3.1.1. Flow regimes

Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991a) and Ramaprian & Tu (Reference Ramaprian and Tu1983) used dimensional analysis and examined the similarity laws of oscillatory and pulsatile pipe flows, respectively. They considered that the OBL flows can be categorized into four regimes based on three length scales: a geometrical length scale based on the diameter of the pipe $R$, an inertia length scale $\delta _{t}= u_{*} / \omega$ and a viscous length scale $\delta _{v}= \nu / u_{*}$. It is worth noting that the Stokes length scales with the geometric mean of inertia and viscous length scales ($\delta \sim \sqrt {\delta _{t}\delta _{v}}$). Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991a) showed the dimensional necessity for a logarithmic layer to exist when two or more of the scales $R$, $\delta _{t}$ and $\delta _{v}$ are widely separated.

Based on the above scales, four different cases of oscillatory pipe flows are defined (Akhavan et al. Reference Akhavan, Kamm and Shapiro1991a): (a) Case I, the pipe diameter-limited, ‘quasi-steady’ turbulent behaviour for which $\delta _{t}\gg R \gg \delta _{v}$ (i.e. $u_{*}/(\omega R)\gg 1$, $Ru*/\nu \gg 1$), where the flow behaves in a quasi-steady way and a universal logarithmic law is valid; (b) Case II, which can in a way be considered as a special version of Case I for which $\delta _{t}\sim R \gg \delta _{v}$ (i.e. $u_{*}/(\omega R)\sim 1$, $u_{*} R/\nu \gg 1$), for which the flow obeys a modified version of the log-law where the universal slope expressed by von Kármán constant $\kappa$ may be constant ($\kappa =0.41$). However, the value of constant $A$ varies over time ($A(\omega t)=f(u_{*}/(R \omega ))$); (c) Case III, for which $R\gg \delta _{t} \gg \delta _{v}$ (i.e. $u_{*}/(\omega R)$, $u_{*}^{2}/(\omega \nu ) \gg 1$) and a logarithmic law is valid for $y<\delta _{t}$. However, in the outer layer, where $y/\delta _{t} \rightarrow \infty$ (i.e. $\delta _t=u_{*}/\omega \rightarrow 0$), the flow behaves in an ‘inviscid way’ similar to the case when $u_{*}\rightarrow 0$ (assuming that $\omega$ is finite). The mean velocity and turbulent moments profiles depend only on $R$ and $\omega$ values; (d) Case IV, which again can be considered to be a special version of Case III, for which $R \gg \delta _{t} \sim \delta _{v}$ (i.e. $u_{*}/(\omega R)$, $u_{*}^{2}/(\omega \nu ) \sim 1$) and a logarithmic profile is once again valid with $A_{s}$ varying over the cycle. Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991a) presented results of pipe flow of case II. Because coastal/wave flow conditions are of interest, flows in the current study belong to the non-diameter-limited cases III and IV but for a closed channel. Considering half the height of the channel (or the hydraulic radius of the channel) as equivalent to $R$, $R \gg u_{*}/\omega$ (or $u_{*}/(\omega R) \gg 1$ except from the shear stress reversal when $u_{*}$ is zero) for all the flows considered in the present study.

The structure of the OBLs was examined by Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) for a wide range of $ {\textit {Re}}_{\delta }$ ($ {\textit {Re}}_{\delta }$ between 257 and 3464). Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) noticed a distinct difference in the boundary layer structure for $Re_{\delta }$ of 762 (expressed in the original work as $Re_{w}=2.9\times 10^{5}$). This flow exhibited an intermittent turbulent behaviour for which the logarithmic distribution, $u^{+}=(1/\kappa ) \ln y^{+} +5.1$, was valid after $\omega t=6{\rm \pi} /9$ ($120^{\circ }$). An explanation for this different behaviour, given by the authors, indicated that the flow experiences transitional conditions for most of its period. However, no detailed explanation was given about the effect of $ {\textit {Re}}_{\delta }$ on the flow structure and consequently its effect on the bed shear stress especially at the transition from the laminar to transitional and to turbulent flow regime. Hino et al. (Reference Hino, Sawamoto and Takasu1976) studied an OBL for $ {\textit {Re}}_{\delta }$ of 876 and $R/\delta$ of 12.8; however once again, the effect of $Re_{\delta }$ variation was not clearly shown as only results from a single flow case were presented. Recently, Kaptein et al. (Reference Kaptein, Duran-Matute, Roman, Armenio and Clercx2019) used large-eddy simulation to examine the effect of the $h/\delta$ ratio (where $h$ is the height of their domain representing the water depth on oscillatory flows over a flat plate) on the phase difference between free stream velocity and bed shear stress maxima. Their results showed that for $h/\delta \geq 40$, velocity, turbulent characteristic and bed shear stress results converged to those for $h/\delta \rightarrow \infty$. In the present study $R/\delta$ is of the order of 250, which was consider large enough to represent the coastal boundary layer conditions for which $R/\delta \rightarrow \infty$.

