3.1.1. Friction coefficient $f_{w}$

The friction coefficient $f_{w}$, defined as $f_{w}=2\tau _{max}/\rho U_{o}^{2}$ is a measure of the maximum ensemble-average bed-shear stress $\tau _{max}$ over the period. The estimation of the values of $f_{w}$ and the development of graphs for its prediction for various wave conditions have been the focus of the early works by Kajiura (Reference Kajiura1964), Yalin & Russell (Reference Yalin and Russell1966), Jonsson (Reference Jonsson1966), Riedel, Kamphuis & Brebner (Reference Riedel, Kamphuis and Brebner1973) and Kamphuis (Reference Kamphuis1975). These studies highlighted the importance of the flow regime on the friction coefficient (Kajiura Reference Kajiura1964; Jensen et al. Reference Jensen, Sumer and Fredsøe1989; Sarpkaya Reference Sarpkaya1993). Figure 4 summarizes the data from the literature combined with the results of the present analysis. Note that the $f_{w}$ data are plotted both with respect to ${\textit {Re}}_{\delta }$ and ${\textit {Re}}_{w}$, which is a different Reynolds number, defined using half of the oscillation amplitude (note the relationship between the two Reynolds numbers, ${\textit {Re}}_{w}={\textit {Re}}_{\delta }^{2}/2$). The analytical solution and the theoretical solution by Fredsøe (Reference Fredsøe1984) are also plotted. In figure 4, a plot using a linear axis is also included for the range between ${\textit {Re}}_{\delta }=200$ and ${\textit {Re}}_{\delta }=2000$. This plot shows the spread of the reported $f_{w}$ values in the transitional regime. Differences as high as 100 % are observed in some cases. These discrepancies may be relevant to the difficulties that experimentalists face when trying to measure the bed-shear stress in unsteady flows as well as the crisis that seem to take place near the critical Reynolds number. The DNS results of the present study are in good agreement with the previous experimental results by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983), Jensen et al. (Reference Jensen, Sumer and Fredsøe1989), Sarpkaya (Reference Sarpkaya1993) and Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010) and the DNS results of Spalart & Baldwin (Reference Spalart and Baldwin1989), as well as the experimental observations of Mier et al. (Reference Mier, Fytanidis and García2021). A detailed comparison between the DNS results and the experimental observations is included in table 2 of Appendix B. Data from Kamphuis (Reference Kamphuis1975) underestimate the friction by a factor of $20$%. In general, for the data in the literature a deviation from the laminar prediction starts being observed for ${\textit {Re}}_{\delta }>600$. For values ${\textit {Re}}_{\delta }\ge 1036$ the present data are in good agreement with the observations of Sarpkaya (Reference Sarpkaya1993) and Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010). The theoretical prediction by Fredsøe (Reference Fredsøe1984) seems to predict well the experimental results for ${\textit {Re}}_{\delta }\ge 3000$, which is reasonably close to the ${\textit {Re}}_{\delta _{cr3}}$ value of $3460$ proposed by Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) as the threshold value for the fully turbulent regime.

Figure 4.

Friction coefficient as a function of ${\textit {Re}}_{\delta }$ and ${\textit {Re}}_{w}$.

Table 2.

Friction factor $f_{w}$ and phase difference ${\rm \Delta} \phi$ (in degrees) computations: comparison between experimental observations and DNS results. Indices A, B and C correspond to the computations using the small, medium and large domain sizes, respectively.

