4.1. FM in the streamwise velocity component

The values of the ratio $K_{np}$ as a function of the wall-normal coordinate, $y^+$, are shown in figure 5 for visibility networks built from the streamwise velocity, $u$. We recall that $K_{np}>1$ indicates a higher frequency under intervals of positive large-scale velocity than under negative ones, and vice versa for $K_{np}<1$.

Figure 5. Large-scale conditional average degree ratio, $K_{np}$, as a function of the wall-normal coordinate, $y^+$, for streamwise velocity, $u$, extracted from (a) the channel flow DNS at different ${\textit {Re}}_\tau$ and (b) the boundary layer experiment. The inset in (a) shows the $K_{np}$ behaviour for the two channel flows as a function of $y/h$. In (b), $K_{np}(y^+)$ is shown for the spatial series obtained through the classical (CTH) and modified Taylor's hypothesis (MTH) as green and cyan plots, respectively; for comparison, the behaviour for the channel flow at ${\textit {Re}}_\tau \approx 5200$ is also reported in black. Moreover, the boundary layer intermittency region is highlighted in (b) as a shaded grey region. Angular brackets indicate averaging over time and spanwise direction in (a) and over three different realizations in (b). The results for synthetic velocity signals with shuffled phases are also shown.

Figure 5(a) shows $K_{np}$ for the spatial series of the two channel flows DNS at ${\textit {Re}}_\tau \approx 5200$ (black) and ${\textit {Re}}_\tau =1000$ (red), while figure 5(b) illustrates the $K_{np}$ behaviour for the time series of the boundary layer (green). Values of $K_{np}>1$ were detected close to the wall for all configurations, while moving away from the wall, $K_{np}$ became smaller than $1$, which indicated a reversal in the scale-interaction mechanism, i.e. higher frequencies were detected during $u_{LS}<0$ than for $u_{LS}>0$. The overall behaviours shown in figure 5 were in accordance with previous works on scale interaction in wall-bounded turbulence, which have indicated that a higher (amplitude and) frequency of the small scales is found for positive $u_{LS}$ intervals in regions close to the wall, while a reversal mechanism occurs far from the wall (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012; Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015, Reference Baars, Hutchins and Marusic2017; Pathikonda & Christensen Reference Pathikonda and Christensen2017; Awasthi & Anderson Reference Awasthi and Anderson2018; Tang & Jiang Reference Tang and Jiang2018; Pathikonda & Christensen Reference Pathikonda and Christensen2019). However, the behaviour of $K_{np}(y^+)$ for the two set-ups also highlights peculiar features of large-to-small scale FM that deserve further investigations.

First, we compared the results for the two channel flows at different Reynolds numbers (red and black lines in figure 5a). A similar trend of $K_{np}(y^+)$ was found for both channels, but the intensity of the FM (close to the wall) was larger for ${\textit {Re}}_\tau \approx 5200$ than for ${\textit {Re}}_\tau =1000$, thus clearly showing that the effect of higher Reynolds numbers was to increase the FM mechanism in the near-wall region, as a consequence of the increasing magnitude of the large-scale fluctuations with increasing ${\textit {Re}}_\tau$. Similarly, away from the wall, the reversal mechanism of scale interaction was strengthened for the higher-Reynolds-number DNS. Furthermore, in figure 5(a) and also in figure 5(b) for the boundary layer, we observed a peak of $K_{np}$ at $y^+\approx 10$: this peak was shown to be related to strong sweep-like events, a phenomenon referred to as ‘splatting’ in which the large scales transport the high-intensity small scales towards the wall below the buffer layer (Agostini & Leschziner Reference Agostini and Leschziner2014; Agostini et al. Reference Agostini, Leschziner and Gaitonde2016). Thus, being the highest small-scale intensity detected in the buffer layer (Agostini et al. Reference Agostini, Leschziner and Gaitonde2016), the strongest FM is revealed by a peak in $K_{np}$, which then represents a sensitive metric to local changes in the flow dynamics. Eventually, it was remarkable to observe in figure 5(a) the near-wall agreement of $K_{np}$ plotted against $y^+$ between the two channel flows at different Reynolds numbers, as the near-wall dynamics is related to near-wall cycle whose characteristic scales are fixed in wall units (see also a discussion on characteristic near-wall spatial and temporal scales in § 4.4).

The behaviour of $K_{np}$ obtained from the time series of the turbulent boundary layer is shown in figure 5(b). When time series are considered, a convection velocity, $U_c$, has to be defined to apply Taylor's hypothesis in filtering the large and small scales. Typically, $U_c$ is set equal to the local mean velocity, $U$, and in the following, this assumption will be referred to as the CTH. Here, $K_{np}$ obtained through the CTH is displayed as a function of $y^+$ in green in figure 5(b); although a similar behaviour with respect to the channel flows is observed (e.g. the black line in figure 5b), there is a significant vertical shift when time series are employed. It is worth noting that, because the local mean velocity, $U$, does not depend on time, the temporal structure of the time series is preserved when the CTH is applied. This implies that the application of any technique (including NVG) to study the scale interaction from a time series is the same as from the corresponding spatial series (i.e. obtained through the CTH), because ${\rm \Delta} x\propto {\rm \Delta} t$.