3.1.2. Laminar flow

To test the accuracy of our measurements, the lowest $ {\textit {Re}}_{\delta }$ case was examined (experiment 1 with $ {\textit {Re}}_{\delta }=254$) and it was compared with an analytical solution. The velocity profile for the laminar regime can be calculated using the following analytical solution:

(3.1)\begin{equation} u(y,\omega t)=U_{o}(\sin(\omega t) - \textrm{e}^{{-}y/\delta}\sin(\omega t - y / \delta)) \end{equation}

by differentiating (3.1) and using the definition of viscous shear stress ($\tau = \rho \nu \partial u / \partial y$) we can estimate the shear stress variation as $\tau (y, \omega t)=\sqrt {2} \rho ({U_{o}^{2}}/{ {\textit {Re}}_{\delta }})\,\textrm {e}^{-y/\delta }\sin (\omega t - y / \delta + {\rm \pi}/4)$ and the wall shear stress $\tau _{b}$ can easily be calculated for $y=0$ as

(3.2)\begin{equation} \frac{\tau_{b}}{\rho}=\sqrt{2} \frac{U_{o}^{2}}{ {\textit{Re}}_{\delta}}\sin(\omega t + {\rm \pi}/4) \end{equation}

In figures 3(a) and 3(b), the analytical profiles for various time instances are plotted for the acceleration and deceleration phases, respectively, together with the experimental observations. The comparison between the analytical and experimental values agrees well. In addition, to evaluate the symmetry of the imposed oscillation from the pistons of the experimental facility, a comparison between the positive and negative parts of the cycle was conducted. Such comparison of these profiles is shown in figure 3(c), in which the measurement of the negative part of the period is multiplied by $-$1.0. No significant bias or skewness towards the positive or negative direction was observed in our measurements. Finally, figure 3(d) shows a comparison with the bed shear stress measurements, estimated as $\tau _{b} = \rho \nu \partial u/\partial y |_{b}$. Once again, the experimental results agree well with the analytical solution above.

Figure 3. Comparison of measurements against analytical solution for laminar regime (Test 1, $ {\textit {Re}}_{\delta }=254$): (a) streamwise velocity profiles during acceleration, $\circ$ measurements, (—–, black) analytical solution; (b) streamwise velocity profiles during deceleration; (c) comparison between positive and negative parts of the period, $\circ$ measurements in the positive part, $\times$ measurement in the negative part multiplied by $-$1.0, (—–, black) analytical solution; (d) measurements of bed shear stress $\tau _{b}$ and free stream velocity $U_{\infty}$, $\circ$ measurements of bed shear stress, ($\bullet$, grey) measurements of free stream velocity, (—–, black) analytical solution for bed shear stress $\tau _{b}$, – – – $U_{\infty}(t)=U_{o}sin(\omega t)$.

3.1.3. Transitional flow

In their work, Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) examined the flow structure for a flow with $ {\textit {Re}}_{\delta }= 876$. They presented data for the mean flow and turbulence characteristics for this Reynolds number but owing to the fact that only a single flow was analysed, the change of the mean flow characteristics as $ {\textit {Re}}_{\delta }$ increased and the flow went through a transition remains unknown. In figure 4, the ensemble average velocity profiles for three characteristic instances of the period (${\rm \pi} /4$, ${\rm \pi} /2$ and $3{\rm \pi} /4$) are shown for all the examined flows. In figure 4(a–c), the velocity profiles are presented in wall units (where $y^{+}=u_{*}y/\nu$, $u^{+}=\bar {u}/u_{*}$ and $u_{*}$ is the shear velocity $u_{*}=\sqrt (\overline {\tau _{b}}/\rho )$). The orange dashed lines show the fit of a logarithmic profile similar to the ‘universal log-law’ for turbulent equilibrium boundary layers. Figures 4(d–f) and 4(g–i) show the velocity defect normalized using the free stream velocity $U_{\infty}$ and shear velocity, respectively. The arrows show the general trends of the velocity profiles. Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) have shown that for high enough $ {\textit {Re}}_{\delta }$ values the velocity profiles should approach the universal logarithmic-law for a smooth wall:

(3.3)\begin{equation} U^{+}=\frac{1}{\kappa}\ln \left(y^{+}\right)+ A_{s} \end{equation}

with $\kappa \approx 0.41$ and $A_{s}\approx 5.1$. For equilibrium boundary layers, (3.3) is valid only for the part of the velocity close to the wall, while far from the wall additional adjusting parameters need to be used to describe the velocity profile, e.g. law of the wake (Krug, Philip & Marusic Reference Krug, Philip and Marusic2017; Jimenez Reference Jimenez2018). Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991a) showed that for $ {\textit {Re}}_{\delta }$ in the transitional regime (when $u_{*}/\omega \nu \sim 1.$) (3.3) is modified to $U^{+}=({1}/{\kappa })\ln (y^{+})+ A_{s}(\omega t)$, $A_{s}$ changes for different phases of the period. Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) also showed that $A_{s}$ varies for a transitional flow ($ {\textit {Re}}_{\delta }=873$). In figures 4(b), 4( f) and 4(i), and to an extent in figures 4(a) and 4(h), it can be observed that the mean profile in the transitional flows and especially for $ {\textit {Re}}_{\delta } = 763$ (experiment 5) deviate significantly from both the logarithmic profiles, which are observed for higher $ {\textit {Re}}_{\delta }$ cases, and the laminar profiles. However, as $ {\textit {Re}}_{\delta }$ increases there is a clear trend towards the equilibrium logarithmic law ($A_{s} \approx 5.1$ in (3.3)). The arrows in figure 4(c–i) show this transition.