The variation of shear stress over the period can be seen in figure 5. For ${\textit {Re}}_{\delta }=552$, it is shown that the bed-shear stress of the numerical results agree well with the prediction of the analytical solution of the laminar regime, $\tau _{b}/\rho U_{o}^{2}=\sqrt {2}/{\textit {Re}}_{\delta } \sin (\omega t + {\rm \pi}/4)$ (Batchelor Reference Batchelor1967). Indeed, a ${\rm \pi} /4$ radians phase lead ${\rm \Delta} \phi$ of the bed-shear stress maximum with respect to the free-stream velocity has been observed. For ${\textit {Re}}_{\delta }=671$, two peaks can be observed, one during the acceleration phase, which can be linked to the laminar flow regime, and one weaker peak during the deceleration phase. The first peak happens closer to $\omega t = {\rm \pi}/2$ compared with the ${\rm \Delta} \phi$ value of ${\textit {Re}}_{\delta }=552$ (${\rm \Delta} \phi =37^{\circ }$ or 0.65 radians). Mier et al. (Reference Mier, Fytanidis and García2021) associated the second peak with the transition to turbulence. As ${\textit {Re}}_{\delta }$ increases further, this second peak increases in magnitude and eventually the first, ‘laminar’ peak, vanishes due to the enhanced effect of the ‘turbulent’ peak for ${\textit {Re}}_{\delta } \ge 1036$. It is also important to note here that, for ${\textit {Re}}_{\delta }=763$, the bed-shear stress maximum happens during the deceleration phase, i.e. ‘lagged’ with respect to the free-stream velocity maximum. A similar behaviour has been observed for ${\textit {Re}}_{\delta }=819$ although the peak of the bed-shear stress seems to take place earlier during the deceleration phase compared with ${\textit {Re}}_{\delta }=763$. This is consistent with the experimental observations by Mier et al. (Reference Mier, Fytanidis and García2021). After a close inspection of the ensemble-averaged bed-shear stress measurement by Hino et al. (Reference Hino, Sawamoto and Takasu1976) it is easy to conclude that they had also observed negative phase differences for a flow of ${\textit {Re}}_{\delta }=876$ (see p. 373, figure 10 in their work). Similarly, observations have been made in the literature both experimentally (Jensen et al. Reference Jensen, Sumer and Fredsøe1989; Carstensen et al. Reference Carstensen, Sumer and Fredsøe2010) and numerically (Spalart & Baldwin Reference Spalart and Baldwin1989; Ozdemir et al. Reference Ozdemir, Hsu and Balachandar2014; Bettencourt & Dias Reference Bettencourt and Dias2018; Ebadi et al. Reference Ebadi, White, Pond and Dubief2019). A more extended presentation and summary of these works can be found in Mier et al. (Reference Mier, Fytanidis and García2021). However, all these studies have not stressed the need to revise the phase difference diagram that is included in the classic papers by Jensen et al. (Reference Jensen, Sumer and Fredsøe1989), Sarpkaya (Reference Sarpkaya1993) and Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010), among others.

Figure 5.

Normalized bed-shear stress variation for various ${\textit {Re}}_{\delta }$.

A revised version of the classic phase lead diagram (here plotted as phase difference) is plotted in figure 6. In addition to the data of the present study and the experimental measurements by Mier et al. (Reference Mier, Fytanidis and García2021), the measurements done by Hino et al. (Reference Hino, Sawamoto and Takasu1976), Jensen et al. (Reference Jensen, Sumer and Fredsøe1989), Sarpkaya (Reference Sarpkaya1993) and Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010) and the DNS results by Spalart & Baldwin (Reference Spalart and Baldwin1989), Ozdemir et al. (Reference Ozdemir, Hsu and Balachandar2014) and Bettencourt & Dias (Reference Bettencourt and Dias2018) are also plotted for comparison. Note that the phase difference ${\rm \Delta} \phi$ values that are plotted here (e.g. for the case of Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) and Jensen et al. (Reference Jensen, Sumer and Fredsøe1989)) are those as read after processing the bed-shear stress signals rather than the positive values reported earlier. From figure 6 it can be observed that there is a clear threshold value ${\textit {Re}}_{\delta }{\sim} 750$ after which the data of the current study and the literature show negative phase difference. Small differences and discrepancies regarding the absolute value of the ${\rm \Delta} \phi$ can be attributed to the challenges associated with the measurement of bed-shear stress under oscillatory conditions in general and the different temporal resolutions at which the bed-shear stresses were measured. Some dependencies of the numerical results on the height of the domain, which have been reported by Kaptein et al. (Reference Kaptein, Duran-Matute, Roman, Armenio and Clercx2019), may also explain some of the variations of the numerical results. Kaptein et al. (Reference Kaptein, Duran-Matute, Roman, Armenio and Clercx2019) used LES to examined the effect of $h/\delta$ ratio (where $h$ is the height of the domain). They concluded that, for $h/\delta \geq 40$, the velocity, the turbulence characteristics and bed-shear stress results converge to those for $h/\delta \rightarrow \infty$, which is the focus of the present study. The data presented here show no significant effect on the height of the domain. The final results, however, were extracted in a domain with $h/\delta =50$, which is above the height-limited conditions reported by Kaptein et al. (Reference Kaptein, Duran-Matute, Roman, Armenio and Clercx2019). Finally, the consistent presence of negative phase differences in the literature shows in a convincing way that phase lag does exist in the intermittently turbulent regime for $750\le {\textit {Re}}_{\delta } \le 1000$.

