3.1 Ensemble-averaged wall shear stress and velocity profiles

In figure 3(a,b) the time development of the ensemble-averaged wall shear stress $\unicode[STIX]{x1D70F}_{0}$ is shown for both turbulent and laminar flow. During the first stage of the positive half-cycle, because of the large flow acceleration, the vorticity produced at the wall has a short time to spread, leading to a large near-wall velocity gradient (Nielsen & Callaghan Reference Nielsen and Callaghan2003). Consequently, the positive peak of the wall shear stress is larger than the negative one, as shown in figure 3(a,b). The asymmetry of the wall shear stress can be evaluated through the asymmetry parameter $A_{\unicode[STIX]{x1D70F}_{0}}$ defined as:

where $\unicode[STIX]{x1D70F}_{0max}$ and $\unicode[STIX]{x1D70F}_{0min}$ are the maximum and the minimum values of the wall shear stress during the cycle. Another measure for the asymmetry is given by $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ which is defined as follows:

Table 1 shows that $A_{\unicode[STIX]{x1D70F}_{0}}$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ have similar trends as functions of the flow parameters, even though there are quantitative differences between themselves. It can be observed that the asymmetry increases both with acceleration skewness and Reynolds number, however, in laminar flow it is larger than in turbulent flow. This finding shows that a Reynolds number threshold, below which the asymmetry decreases with $R_{\unicode[STIX]{x1D6FF}}$ and above which it increases, exists for Reynolds number smaller than those considered in table 1, thus suggesting that the asymmetry has a minimum. Following these results, the effect of the Reynolds number on the asymmetry of the wall shear stress in the range between the laminar regime and $R_{\unicode[STIX]{x1D6FF}}=1100$ was investigated. A series of 10 numerical simulations was therefore performed for $R_{\unicode[STIX]{x1D6FF}}<1100$ and $\unicode[STIX]{x1D6FD}=0.75$ in order to describe the trend of the asymmetry. For all these simulations the same computational domain and grid size adopted for $R_{\unicode[STIX]{x1D6FF}}=1100$ were employed. The ensemble average was computed by phase averaging over 35 cycles.

Figure 3. Time development of the ensemble-averaged wall shear stress during a cycle. (a) $R_{\unicode[STIX]{x1D6FF}}=1100$ , (b) $R_{\unicode[STIX]{x1D6FF}}=1414$ . Thin lines: laminar flow; thick lines: turbulent flow.

In figure 4(a), where the trends of $A_{\unicode[STIX]{x1D70F}_{0}}$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ are shown, it can be observed that when the Reynolds number decreases below 1100, $A_{\unicode[STIX]{x1D70F}_{0}}$ decreases and attains the minimum at approximately $R_{\unicode[STIX]{x1D6FF}}=925$ . On the other hand $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ does not show important variations as long as $R_{\unicode[STIX]{x1D6FF}}$ is larger than approximately 900. A decrease of $R_{\unicode[STIX]{x1D6FF}}$ below 900 causes an increase of both $A_{\unicode[STIX]{x1D70F}_{0}}$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ . For $R_{\unicode[STIX]{x1D6FF}}=650$ the flow is found to be laminar, therefore the asymmetries take the values reported in table 1 for laminar flow. In order to illustrate why the asymmetry in turbulent flow may be much smaller than in laminar flow, figure 4(b) shows the trend of the wall shear stress for Reynolds numbers 750, 800, 925 and 1050 along with the laminar wall shear stress. The quantity $\unicode[STIX]{x1D70F}_{0}\times R_{\unicode[STIX]{x1D6FF}}$ in figure 4(b) is the wall shear stress scaled by $\unicode[STIX]{x1D71A}^{\ast }\unicode[STIX]{x1D708}^{\ast }u_{max}^{\ast }/\unicode[STIX]{x1D6FF}^{\ast }$ , which use turns out to be advantageous since all the curves in laminar flow collapse on each other. At $R_{\unicode[STIX]{x1D6FF}}=750$ the flow is already in the intermittently turbulent regime, hence, turbulence develops during the decelerating phase of the positive half-cycle, causing the appearance of a secondary peak in the wall shear stress. Turbulence also develops during the decelerating phase of the negative half-cycle, hence a small secondary peak also appears at approximately $t=5.5$ . These peaks do not exceed the main ones, therefore the asymmetry $A_{\unicode[STIX]{x1D70F}_{0}}$ still depends on the main ones. For such a low Reynolds number the main peaks of $\unicode[STIX]{x1D70F}_{0}$ are slightly smaller than in laminar flow and to a greater extent this occurs in the positive half-cycle, therefore a decrease of $A_{\unicode[STIX]{x1D70F}_{0}}$ with respect to the laminar flow is observed. When the Reynolds number increases up to 800, the secondary peak of the first half-cycle becomes close to the main one but in the negative half-cycle it exceeds the main one, therefore a decrease of $A_{\unicode[STIX]{x1D70F}_{0}}$ with respect to $R_{\unicode[STIX]{x1D6FF}}=750$ occurs. The smaller decrease of $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ with respect to $A_{\unicode[STIX]{x1D70F}_{0}}$ is due to the increase of the second peak in the positive half-cycle which provides a positive contribution to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ but it does not affect $A_{\unicode[STIX]{x1D70F}_{0}}$ . A further increase of the Reynolds number causes an increase of the secondary peaks which finally become the only one of each half-cycle. In particular, for $R_{\unicode[STIX]{x1D6FF}}=925$ a large decrease of the asymmetry with respect to that in laminar regime is detected. The proximity of the two peaks, combined with the circumstance that $\unicode[STIX]{x1D70F}_{0max}$ in turbulent flow is substantially larger than in laminar flow, explains such a result (see (3.1)).

Figure 4. Asymmetry parameters $A_{\unicode[STIX]{x1D70F}_{0}}$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ as functions of the Reynolds number (a); ensemble-averaged wall shear stress (b). $\unicode[STIX]{x1D6FD}=0.75$ .

When $R_{\unicode[STIX]{x1D6FF}}$ takes larger values, the onset of turbulence shifts back in time during the cycle. Hence, in the positive half-cycle the peak of the wall shear stress occurs when the laminar wall shear stress is larger, thus causing an increase of $\unicode[STIX]{x1D70F}_{0max}$ with respect to lower $R_{\unicode[STIX]{x1D6FF}}$ . In the negative half-cycle the trend of the laminar wall shear stress is rather constant near the peak, therefore the magnitude of $\unicode[STIX]{x1D70F}_{0min}$ increases to a lesser extent with respect to $\unicode[STIX]{x1D70F}_{0max}$ . Overall, these effects induce an increase of the asymmetry when $R_{\unicode[STIX]{x1D6FF}}$ increases beyond a value of approximately 925. It is much more difficult to explain the trend of $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ especially in the range $R_{\unicode[STIX]{x1D6FF}}=900{-}1100$ , where important qualitative differences with respect to $A_{\unicode[STIX]{x1D70F}_{0}}$ are observed. Such differences are, however, easy to accept as $A_{\unicode[STIX]{x1D70F}_{0}}$ only depends on the peaks of the wall shear stress while $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D70F}_{0}}$ depends on the overall trend. The increase of the asymmetry with $R_{\unicode[STIX]{x1D6FF}}$ qualitatively agrees with the experimental findings of van der A et al. (Reference van der A, O’Donoghue, Davies and Ribberink2011).