Figure 4. Reynolds number effect. Ensemble average velocity profiles in wall units for: (a) $\omega t={\rm \pi} /4$; (b) $\omega t={\rm \pi} /2$; (c) $\omega t=3{\rm \pi} /4$. Ensemble average velocity defect profile normalized with free stream velocity $U_{\infty }$ and $\delta$ for: (d) $\omega t={\rm \pi} /4$; (e) $\omega t={\rm \pi} /2$; ( f) $\omega t=3{\rm \pi} /4$. Ensemble average velocity defect profile normalized with shear velocity $u_{*}$ for:(h) $\omega t={\rm \pi} /4$; (i) $\omega t={\rm \pi} /2$; (g) $\omega t=3{\rm \pi} /4$. Dashed orange lines show logarithmic fit for the cases with $ {\textit {Re}}_{\delta } \geq 763$. The arrows show the increasing $ {\textit {Re}}_{\delta }$ path.

To evaluate the fit of the logarithmic profiles, the log-law diagnostic function $\varXi$ ($\varXi =y^{+}({\partial \bar {{u}}^{+}}/{\partial y^{y}})$) is plotted in figure 5 for three $ {\textit {Re}}_{\delta }$ (763, 937 and 1315) for $\omega t={\rm \pi} /2$ to $5{\rm \pi} /6$. The $\varXi$ function should approach $1/\kappa$ for zero-pressure gradient boundary layers in regions where the log-law occurs (Nagib, Chauhan & Monkewitz Reference Nagib, Chauhan and Monkewitz2007). In addition to the equilibrium value $1/\kappa$, the $1/\kappa (\omega t)$ values are also plotted for each profile. It can be seen that for $ {\textit {Re}}_{\delta }=763$ (experiment 5) the part of the profile where a logarithmic equation may fit is smaller compared with the higher $ {\textit {Re}}_{\delta }$ cases. For this flow, the slope of the logarithmic profile will be larger than 1/0.41. As $ {\textit {Re}}_{\delta }$ increases to 937 and 1315 we can observe that the log profile slope $1/\kappa (\omega t)$ starts to approach 1/0.41 for parts of the deceleration. Furthermore, the region where a logarithmic profile may fit increases in size. Finally, for $ {\textit {Re}}_{\delta }=1315$ the profiles seem to agree well with the $1/0.41$ slope, although the slope becomes smaller towards $\omega t = 5{\rm \pi} /6$. It is important to note that in our work the use of $\kappa$ (velocity profile's slope) and $A_{s}$ (velocity profile's intersect) for the parts of the flow that are not in equilibrium (e.g. the values for $\omega t<{\rm \pi} /2$) is merely to provide us with a diagnostic parameter for the development of a true logarithmic profile. This same approach has been used in the past specifically for the case of OBL flows by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) (figures 7 and 9 in their original work) and by Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991a) (figures 19 and 23 in their original work). The values of $\kappa$ and $A_{s}$ are obtained by fitting the logarithmic law in a region of approximately $30\le y^{+} \le 150$. The region where a logarithmic layer exists varies over time and for different $ {\textit {Re}}_{\delta }$ values. However, the region of the fit was chosen with the aims to maximize the region of the fitting but also to avoid the wake effects (Krug et al. Reference Krug, Philip and Marusic2017).

Figure 5. Log-law diagnostic function $\varXi$ during the deceleration phase for $ {\textit {Re}}_{\delta }$ values of $763$, $937$ and $1315$.

Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991a) argued that $A_{s}$ should approach an equilibrium value for oscillatory pipe flows when $u_{*}^2/\omega \nu \gg 1$ (and $u_{*}/\omega R \ll 1$). Their analysis did not include cases for $u_{*}^2/\omega \nu \gg 1$. Instead they referred to the works of Mizushina, Maruyama & Shiozaki (Reference Mizushina, MARUYAMA and Shiozaki1974) and Ramaprian & Tu (Reference Ramaprian and Tu1983) who examined conditions of $u_{*}^2/\omega R \approx 0.1$ and $u_{*}^2/\omega \nu \approx 100$. The present analysis extends significantly the ranges of Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991a), Mizushina et al. (Reference Mizushina, MARUYAMA and Shiozaki1974) and Ramaprian & Tu (Reference Ramaprian and Tu1983).