Figure 6.

Phase difference ${\rm \Delta} \phi$ diagram as a function of ${\textit {Re}}_{\delta }$ and ${\textit {Re}}_{w}$.

The DNS results and the ${\rm \Delta} \phi$ measurements by Mier et al. (Reference Mier, Fytanidis and García2021) are also summarized in table 2 (Appendix B). To ensure that the bed-shear stress signal will be captured in maximum detail, the bed-shear stress was computed while running the simulations at every time step, which had a significantly smaller size than $1^{\circ }$ and was determined in a way that the Courant–Friedrichs–Lewy number was less than 0.45 for all portions of the cycle. The results presented here are the ensemble-average results of these extracted values. Note that the symmetry between $\omega t$ and $\omega t + {\rm \pi}$ was also used to enhance the collected statistics.

3.1.2. Flow structures

In figure 7, time variation of some characteristic dimensionless Reynolds numbers are reported, which will allow for the comparison of the examined OBL flows with previous studies from the literature. In figure 7 a, the shear Reynolds number ${\textit {Re}}_{\delta _{*}}=u_{*}\delta /\nu$ as introduced by Sarpkaya (Reference Sarpkaya1993) is presented. In figures 7(b) and 7(c), the classic Reynolds numbers that are usually used in studies of developing boundary layers are plotted, namely ${\textit {Re}}_{\theta }=U_{\infty }\theta /\nu$, defined using the momentum thickness $\theta$ and the mean free-stream velocity $U_{\infty }$ (with $U_{\infty }(\omega t) = U_{o}\sin (\omega t)$), and ${\textit {Re}}_{\tau }=u_{*}y_{max}/\nu$ defined using the ensemble-average $u_{*}$ (which is also varying with $\omega t$) and the thickness of the boundary layer $y_{max}$ defined in agreement with the conventions by Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) and Mier et al. (Reference Mier, Fytanidis and García2021) as the height for which the shear stress becomes zero $\bar {{\tau }}=0$.

Figure 7.

Variation of (a) ${\textit {Re}}_{\delta *}$, (b) ${\textit {Re}}_{\theta }$ and (c) ${\textit {Re}}_{\tau }$ over the period.

The displacement thickness and the momentum thickness are computed as

(3.2)$$\begin{gather} \delta_{*}(\omega t)=\int_{0}^{y_{max}}\frac{U_{\infty}(\omega t)- \bar{u}(y,\omega t)}{U_{\infty}(\omega t)}{\textrm{d}y} \end{gather}$$

(3.3)$$\begin{gather}\theta(\omega t)=\int_{0}^{y_{max}} \frac{\bar{u}(y,\omega t)}{U_{\infty}(\omega t)} \left(\frac{U_{\infty}(\omega t)-\bar{u}(y,\omega t)}{U_{\infty}(\omega t)} \right) {\textrm{d}y}. \end{gather}$$

Note that, for the case of oscillatory flows, the definitions of (3.2) and (3.3) can lead to negative $\delta _{*}$ and $\theta$ values during the acceleration phase. This is the result of the characteristic near-bed overshoots of the velocity that can be observed in oscillatory flows. The parts of the flow when overshoots occur are excluded from the present analysis since there is no clear physical meaning in negative momentum and displacement thickness values.

