark concerning the appearance of a well-defined $-5/3$ power-law spectrum in the near field, where the turbulent flow is still developing and suffers a subsequent deterioration in the $-5/3$ range. First, the corresponding velocity fluctuations are highly intermittent, which can be roughly resolved into two parts, one being the large-scale vortex shedding and the other being the occurrence of the extreme events. As has already been confirmed in § 3, the formation of the extreme events is not necessarily directly related to the local Reynolds number. This suggestion is in accord with the previous findings that a well-defined $-5/3$ power law can be found at the location where the local Reynolds number is very small. Similar conclusions have been drawn by previous researchers. It was pointed out by Kraichnan (Reference Kraichnan1974) that ‘Kolmogorov’s 1941 theory has achieved an embarrassment of success. The $-5/3$ -spectrum has been found not only where it reasonably could be expected but also at Reynolds numbers too small for a distinct inertial range to exist and in boundary layers and shear flows…’. A similar but more recent statement is as follows: ‘Conditions for the occurrence of a $-5/3$ spectrum appear to be more “forgiving” than those initially outlined by Kolmogorov’ (Isaza et al. Reference Isaza, Salazar and Warhaft2014).

It is the large-scale vortex shedding and the occurrence of the intermittent fluctuations instead of the local Reynolds number $Re_{\unicode[STIX]{x1D706}}$ that actually determine the range of the $-5/3$ power law. Our current work contributes to the understanding of the appearance of the $-5/3$ power-law scaling in the near field of a spatially developing turbulent flow (see, for example, Gomes-Fernandes et al. Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2014; Laizet et al. Reference Laizet, Nedić and Vassilicos2015b ; Paul et al. Reference Paul, Papadakis and Vassilicos2017; Portela et al. Reference Portela, Papadakis and Vassilicos2017). Finally, it should be stressed that the frequency range over which the spectrum has an exponent close to $-5/3$ is defined by two distinctly different behaviours corresponding to large-scale vortex shedding and occurrence of the extreme events. This finding obviously disobeys the predictions given by Kolmogorov (Reference Kolmogorov1941a ,Reference Kolmogorov b ,Reference Kolmogorov c ). It must be pointed out that in this work, we confirm that the extreme events and the $-5/3$ energy spectrum at $X/T_{0}=6$ are closely related. One cannot claim that it is the extreme events that are causing the $-5/3$ energy spectrum.

4.3 Second-order structure function

In this subsection, we address the final question raised in the introduction section, that is, the shape of the second-order structure function in the flow region where the extreme events can be identified. Thus, we explore the second-order structure function $\langle \unicode[STIX]{x1D6FF}u^{\prime 2}(\unicode[STIX]{x0394}t)\rangle$ with $\unicode[STIX]{x1D6FF}u^{\prime }(\unicode[STIX]{x0394}t)$ being $u^{\prime }(t+\unicode[STIX]{x0394}t)-u^{\prime }(t)$ at the two extensively investigated locations (i.e. $X/T_{0}=6$ and 26). According to our definition, the second-order structure function $\langle \unicode[STIX]{x1D6FF}u^{\prime 2}(\unicode[STIX]{x0394}t)\rangle$ depends on the streamwise location $X$ and time difference $\unicode[STIX]{x0394}t$ . According to the Kolmogorov phenomenology, in the so-called ‘inertial subrange’ we have the well-known $2/3$ power law (i.e. equivalent to the $-5/3$ energy spectrum in frequency space) albeit the range of the $2/3$ power law in real space is always much smaller than the frequency range over which the $-5/3$ power law can be found (see, for example, Tennekes & Lumley Reference Tennekes and Lumley1972; Frisch Reference Frisch1995; Pope Reference Pope2000). It is worth mentioning that structure functions considered here are actually averaged structure functions.

Figure 17. Second-order structure functions $\langle \unicode[STIX]{x1D6FF}u^{\prime 2}(\unicode[STIX]{x0394}t)\rangle$ at $X/T_{0}=6$ and 26 along the centreline. The vertical dashed line corresponds to half-vortex shedding cycle (i.e. the phase difference between $u^{\prime }(\unicode[STIX]{x0394}t)$ and $u^{\prime }(t+\unicode[STIX]{x0394}t)$ is $\unicode[STIX]{x03C0}$ ).