3.3.1. Bed shear stress and friction coefficient

Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) presented time series of the bed shear stress variation over an oscillation period for a wide range of flows. Also included were the experimental observations by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) and the direct numerical simulation results of Spalart & Baldwin (Reference Spalart, Baldwin and André1989). The purpose of the present work is to examine in more detail the behaviour of bed shear stress in the transitional regime (and especially for $550 \le {\textit {Re}}_{\delta } \le 1000$), for which only limited data are available in the literature, i.e. by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) ($ {\textit {Re}}_{\delta }=876$), Spalart & Baldwin (Reference Spalart, Baldwin and André1989) ($ {\textit {Re}}_{\delta }=800$ and $1000$) and Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) ($ {\textit {Re}}_{\delta }=762$). In this regime some inconsistencies have also been noticed in the literature regarding the phase when the maximum bed shear stress occurs with respect to the maximum free stream velocity (${\rm \Delta} \phi$ in figure 3) (see § 3.3.2). In the present study the bed shear stress is estimated using the following:

(3.6)\begin{equation} \frac{\overline{\tau}_{b}}{\rho}=\nu\frac{\partial \bar{u}}{\partial y} - \overline{{u}'{v}'} \end{equation}

The gradient of the ensemble-average velocity is typically estimated over the 3–4 nearest points (which typically are within a distance of less than 0.2 mm from the wall) to ensure accurate estimation of the gradient. Also, for all the examined flows the second term of (3.6) is nearly zero; this arises from the fact that the first points of measurement are usually inside the viscous sublayer. This is typically the case for both unidirectional (e.g. Nezu & Nakagawa Reference Nezu and Nakagawa1993) as well as oscillatory flows (e.g. see the DNS results of Spalart & Baldwin (Reference Spalart, Baldwin and André1989) or the experimental observations by Hino et al. Reference Hino, Kashiwayanagi, Nakayama and Hara1983).

Another way to estimate the bed shear stress is by using the integral of the momentum equation (Hino et al. Reference Hino, Kashiwayanagi, Nakayama and Hara1983; Jensen et al. Reference Jensen, Sumer and Fredsøe1989):

(3.7)\begin{equation} \left(\frac{\bar{{\tau}}_{b}}{\rho} \right)=\left(\frac{\bar{{\tau}}}{\rho} \right)_{wall}= \int_{0}^{D}\frac{\partial}{\partial t}\left(U_{\infty}(\omega t)-\bar{u}(y,\omega t)\right){\textrm{d}y} \end{equation}

Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) used half the height of the cross-section as distance $D$, while Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) used the boundary layer thickness $y_{max}$ (see appendix B in the paper by Jensen Reference Jensen1989). In the present work, the approach of Jensen (Reference Jensen1989) was adopted.

Throughout the present work, $\tau _{b}$ results were obtained using (3.6), because that method is better suited than (3.7) considering the type of measurements performed (point-wise LDV measurements close to the wall, even inside the viscous sublayer). Some discrepancies between the computed values using (3.6) and (3.7) are consistent with the observations by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) and Jensen (Reference Jensen1989) for OBL flows and by Coles (Reference Coles1956) for unidirectional boundary layers. The latter argued that the momentum integral equation may introduce large errors for flow under a pressure gradient, especially close to flow reversal. Although the main results of the present analysis do not seem to be sensitive to the choice between (3.6) and (3.7), (3.6) has been adopted for the rest of the analysis.

For comparison, the normalized bed shear stress ($\overline {\tau _{b}}/\overline {\tau }_{b_{max}}$) computed using (3.6) and (3.7) are shown in figure 8, where $\overline {\tau }_{b_{max}}$ is the maximum of the ensemble-average bed shear stress. In addition, the corresponding results by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) and Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) are plotted. These studies had examined flows with slightly different $ {\textit {Re}}_{\delta }$ compared with those in the present analysis. These $ {\textit {Re}}_{\delta }$ values are shown in figure 8 in grey. From the results, it becomes obvious that two peaks, one associated with the laminar regime ($\circ$) and one associated with the intermittent-turbulent/turbulent regime ($\bullet$), exist. Depending on the $ {\textit {Re}}_{\delta }$, one of the two peaks becomes larger. The absolute maximum is also shown in figure 8 by (${\bigcirc}$, grey). For low $ {\textit {Re}}_{\delta }$ values ($ {\textit {Re}}_{\delta } < 552$) only a single peak exists. The second peak which is related also to the transition to turbulence starts to occur for $ {\textit {Re}}_{\delta }=552$. This is consistent with the experimental observations by Fishler & Brodkey (Reference Fishler and Brodkey1991) and Hino et al. (Reference Hino, Sawamoto and Takasu1976) who measured significant turbulent bursts during the deceleration phase in flows of similar $ {\textit {Re}}_{\delta }$. For $ {\textit {Re}}_{\delta }$ of $671$ the second ‘turbulent’ peak increases but still remains small compared with the ‘laminar’ peak. It is at $ {\textit {Re}}_{\delta }=763$ when the ‘turbulent’ peak becomes larger than the ‘laminar’ peak. This behaviour of a gradually increasing second peak continues until $ {\textit {Re}}_{\delta }=1036$. For $ {\textit {Re}}_{\delta } > 1036$ the ‘laminar’ peak is absorbed by the strength of the ‘turbulent’ peak.

Figure 8. Normalized bed shear stress $|\overline {\tau }_{b}|/\overline {\tau }_{b_{max}}$ for various $ {\textit {Re}}_{\delta }$. The peaks associated with the laminar regime are shown with $\circ$ and the peaks associated with the ‘turbulent’/‘transitional’ regime are shown with $\bullet$. The maximum of the two peaks is shown with (${\bigcirc}$, grey). The $ {\textit {Re}}_{\delta }$ values for the data by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) and Jensen (Reference Jensen1989) are shown in grey text.