In figure 5(a,b,c) the ensemble-averaged velocity profiles are shown for $\unicode[STIX]{x1D6FD}=0.60$ , 0.75 and 0.85 respectively. In these figures the velocity profiles are plotted at the phases $t=0$ , 2.35, 3.14, 3.92 and at two additional phases depending on the $\unicode[STIX]{x1D6FD}$ parameter and characterized by the maximum and the minimum value respectively of the free stream velocity during the cycle. Comparing figure 5(a–c) we observe that the distance $y_{max}$ from the wall of the position of maximum velocity decreases with $\unicode[STIX]{x1D6FD}$ as a result of the increased acceleration. On the other hand, the distance $y_{min}$ of the position of minimum velocity increases with $\unicode[STIX]{x1D6FD}$ . Such an increase occurs because in the negative half-cycle the vorticity has a longer time to spread far from the wall when $\unicode[STIX]{x1D6FD}$ increases (Nielsen & Callaghan Reference Nielsen and Callaghan2003). Table 2 shows the distances $y_{max}$ and $y_{min}$ , which can be assumed as a measure of the thickness of the boundary layer. It can be noted that as the Reynolds number increases both $y_{max}$ and $y_{min}$ increase. This result is somewhat expected as generally turbulent diffusivity increases with $R_{\unicode[STIX]{x1D6FF}}$ . The data in table 2 also allow for assessment of the thickness of the boundary layer in turbulent flow and to compare it with that in laminar flow.

Figure 5. Ensemble-averaged velocity profiles at different phases during the period (a)  $\unicode[STIX]{x1D6FD}=0.6$ , (b) $\unicode[STIX]{x1D6FD}=0.75$ , (c) $\unicode[STIX]{x1D6FD}=0.85$ . Continuous line: $R_{\unicode[STIX]{x1D6FF}}=1100$ ; dashed line: $R_{\unicode[STIX]{x1D6FF}}=1414$ .

Figure 6. Amplitudes $|\langle u\rangle _{1}|$ , $|\langle u\rangle _{2}|$ and $|\langle u\rangle _{3}|$ (a,c,e) and phases $\unicode[STIX]{x1D719}_{1}$ , $\unicode[STIX]{x1D719}_{2}$ and $\unicode[STIX]{x1D719}_{3}$ (b,d,f) of the first three harmonic components of the ensemble-averaged velocity profiles for $R_{\unicode[STIX]{x1D6FF}}=1100$ .

In figure 6 the amplitude $|\langle u\rangle _{n}|$ and the phase $\unicode[STIX]{x1D719}_{n}$ of the first three harmonic components of the ensemble-averaged velocity are plotted. Far from the wall, where the flow is irrotational, the amplitudes are given by (2.1). Within the boundary layer the amplitudes vary and vanish at the bottom where the no-slip condition is satisfied. A comparison among the first components in figure 6(a), shows that the smaller is $\unicode[STIX]{x1D6FD}$ the larger is the component. For the second and the third harmonic components (figure 6 c,e), the comparison shows the opposite. The trends along the $y$ -axis of the first and of the second component are qualitatively similar to each other both in terms of amplitudes and phases for all the considered values of $\unicode[STIX]{x1D6FD}$ (figure 6 a–d). On the other hand, the third component shown in figure 6(e) exhibits large fluctuations near the bottom which increase as $\unicode[STIX]{x1D6FD}$ decreases. Analogous fluctuations can also be observed in figure 6(f) as regards the phase. Present results have several points in common with those reported in figure 5 of van der A et al. (Reference van der A, O’Donoghue, Davies and Ribberink2011), including the large oscillations of the profile of the harmonic components for low values of $\unicode[STIX]{x1D6FD}$ . Analogous results are observed for $R_{\unicode[STIX]{x1D6FF}}=1414$ .

Figure 7. Ensemble-averaged velocity profiles in semi-log plot during the period for $R_{\unicode[STIX]{x1D6FF}}=1100$ . Thin dashed line: $\langle u\rangle ^{+}=\log (y^{+})/0.41+5$ .

In figure 7 the ensemble-averaged velocity profiles in semi-log plot are shown at 16 phases for $R_{\unicode[STIX]{x1D6FF}}=1100$ . Note that the logarithm of the absolute value of the velocity is plotted, therefore near flow reversal, a discontinuity in the first derivative may appear such as at $t=0$ and $t=7\unicode[STIX]{x03C0}/8$ . In this figure the length is expressed in wall units ( $y^{+}=yR_{\unicode[STIX]{x1D6FF}}\sqrt{\unicode[STIX]{x1D70F}_{0}}$ ) while $\langle u\rangle ^{+}$ is the ensemble-averaged velocity made dimensionless by means of the instantaneous friction velocity ( $\langle u\rangle ^{+}=\langle u\rangle /\sqrt{\unicode[STIX]{x1D70F}_{0}}$ ). As shown in figure 7, logarithmic layers are not detected in the early stage of the accelerating phase. A logarithmic layer emerges between $\unicode[STIX]{x03C0}/8$ and $\unicode[STIX]{x03C0}/4$ for $\unicode[STIX]{x1D6FD}=0.85$ and between $\unicode[STIX]{x03C0}/4$ and $3\unicode[STIX]{x03C0}/8$ for $\unicode[STIX]{x1D6FD}=0.75$ and $\unicode[STIX]{x1D6FD}=0.6$ . During the decelerating phase the logarithmic velocity profile persists up to $5\unicode[STIX]{x03C0}/8$ and completely disappears near flow reversal. In the negative half-cycle a logarithmic layer emerges first for $\unicode[STIX]{x1D6FD}=0.60$ at about $t=3\unicode[STIX]{x03C0}/2$ and later even for $\unicode[STIX]{x1D6FD}=0.75$ and $\unicode[STIX]{x1D6FD}=0.85$ . Therefore, in the negative half-cycle the situation is reversed with respect to the positive one. Once again the logarithmic layer persists during most of the decelerating phase. Figure 7 shows that the equation $\langle u\rangle ^{+}=\log (y^{+})/k+A$ , with $k=0.41$ and $A=5$ , is in a fairly good agreement with present results where a logarithmic layer is detected. For a sinusoidally oscillating flow Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) reported that the constants of the log-law at $R_{\unicode[STIX]{x1D6FF}}=3464$ are $k=0.4$ and $A=5$ , while Salon et al. (Reference Salon, Armenio and Crise2007) for $R_{\unicode[STIX]{x1D6FF}}=1790$ reported $k=0.41$ and $A$ phase dependent, with $A=5$ for $2\unicode[STIX]{x03C0}/3<t<5\unicode[STIX]{x03C0}/6$ . Therefore, present results are consistent with those found by previous investigations on unsteady flows. Figure 8 shows the semi-log plot of the velocity profiles for $R_{\unicode[STIX]{x1D6FF}}=1414$ . It can be noted that an increase of $R_{\unicode[STIX]{x1D6FF}}$ shifts back the phase of occurrence of a logarithmic layer. Compare for example figures 7 and 8 at $t=\unicode[STIX]{x03C0}/4$ .

Figure 8. Ensemble-averaged velocity profiles in semi-log plot during the period for $R_{\unicode[STIX]{x1D6FF}}=1414$ . Thin dashed line: $\langle u\rangle ^{+}=\log (y^{+})/0.41+5$ .

3.2 Turbulent structures

Flow visualizations of the near-wall region of a steady turbulent boundary layer (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967; Corino & Brodkey Reference Corino and Brodkey1969) have shown that it is characterized by low- and high-speed streaks of fluid, mostly aligned with the streamwise direction and having a mean spacing of approximately $100\unicode[STIX]{x1D708}^{\ast }/u_{\unicode[STIX]{x1D70F}}^{\ast }$ , where $u_{\unicode[STIX]{x1D70F}}^{\ast }$ is the friction velocity. These turbulent structures are now recognized to be ubiquitous features of wall turbulent flows and essential in the process of generation and sustenance of turbulence (Schoppa & Hussain Reference Schoppa and Hussain2002). Low-speed streaks have also been observed in sinusoidally oscillating boundary layers by Fishler & Brodkey (Reference Fishler and Brodkey1991), Costamagna et al. (Reference Costamagna, Vittori and Blondeaux2003) and Salon et al. (Reference Salon, Armenio and Crise2007). As shown in the following, the time development of the wall shear stress is strongly affected by the dynamics of the streaks, thus any insight on these turbulent structures contributes to a better understanding of the spatial and temporal distribution of the turbulent skin friction.