The overestimation of modulation parameters when the time series and CTH are used has been previously observed for AM in jet (Fiscaletti et al. Reference Fiscaletti, Ganapathisubramani and Elsinga2015), mixing layer (Fiscaletti et al. Reference Fiscaletti, Attili, Bisetti and Elsinga2016) and turbulent boundary layer flows (Yang & Howland Reference Yang and Howland2018). Specifically, Yang & Howland (Reference Yang and Howland2018) reported a distortion of the spatial series when the CTH is used and suggested employing a convection velocity, $U_c$, defined as $U_c(t)=U+\alpha u(t)$, where $u(t)=u_{LS}(t)+u_{SS}(t)$ is the fluctuating component of the streamwise velocity and $\alpha =O(1)$ is a proportionality constant. The correction proposed by Yang & Howland (Reference Yang and Howland2018) is based on the rationale that the sampling time step has to be scaled using the local viscous scales, so that the small-scale activity is enhanced (or reduced) where the wall shear stress is high (or low) owing to an increase (decrease) in the local friction velocity (Yang & Howland Reference Yang and Howland2018). However, variations in the (fluctuating) wall shear stress are mainly induced by variations into large-scale fluctuations, $u_{LS}(t)$, rather than $u(t)$, as observed by Yang & Howland (Reference Yang and Howland2018) and reported in the previous literature (e.g. Zhang & Chernyshenko Reference Zhang and Chernyshenko2016; Baars et al. Reference Baars, Hutchins and Marusic2017) (see a more detailed discussion about the relation between the near-wall small scales and wall shear stress in § 4.2). Therefore, in this work, we exploited the time varying formulation by Yang & Howland (Reference Yang and Howland2018) but only accounted for the large-scale component of $u(t)$, namely

(4.1)\begin{equation} U_c(t)=U+\alpha u_{LS}(t). \end{equation}

In what follows, we will refer to the application of (4.1) as the convection velocity as the MTH, where we selected $\alpha =0.8$ (which has been proven to be a suitable value). It should be noted that a correction based on $u_{LS}$ arguments was also discussed by Fiscaletti et al. (Reference Fiscaletti, Ganapathisubramani and Elsinga2015) for jet and boundary layer flows to compensate for an AM overestimation. Moreover, it should be noted that, different form the CTH, the structure of $u(x)$ is different from the structure of $u(t)$ for the MTH case, because $U_c(t)$ is not a constant, thus leading to a non-uniform spacing of the spatial series.

The $K_{np}$ behaviour for the MTH is shown in cyan in figure 5(b): the overestimation of $K_{np}$ is compensated and its values are much more comparable to the spatial series obtained from DNS of the channel flow at ${\textit {Re}}_\tau \approx 5200$ (black line in figure 5b). This analysis confirms the applicability of the time-dependent correction of Yang & Howland (Reference Yang and Howland2018) for AM when time series are employed, and extends such a correction to the study of FM through (4.1). In particular, we stress that $u_{LS}(t)$ represents a more suitable choice than $u(t)$ in (4.1) because it is assumed that fluctuations in the large-scale velocity, $u_{LS}(t)$, drive the variations in the friction velocity affecting the behaviour of the small scales (see relation (4.2) and the accompanying discussion). In this regard, the relation (4.1) leads to scaling arguments that are in good agreement with the quasi-steady quasi-homogeneous theory, as will be discussed in § 4.2.

One additional feature emerging from figure 5 concerns the reversal in the modulation mechanism from the wall proximity to the outer flow. In fact, while figure 5 shows a continuous decrease of $K_{np}$ for $y^+>10$ and $y/h \lesssim 0.5$, an almost absence of FM was previously found in the log region by means of other techniques (where a reversal of the FM was only detected in the proximity of the end of the boundary layer) (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012; Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015). We point out that the $K_{np}$ behaviour in figure 5 resembles the decreasing behaviour of the AM parameters (e.g. see Mathis et al. Reference Mathis, Hutchins and Marusic2009a,Reference Mathis, Monty, Hutchins and Marusicb), with a reversal of $K_{np}$ from the near-wall (where $K_{np}>1$) towards the outer region (where $K_{np}<1$). In principle, if the small scales are both amplitude and frequency modulated, one could expect that – following the Newtonian principle that to the same natural effects we must, as far as possible, assign the same causes – a similar underlying mechanism is at play for both AM and FM. This can justify the similarity between the $K_{np}(y^+)$ behaviour, which quantifies FM, and the widely reported behaviour of AM parameters, either in internal or external wall-bounded turbulent flows. Therefore, both AM and FM result from a common underlying phenomenon, for which both the amplitude and frequency of the small scales are concurrently affected by negative or positive large-scale fluctuations at different wall-normal locations.