It is important also to comment on the time instance when the maximum shear stress occurs. The ‘turbulent’ peaks start to occur towards the middle of the deceleration phase. As the values of these peaks increase, the absolute maximum of the bed shear stress starts occurring earlier during the deceleration phase. In other words, the bed shear stress maximum ‘lags’ with respect to the free stream velocity maximum (which takes place always at $\omega t = {\rm \pi}/2$). Although this behaviour has been observed in the literature, no detailed analysis has ever been performed to examine the presence of this phase lag and how the phase difference changes in the transitional regime. This was the main motivation for the present study. The authors suggest that the ‘phase lead’ diagram of Jensen et al. (Reference Jensen, Sumer and Fredsøe1989), which is included in many classic textbooks on coastal engineering and coastal boundary layers (e.g. Fredsøe & Deigaard Reference Fredsøe and Deigaard1992), needs to be revised to take into consideration the presence of the phase lag. More about this point will be discussed in § 3.3.2.

The behaviour of the bed shear stress time series, presented in figure 8, is consistent with the published values of Reynolds number separating the different OBL flow regimes. In the introduction, the disturbed laminar regime was defined as a regime in which the flow behaves like in the laminar regime, but small perturbations are superimposed on the OBL flow. This kind of linear instability-related disturbances (Carstensen et al. Reference Carstensen, Sumer and Fredsøe2010) are not sufficiently strong to alter the mean velocity profiles. Figure 3 showed the excellent agreement between our measurements for $ {\textit {Re}}_{\delta }=254$ and the laminar solution. These linear instability-related features are extremely difficult to be captured using the applied pointwise measurement technique (i.e. LDV). However, it is worth noting that the second ‘turbulent peak’ of the bed shear stress starts to appear for $ {\textit {Re}}_{\delta }=552$, which is very close to the threshold value for the intermittently turbulent regime (Pedocchi et al. Reference Pedocchi, Cantero and García2011; Ozdemir et al. Reference Ozdemir, Hsu and Balachandar2014).

In addition to bed shear stress variation over a period, also of interest is the maximum bed shear stress and its variation as a function of $ {\textit {Re}}_{\delta }$. Numerous studies in the literature deal with the estimation of the maximum bed shear stress over the period, usually expressed in terms of the friction factor $f_{w}$ (e.g. Jonsson Reference Jonsson1966; Kamphuis Reference Kamphuis1975; Sarpkaya Reference Sarpkaya1993), where $f_{w}=2({\overline {\tau }_{b_{max}}}/{\rho })/U_{o}^{2}$. Effects of roughness height, which is usually expressed using the relative ratio $\alpha /k_{s}$ (e.g. Jonsson Reference Jonsson1966; Kamphuis Reference Kamphuis1975), and flow regime using a form of $Re_{*}=u_{*}k_{s}/\nu$ (e.g. Pedocchi & García Reference Pedocchi and García2009a) have also been examined (note that here $k_{s}$ is an effective Nikuradse roughness, usually estimated using a characteristic bed particle diameter, García Reference García2008).

In figure 9 the friction factor $f_{w}$ as a function of $ {\textit {Re}}_{\delta }$ is plotted together with data from previous studies in the literature. A second abscissa axis is added showing the values of $ {\textit {Re}}_{w}$ arising from the fact that some works have defined the Reynolds number using half of the oscillation amplitude ($ {\textit {Re}}_{w}= {\textit {Re}}_{\delta }^{2}/2$). The prediction of the laminar solution and the semi-empirical theoretical solution of Fredsøe (Reference Fredsøe1984) are also shown. In general, a good agreement is observed between the data from this study and the experimental and theoretical results in the literature. Data of Kamphuis (Reference Kamphuis1975) seem to underestimate the friction coefficient (by a factor of $\sim 20\,\%$) compared with the rest of the datasets. The observed $f_{w}$ results are reasonably close to the measurements of 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, Baldwin and André1989). For higher $ {\textit {Re}}_{\delta }$ values ($ {\textit {Re}}_{\delta } > 1123$) the results are close to the experimental observations of Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) and Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010) but start to deviate from the observations by Sarpkaya (Reference Sarpkaya1993).

Figure 9. Friction factor $f_{w}$ as a function of $ {\textit {Re}}_{\delta }$ and $ {\textit {Re}}_{w}$.