In figure 9 the fluctuation $u^{\prime }$ of the streamwise velocity component, computed by subtracting to the local velocity $u$ the streamwise velocity averaged in the $(x,z)$ plane, is shown at four phases during the first half-cycle, in the plane placed at $y=0.16$ , for $R_{\unicode[STIX]{x1D6FF}}=1100$ and $\unicode[STIX]{x1D6FD}=0.60$ , 0.75 and 0.85. In terms of wall units, the position of the plane corresponds to $y^{+}\approx 10$ at the phase of the peak of $\unicode[STIX]{x1D70F}_{0}$ . Figure 9 shows that no streaks can be detected at the beginning of the cycle. Focusing attention on $\unicode[STIX]{x1D6FD}=0.60$ (a–d), the streaks emerge between $t=0$ and $t=\unicode[STIX]{x03C0}/8$ and in the interval $\unicode[STIX]{x03C0}/8-\unicode[STIX]{x03C0}/4$ they attain a well-developed state, spanning the entire fluid domain. Between $t=\unicode[STIX]{x03C0}/4$ and $t=3\unicode[STIX]{x03C0}/8$ the streaks begin to meander and interact among themselves until they break down and split into short segments, which mostly remain aligned with the streamwise direction. The onset of breaking occurs during the accelerating phase, i.e. before the peak of the free stream velocity. The breaking propagates in space causing a rapid increase of the plane-averaged wall shear stress. When $\unicode[STIX]{x1D6FD}$ increases both the phase of appearance and breaking of the streaks are shifted back. The panel in figure 9, for $\unicode[STIX]{x1D6FD}=0.75$ and $t=\unicode[STIX]{x03C0}/4$ , shows that breaking down of streaks is in progress in the fluid domain, since it is occurring in certain areas of the domain while outside these areas the streaks still show an elongated shape. The low-speed streaks disappear near the flow reversal and reappear during the accelerating phase of the negative half-cycle (not shown). Both the reappearance and the breakdown occur earlier for lower values of $\unicode[STIX]{x1D6FD}$ , therefore, in this regard, in the negative half-cycle the situation is reversed with respect to the positive one.

Figure 9. Low-speed streaks for $R_{\unicode[STIX]{x1D6FF}}=1100$ in the plane $y=0.16$ . Darker areas denote the low-speed streaks. (a–d) $\unicode[STIX]{x1D6FD}=0.60$ ; (e–h) $\unicode[STIX]{x1D6FD}=0.75$ ; (i–l) $\unicode[STIX]{x1D6FD}=0.85$ .

There is a general consensus in the scientific community that the streaks are generated by a sequence of streamwise vortices, which follow each other and pump low-speed fluid away from the wall and high-speed fluid towards the wall. While these vortices are advected downstream, they leave behind a streak of low-speed fluid (Robinson Reference Robinson1991; Hamilton, Kim & Waleffe Reference Hamilton, Kim and Waleffe1995; Moin & Mahesh Reference Moin and Mahesh1998; Schoppa & Hussain Reference Schoppa and Hussain2002). Within this framework, theories have been developed to describe the regeneration mechanism of turbulence according to which the low-speed streaks instability generates new vortical structures which in turn sustain the streaks (Schoppa & Hussain Reference Schoppa and Hussain2002).

In order to gain insights about the process that leads to the streak appearance and breakdown in the flow under investigation, the low-speed regions along with the vortex structures are shown in a three-dimensional plot in figure 10 at six phases during the first half-cycle for $R_{\unicode[STIX]{x1D6FF}}=1100$ and $\unicode[STIX]{x1D6FD}=0.75$ . The vortex cores are identified by means of the method of Jeong & Hussain (Reference Jeong and Hussain1995), according to which they are localized in the regions characterized by two negative eigenvalues of the symmetric tensor $\unicode[STIX]{x1D63F}^{2}+\unicode[STIX]{x1D734}^{2}$ , where $\unicode[STIX]{x1D63F}$ and $\unicode[STIX]{x1D734}$ denote the symmetric and antisymmetric parts of the velocity gradient tensor respectively. Therefore, negative values of the second eigenvalue $\unicode[STIX]{x1D706}_{2}$ ( $\unicode[STIX]{x1D706}_{1}>\unicode[STIX]{x1D706}_{2}>\unicode[STIX]{x1D706}_{3}$ ) identify the vortex cores.

Figure 10. Evolution of the low-speed streaks and of the vortex structures during the first half-cycle for $R_{\unicode[STIX]{x1D6FF}}=1100$ and $\unicode[STIX]{x1D6FD}=0.75$ . Darker regions denote the low-speed streaks; grey regions are the isosurfaces of second eigenvalue $\unicode[STIX]{x1D706}_{2}$ and denote the vortex cores; (a–e):  $\unicode[STIX]{x1D706}_{2}=-0.007$ , $u^{\prime }=-0.08$ ; (f) $\unicode[STIX]{x1D706}_{2}=-0.04$ , $u^{\prime }=-0.16$ ; (a) $3\unicode[STIX]{x03C0}/64$ , (b) $5\unicode[STIX]{x03C0}/64$ , (c)  $\unicode[STIX]{x03C0}/8$ (d)  $3\unicode[STIX]{x03C0}/16$ , (e) $13\unicode[STIX]{x03C0}/64$ , (f) $\unicode[STIX]{x03C0}/4$ .

Figure 10(a) shows the turbulent structures a short time after the free stream velocity reverses its direction. Several vortices, remnant of the previous half-cycle and devoid of any type of organization, can be observed along with blobs of fluid characterized by low streamwise velocity. Because of the shear of the mean flow, several vortex structures are tilted towards the $x$ direction and stretched, causing an increase of the streamwise vorticity component. These vortices affect the wall-normal distribution of streamwise momentum through the ejection of slow fluid away from the wall, according to the previously described mechanism. Thus for example a newly formed streak with the tip at about $(x,z)\approx (10,15.3)$ can be observed in figure 10(a). As the flow evolves, the streaks and the streamwise vortices grow and strengthen as shown in figure 10(b), where the previously mentioned streak has the tip at approximately $(x,z)\approx (30,15.3)$ . The growth occurs through a mechanism that feeds both the vortices and the streaks. Because of the $y$ gradient of the $u$ velocity component, the wall-normal vorticity induced by the presence of the streaks is tilted in the $x$ direction, thus it feeds the streamwise vortex structure. However, as shown by the analysis of Schoppa & Hussain (Reference Schoppa and Hussain2002), the main contribution to the production of streamwise vortices comes from the stretching of sheets of streamwise vorticity because of $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ velocity gradient induced by the streak waviness in $(x,z)$ planes. These authors also identified a ‘streak transient growth mechanism’ able to amplify $x$ -dependent disturbances and thus able to produce large streamwise velocity gradients. Once the streamwise vortices have grown substantially, in turn they sustain the streaks through the ejection mechanism. This explains why in figure 10(c) the streaks are surrounded by vortices. These turbulent structures gradually develop during the accelerating phase up to $t\approx 3\unicode[STIX]{x03C0}/16$ when an explosive growth of both streaks and vortices accompanied by a random motion occurs (see figure 10 d). More specifically, the low-speed streaks in the left part of figure 10(d) ( $0<z<12$ ) undergo a wavy motion followed by the development of several vortices, some of them being quasi-longitudinal in shape while others have the shape of arches that wrap around the low-speed streaks. The ensemble of all these turbulent structures is called a turbulent spot and has been studied experimentally by Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010) for the case of a Stokes layer. According to these experiments, the turbulent spots are areas characterized by large turbulent fluctuations, separated by the rest of the space by sharp fronts. The authors found that the spots first emerge in the form of two longitudinal streaks which then break into small turbulent structures. Figure 10(c,d) show that a similar phenomenon occurs in this flow as the turbulent spot appears in $0<z<12$ where two low-speed streaks are initially present in figure 10(c). Breaking of the low-speed streaks is then associated with the emergence of turbulent spots. Here, this can be better demonstrated by a comparison between figures 9 and 10. In figure 10(c) no turbulent spots are detected, hence in figure 9(f), at $t=\unicode[STIX]{x03C0}/8$ , the streaks are still intact. The turbulent spot that for the first time appears in figure 10(d), affects almost the entire fluid domain in figure 10(f). In figure 9(g), at the same instant of figure 10(f) ( $t=\unicode[STIX]{x03C0}/4$ ), the streaks are broken down over a large part of the domain.