Specifically, concerning the reversal coordinate (i.e. the $y^+$ location where $K_{np}$ switches from $K_{np}>1$ to $K_{np}<1$), for AM, the reversal typically occurs in the middle of the log region, $y^+\approx 3.9 {\textit {Re}}_\tau ^{1/2}$ (Mathis et al. Reference Mathis, Hutchins and Marusic2009a,Reference Mathis, Monty, Hutchins and Marusicb; Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012; Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015). For FM, it is still not clear whether a modulation reversal occurs in the log region (as shown in figure 5) or if it is only limited to the wake region (Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015). From figure 5, we can conclude that the FM mechanism is indeed limited to a near-wall region up to approximatively $y^+= 100$, which is consistent with the analysis by Ganapathisubramani et al. (Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012). Nevertheless, figure 5 also shows that the reversal $y^+$ location increases with the Reynolds number. In fact, we find that when the spatial series are focused both from channel DNSs and the time series by the MTH, the reversal coordinates (i.e. $y^+\approx 35, 75, 145$ for the channel and boundary layer flows at ${\textit {Re}}_\tau =1000, 5186, 14\,750$, respectively) scale as $y^+\approx 1.15{\textit {Re}}_\tau ^{1/2}$ (the power-law fit gives an exponent of $0.5$ with an $R^2=0.985$). This scaling has the same functional relationship found for AM, i.e. $y^+\approx 3.9 {\textit {Re}}_\tau ^{1/2}$, but with a different proportionality constant. In particular, the ${\textit {Re}}_\tau ^{1/2}$ trend is reminiscent of the scaling of the outer peak position as a function of Reynolds number (Mathis et al. Reference Mathis, Hutchins and Marusic2009a), thus strengthening the underlying connection of FM with the change in large-scale features.

Another notable aspect discernible in figure 5 is the V-like shape of $K_{np}$ in the outer region of the channel flow, i.e. around $y/h\approx 0.5$ ($y^+\approx 2500$). In the literature, the increase, which gives the V-like shape, of AM of the small scales (which are representative of fine-scale turbulent motion) can be observed close to the channel centre, e.g. in Chung & McKeon (Reference Chung and McKeon2010) (see figure 4 therein) for the streamwise velocity, $u$, or in Yao et al. (Reference Yao, Huang and Xu2018) (see figure 3(c) therein) for the $v$ and $w$ components. However, to the best of our knowledge, this peculiar increase of the modulation parameter in the channel flow has not been explicitly discussed so far. Here we propose an interpretation based on the insights gained from turbulent boundary layers.

Previous analyses, in fact, have highlighted that the preferential arrangement of the small scales in the wake region of turbulent boundary layers is mainly affected by intermittency, namely the presence of bulges of turbulent and non-turbulent flow (Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015, Reference Baars, Hutchins and Marusic2017). However, figure 5(a) shows a similar V-like behaviour – as for previous results on AM – also in the outer region of the channel flows, despite the absence of a turbulent/non-turbulent region in the channel flow (that is, an internal flow). In particular, the V-like shape for the channel at ${\textit {Re}}_\tau =1000$, although less evident than at ${\textit {Re}}_\tau \approx 5200$, consistently occurs at $y/h\approx 0.5$, as highlighted in the inset of figure 5(a). Here we suggest that, similar to the effect of intermittency in boundary layer flows, the preferential arrangement in the proximity of the channel centreline could be affected by an alternating occurrence of high- and low-rotational fluid motion above the head of large- or very-large-scale structures. This phenomenon would lead to the increase of both the AM and FM parameters toward the channel centreline. Although the clarification of this issue goes beyond the aim of this work, we do believe it deserves future investigations, as the channel flow set-up is much less considered for scale interaction analyses than the turbulent boundary layers (for which high ${\textit {Re}}_\tau$ data are much more readily available from experimental measurements).

With the aim to ensure that the behaviour of $K_{np}(y^+)$ described so far is the result of an intrinsic flow phenomenon, rather than an artefact arising from the network representation, figure 5 shows the results for random-phase signals. Through a randomization of the phase of velocity Fourier coefficients, the energy spectral density and the turbulence intensity do not change, but all phase information is lost. Hence, following Mathis et al. (Reference Mathis, Hutchins and Marusic2009a), first the signals of $u$ were phase-scrambled, then the large-scale component, $u_{LS}$, was extracted from the new random-phase signal (the amplitude spectrum was not changed), and eventually the degree was calculated from the full random-phase signal and conditioned to the sign of the random-phase $u_{LS}$. Figure 5 shows that in both the channel flow and boundary layer set-ups, the characteristic behaviour of $K_{np}$ for turbulent signals (black curves) disappeared for the random-phase velocity signals (blue curves). As previously reported (e.g. see Chung & McKeon Reference Chung and McKeon2010), phase relationships between the large and small scales play an important role in the characterization of scale interaction, specifically on scale modulation; thus, if any realistic phase information is lost, modulation effects will disappear as well.