3.3.2. Phase difference

Hino et al. (Reference Hino, Sawamoto and Takasu1976) measured the near bed velocity time series and showed that the velocities experience significant fluctuations during the deceleration phase. These spikes increase in magnitude as $ {\textit {Re}}_{\delta }$ increases. The moment during the period when these spikes in the velocity signal start to appear also varies with $ {\textit {Re}}_{\delta }$, starting earlier for higher $ {\textit {Re}}_{\delta }$ and moving towards the end of the acceleration phase (see Hino et al. Reference Hino, Sawamoto and Takasu1976, pp. 200–201, figures 6, 7 and 8). These velocity fluctuations are associated with a peak in the phase-averaged bed shear stress that follows a similar peak in r.m.s. fluctuations. In fact, this behaviour was also shown in the ensemble-average wall shear stress measurements by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983, p. 373, figure 10). Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) measured the bed shear stress variation over the circle of an oscillation. From their analysis, the fluctuation of bed shear stress can be used to determine the inception of turbulence. Starting from the laminar regime and as $ {\textit {Re}}_{\delta }$ increases, bed shear stress fluctuations start appearing at the deceleration point near bed shear stress reversal. In the transitional regime, these fluctuations increase in magnitude and appear earlier during the period as flow Reynolds number increases. From figure 9 in the paper by Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) it is clear that actually the maximum bed shear stress occurs after the maximum velocity instance for the case of $ {\textit {Re}}_{\delta }=726$ (this value corresponds to $ {\textit {Re}}_{w}$ of $2.9\times 10^5$ based on the different Reynolds number $ {\textit {Re}}_{w}$ adopted by Jensen et al. Reference Jensen, Sumer and Fredsøe1989). However, it is worth pointing out that both Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) and Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) did not comment on the presence of a phase lag in their results. Instead, they reported only the laminar peak of the bed shear stress, as it is shown in figure 1. Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010) studied coherent structures development in oscillatory flows with gradually increasing oscillation amplitude but constant period. In their study, they conducted a comprehensive analysis of coherent structures by means of flow visualization while the effect of these structures on bed shear stress was examined quantitatively using bed shear stress measurements with a hot-film probe. Despite the fact that their measurements were mainly instantaneous, similar conclusions regarding the phase difference ${\rm \Delta} \phi$ can be drawn after careful inspection of their bed shear stress measurements. In figures 15 and 16 of their work, the instantaneous bed shear stress measurements are plotted. The range of the flows is for $ {\textit {Re}}_{\delta }$ values between 616 and 1288 for figure 15 (these values correspond to $ {\textit {Re}}_{w}$ between $1.9\times 10^5$ and $8.3 \times 10^5$, based on the different Reynolds number adopted by Carstensen et al. Reference Carstensen, Sumer and Fredsøe2010), and between 1549 and 3162 for figure 16. Close examination of these instantaneous data shows that the actual maximum bed shear stress is delayed by approximately 45$^{\circ}$ for $ {\textit {Re}}_{\delta }=616$. This phase lag between maximum bed shear stress and maximum velocity decreases as $ {\textit {Re}}_{\delta }$ increases (equalling 734, 812, 892 and 969). As the Reynolds number further increases, it becomes difficult to exactly evaluate the time instance when the maximum bed shear stress is reached; however, it can still be seen that the phase of the maximum bed shear stress shifts closer to the maximum velocity instance. The above observations also motivated the present experimental analysis.

Previous numerical studies have also shown the presence of a phase lag at the intermittent turbulent regime (Spalart & Baldwin Reference Spalart, Baldwin and André1989; Vittori & Verzicco Reference Vittori and Verzicco1998; Costamagna et al. Reference Costamagna, Vittori and Blondeaux2003; Bettencourt & Dias Reference Bettencourt and Dias2018). Figure 2 in the paper by Spalart & Baldwin (Reference Spalart, Baldwin and André1989) shows that for $ {\textit {Re}}_{\delta }=600$ there is a ‘phase lead’ of the bed shear stress with respect to free stream velocity while bed shear stress lags with respect to the free stream velocity for $ {\textit {Re}}_{\delta }=800$. This means that there is a threshold value for $ {\textit {Re}}_{\delta }$ for which the phase difference between bed shear stress and free stream velocity maximum shifts to negative values. In figure 3 from the paper by Spalart & Baldwin (Reference Spalart, Baldwin and André1989) it is also shown that the phase lag decreased for a higher $ {\textit {Re}}_{\delta }=1000$. Similar values of phase difference have been obtained using DNS by Vittori & Verzicco (Reference Vittori and Verzicco1998) (figure 19 in their paper for $ {\textit {Re}}_{\delta }=1000$), Costamagna et al. (Reference Costamagna, Vittori and Blondeaux2003) (figure 5 in their paper for $ {\textit {Re}}_{\delta }$ 740 and 1120), and one-dimensional modelling of oscillatory boundary layer flows by Hanjalić, Jakirlić & Hadžić (Reference Hanjalić, Jakirlić, Hadžić and Durst1995) and Cotton et al. (Reference Cotton, Craft, Guy and Launder2001).