In addition to the turbulent spots, Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010) detected two-dimensional vortex structures, denoted as vortex tubes, produced as a result of the flow instability due to an inflection point in the laminar velocity profile (Carstensen et al. Reference Carstensen, Sumer and Fredsøe2010; Scandura Reference Scandura2013). An inflection point is also apparent in the mean velocity profiles of the turbulent flow shown in figure 5, however, vortex tubes have not been observed here. The reason for this result is most likely due to the fact that in turbulent flow, the wall-normal gradient of $\langle u\rangle$ at the inflection point is much smaller than in laminar flow, therefore, the instability is not triggered.

Figure 11 shows the fluctuating velocity components $(u^{\prime },v^{\prime },w^{\prime })$ along the spanwise direction, at $x=12.5$ and $y=0.16$ , for $\unicode[STIX]{x1D6FD}=0.75$ , $R_{\unicode[STIX]{x1D6FF}}=1100$ and $t=\unicode[STIX]{x03C0}/8$ , when low-speed streaks are present in the flow. Figure 11(a) is cropped from figure 9, for $\unicode[STIX]{x1D6FD}=0.75$ and $t=\unicode[STIX]{x03C0}/8$ , and shows the streaks in the range $12<x<13$ . The vertical velocity component is very low at $y=0.16$ , therefore in figure 11 it has been magnified by 10 times in order to make it comparable with the others. The trend of $u^{\prime }$ shows nine low-speed streaks which mostly occur in concomitance with positive values of $v^{\prime }$ . This shows that the low-speed streaks are generated by an upward motion of slow fluid, in agreement with the previously described mechanism that involves streamwise vortices. Unlike the wall-normal component, $w^{\prime }$ does not appear to be well correlated with $u^{\prime }$ . However, it can be observed that close to each position where the vertical velocity vanishes, the spanwise velocity has a local maximum or a local minimum. This is a feature of the flow field generated by a streamwise vortex, as can be easily checked.

Figure 11. (a) Low- and high-speed streaks in the range $12<x<13$ ; darker areas denote low-speed streaks: see figure 9 for the contour map legend. (b) Trend of the velocity fluctuations along the spanwise direction at $x=12.5$ and $y=0.16$ . $R_{\unicode[STIX]{x1D6FF}}=1100$ , $\unicode[STIX]{x1D6FD}=0.75$ , $t=\unicode[STIX]{x03C0}/8$ .

Studies concerning steady turbulent boundary layers and channel flows have shown that the spacing $\unicode[STIX]{x1D6E5}^{+}$ between low-speed streaks ranges from 80 to 120 wall units (Pope Reference Pope2000). For a sinusoidally oscillating flow Costamagna et al. (Reference Costamagna, Vittori and Blondeaux2003) reported $\unicode[STIX]{x1D6E5}^{+}=127$ at $R_{\unicode[STIX]{x1D6FF}}=800$ while Salon et al. (Reference Salon, Armenio and Crise2007) reported $\unicode[STIX]{x1D6E5}^{+}=110$ at $R_{\unicode[STIX]{x1D6FF}}=1790$ . Of course, because of the flow unsteadiness, the spacing between low-speed streaks varies during the cycle. In this study $\unicode[STIX]{x1D6E5}^{+}$ has been determined by computing the spanwise autocorrelation function of the fluctuating streamwise velocity component $u^{\prime }$ . An estimate of the low-speed streak spacing is given by two times the separation at which the correlation attains the minimum. In figure 12 the streaks spacing expressed in wall units $(\unicode[STIX]{x1D6E5}^{+}=\unicode[STIX]{x0394}\sqrt{\unicode[STIX]{x1D70F}_{0}}R_{\unicode[STIX]{x1D6FF}})$ , computed by phase averaging over all the available cycles, is shown when low-speed streaks are detected, even though they are broken into short segments, not spanning the entire fluid domain. As shown in figure 9, no low-speed streaks are observed in a neighbourhood of the flow reversal, therefore in figure 12 no data are reported at $t=0$ and $t=\unicode[STIX]{x03C0}$ . Figure 12(a) shows that at $t=\unicode[STIX]{x03C0}/8$ the spacing increases both for $\unicode[STIX]{x1D6FD}=0.60$ and $\unicode[STIX]{x1D6FD}=0.75$ . Such an increase is due to the merging of neighbouring streaks. An analogous increase of $\unicode[STIX]{x1D6E5}^{+}$ is not observed for $\unicode[STIX]{x1D6FD}=0.85$ because of the large acceleration that characterizes this case, which anticipates the break up of the streaks. After breaking, the spacing takes values within $100-110$ up to $t\approx 3\unicode[STIX]{x03C0}/4$ . For $R_{\unicode[STIX]{x1D6FF}}=1414$ (figure 12 b) no increase of the streak spacing is observed at the beginning of the positive half-cycle as a larger Reynolds number shifts back the onset of breaking. Even for $R_{\unicode[STIX]{x1D6FF}}=1414$ the spacing is within 100–110 just after the streaks break. The spacing however increases towards the end of the decelerating phase because of the coalescence of neighbouring streaks, thus assuming a value of approximately 125 at $t=3\unicode[STIX]{x03C0}/4$ . This phenomenon is more important for conditions characterized by a slower deceleration such as $\unicode[STIX]{x1D6FD}=0.75,0.85$ .

Figure 12. Mean spacing between two low-speed streaks during the period. (a)  $R_{\unicode[STIX]{x1D6FF}}=1100$ ; (b) $R_{\unicode[STIX]{x1D6FF}}=1414$ .

In the negative half-cycle the streaks can be already observed at $t=9\unicode[STIX]{x03C0}/8$ with a large spacing in between that persists for a longer time in comparison to the positive half-cycle. This result is due to the low acceleration of the negative half-cycle which leaves more time to the streaks before reaching the instability condition which causes them to break. After breaking has occurred the spacing decreases up to a value of approximately 110. In summary, when the streaks are broken into short segments the spacing is in the range of 100–110 wall units, while in the other phases the spacing ranges between 100–200 wall units.

3.3 Turbulent kinetic energy and Reynolds stress

The contour plot of the turbulent kinetic energy $k=(\langle u^{^{\prime }2}\rangle +\langle v^{^{\prime }2}\rangle +\langle w^{^{\prime }2}\rangle )/2$ in the $(t,y)$ plane is shown in figure 13. It can be noted that during the accelerating phase of the positive half-cycle the turbulent kinetic energy becomes large near the wall first and then away from it. This is a consequence of the closeness to the wall of the peak of production of turbulent kinetic energy, as illustrated in figure 14. Indeed, large values of $k$ appear away from the wall only after convection and diffusion have acted for a sufficiently long time. The intense production of $k$ that occurs during the accelerating phase, generates both large time derivatives and large $y$ gradients near the wall. During the decelerating phase, because of the decrease of production, both the time derivative and the $y$ gradient become relatively small.

Figure 13. Turbulent kinetic energy $k$ as a function of the time $t$ and of the $y$ coordinate. Contour interval: $10^{-3}$ . (a,c,e) $R_{\unicode[STIX]{x1D6FF}}=1100$ ; (b,d,f) $R_{\unicode[STIX]{x1D6FF}}=1414$ ; (a,b) $\unicode[STIX]{x1D6FD}=0.60$ ; (c,d) $\unicode[STIX]{x1D6FD}=0.75$ ; (e,f) $\unicode[STIX]{x1D6FD}=0.85$ .