From figure 7 it can be observed that ${\textit {Re}}_{\theta }$ and ${\textit {Re}}_{\tau }$ continue to grow during the deceleration phase, despite the fact that both $U_{\infty }$ and the shear velocity may decrease. This is explained later in figure 12(b), which shows that both $y_{max}$ and $\theta$ continue growing for a portion of the deceleration phase.

OBL flow structures have been examined in the past both experimentally (Sarpkaya Reference Sarpkaya1993; Carstensen et al. Reference Carstensen, Sumer and Fredsøe2010) and numerically (Verzicco & Vittori Reference Verzicco and Vittori1996; Costamagna et al. Reference Costamagna, Vittori and Blondeaux2003; Mazzuoli et al. Reference Mazzuoli, Vittori and Blondeaux2011; Ozdemir et al. Reference Ozdemir, Hsu and Balachandar2014; Xiong et al. Reference Xiong, Qi, Gao, Xu, Ren and Cheng2020). Analyses have been also extended to rough bed OBL flows by others (Carstensen, Sumer & Fredsøe Reference Carstensen, Sumer and Fredsøe2012; Ghodke & Apte Reference Ghodke and Apte2016; Mujal-Colilles et al. Reference Mujal-Colilles, Christensen, Bateman and Garcia2016; Mazzuoli & Vittori Reference Mazzuoli and Vittori2019). It is already established that turbulent spots (TS), which are highly energetic $\varLambda$-shaped flow structures, exist for OBL flows of the intermittently turbulent regime such as those examined in the present work. The presence of these structures marks the onset of turbulence (Carstensen et al. Reference Carstensen, Sumer and Fredsøe2010) and are known to alter the wall-shear stress signals (Vittori & Verzicco Reference Vittori and Verzicco1998; Carstensen et al. Reference Carstensen, Sumer and Fredsøe2010; Mazzuoli et al. Reference Mazzuoli, Vittori and Blondeaux2011). TS are also observed in unidirectional developing boundary layers (Cantwell, Coles & Dimotakis Reference Cantwell, Coles and Dimotakis1978; Perry, Lim & Teh Reference Perry, Lim and Teh1981; Park et al. Reference Park, Wallace, Wu and Moin2012; Wu et al. Reference Wu, Moin, Wallace, Skarda, Lozano-Durán and Hickey2017) and are usually associated with bypass transition to turbulence (Schlatter et al. Reference Schlatter, Brandt, De Lange and Henningson2008; Wu et al. Reference Wu, Moin, Wallace, Skarda, Lozano-Durán and Hickey2017).

In figure 8, the coherent structures visualized using the $\lambda _{2}$ criterion introduced by Jeong & Hussain (Reference Jeong and Hussain1995) are shown for ${\textit {Re}}_{\delta }=763$ and at $\omega t={\rm \pi} /2$, $7{\rm \pi} /12$ and $2{\rm \pi} /3$. The isosurfaces of $\lambda _{2}=-0.0025$ are plotted, coloured using the velocity magnitude $u/U_{o}$. In figure 8(a), the generation of hairpin packets that are randomly distributed in space is shown. These packets, generated towards the end of the acceleration phase, later grow and form what looks like the TS (figure 8b) described by Wu et al. (Reference Wu, Moin, Wallace, Skarda, Lozano-Durán and Hickey2017), until these TS merge later, during the deceleration phase (figure 8c). The fact that TS keep growing and merging for as long as $45^{\circ }$ (${\rm \pi} /4$ radians) during the deceleration phase is another sign that the OBL keeps developing during parts of the deceleration phase until near-bed flow reversal occurs. The final state of what looks like the final developed stage of the flow is shown in figure 9 for $\omega t = 3{\rm \pi} /2$, during which hairpin vortices have occupied most of the simulation domain. It is also instructional to notice the variation of bed-shear stress during this period (also plotted in figures 8 and 9). Indeed, as the flow structures start developing, the bed-shear stress continues to increase during the deceleration phase until $\omega t= 2{\rm \pi} /3$. Then, the bed-shear stress starts decreasing again under the effect of deceleration. This late transition of the flow to an increased energy state is the reason for the phase lag in the bed-shear stress signal that is shown in figures 5 and 6.