The NVG-based approach is demonstrated to be reliable in capturing FM in turbulent velocity signals, and sensitive to phase randomization. Moreover, the behaviour of $K_{np}(y^+)$ for both the channel and boundary layer is found to be robust under different cut-off wavelengths (used to extract $u_{LS}$ from the full signal $u$), as discussed in Appendix B. Further insights into the FM of the streamwise velocity will be presented in § 4.2 focusing on the near-wall region, i.e. where large-to-small scale modulation essentially takes place.

4.2. Scale interaction in the near-wall region

The presence of a near-wall modulation, whose intensity increases with the Reynolds number, has posed a challenge to the classical view on the universality of near-wall turbulence, i.e. the independence of the near-wall statistics (scaled in wall units) to the Reynolds number when this is sufficiently large. Because large-scale structures affect the behaviour of the wall shear stress (Mathis et al. Reference Mathis, Marusic, Chernyshenko and Hutchins2013), the classical universality hypothesis has been recently replaced with the hypothesis that statistics have to be normalized by considering the large-scale skin friction, $\tau _{LS}$, rather than the mean skin friction, $\tau _w$ (Zhang & Chernyshenko Reference Zhang and Chernyshenko2016; Chernyshenko Reference Chernyshenko2020). This hypothesis is referred to as the quasi-steady quasi-homogeneous (QSQH) hypothesis, because the temporal and spatial variations of the large-scale structures are much slower than variations of the near-wall turbulence (Zhang & Chernyshenko Reference Zhang and Chernyshenko2016).

The aim of this section is (i) to provide the proportionality relationships between $u_{LS}$ and the (temporal or spatial) frequency of the small scales as expected from the QSQH hypothesis; and (ii) to test the validity of such relationships by means of the network degree centrality. In particular, we will focus on velocity signals extracted at $y^+=15$, which corresponds to the $y^+$ value of maximum $K_{np}$ in figure 5, thus being a representative wall-normal coordinate of the near-wall region. This choice is also related to the fact that, evidenced by Zhang & Chernyshenko (Reference Zhang and Chernyshenko2016) and Agostini & Leschziner (Reference Agostini and Leschziner2019), the validity of the QSQH hypothesis is found to be rather accurate only in a narrow region close to the wall, that is, $y^+<70-80$.

According to the QSQH hypothesis, variations in the large-scale velocity, $u_{LS}$, induce proportional variations in the large-scale skin friction, $\tau _{LS}$, namely $(\tau _w+\tau _{LS}) \propto (U+u_{LS})$, where $U$ is the local mean velocity. Because, by definition, $U_\tau =\sqrt {\tau _w/\rho }$ (where $\rho$ is the fluid density), the effect of $u_{LS}$ on $\tau _{LS}$ can be stated in terms of velocities as

(4.2)\begin{equation} (U_\tau+u_{\tau,LS})\propto \sqrt{(\tau_w+\tau_{LS})} \propto \sqrt{(U+u_{LS})}, \end{equation}

where $u_{\tau ,LS}$ is the fluctuating (i.e. zero-mean) large-scale component of the friction velocity (Baars et al. Reference Baars, Hutchins and Marusic2017). Owing to near-wall modulation, a quasi-steady or quasi-homogeneous variation of the friction velocity affects also the (amplitude and) frequency of the small scales. Specifically, $u_{LS}>0$ events induce $u_{\tau ,LS}>0$ (see relation (4.2)) that, in turn, produces an increase of the small-scale instantaneous spatial or temporal frequency according to the FM mechanism; and vice versa for $u_{LS}<0$ (Baars et al. Reference Baars, Hutchins and Marusic2017).

In the near-wall region, the spatial scales are supposed to have a constant characteristic length when normalized in wall units (e.g. see the inner spectral peak for $\lambda _x^+=\lambda _x U_\tau /\nu ={\rm const.}\approx 1000$ in figure 3a). Therefore, spatial scales are related to $u_{\tau ,LS}$ variations as $\lambda _x (U_\tau +u_{\tau ,LS})={\rm const.}$, namely $\lambda _x\propto 1/(U_\tau +u_{\tau ,LS})$. Because the spatial frequency (i.e. wavenumber), $\kappa _x$, is related to the spatial scales as $\kappa _x\propto \lambda _x^{-1}$, by using (4.2), we obtain the following scaling relations:

(4.3a,b)\begin{equation} \lambda_x\propto {(U+u_{LS})}^{{-}1/2},\quad \kappa_x\propto {(U+u_{LS})}^{1/2}. \end{equation}

In the case of a time series, the temporal frequency, $f$, is related to the spatial scales as $f=U_c/\lambda _x$, where $U_c$ is the convection velocity. Assuming that $U_c$ scales in wall units (Baars et al. Reference Baars, Hutchins and Marusic2017), namely $U_c\propto (U_\tau +u_{\tau ,LS})$, the temporal frequency is eventually expected to scale as

(4.4)\begin{equation} f\propto (U_\tau+u_{\tau,LS})^2\propto (U+u_{LS}). \end{equation}

Therefore, the relations (4.3a,b) and (4.4) represent the expected scaling of the spatial and temporal frequency, respectively, according to the QSQH hypothesis.