The peaks of the turbulent kinetic energy fall in the range $y=0.2{-}0.3$ , and precede the peaks of the free stream velocity, both during the positive and the negative half-cycle. In the positive half-cycle the larger is $\unicode[STIX]{x1D6FD}$ the sooner the maximum of $k$ occurs, while the opposite is observed in the second half-cycle. An increase of $\unicode[STIX]{x1D6FD}$ brings the maximum of the negative half-cycle closer to that of the next positive half-cycle. For $R_{\unicode[STIX]{x1D6FF}}=1100$ the peaks of the turbulent kinetic energy range from 0.0134 to 0.0157, while for $R_{\unicode[STIX]{x1D6FF}}=1414$ they range from 0.0094 to 0.0139. The ratio of the first peak to the second one increases both with acceleration skewness and Reynolds number and ranges from 1.04 to 1.48. Even though the maximum of $k$ occurs in the positive half-cycle, during the negative half-cycle the turbulent kinetic energy spreads at larger distances from the wall than in the positive half-cycle. This result is explained by the longer duration of the acceleration phase of the negative half-cycle, which leaves more time for the turbulent kinetic energy to spread away from the wall.

A quantitative assessment of the mechanisms affecting the turbulent kinetic energy can be provided by means of the turbulent kinetic energy budget, which in dimensionless form reads:

where $s_{j,\ell }=(\unicode[STIX]{x2202}u_{j}^{\prime }/\unicode[STIX]{x2202}x_{\ell }+\unicode[STIX]{x2202}u_{\ell }^{\prime }/\unicode[STIX]{x2202}x_{j})/2$ . The terms contributing to the right-hand side of (3.3) from left to right are: production of turbulent kinetic energy, turbulent convection, pressure transport, viscous diffusion and dissipation $\unicode[STIX]{x1D716}$ . In figure 14, where such terms are shown at 16 phases during the cycle for $R_{\unicode[STIX]{x1D6FF}}=1100$ and $\unicode[STIX]{x1D6FD}=0.75$ , it can be observed that all these contributions are important in a thin layer near the wall, approximately $1.5\unicode[STIX]{x1D6FF}^{\ast }$ thick. The pressure transport term is always much smaller than the other terms, while viscous diffusion and turbulent convection are both important and of the same order of magnitude, except near the wall, where the turbulent convection vanishes. In order to provide a comprehensive picture of the time variation during the period of production and dissipation of the turbulent kinetic energy, (3.3) have been integrated between $y=0$ and $y=L_{y}$ and the results have been reported in figure 15(a,b) as a function of time. Here $I_{p}$ and $I_{\unicode[STIX]{x1D716}}$ denote the integral of turbulence production and dissipation respectively, which are the only two terms, belonging to the right-hand side of (3.3), that do not vanish after integration. Figure 15(a) shows that turbulence production and dissipation rapidly increase at the beginning of the cycle, first for large $\unicode[STIX]{x1D6FD}$ . During the decelerating phase $I_{p}$ and $I_{\unicode[STIX]{x1D716}}$ decay and both take low values in the central part of the period. In the negative half-cycle they increase again but, in contrast to the positive one, such an increase is more rapid for lower values of $\unicode[STIX]{x1D6FD}$ . The increase of the Reynolds number expands the fraction of the cycle in which production and dissipation are both important, as it can be observed by comparing figure 15(a) with figure 15(b).

Figure 14. Terms of the turbulent kinetic energy budget at 16 phases during the wave cycle for $\unicode[STIX]{x1D6FD}=0.75$ and $R_{\unicode[STIX]{x1D6FF}}=1100$ .

Figure 15. (a,c) $R_{\unicode[STIX]{x1D6FF}}=1100$ , (b,d) $R_{\unicode[STIX]{x1D6FF}}=1414$ . (a,b) Production and dissipation of turbulent kinetic energy integrated along the $y$ direction. Here $I_{p}$ and $I_{\unicode[STIX]{x1D716}}$ denote the integral of production and dissipation respectively. (c,d) The lines denote $I_{p}+I_{\unicode[STIX]{x1D716}}$ while the markers denote the time derivative of the turbulent kinetic energy integrated along the $y$ direction $\unicode[STIX]{x2202}I_{k}/\unicode[STIX]{x2202}t$ .

In figure 15(c,d) the quantity $I_{\unicode[STIX]{x1D716}}+I_{p}$ is shown along with the trend of the time derivative of the turbulent kinetic energy integrated along the $y$ direction $\unicode[STIX]{x2202}I_{k}/\unicode[STIX]{x2202}t$ . Because of the form of (3.3) and of the boundary conditions, $\unicode[STIX]{x2202}I_{k}/\unicode[STIX]{x2202}t$ must be equal to $I_{\unicode[STIX]{x1D716}}+I_{p}$ . Indeed, figure 15(c,d) shows that these quantities are in a fairly good agreement. The small discrepancies arise from unavoidable artificial dissipative effects due to the numerical scheme adopted for the time advancement of the governing equations. Figure 15(c) shows that during the accelerating phase of the positive half-cycle $\unicode[STIX]{x2202}I_{k}/\unicode[STIX]{x2202}t$ increases rapidly, attains a maximum and then decreases up to a minimum. The rapid increase begins when the low-speed streaks still have an elongated shape (see figure 9), while the maximum occurs near the onset of streaks breaking. Prior to this event, the flow quantities, such as the wall shear stress, are scarcely affected by the turbulence. After the streaks begin to break, small-scale vortices characterized by a large dissipation of turbulent kinetic energy are generated, therefore the time derivative of the spatially integrated turbulent kinetic energy rapidly decreases.

In the negative half-cycle the trend of $\unicode[STIX]{x2202}I_{k}/\unicode[STIX]{x2202}t$ shows differences with respect to the first half-cycle due to the acceleration skewness. For example, the first peak is smaller than that in the positive half-cycle, but later larger values than those of the positive half-cycle occur. Figure 15(d) shows that for $R_{\unicode[STIX]{x1D6FF}}=1414$ the value of $\unicode[STIX]{x2202}I_{k}/\unicode[STIX]{x2202}t$ related to the first maximum and those at later phases in the same half-cycle are closer to each other than those at $R_{\unicode[STIX]{x1D6FF}}=1100$ . The anticipation of streaks breaking during the cycle due to the larger Reynolds number explains such a result as $\unicode[STIX]{x2202}I_{k}/\unicode[STIX]{x2202}t$ has a shorter time to grow.

Insights about the distribution of the turbulent kinetic energy among the different spatial scales and for the three velocity components are given in figure 16 where the one-dimensional longitudinal and transverse energy spectra along the $x$ direction at $y=1$ are shown for $R_{\unicode[STIX]{x1D6FF}}=1414$ and $\unicode[STIX]{x1D6FD}=0.75$ . The equations of the Kolmogorov spectra that characterize the inertial sub-range of a statistically stationary homogeneous isotropic turbulence (Pope Reference Pope2000) are also shown in figure 16. Such equations have the following dimensionless form:

where $c\approx 0.49$ and $c_{1}\approx 0.65$ . The dissipation $\unicode[STIX]{x1D716}$ is evaluated at the specified $y$ position. Figure 16 shows that the spectra match the Kolmogorov spectra within a short range of wavenumbers. The shortness of such a range is justified by the relatively low Reynolds number. For $E_{11}$ the agreement with (3.4) is fairly good for all the phases shown in figure 16, while for $E_{22}$ and $E_{33}$ an agreement occurs at the phases of well-developed turbulence ( $\unicode[STIX]{x03C0}/4,3\unicode[STIX]{x03C0}/8$ ), but even at $t=0$ for $E_{22}$ and during the deceleration for $E_{33}$ . It is interesting to note that according to (3.4), where an inertial sub-range is present, the spectra $E_{22}$ and $E_{33}$ are equal to each other, therefore significant differences between them may arise only for wavenumber that are quite far away from the inertial sub-range. Figure 16 also shows that at low wavenumbers the longitudinal spectrum $E_{11}$ is larger than the $E_{22}$ and $E_{33}$ spectra. For large $k_{x}$ $E_{11}$ is still the largest, apart from $t=0$ and $t=7\unicode[STIX]{x03C0}/8$ where $E_{33}$ is slightly larger than $E_{11}$ . A comparison between $E_{33}$ and $E_{22}$ shows that for small $k_{x}$ the former is larger than the latter, while at large wavenumbers they are close to each other. Therefore, it emerges that $E_{22}$ takes smaller values than the other spectra, especially close to the phases of flow reversal. The spectra do not change significantly in the interval $\unicode[STIX]{x03C0}/4-3\unicode[STIX]{x03C0}/8$ , thus suggesting that despite the flow unsteadiness, the contribution to the turbulent intensity from the different spatial scales is rather constant in time during this range of phases.