Figure 8.

Development of flow structures. Coherent structures visualized using $\lambda _{2}=-0.0025$ and coloured using the normalized velocity magnitude $u/U_{o}$ at (a) $\omega t = {\rm \pi}/2$ (b) $\omega t = 7{\rm \pi} /12$ and (c) $\omega t = 2{\rm \pi} /3$.

Figure 9.

Development of flow structures. Coherent structures visualized using $\lambda _{2}=-0.005$ and coloured with the normalized velocity magnitude $u/U_{o}$ at $\omega t = 3 {\rm \pi}/ 4$.

In agreement with the work of Wu et al. (Reference Wu, Moin, Wallace, Skarda, Lozano-Durán and Hickey2017), the TS are generated randomly in space and time. However, it is clear that, as ${\textit {Re}}_{\delta }$ increases, the TS are generated earlier during the period (Carstensen et al. Reference Carstensen, Sumer and Fredsøe2010). These OBL flow features have been examined in the past by Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010), who observed their presence for as low as ${\textit {Re}}_{\delta }=550$ (${\textit {Re}}_{w}=1.5\times 10^{5}$ in the original work). In the present analysis, TS started appearing for ${\textit {Re}}_{\delta }=671$, since ${\textit {Re}}_{\delta }=552$ behaves in a laminar fashion. Sarpkaya (Reference Sarpkaya1993) had identified a critical ${\textit {Re}}_{\delta *}$ value between 20 and 28 for which the high–low velocity streaks appear and disappear periodically. This threshold value of (${\textit {Re}}_{\delta *}{\sim} 24$) is plotted with an orange dotted line in figure 7(a). This threshold value is in good qualitative agreement with our observations, which show ${\textit {Re}}_{\delta *}>24$ for part of the period of the intermittently turbulent flows examined here. Park et al. (Reference Park, Wallace, Wu and Moin2012) also observed that TS may appear as ‘individual’ for ${\textit {Re}}_{\theta }{\sim} 300$ and as ‘merged’ for ${\textit {Re}}_{\theta }{\sim} 500$ (plotted in figure 7(b) using blue and orange dashed lines, respectively). For the flow presented here (${\textit {Re}}_{\delta }=763$), ${\textit {Re}}_{\theta }$ reaches a value of 436 for the state which is shown in figure 9. This value, although close enough to the ${\textit {Re}}_{\theta }{\sim} 500$ by Park et al. (Reference Park, Wallace, Wu and Moin2012), represents a state when the flow characteristics mimic those of fully turbulent flow. As shown in figure 10, this is the part of the flow when a velocity profile that resembles the characteristics of the logarithmic profile can be observed. Later, we will show that the actual slope and intersect of this velocity profile differ from those of the equilibrium BL. Figure 10 also includes the laminar solution and the logarithmic profile by VanDriest (Reference VanDriest1956)

(3.4)\begin{equation} \bar{{u}}^{+} =2\int_{0}^{y^{+}}\frac{{\textrm{d}y}^{+}}{1+[1+4\kappa^{2}y^{{+}2} (1-\exp({-}y^{+}/A_{v}))^{2}]^{{1}/{2}}}, \end{equation}

where $\kappa =0.41$ and $A_{v}=26$. This profile agrees well with (3.1) for $y^{+}\ge 30$. A logarithmic fit to the data is also plotted with orange dashed lines. The LDV data by Mier et al. (Reference Mier, Fytanidis and García2021) for the same ${\textit {Re}}_{\delta }$, data by Hino et al. (Reference Hino, Sawamoto and Takasu1976) for ${\textit {Re}}_{\delta }=876$ and Spalart & Baldwin (Reference Spalart and Baldwin1989) and Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) for ${\textit {Re}}_{\delta }=1000$ are plotted for comparison. It is shown that a logarithmic profile is observed for $\omega t =2{\rm \pi} /3$ and $3{\rm \pi} /4$. These are the instances during the period when $\tau _{b}$ and ${\textit {Re}}_{\theta }$ (also plotted in figure 10) are also increased. In addition, the shape factor $H$, defined as the ratio between displacement and momentum thickness ($H=\delta _{*}/\theta$) approaches a constant value. This value is slightly larger than the $H{\sim} 1.4$ that was observed by Mier et al. (Reference Mier, Fytanidis and García2021) as the converged value at the fully turbulent stage. As shown later, this happens due to an incomplete transition, in a narrow sense.