As discussed in § 2.3, the degree centrality represents a measure of the instantaneous wavelength or a temporal period. Therefore, it is expected that the degree, $k$, scales as $k\propto \lambda _x$ for the channel flow (in which spatial series are mapped into NVGs), and $k\propto 1/f$ for the boundary layer (in which time series are analysed). If the degree is indeed an effective parameter to quantify FM, $k$ should then be proportional to $(U+u_{LS})^{\beta _{u,x}}$ in the channel flow and $(U+u_{LS})^{\beta _{u,f}}$ in the boundary layer. Following the aforementioned scaling arguments (i.e. relations (4.3a,b) and (4.4)), the two exponents that verify the QSQH hypothesis should be equal to $\beta _{u,x}=-0.5$ and $\beta _{u,f}=-1$.

To test the $\beta _{u,x}$ and $\beta _{u,f}$ scaling, we conditionally averaged the degree centrality values (computed from NVGs of the full velocity signals $u(x)$) to the $u_{LS}$ values at $y^+\approx 10$. In particular, the $u_{LS}$ values were first divided into uniformly-binned intervals in the range between $min[u_{LS}] \hbox{ and } max[u_{LS}]$. Then, for each visibility network (i.e. each signal), nodes $i$, for which $u_{LS}(i)$ belongs to a specific bin, were selected, and the corresponding degree values, $k_i$, were averaged for that specific bin. By extending the averages to all $u_{LS}$ bins, the conditional average, $(\bar {k}\,|\,u_{LS})$, was obtained, where the overbar indicates an average over a set of nodes. When plotting $(\bar {k}\,|\,u_{LS})$, the $u_{LS}$ value representative of each bin was chosen as the middle value of the bin.

The behaviour of $(\bar {k}\,|\,u_{LS})$ as a function of $u_{LS}$ revealed the scaling between degree-based frequency variations and large-scale velocity variations. Such conditional degree averages are shown in figure 6 for the channel flow at ${\textit {Re}}_\tau \approx 5200$ (figure 6a) and the boundary layer (figure 6b), as a function of $(u_{LS}/U+1)$, where $U=U(y^+=10)$ is constant and $(u_{LS}/U+1)$ values equal to $1$ correspond to large-scale zero-crossing points ($u_{LS}=0$). We find a scaling of the conditioned degree, which follows a power-law with the best-fit exponent $\beta _{u,x}=-0.48$ for the channel flow (spatial data) and $\beta _{u,f}=-1$ for the boundary layer when the local mean velocity is used in the Taylor's hypothesis (CTH). These exponent values are in excellent agreement with the expected values of $-0.5$ and $-1$. We recall that, for the CTH case, because the convection velocity is constant and equal to the local mean velocity $U$, the scaling exponent obtained is representative of a (temporal) frequency, thus $\beta _{u,f}=-1$. However, when the MTH is employed (4.1), the structure of the spatial series (obtained from the corresponding time series) significantly changes and scaling arguments are therefore congruent with the DNS spatial series. Accordingly, the $(\bar {k}\,|\,u_{LS})$ scaling for the MTH case in figure 6(b) produces an exponent which is close to $-0.5$, as expected from the spatial series. Finally, we mention that an exponent $\beta _{u,x}=-0.5$ is found for the channel flow at ${\textit {Re}}_\tau = 1000$ with an $R^2\approx 0.92$ when the cut-off filter is set to $\lambda _{x,c}^+=2500$, while for larger $\lambda _{x,c}^+$ values, a poorer fitting is observed, likely as a result of the limited scale separation for this set-up.

Figure 6. Average degree conditioned to the $u_{LS}$ values as a function of the normalized $u_{LS}$ deviation, for $u$ signals at $y^+\approx 10$ in (a) the turbulent channel flow and (b) the turbulent boundary layer (angular brackets indicate the average over time and homogeneous directions). In (b), the scaling for the spatial series obtained through the CTH and MTH are shown as black lines. The power-law fitting curves are shown as dashed lines, together with the exponent of the fitting and the coefficient of determination, $R^2$, for both set-ups. Light-blue dashed lines correspond to the expected scaling trends for the spatial data. The intervals of $u_{LS}/U+1$ in abscissa cover, for each set-up, a range from the $5$th to the $95$th percentile of all $u_{LS}$ at the selected vertical coordinate $y^+\approx 10$.