Figure 16. Energy spectra along the streamwise direction at six phases during the positive half-cycle for $R_{\unicode[STIX]{x1D6FF}}=1414$ and $\unicode[STIX]{x1D6FD}=0.75$ . The straight lines are described by (3.4).

In figure 17 the space–time contour plot of the Reynolds stress $\langle u^{\prime }v^{\prime }\rangle$ during the flow cycle is shown. Here comments similar to those already made for the turbulent kinetic energy could be repeated. The peaks of the Reynolds stress fall in the range $y=0.44{-}0.70$ and precede those of the free stream velocity both during the positive and the negative half-cycle. The negative peaks range from $-$ 0.0028 to $-$ 0.0024 while the positive ones range from 0.0022 to 0.0024. In the negative half-cycle large absolute values of the Reynolds stress are found at distances further away from the wall compared to those in the positive half-cycle, similarly to the turbulent kinetic energy shown in figure 13.

Figure 17. Reynolds stress $\langle u^{\prime }v^{\prime }\rangle$ as a function of the time $t$ and of the $y$ coordinate. Black lines: positive values, grey lines: negative values. Contour interval: $2\times 10^{-4}$ . (a,c,e)  $R_{\unicode[STIX]{x1D6FF}}=1100$ ; (b,d,f) $R_{\unicode[STIX]{x1D6FF}}=1414$ ; (a,b) $\unicode[STIX]{x1D6FD}=0.60$ ; (c,d) $\unicode[STIX]{x1D6FD}=0.75$ ; (e,f) $\unicode[STIX]{x1D6FD}=0.85$ .

One of the important consequences of acceleration skewness is that the period average of the Reynolds stress $\overline{\langle u^{\prime }v^{\prime }\rangle }$ does not vanish. In figure 18(a–c), where the period-averaged Reynolds stress profiles are shown, it can be observed that the magnitude of $\overline{\langle u^{\prime }v^{\prime }\rangle }$ is large near the wall and rapidly decreases as $y$ increases. In the presence of a non-vanishing $\overline{\langle u^{\prime }v^{\prime }\rangle }$ , the balance of the period-averaged momentum is satisfied thanks to the viscous stress generated by a steady streaming in the boundary layer (Scandura Reference Scandura2007). The trend of the steady streaming is shown in figure 18(d–f). It can be observed that for the smallest value of acceleration skewness the difference between the two Reynolds numbers is not significant, while it becomes more apparent when the acceleration skewness is larger. Interestingly, for large $\unicode[STIX]{x1D6FD}$ an increase of the Reynolds number causes a decrease both of the period-averaged Reynolds stress and of the steady streaming. On the basis of this result one might infer that for very large Reynolds numbers the steady streaming vanishes, as in a laminar flow. However, to draw plausible conclusions in this regard, data at much larger Reynolds number than those here considered are necessary.

Figure 18. (a–c) Period average of the Reynolds stress $\overline{\langle u^{\prime }v^{\prime }\rangle }$ ; (d–f) period average of the ensemble-averaged streamwise velocity $\overline{\langle u\rangle }$ .

3.4 Wall shear stress statistics

In § 3.1 the wall shear stress has been analysed mainly focusing on its ensemble average. Here the analysis is extended with the aim of gaining insights about its spatial distribution and statistics. In figure 19 the spatial distribution of the streamwise and spanwise components of the wall shear stress fluctuations for $R_{\unicode[STIX]{x1D6FF}}=1100$ and $\unicode[STIX]{x1D6FD}=0.75$ are shown at four phases during the positive half-cycle. The fluctuations along the streamwise direction $\unicode[STIX]{x1D70F}_{x}^{\prime }$ are computed by subtracting from the wall shear stress $\unicode[STIX]{x1D70F}_{x}$ its spatial average, while the fluctuations in the spanwise direction are equal to the wall shear stress $\unicode[STIX]{x1D70F}_{z}$ , since the spatial average vanishes. Figure 19(a,b) show small areas characterized by large fluctuations, outside of which the wall shear stress takes values close to those in laminar flow. A comparison between figure 10(d–f) ( $0<z<12$ ) and figure 19(a–c) respectively, shows that within a turbulent spot the wall shear stress fluctuations are large. Moreover, comparing figure 19(c,g) with figure 9 for $\unicode[STIX]{x1D6FD}=0.75$ and $t=\unicode[STIX]{x03C0}/4$ , it can be observed that large fluctuations of the wall shear stress are detected where breaking of the low-speed streaks has occurred. Therefore, the areas characterized by large fluctuations shown in figure 19 are the imprint of the turbulent spot and they also correspond to the areas where the low-speed streaks are broken into short segments.

Figure 19. Wall shear stress fluctuations for $R_{\unicode[STIX]{x1D6FF}}=1100$ and $\unicode[STIX]{x1D6FD}=0.75$ . (a,e) $t=3\unicode[STIX]{x03C0}/16$ ; (b,f) $t=13\unicode[STIX]{x03C0}/64$ ; (c,g) $t=\unicode[STIX]{x03C0}/4$ ; (d,h) $t=9\unicode[STIX]{x03C0}/32$ . Streamwise wall shear stress fluctuations $\unicode[STIX]{x1D70F}_{x}^{\prime }:$ (a) (min, max) $=$ ( $-$ 0.0012, 0.0042); (b) ( $-$ 0.0015, 0.0097); (c) ( $-$ 0.0027, 0.0156); (d) ( $-$ 0.0033, 0.0125). Spanwise wall shear stress $\unicode[STIX]{x1D70F}_{z}$ : (e) (min, max) $=$ ( $-$ 0.002, 0.0019); (f) ( $-$ 0.0045, 0.0068); (g) ( $-$ 0.0058, 0.0041); (h) ( $-$ 0.0074, 0.0074). Black lines: positive values; grey lines: negative values. Contour interval: $5\times 10^{-4}$ . The fluid oscillates along the $x$ direction.

A comparison between figure 3(a) for $\unicode[STIX]{x1D6FD}=0.75$ and figure 10 shows that the rapid increase of the ensemble-averaged wall shear stress begins with the appearance of the turbulent spots ( $t\approx 3\unicode[STIX]{x03C0}/16$ ) and ends when they spread in the whole fluid domain, which occurs at approximately $t=9\unicode[STIX]{x03C0}/32$ . Since within the turbulent spots the fluctuations of the wall shear stress are large, in the presence of a wall made up by loose material, the mobilization of clusters of sediments has a high probability to occur. However, defining the condition of incipient motion of such sediments would be particularly difficult because the turbulent spots are randomly distributed in the space and they do not appear exactly at the same phase in every cycle. Concerning the spatial distribution of the spanwise wall shear stress, from figure 19 it seems to emerge that $\unicode[STIX]{x1D70F}_{z}$ takes significant values mainly where $\unicode[STIX]{x1D70F}_{x}^{\prime }$ is positive. This issue will be addressed in the following by means of a more objective approach.