Figure 10.

Ensemble-average velocity profiles $\overline {{u}^{+}}$ and logarithmic fit for ${\textit {Re}}_{\delta }=763$ (case $763B$).

Figure 10 shows a general good agreement between the DNS results and the experimental observations, with the exception of results for $\omega t=5{\rm \pi} /6$, which is an instant really close to the near-bed flow reversal when $\tau _{b}=0$. Thus, these deviations which are magnified due to the low $u_{*}$ values are attributed to the challenges associated with the measurement of $\tau _{b}$ values that are close to zero. Also, the comparison with the results by Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) and Spalart & Baldwin (Reference Spalart and Baldwin1989) shows that a logarithmic profile occurs earlier for higher ${\textit {Re}}_{\delta }$ values. Finally, the figure shows some deviations between the results presented here and the observations by Hino et al. (Reference Hino, Sawamoto and Takasu1976). These deviations had been reported earlier by Jensen (Reference Jensen1989), who also observed that his velocity profiles differ from those by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) and highlighted the challenges of experimental works to estimate near-bed velocity profiles in OBL flows.

Figure 11 show the corresponding results for ${\textit {Re}}_{\delta }=1315$. Again, an excellent agreement between the numerical and experimental results is observed (with the same deviations at $\omega t {\sim} 5{\rm \pi} /6$ when $\tau _{b} \rightarrow 0$). As ${\textit {Re}}_{\delta }$ increases, the part of the circle for which a logarithmic layer is observed increases. In addition, ${\textit {Re}}_{\theta }$ values get larger earlier and reach higher maximum values. A comparison of the slope and intersect values of the logarithmic fit shows values reasonably close to those of a fully turbulent unidirectional ZPGBL.

Figure 11.

Ensemble-average velocity profiles $\overline {{u}^{+}}$ and logarithmic fit for ${\textit {Re}}_{\delta }=1315$ (case $1315B$).

Shape factor $H$ values also get close to the threshold value of $H{\sim} 1.4$ reported by Mier et al. (Reference Mier, Fytanidis and García2021) for these flows in the regions where the logarithmic self-similar profile exists. Another diagnostic parameter of interest is the defect shape factor $G$, defined as

(3.5)\begin{equation} G=\frac{\int_{0}^{y_{max}}(U_{\infty}^{+}-u^{+})^{2}{\textrm{d}y}}{\displaystyle \int_{0}^{y_{max}}(U_{\infty}^{+}-u^{+}){\textrm{d}y}}. \end{equation}

Note the relationship between $G$ and $H$, $G=(H-1)/(H\sqrt {c_{f}/2})$ where $c_{f}=2(u_{*}/U_{\infty })$ is the time varying skin-friction coefficient (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017). Mellor & Gibson (Reference Mellor and Gibson1966) showed that $G$ values become constant in near-equilibrium conditions for flows with constant Clauser parameter $\beta$. Here, despite the fact that $\beta$ values are not constant, we will evaluate the values of $H$ and $G$ that correspond to the part of the flows for which a self-similar velocity profile is observed.