The observations in the present study for the phase difference ${\rm \Delta} \phi$ between free stream and bed shear stress maxima are plotted together with other data in the literature in figure 10. It is worth noting that all the data from the literature that showed phase lag are plotted with the appropriate negative ${\rm \Delta} \phi$ values. The prediction of the analytical (${\rm \Delta} \phi = {{\rm \pi} }/{4}$) and the theoretical solution of Fredsøe (Reference Fredsøe1984) are also shown. The results found in the literature seem to agree reasonably well with the observations in this study. For flows in the laminar regime, bed shear stress maxima seem to lead the free stream velocity maxima by the standard ${\rm \pi} /4$ rads. As the $ {\textit {Re}}_{\delta }$ increases further and the flows approach the end of the ‘disturbed-laminar’ and the beginning of the ‘intermittent turbulent’ regime, this phase lead decreases (note that after $ {\textit {Re}}_{\delta }=550$ the second/‘turbulent’ peak of the bed shear stress is increasing). At a threshold value of $ {\textit {Re}}_{\delta }=763$ a phase lag, i.e. negative ${\rm \Delta} \phi$, is observed as the second ‘transition to turbulence’-related peak becomes larger. This peak happens earlier and earlier as the $ {\textit {Re}}_{\delta }$ value increases, until it turns to positive values again for $ {\textit {Re}}_{\delta }\sim 1000$. For $ {\textit {Re}}_{\delta } > 1450$ the phase difference seems to be predicted well using the theoretical solution of Fredsøe (Reference Fredsøe1984). In the authors opinion, this modified diagram is the main contribution from the work presented herein and it has important implications in the fields of coastal engineering, sediment transport and morphodynamics. Of relevance to the analysis of the phase lag is the second burst of sediment entrainment, which is commonly observed in oscillatory sheet flows (Ribberink et al. Reference Ribberink, Dohmen-Janssen, Hanes, McLean and Vincent2000, Reference Ribberink, van der Werf, O'Donoghue and Hassan2008; Nielsen, van der Wal & Gillan Reference Nielsen, van der Wal and Gillan2002). Similar sediment entrainment bursts during the deceleration phase have been observed in time-varying flows by Admiraal, García & Rodriguez (Reference Admiraal, García and Rodriguez2000). In the following section, turbulence parameters will be presented in an effort to elucidate the changes of the flow structure that are associated with the phase difference between bed shear stress and free stream velocity.

Figure 10. Phase difference ${\rm \Delta} \phi$ as a function of $ {\textit {Re}}_{\delta }$ and $ {\textit {Re}}_{w}$.

3.4.1. Experiment 5 – $ {\textit {Re}}_{\delta }=763$

The ensemble-average velocity profiles for every $\omega t={\rm \pi} /12$ normalized with the maximum velocity $U_{o}$ are plotted in figure 11. Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) examined a flow of similar $ {\textit {Re}}_{\delta }$ ($762$); however, they measured only bed shear stress values. Thus, the mean flow measurements were compared with the closest experimental data from the literature; those of Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) for $ {\textit {Re}}_{\delta }=876$. The laminar solution is also plotted for reference (note that experiment 5 is not in the laminar regime). The vertical coordinates $y$ are normalized using $\delta$. Data of Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) are reported in ${\rm \pi} /32$ intervals, which do not match exactly with the data presented here. Thus, the corresponding data that match exactly with the time instances of Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) are shown in grey. The measured profiles agree well with the laminar solution during the acceleration phase (for $\omega t \ge {\rm \pi}/6$). This can be explained as a result of flow laminarization owing to the severe favourable pressure gradient that the flow experiences during acceleration. It is also in agreement with previous observations by Merkli & Thomann (Reference Merkli and Thomann1975), Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983), Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991a), Akhavan et al. (Reference Akhavan, Kamm and Shapiro1991b) and Carstensen et al. (Reference Carstensen, Sumer and Fredsøe2010). The velocity profiles start deviating from the laminar solution after $\omega t = 2 {\rm \pi}/3$ when turbulence increases under adverse pressure gradient. Measurements by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) seem to agree well with our observations far from the wall, where $y>\delta$. However, close to the wall ($y/\delta <1$) the results deviate from one another. The results become closer towards the end of deceleration ($\omega t \ge 3{\rm \pi} /4$).

Figure 11. Ensemble-average velocity profiles $\bar {{u}}$ normalized using the maximum velocity $U_{o}$ for $ {\textit {Re}}_{\delta }=763$.

The ensemble-average velocity profiles are plotted using wall units in figure 12. For comparison, the results by Hino et al. (Reference Hino, Kashiwayanagi, Nakayama and Hara1983) ($ {\textit {Re}}_{\delta }=876$), Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) and Spalart & Baldwin (Reference Spalart, Baldwin and André1989) ($ {\textit {Re}}_{\delta }=1000$) are also shown. In addition, the laminar solution for $ {\textit {Re}}_{\delta }=763$ is plotted in wall units. Furthermore, the velocity profile by VanDriest (Reference VanDriest1956) is plotted:

(3.8)\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 $A_{v}=26$. Equation (3.8) agrees well with the equilibrium logarithmic law in the range of $y^{+} \ge 30$. The logarithmic fits are also plotted using dashed orange lines.

Figure 12. Ensemble-average velocity profiles $\overline {{u}+}$ and logarithmic fit for $ {\textit {Re}}_{\delta }=763$. For comparison the equilibrium logarithmic law by VanDriest (Reference VanDriest1956) (where $\kappa =0.41$ and $A_{v}=26$) is also plotted.