The conditional averages were performed here by using uniformly-binned intervals of $u_{LS}$, whereas Baars et al. (Reference Baars, Hutchins and Marusic2017), by adopting a variable-interval scheme for conditional averages, reported a scaling of approximatively $0.8$ (instead of $1$) for the frequency in turbulent boundary layers over a wide range of ${\textit {Re}}_\tau$. They indicated that the discrepancy in the expected exponent might be caused by an inaccurate assumption that the small scales are convected at a fixed inner-scaled velocity. However, here we show that the expected scaling for $f$ is still obtained by assuming $U_c\propto (U_\tau +u_{\tau ,LS})$, which suggests that the discrepancy in the fitting in Baars et al. (Reference Baars, Hutchins and Marusic2017) might be related to different methodological arguments.

The relevance of the scaling shown in figure 6 is twofold. First, it demonstrates that, similarly to the near-wall AM (Baars et al. Reference Baars, Hutchins and Marusic2017), the near-wall FM agrees with the quasi-steady quasi-homogeneous hypothesis. Second, the outcomes of figure 6 further validate the capability of the visibility-based approach, which relies on the degree centrality, to capture FM in wall-bounded turbulence, as well as the validity of the MTH (4.1) in converting a time series into a spatial series.

4.3. Analysis of the spanwise and wall-normal velocity components

The application of the NVG approach to the spatial series of wall-normal, $v(x)$, and spanwise, $w(x)$, velocity from the DNS of the turbulent channel flows is here reported. Figure 7(a) shows the ratio $K_{np}$ for the $v$ (black and green curves) and $w$ (red and purple curves) components as a function of $y^+$ for the two channel flows at different ${\textit {Re}}_\tau$. Here, $K_{np}>1$ is found in the near-wall region, which indicates a positive frequency modulation of the large scales on the small scales of $v$ and $w$. This result is consistent with the AM investigations, which show a similar modulating effect of the large scales on the small scales of the three velocity components (Hutchins & Marusic Reference Hutchins and Marusic2007b; Talluru et al. Reference Talluru, Baidya, Hutchins and Marusic2014; Yao et al. Reference Yao, Huang and Xu2018; Wu et al. Reference Wu, Christensen and Pantano2019). In particular, the trends shown in figure 7(a) are similar to that reported in figure 5(a) for the $u$ signals, although lower $K_{np}$ values are obtained from $v$ and $w$. Moreover, as for $u$, the FM of the small scales of $v$ and $w$ is weaker at lower Reynolds number, because smaller $K_{np}$ values are observed in the near-wall region for ${\textit {Re}}_\tau =1000$.

Figure 7. (a) The $K_{np}$ ratio as a function $y^+$ for the wall-normal and spanwise velocity components, $v$ and $w$, extracted from the two channel-flow DNSs, together with the corresponding $K_{np}$ values for random-phase signals. The respective Reynolds number value of the DNS is reported within brackets in the legend. Angular brackets indicate averaging over time and the spanwise direction. (b) Average degree conditioned to the $u_{LS}$ values as a function of the normalized $u_{LS}$ deviation for $v$ and $w$ signals at $y^+\approx 10$ in the turbulent channel flow at ${\textit {Re}}_\tau \approx 5200$. The power-law fitting curves are shown as cyan and green dashed lines for the $v$ and $w$ cases, respectively, together with the exponents of the fitting and the coefficients of determination, $R^2$.

Furthermore, similar to the $u$ component, an almost constant $K_{np}\approx 1$ behaviour is found for the network built from the random-phase $v$ and $w$ signals (see orange and blue curves in figure 7(a) referring to ${\textit {Re}}_\tau \approx 5200$ as a representative case), which confirms the ability of the degree to capture phase information from the (full) signal. It should be noted that, for the sake of consistency, a unique phase shuffling was performed in this case for the three velocity components. As the degree centrality from the $v$ and $w$ signals is conditionally averaged on $u_{LS}$, the phase of the streamwise velocity signal was extracted and randomly shuffled, so that the random-phase $u_{LS}$, $v$ and $w$ signals were obtained via the respective (non-shuffled) amplitudes but with the same random phases.

Although spectral peak separation in the spectrograms of the transversal velocity components is less evident than for the streamwise velocity, the generation and amplification of small-scale motions (i.e. fine-scale vortices) of all the three velocity components are strongly connected with large-scale events (Hutchins & Marusic Reference Hutchins and Marusic2007b). Within this perspective, the wall-normal and spanwise velocities are expected to be modulated in the near-wall region by following the QSQH hypothesis in a similar way as the streamwise component, $u$. However, while results for AM of the three velocity components (Talluru et al. Reference Talluru, Baidya, Hutchins and Marusic2014; Agostini & Leschziner Reference Agostini and Leschziner2019; Chernyshenko Reference Chernyshenko2020) and scaling arguments for the AM and FM of the $u$ component (Baars et al. Reference Baars, Hutchins and Marusic2017) have been provided, as far as we know, similar scaling arguments (as in Baars et al. (Reference Baars, Hutchins and Marusic2017), figure 9 therein) for $v$ and $w$ have not been pursued for FM to date.