Figure 12 summarizes the results of the examined flows for the ${\textit {Re}}_{\delta }$ in the intermittently turbulent regime. The temporal variations of the skin-friction coefficient $c_{f}$, normalized boundary-layer thickness $y_{max}/\delta$, displacement thickness $\delta _{*}/\delta$ and momentum thickness $\theta /\delta$ are plotted together with the values of $H$, $G$ and the fitted values of the slope $\kappa$ and the intersect $A_{s}$ of the velocity profiles. From figure 12, it becomes clear that, as ${\textit {Re}}_{\delta }$ increases, the maximum value of the boundary-layer thickness increases and the diagnostic shape factors $H$ and $G$ reach nearly constant values earlier during the period. At the same time, the values of the slope and intersect of the velocity reach values close to the well-accepted values of the von Kármán constant $\kappa =0.41$ and $A_{s}=5.1$ (Krug et al. Reference Krug, Philip and Marusic2017; Jiménez Reference Jiménez2018). It is important to stress here the log-law slope and intersect $\kappa$ and $A_{s}$ were used across the period solely as diagnostic parameters for defining the part of the period that is ‘near equilibrium’ (when $\kappa$ and $A_{s}$ get the standard values). This same approach has been used in the previous works by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) (see figures 7 and 9 in their original manuscript) and Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991b) (figures 19 and 23 in their original work), Ebadi et al. (Reference Ebadi, White, Pond and Dubief2019) (figure 3 in their original work) and Kaptein et al. (Reference Kaptein, Duran-Matute, Roman, Armenio and Clercx2020) (figures 7 and 8 in their original manuscript).

Figure 12 shows the path towards a transition to a state that mimics well the characteristics of the fully turbulent regime. This is illustrated for the examined flow range of ${\textit {Re}}_{\delta }$ $671$ and $1315$. Here, ${\textit {Re}}_{\delta }=671$ experiences a late transition that never actually gets to a stage that qualitatively looks similar to turbulent. The slope and intersect of the velocity profile deviate significantly from these of equilibrium for the whole period. Also, diagnostic parameters $H$ and $G$ do not get close to the threshold values of $H{\sim} 1.4$ observed by Mier et al. (Reference Mier, Fytanidis and García2021) and $G{\sim} 7.3 - 7.4$ observed in figure 12(d). For ${\textit {Re}}_{\delta }=763$, which is the threshold value for phase lag to exist (figure 6), the quantities start approaching the behaviour of the fully turbulent regime during the deceleration phase; a logarithmic velocity profile is observed (see also figure 10), $H$ and $G$ values get close to the equilibrium values and $\kappa$ and $A_{s}$ values advance towards the equilibrium values. In addition, the characteristic increase of the skin-friction coefficient $c_{f}$ takes place during the deceleration phase. As shown later, the absolute values of $\kappa$ and $A_{s}$ do not actually reach the 0.41 and 5.1 values but they start getting close. A similar behaviour is observed for ${\textit {Re}}_{\delta }=819$. The transition starts earlier, but again, $\kappa$ and $A_{s}$ values do not converge to the equilibrium values, although the near-equilibrium conditions allow the formation of a logarithmic profile during the deceleration phase. For ${\textit {Re}}_{\delta }=1036$, the behaviour starts changing, with the diagnostic parameters $H$ and $G$ reaching the equilibrium values earlier and the equilibrium $\kappa$ and $A_{s}$ values start appearing for parts of the circle. This becomes even more convincing for ${\textit {Re}}_{\delta }=1315$. However, as the flow experiences a severe adverse pressure gradient (i.e. $\beta \ge 1$) the flow starts deviating from these equilibrium conditions. This finding expands slightly the ${\textit {Re}}_{\delta } \ge 1750$ limit for OBL behaviour that mimics the fully turbulent regime observed by Sarpkaya (Reference Sarpkaya1993) and Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) and Salon et al. (Reference Salon, Armenio and Crise2007). It seems that it is possible to observe near-equilibrium conditions that match the universal velocity slope and intersect ($\kappa =0.41$ and $A_{s}=5.1$) for as low as ${\textit {Re}}_{\delta }{\sim} 1000$.

Figure 12.

Effect of ${\textit {Re}}_{\delta }$ on the variation over $\omega t$ of (a) $c_{f}$ (b) $y_{max}/\delta$, $\delta _{*}/\delta$ and $\theta /\delta$, (c) $H$, (d) $G$, (e) $\kappa$ and ( f) $A_{s}$.