In addition to the velocity profiles, the measured normalized wall shear stress $\tau _{b}/\tau _{b_{max}}$, the shape factor $H$ and the $ {\textit {Re}}_{\theta }$ values are also shown for each $\omega t$. In this plot, the effect of velocity profile on the aforementioned parameters is shown. During the acceleration phase, the velocity profile agrees well with the laminar solution. Significant deviations between the measured velocity profiles and the laminar solution exist after $\omega t \ge 7{\rm \pi} /12$. At that time, an enhanced shear stress causes the $u^{+}$ values to decrease and start approaching the logarithmic law. At the same instance, $H$ starts approaching $1.4$ and $ {\textit {Re}}_{\theta }= 287$. It can be observed that the higher $ {\textit {Re}}_{\delta }$ flows of Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) and Spalart & Baldwin (Reference Spalart, Baldwin and André1989) approach the equilibrium logarithmic law earlier, towards the end of acceleration. Later, during the deceleration phase ($\omega t = 2 {\rm \pi}/3$ and $5{\rm \pi} /6$) the profiles agree well with the logarithmic law, until they start deviating again near the bed shear stress reversal ($\omega t = 12{\rm \pi} /12$).

The streamwise r.m.s. fluctuations are plotted in figure 13, normalized using the shear velocity $u_{*}$. For comparison, the measurements of Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) for a fully turbulent flow ($ {\textit {Re}}_{\delta }=3464$) are shown. In addition, some unidirectional zero pressure gradient boundary layer flow results from the DNS analysis by Spalart (Reference Spalart1988) and Schlatter & Örlü (Reference Schlatter and Örlü2010) are shown. Laminarization during the acceleration phase reduces significantly the $\sqrt {\overline {{u}'^2}}^{+}$ values and thus, as it may be expected, the values deviate significantly from the observations of fully turbulent flows by Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) and the numerical results for the unidirectional flows. After $\omega t= 2{\rm \pi} /3$ the results approach the profiles of Spalart (Reference Spalart1988) and Schlatter & Örlü (Reference Schlatter and Örlü2010) regardless of the fact that $ {\textit {Re}}_{\theta }(\omega t)$ is still 0.6 times smaller compared with the $ {\textit {Re}}_{\theta }=677$, which is the lowest value that is shown in figure 13. Once again, this phase corresponds to $H$ values close to 1.4. Later, close to the bed shear stress reversal, the $\sqrt {\overline {{u}'^2}}^{+}$ values start deviating from the turbulent unidirectional boundary layers profiles.

Figure 13. Ensemble-average streamwise r.m.s. profiles $\sqrt {\overline {{u}'^2}}^{+}$ for $ {\textit {Re}}_{\delta }=763$.

The vertical and spanwise r.m.s. fluctuations are plotted in figures 14 and 15. The non-dimensionalization remains the same (wall units) and the experimental data of fully turbulent OBL and the numerical results for unidirectional boundary layers are again used for comparison. The analysis results in similar conclusions; the turbulence statistics are reduced during the acceleration phase, when flow laminarization occurs, and approach the fully turbulent profiles during part of the deceleration phase ($7 {\rm \pi}/12 < \omega t \le 11{\rm \pi} /12$). It is worth noting here that the agreement with the equilibrium boundary layer behaviour occurs after the instance when the peak of the bed shear stress occurs. This peak seems to be associated with the transition to turbulence, because the turbulence statistics and the mean velocity profile approach those of fully developed turbulent flow after this ‘turbulent’ peak.

Figure 14. Ensemble-average vertical r.m.s. profiles $\sqrt {\overline {{v}'^2}}^{+}$ for $ {\textit {Re}}_{\delta }=763$ (the legend is the same as in figure 13).

Figure 15. Ensemble-average spanwise r.m.s. profiles $\sqrt {\overline {{w}'^2}}^{+}$ for $ {\textit {Re}}_{\delta }=763$ (the legend is the same as in figure 13).

3.4.2. Experiment 10 – $ {\textit {Re}}_{\delta }=1315$

The mean velocity profiles in wall units for experiment 10 are shown in figure 16. The logarithmic fit is plotted with an orange dashed line. For comparison, the laminar solution and the universal log-law for the fully turbulent flow (3.8) are also plotted. In addition, measurements by Jensen et al. (Reference Jensen, Sumer and Fredsøe1989) for $ {\textit {Re}}_{\delta }=3436$ are also plotted. Compared with experiment 5, experiment 10 exhibits a behaviour that mimics closer that of fully developed turbulent flow for a larger portion of the period. At the beginning of the acceleration phase, the profile again deviates from the universal log-law. The velocity profile approaches the log-law only towards the end of the acceleration phase. At that time, the shape factor $H$ approaches 1.4 and $ {\textit {Re}}_{\theta }$ shows values larger than 346. During the deceleration phase the velocity profiles agree with the log-law, although small variations of $\kappa$ and $A_{s}$ values do exist compared with the 0.41 and 5.1 values. Such variations are attributed to the adverse pressure gradient effect. The turbulent case of Jensen (Reference Jensen1989) shows less sensitivity to the favourable pressure gradient and matches the log-law over a larger portion of the acceleration phase (for $\omega t \ge {\rm \pi}/12$). The r.m.s. of the streamwise, vertical and spanwise fluctuations are plotted in figures 17, 18 and 19 in wall units. The unidirectional DNS data of Spalart (Reference Spalart1988) and Schlatter & Örlü (Reference Schlatter and Örlü2010) are again included for comparison together with the measurements by Jensen et al. (Reference Jensen1989).

Figure 16. Ensemble-average velocity profiles $\overline {{u}}^{+}$ and logarithmic fit for $ {\textit {Re}}_{\delta }=1315$.