By analogy with the modulation of the $u$ component, the conditional average degree, $(\bar {k}\,|\,u_{LS})$, is evaluated as a function of $(u_{LS}/U+1)$ at $y^+\approx 10$ for NVGs built from $v(x)$ and $w(x)$ signals. The conditional average degree and the corresponding fitting are shown in figure 7(b) for the ${\textit {Re}}_\tau \approx 5200$ set-up and confirm the power-law modulation effect of the large scales in the near-wall region even for the other velocity components, namely $k\propto (U+u_{LS})^{\beta _{v,x}}$ and $k\propto (U+u_{LS})^{\beta _{w,x}}$. However, while, for the $u$ component, the exponent of the power-law was $\beta _{u,x}\approx -0.5$, a weaker scale interaction effect was found for the $v$ and $w$ components, being $\beta _{v,x}\approx -0.3$ and $\beta _{w,x}\approx -0.3$, which are both smaller (in modulus) than $\beta _{u,x}$. This outcome is consistent with the smaller $K_{np}$ values for $v$ and $w$ (see figure 7a) than for $u$ (see figure 5a), which indicates a weaker FM in the near-wall region for the transversal velocity components.

The power-law relationships found for $u$, $v$ and $w$ suggest that, although the intensity of modulation is different for each velocity component, the response of the small-scales of $v$ and $w$ exhibits a functional relation qualitatively analogous to the response of $u$. In general, there could be several factors playing a role in the scale-interaction mechanisms (such as the direction of the large scales as discussed by Chernyshenko Reference Chernyshenko2020), but we can conclude that the QSQH hypothesis is valid for all velocity components, although a more refined description is required for the transversal components, $v$ and $w$.

The results shown in this section reveal that the three velocity components are all affected by a large-scale FM in the near-wall region, where an increase of the local (spatial) frequency is observed under $u_{\tau ,LS}>0$ periods induced by positive $u_{LS}$ events. In particular, we provide novel insights on FM for the $v$ and $w$ components, which have been investigated less than $u$, in terms of FM intensity for the spatial series (figure 7a) and scaling arguments on the QSQH hypothesis (figure 7b).

4.4. Time and space shifting in FM

To conclude our analysis, we provide results on the investigation of the time- and space-shifted FM, as quantified by $K_{np}$. We recall that a lead of the small-scale amplitude was found with respect to the large scales in the near-wall region of turbulent boundary layers, while a small-scale lag is found above the reversal coordinate (Bandyopadhyay & Hussain Reference Bandyopadhyay and Hussain1984; Guala et al. Reference Guala, Metzger and McKeon2011). Concerning FM, a lead of the small-scale frequency with respect to the large scales was found in the near-wall region but, different from AM, scattered behaviours were found far from the wall (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012; Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015).

To address this issue, in figure 8, we show the conditionally average degree, $K_{np}$, as a function of $y^+$ and the spatial delay, ${\rm \Delta} r_x^+$, where for the time series, it holds the Taylor's hypothesis ${\rm \Delta} r_x^+=-U_c{\rm \Delta} t^+$ (the minus sign highlights the opposite direction of the reference systems between the fixed-point time series and spatial series). Therefore, the formulation of $K_{np}$ reported in (2.3a–c) is extended to account for spatial shifting, ${\rm \Delta} r_x^+$, as $K_{np}({\rm \Delta} r_x)=K_n({\rm \Delta} r_x)/K_p({\rm \Delta} r_x)$, with

(4.5)\begin{equation} \left.\begin{gathered} K_n({\rm \Delta} r_x)=\frac{1}{N_{neg}({\rm \Delta} r_x)} \sum_{j=1}^N {\left(k(x_j-{\rm \Delta} r_x)|u_{LS}(x_j)<0\right)},\\ K_p({\rm \Delta} r_x)=\frac{1}{N_{pos}({\rm \Delta} r_x)} \sum_{j=1}^N{\left(k(x_j-{\rm \Delta} r_x)|u_{LS}(x_j)>0\right)}. \end{gathered}\right\} \end{equation}

Positive or negative ${\rm \Delta} r_x^+$ values indicate in (4.5) a lag or lead, respectively, of the degree, $k$, with respect to $u_{LS}$ in the conditional averages of (2.3a–c) (the results in figure 5 correspond to ${\rm \Delta} r_x^+={\rm \Delta} t^+=0$). If the CTH is employed for the turbulent boundary layer time series (figure 8a), a slight lead of $K_{np}$ with respect to $u_{LS}$ (i.e. high $K_{np}$ values are at ${\rm \Delta} r_x^+<0$ but close to ${\rm \Delta} r_x^+=0$) is observed for $y^+\lesssim 15$. However, a more substantial lead is observed for larger $y^+$ coordinates in the near-wall region up to $y^+\approx 100$, in agreement with previous analyses (Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015), while a lag of $K_{np}$ with respect to large scales is detected for $y^+>rsim 100$. A clearer picture is obtained when the MTH is exploited (figure 8b). Significant lead of $K_{np}$ with respect to $u_{LS}$ is found in the whole near-wall region (including the wall proximity, $y^+\lesssim 15$), while the lag for $y^+>rsim 100$ is less evident and a lead is recovered for larger $y^+$ values (see blue contours in figure 8b). Eventually, no clear patterns are observed in the intermittent regions ($y^+>rsim 5\times 10^3$).