The statistical parameters here considered to characterize the wall shear stress are: the relative intensity $R_{x}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}/\unicode[STIX]{x1D70F}_{0}$ , where $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}=\langle \unicode[STIX]{x1D70F}_{x}^{^{\prime }2}\rangle ^{1/2}$ is the r.m.s. of the wall shear stress fluctuations, the skewness coefficient $S_{x}=\langle \unicode[STIX]{x1D70F}_{x}^{^{\prime }3}\rangle /\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}^{3}$ and the flatness coefficient $F_{x}=\langle \unicode[STIX]{x1D70F}_{x}^{^{\prime }4}\rangle /\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}^{4}$ along with the relative intensity $R_{z}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{z}}/\unicode[STIX]{x1D70F}_{0}$ and the flatness coefficient $F_{z}=\langle \unicode[STIX]{x1D70F}_{z}^{^{\prime }4}\rangle /\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{z}}^{4}$ of the spanwise component of the wall shear stress. In the following the time development of these statistics is examined in the light of the low-speed streaks dynamics previously discussed and of knowledge about these statistics for steady flows. Figure 20 shows the trend of the relative intensity $R_{x}$ during a period. Both in the positive and in the negative half-cycle a time interval exists in which $|R_{x}|$ shows weak variations around a value close to 0.4. Such a time interval begins when the breakage of the low-speed streaks has occurred in all the fluid domain and ends near the flow reversal, when $R_{x}$ diverges because of the vanishing of the wall shear stress. Comparing figure 20(a) with figure 20(b), it can be noted that when the Reynolds number increases the relative intensity is fairly constant in a longer time interval, while the value of the constant does not appreciably change. In the negative half-cycle (figure 20 a), just after the flow reversal, another range of phases characterized by weak variations of $|R_{x}|$ around a value of 0.15 is observed. More specifically, for $\unicode[STIX]{x1D6FD}=0.85$ and $t$ in the range 3.3–4.2, $|R_{x}|$ varies within 0.12–0.16. Such a time interval corresponds to the phases in which the low-speed streaks are substantially straight and span all the computational domain, as shown in figure 9. Figure 20(b) shows that the width of this time interval decreases with Reynolds number as a larger portion of the cycle is occupied by well-developed turbulence, characterized by low-speed streaks that have already undergone the breaking process. A time interval with the previous characteristics also exists during the positive half-cycle, but it is very short because of the high acceleration, which quickly leads to the streaks instability and breaking. In the time that elapses between the two mentioned time intervals of rather constant value of $R_{x}$ , the relative intensity increases, reaches a maximum and then stabilizes onto a value of about 0.4. Values of $R_{x}$ close to 0.4 have also been reported by investigations on steady turbulent flows (Alfredsson & Johansson Reference Alfredsson and Johansson1988; Durst, Jovanovic & Sender Reference Durst, Jovanovic and Sender1995; Obi et al. Reference Obi, Inoue, Furukawa and Masuda1996). Therefore, it is expected that the turbulence exhibits characteristics similar to those of a steady flow when values of $R_{x}$ close to 0.4 are observed during a time interval.

Figure 20. Relative intensity $R_{x}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}/\unicode[STIX]{x1D70F}_{0}$ of the wall shear stress along the $x$ direction during the wave period. (a) $R_{\unicode[STIX]{x1D6FF}}=1100$ , (b) $R_{\unicode[STIX]{x1D6FF}}=1414$ .

Figure 21. Relative intensity $R_{z}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{z}}/\unicode[STIX]{x1D70F}_{0}$ of the wall shear stress along the $z$ direction during the wave period. (a) $R_{\unicode[STIX]{x1D6FF}}=1100$ , (b) $R_{\unicode[STIX]{x1D6FF}}=1414$ .

The relative intensity $R_{z}=\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{z}}/\unicode[STIX]{x1D70F}_{0}$ of the wall shear stress in the spanwise direction is shown in figure 21. It can be observed that $R_{z}$ has the same order of magnitude as $R_{x}$ and a similar trend. When the streaks exhibit an elongated shape, $|R_{z}|$ takes values in the range 0.04–0.1, while after streaks breaking has occurred, it takes a value of approximately 0.25.

Figure 22 shows that the skewness $S_{x}$ of the wall shear stress in the streamwise direction is positive during the first half-cycle and negative during the second one. The rapid increase of $S_{x}$ occurs during the final stage of the elongated streaks development, while the peak of $S_{x}$ occurs at the onset of streaks breaking. In the positive half-cycle, when the breaking has occurred in all the fluid domain, $S_{x}$ takes a value of approximately 1.1, then it slowly decreases attaining values slightly smaller than 1 before undergoing a rapid decrease near the flow reversal. On the other hand, in the negative half-cycle $S_{x}$ almost immediately stabilizes to a value close to $-1$ after streaks breaking has occurred. An increase of $R_{\unicode[STIX]{x1D6FF}}$ causes a decrease of the peaks of $S_{x}$ as the phase at which the streak breaking starts shifts back during the cycle, thus $S_{x}$ has a shorter time to grow. Several studies (Alfredsson & Johansson Reference Alfredsson and Johansson1988; Durst et al. Reference Durst, Jovanovic and Sender1995; Obi et al. Reference Obi, Inoue, Furukawa and Masuda1996) reported that in steady turbulent flows the asymmetry of the wall shear stress takes values approximately equal to 1. The agreement of the present results with steady flow data confirms that there is a part of the cycle where the turbulence approximately evolves through a succession of quasi-steady states, as already suggested by the trend of $R_{x}$ .

Figure 22. Skewness coefficient $S_{x}$ of the wall shear stress along the $x$ direction during a wave period. (a) $R_{\unicode[STIX]{x1D6FF}}=1100$ , (b) $R_{\unicode[STIX]{x1D6FF}}=1414$ .

The trend of the flatness coefficient $F_{x}$ is shown in figure 23. It is worth pointing out that the oscillations shown by $F_{x}$ are due to the strong effect on this statistic of large values of the wall shear stress fluctuations. In order to smooth out such oscillations, the ensemble average should be computed on a very large number of flow cycles, at huge computational costs. Therefore, for the flatness coefficient even trends contaminated by spikes have been accepted. As already observed for $S_{x}$ , the rapid increase of $F_{x}$ begins when the streaks still retain an elongated shape in the streamwise direction, while the peak of $F_{x}$ occurs at the onset of the streak breaking. After breaking has occurred in the whole fluid domain, $F_{x}$ takes a values of 5.4. However, afterwards $F_{x}$ slowly decreases and takes a value of approximately 4.7 at $t=\unicode[STIX]{x03C0}/2$ . An increase of the Reynolds number causes a decrease of the peaks in $F_{x}$ for the same reason already explained regarding the skewness. As shown in figure 24, the trend of the flatness coefficient $F_{z}$ is qualitatively similar to $F_{x}$ but $F_{z}$ is larger than $F_{x}$ especially when elongated streaks are present. During these phases $F_{z}$ can take values up to 40 in the positive half-cycle and up to 80 in the negative one. In figure 24 the larger values of $F_{z}$ have been left outside from the range shown along the $y$ -axis in order to focus more closely on the phases of well-developed turbulence, where $F_{z}$ takes values of approximately 6.7. It can be also observed that when the Reynolds number increases the time intervals during which $F_{z}$ is fairly constant increases. This is a common feature of all the statistics that have been examined here.

Figure 23. Flatness coefficient $F_{x}$ of the wall shear stress along the $x$ direction during a wave period. (a) $R_{\unicode[STIX]{x1D6FF}}=1100$ , (b) $R_{\unicode[STIX]{x1D6FF}}=1414$ .

Figure 24. Flatness coefficient $F_{z}$ of the wall shear stress along the $z$ direction during a wave period. (a) $R_{\unicode[STIX]{x1D6FF}}=1100$ , (b) $R_{\unicode[STIX]{x1D6FF}}=1414$ .

Figure 25. Probability density functions (p.d.f.) of the wall shear stress fluctuations along the $x$ direction, $\unicode[STIX]{x1D70F}_{x}^{\prime }/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}$ , at 16 phases during the cycle for $R_{\unicode[STIX]{x1D6FF}}=1100$ and $\unicode[STIX]{x1D6FD}=0.75$ . Continuous lines show the numerical results while dashed lines show the log-normal probability density function that best fits to the numerical results.