Figure 17 shows $\langle \unicode[STIX]{x1D6FF}u^{\prime 2}(\unicode[STIX]{x0394}t)\rangle$ versus $\unicode[STIX]{x0394}tU_{in}/T_{0}$ at $X/T_{0}=6$ and 26 along the centreline. At $X/T_{0}=6$ the expected $2/3$ power law is absent and the turbulent flow exhibits a substantial power-law range with a large exponent close to $1$ in the intermediate range $0.1\lesssim \unicode[STIX]{x0394}tU_{in}/T_{0}\lesssim 3$ instead. In contrast, at the downstream location $X/T_{0}=26$ the second-order structure function acquires an approximate $2/3$ scaling law within the intermediate range $0.3\lesssim \unicode[STIX]{x0394}tU_{in}/T_{0}\lesssim 3$ , which is in accord with Kolmogorov’s prediction. Compared with the upstream location $X/T_{0}=6$ , at $X/T_{0}=26$ the $-5/3$ frequency range is smaller but still a discernible $2/3$ power law can be identified. Thus, the absence of the $2/3$ scaling law at $X/T_{0}=6$ is unlikely to be associated with the squeezing effect introduced by the Fourier transform.

Figure 18. Second-order structure functions of $u_{{<}}^{\prime }$ , $u_{{\sim}}^{\prime }$ and $u_{{>}}^{\prime }$ at (a) $X/T_{0}=6$ and (b) $X/T_{0}=26$ along the centreline. For comparison, the corresponding profile of $u^{\prime }$ is also plotted. From bottom to top, the three horizontal dashed lines indicate $2(U_{rms}^{{<}})^{2}$ , $2(U_{rms}^{{\sim}})^{2}$ and $2(U_{rms}^{{>}})^{2}$ . The vertical dashed lines correspond to half-vortex shedding cycle.

For sufficiently small $\unicode[STIX]{x0394}t$ , the Taylor expansion of the velocity difference is $u^{\prime }(t+\unicode[STIX]{x0394}t)-u^{\prime }(t)=(\unicode[STIX]{x2202}u^{\prime }(t)/\unicode[STIX]{x2202}t)\unicode[STIX]{x0394}t+O(\unicode[STIX]{x0394}t^{2})$ . The second-order structure is therefore $\langle \unicode[STIX]{x1D6FF}u^{\prime 2}(\unicode[STIX]{x0394}t)\rangle =\langle (\unicode[STIX]{x2202}u^{\prime }(t)/\unicode[STIX]{x2202}t)^{2}\rangle \unicode[STIX]{x0394}t^{2}+O(\unicode[STIX]{x0394}t^{3})$ . Figure 17 further shows that for small $\unicode[STIX]{x0394}t$ , the second-order structure functions at $X/T_{0}=6$ and 26 indeed acquire a well-defined power law with an exponent close to 2, implying that in current DNSs the small dissipation scales are well resolved in time.

To shed light on the formation mechanism of the substantial $1$ power law at $X/T_{0}=6$ , the second-order structure functions of the filtered flow fields (i.e. $u_{{>}}^{\prime }$ , $u_{{\sim}}^{\prime }$ and $u_{{<}}^{\prime }$ ) are computed (see figure 18 a). The second-order structure functions at $X/T_{0}=26$ are shown in figure 18(b) as a further comparison. Obviously, at sufficiently large $\unicode[STIX]{x0394}t$ , we have $\langle \unicode[STIX]{x1D6FF}u_{{<}}^{\prime 2}(\unicode[STIX]{x0394}t)\rangle =2(U_{rms}^{{<}})^{2}$ , $\langle \unicode[STIX]{x1D6FF}u_{{\sim}}^{\prime 2}(\unicode[STIX]{x0394}t)\rangle =2(U_{rms}^{{\sim}})^{2}$ and $\langle \unicode[STIX]{x1D6FF}u_{{>}}^{\prime 2}(\unicode[STIX]{x0394}t)\rangle =2(U_{rms}^{{>}})^{2}$ , which is indeed the case as shown in both figures 18(a) and 18(b).