Figure 8. Contour plot of the $K_{np}$ ratio as a function of the wall-normal coordinate, $y^+$, and the temporal or spatial shifting, ${\rm \Delta} t^+$ or ${\rm \Delta} r_x^+$, respectively. Shiftings for the turbulent boundary layer are reported in (a,b) whether the CTH or MTH is used, respectively. The spatial shifting for the turbulent channel flow at ${\textit {Re}}_\tau \approx 5200$ is shown in (c–e) for the three velocity components. Iso-level contours are displayed by using a level-step equal to $0.03$ in (a), $0.01$ in (b) and $0.015$ in (c–e).

Figure 9. First 2500 time instants (out of $10^4$) of the three synthetic modulated signals (shown in black) and the modulating signal (shown in red) for: (a) amplitude modulated signal (shown as AM); (b) frequency modulated signal (shown as FM); (c) amplitude and frequency modulated signal (shown as AM+FM). The inset shows a zoom of the AM+FM modulated signal. (d) Results of the application of the NVG to the synthetic signals in (a–c). Values of $K_{np}$ are shown as box plots, where $q_1$, $q_2$ and $q_3$ are the 25th, median, and 75th percentiles, respectively, while $q_{min}=q_1-1.5(q_3-q_1)$ and $q_{max}=q_1+1.5(q_3-q_1)$, whose values are explicitly indicated at the tips of the whiskers as percentages.

The space-shifted $K_{np}$ values in the turbulent channel flow at ${\textit {Re}}_\tau \approx 5200$ for the three velocity components are displayed in figure 8(c–e). Similar to the turbulent boundary layer, a lead of $K_{np}$ with respect to $u_{LS}$ is found for $y^+<100$, as highlighted by high $K_{np}$ values for ${\rm \Delta} r_x^+<0$. Differences among the $u$, $v$ and $w$ components are detected in proximity of the channel centreline, where $K_{np}$ appears to lead, be in-phase and slightly lag $u_{LS}$ for the $u$ (figure 8c), $v$ (figure 8d), and $w$ components (figure 8e), respectively. It should be noted that the ${\textit {Re}}_\tau \approx 5200$ is here used as a representative set-up for spatial data, and the results for the lower ${\textit {Re}}_\tau$ channel flow are in agreement with the results in figure 8, so they are not shown for the sake of conciseness.

In particular, it is worth highlighting that the maximum $K_{np}$ values in the near-wall region are found at ${\rm \Delta} r_x^+\approx 1000$ for the streamwise velocity in the turbulent boundary layer when the MTH is employed (figure 8b), as well as for all the velocity components in the turbulent channel flow (figure 8c–e). The value ${\rm \Delta} r_x^+\approx 1000$ is in excellent agreement with the characteristic length scale in the near-wall region, being $\lambda _x^+=O(10^3)$ (see inner spectral peak in figure 3a). The equivalent time scale is ${\rm \Delta} t^+={\rm \Delta} r_x^+/U_c^+\approx 100$ (being $U_c^+\approx 10$ in the buffer layer and viscous sublayer), which is the characteristic turnover time of the near-wall cycle. As the small scales are supposed to be actively modulated by the large scales, the time taken for this process to be completed is therefore equivalent to the time scale of the near-wall cycle (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Monty, Chung and Marusic2012).

The results shown in figure 8(a–c) for the streamwise velocity reveal that different convection velocities indeed play a significant role in the FM dynamics, not only in terms of overestimation (as highlighted in figure 5b), but also in terms of spatial delay that, in the near-wall region, is strongly related to the near-wall cycle. Therefore, NVG is again revealed to be a reliable approach for quantifying FM even in the presence of a temporal or spatial shifting. Finally, we note that an important issue about scale interaction is whether the large scales actually cause an increase or decrease of the small-scale activity, as the parameters used so far to quantify AM and FM only show that there is a relation (e.g. a correlation) between the large scales and small scales. Although definite answers to this issue are not still available, our detection of the presence of a significant temporal or spatial delay close to the characteristic time or length scale of the near-wall cycle, in conjunction with the arguments leading to relation (4.2), could provide supporting clues that a causation process is at play. In fact, fluctuations in the large-scale component of the wall shear stress, which affect the small-scale behaviour, appear to be directly caused by the outer large-scale structures rather than being the feature of near-wall processes (Zhang & Chernyshenko Reference Zhang and Chernyshenko2016).