The probability density function of the dimensionless fluctuating wall shear stress $\unicode[STIX]{x1D70F}_{x}^{\prime }/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}$ is shown in figure 25 for $\unicode[STIX]{x1D6FD}=0.75$ and $R_{\unicode[STIX]{x1D6FF}}=1100$ . During the positive half-cycle negative fluctuations have a larger probability of occurrence than positive ones, the opposite is observed during the negative half-cycle. This behaviour is congruent with the trend of $S_{x}$ during the cycle, shown in figure 22. For practical applications, such as those related to sediment transport predictions, it is useful to have an analytical expression describing the probability density of $\unicode[STIX]{x1D70F}_{x}^{\prime }/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}$ at every phase. In this respect the three parameter log-normal distribution fits rather well to the numerical results, as shown in figure 25. The log-normal distribution is given by the following expression:

where $\unicode[STIX]{x1D707}_{y}$ and $\unicode[STIX]{x1D70E}_{y}$ are the mean and the standard deviation of $\log (\unicode[STIX]{x1D70F}_{x}^{\prime }/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}-\unicode[STIX]{x1D70F}_{l})$ respectively and $\unicode[STIX]{x1D70F}_{l}$ is the location parameter. The parameters of (3.5) have been estimated by using the method outlined by Yevjevich (Reference Yevjevich1982). Equating the skewness of the probability distribution to that of the numerical data $S_{x}$ , the following expression for $\exp (\unicode[STIX]{x1D70E}_{y}^{2})$ is obtained:

Then the parameters $\unicode[STIX]{x1D707}_{y}$ and $\unicode[STIX]{x1D70F}_{l}$ are determined as follows:

where $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70F}_{x}^{\prime }/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}}$ and $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}^{\prime }/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}}$ are the mean and the standard deviation of $\unicode[STIX]{x1D70F}_{x}^{\prime }/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{x}}$ respectively. Figure 25 shows that the most important discrepancies between the numerical results and the log-normal distribution occur at $t=\unicode[STIX]{x03C0}/4$ , $t=11\unicode[STIX]{x03C0}/8$ and $t=3\unicode[STIX]{x03C0}/2$ , which fall in the time intervals in which elongated streaks are present. Some discrepancies are also observed near the flow reversal at $t=0$ and $t=7\unicode[STIX]{x03C0}/8$ .

The probability density function of $\unicode[STIX]{x1D70F}_{z}/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{z}}$ shown in figure 26, is symmetric with respect to $\unicode[STIX]{x1D70F}_{z}=0$ , since the probability of exceeding a value $\unicode[STIX]{x1D70F}_{z}>0$ is equal to the probability of not exceeding the value $-\unicode[STIX]{x1D70F}_{z}$ . In figure 24 the existence of time intervals in which the flatness of the wall shear stress is large was observed. This fact is reflected in the distribution of $\unicode[STIX]{x1D70F}_{z}/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{z}}$ , which shows sharp peaks at some phases. Such a result is due to the high probability of finding values of $\unicode[STIX]{x1D70F}_{z}$ in a small neighbourhood of $\unicode[STIX]{x1D70F}_{z}=0$ when breaking of low-speed streaks has not yet occurred over the entire fluid domain. The probability density function of $\unicode[STIX]{x1D70F}_{z}/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{z}}$ can be described with a fairly good approximation by means of (3.9),

which is a modification of the Pearson type VII distribution (Johnson, Kotz & Balakrishnan Reference Johnson, Kotz and Balakrishnan1995). In (3.9) the condition that the variance of $\unicode[STIX]{x1D70F}_{z}/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{z}}$ is equal to 1 is already imposed, $B$ is the beta function and $\unicode[STIX]{x1D6FE}$ is the parameter of the distribution, which must be larger than 3/2. Here, $\unicode[STIX]{x1D6FE}$ has been determined by means of the maximum likelihood method. Figure 26 shows that the numerical results are in a fairly good agreement with (3.9), except at the phases $\unicode[STIX]{x03C0}/4$ and $3\unicode[STIX]{x03C0}/2$ , in which the breaking of streaks is in progress and the probability density exhibits a sharp peak. It is worth highlighting that the flatness provided by (3.9) is finite only if $\unicode[STIX]{x1D6FE}$ is larger than 5/2 and that in figure 26 this condition is violated at $t=\unicode[STIX]{x03C0}/4$ , $t=11\unicode[STIX]{x03C0}/8$ and $t=3\unicode[STIX]{x03C0}/2$ . In order to improve the description of the numerical results at phases $\unicode[STIX]{x03C0}/4$ and $3\unicode[STIX]{x03C0}/2$ , a generalization of the normal distribution can be employed,

where $\unicode[STIX]{x1D6E4}$ is the gamma function and $\unicode[STIX]{x1D6FC}$ is the parameter of the distribution. Equation (3.10) is shown by markers in figure 26 at the phases $\unicode[STIX]{x03C0}/4$ and $3\unicode[STIX]{x03C0}/2$ . It can be observed that the agreement with the numerical data is improved compared to that of (3.9). However, the results at $t=13\unicode[STIX]{x03C0}/8$ show that the shape of the probability density may be rather complicated, indeed at this phase (3.9) provides a good description of the probability density except in a neighbourhood of $\unicode[STIX]{x1D70F}_{z}/\unicode[STIX]{x1D70E}_{z}=0$ , where the probability density shows a sharp peak and could be better described by (3.10).

Figure 26. Probability density functions (p.d.f.) of the wall shear stress $\unicode[STIX]{x1D70F}_{z}/\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D70F}_{z}}$ along the $z$ direction at 16 phases during the cycle for $R_{\unicode[STIX]{x1D6FF}}=1100$ and $\unicode[STIX]{x1D6FD}=0.75$ . Continuous lines show the numerical results while dashed lines show the Pearson distribution type VII (3.9) that best fits to the numerical results. The curves denoted by dots at the phases $\unicode[STIX]{x03C0}/4$ and $3\unicode[STIX]{x03C0}/2$ are described by (3.10).

Figure 27. Trend of the functions $\unicode[STIX]{x1D713}_{1}$ and $\unicode[STIX]{x1D713}_{2}$ (see (3.11)–(3.13)) at 16 phases during the cycle for $R_{\unicode[STIX]{x1D6FF}}=1100$ and $\unicode[STIX]{x1D6FD}=0.75$ ; continuous lines: $\unicode[STIX]{x1D713}_{1}$ ; dashed lines: $\unicode[STIX]{x1D713}_{2}$ .

Finally, we tackle the issue arisen in connection with figure 19, concerning the observation that large values of the spanwise wall shear stress seem to occur simultaneously with positive fluctuations of the streamwise wall shear stress. In order to address this issue, we write the probability density function as follows:

where the vertical bar means ‘conditioned to’. The functions $\unicode[STIX]{x1D713}_{1}$ and $\unicode[STIX]{x1D713}_{2}$ are the contributions to the probability density function of $\unicode[STIX]{x1D70F}_{z}$ of events characterized by $\unicode[STIX]{x1D70F}_{x}^{\prime }>0$ and $\unicode[STIX]{x1D70F}_{x}^{\prime }<0$ respectively. The function $\unicode[STIX]{x1D713}_{1}$ and $\unicode[STIX]{x1D713}_{2}$ are plotted in figure 27 in order to gain insights about the contributions of the two kind of events to the probability density of $\unicode[STIX]{x1D70F}_{z}$ . Figure 27 shows that, in the positive half-cycle, small values of the spanwise wall shear stress occur more frequently in concomitance with negative fluctuations of the streamwise wall shear stress, while large values occur more frequently simultaneously with positive fluctuations. Such behaviour is amplified when the breaking of streaks is in progress, as occurring at phases $\unicode[STIX]{x03C0}/4$ and $3\unicode[STIX]{x03C0}/2$ . Note that in the negative half-cycle the role of the functions $\unicode[STIX]{x1D713}_{1}$ and $\unicode[STIX]{x1D713}_{2}$ is inter-changed with respect to the positive half-cycle. Another result that emerges from figure 27 is that, in the positive half-cycle, the peakedness of the probability density function is mainly due to values of the wall shear stress occurring simultaneously with negative fluctuations of the streamwise wall shear stress, while the opposite is observed in the negative half-cycle.