From Parseval’s theorem, we can derive the following relations:

(4.1) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D6FF}u^{\prime 2}(\unicode[STIX]{x0394}t)\rangle =\langle \unicode[STIX]{x1D6FF}u_{{<}}^{\prime 2}(\unicode[STIX]{x0394}t)\rangle +\langle \unicode[STIX]{x1D6FF}u_{{\sim}}^{\prime 2}(\unicode[STIX]{x0394}t)\rangle +\langle \unicode[STIX]{x1D6FF}u_{{>}}^{\prime 2}(\unicode[STIX]{x0394}t)\rangle & & \displaystyle\end{eqnarray}$$

and

(4.2) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D6FF}u_{{\sim}}^{\prime 2}(\unicode[STIX]{x0394}t)\rangle =\langle \unicode[STIX]{x1D6FF}u_{{\sim},l}^{\prime 2}(\unicode[STIX]{x0394}t)\rangle +\langle \unicode[STIX]{x1D6FF}u_{{\sim},h}^{\prime 2}(\unicode[STIX]{x0394}t)\rangle . & & \displaystyle\end{eqnarray}$$

A brief justification of the two equations can be found in supplementary appendix D. These two equations allow us to quantify the contribution of each velocity components, $u_{{>}}^{\prime }$ , $u_{{\sim}}^{\prime }$ and $u_{{<}}^{\prime }$ , on the profile shape of $\langle \unicode[STIX]{x1D6FF}u^{\prime 2}(\unicode[STIX]{x0394}t)\rangle$ . Figure 18(a) implies that the discrepancy from the $2/3$ scaling is probably associated with the strong vortex shedding. More specifically, for the time difference $\unicode[STIX]{x0394}tU_{in}/T_{0}\simeq 3$ , which defines the end of the $1$ slope, the large-scale velocity components $u_{{>}}^{\prime }$ make an overwhelming contribution to $\langle \unicode[STIX]{x1D6FF}u^{\prime 2}(\unicode[STIX]{x0394}t)\rangle$ .

At $X/T_{0}=6$ and 26, the second-order structure function of $u_{{>}}^{\prime }$ acquire a well-defined 2 power-law scaling in the range $\unicode[STIX]{x0394}tU_{in}/T_{0}\lesssim 3$ . Moreover, the vortex shedding periods are more or less the same (see figures 11 and 17). That is, the time scales of $u_{{>}}^{\prime }$ at $X/T_{0}=6$ and 26 are close to each other. Therefore, the stronger vortex shedding at $X/T_{0}=6$ , as can be seen from the height of the oscillations in figure 18 and peak values of the spectra in figure 11, corresponds to a large coefficient $\langle (\unicode[STIX]{x2202}u^{\prime }(t)/\unicode[STIX]{x2202}t)^{2}\rangle$ , which can significantly contaminate the profile of $\langle \unicode[STIX]{x1D6FF}u^{\prime 2}(\unicode[STIX]{x0394}t)\rangle$ within the range $0.3\lesssim \unicode[STIX]{x0394}tU_{in}/T_{0}\lesssim 3$ (see figure 18 a). In other words, fluid motions within this intermediate range are not sufficiently separated from large-scale motions and can directly ‘feel’ the large-scale vortex shedding. This observation violates Kolmogorov’s universal equilibrium hypothesis. In the range $0.1\lesssim \unicode[STIX]{x0394}tU_{in}/T_{0}\lesssim 0.3$ , however, the interpretation of the appearance of the approximate $1$ power law should not be pushed too far since this exponent lies between the scaling law of $2/3$ and 2. It is difficult to accurately compute the scaling law in the transition region between the $2/3$ power-law range and the dissipation range with an exponent close 2. We again demonstrate that the $-5/3$ energy spectrum found at $X/T_{0}=6$ is not the well-known Kolmogorov $-5/3$ spectrum. On the other hand, at $X/T_{0}=26$ , the occurrence of the $2/3$ scaling of the second-order structure function along with the $-5/3$ energy spectrum appear to be in accord with the Kolmogorov phenomenology.