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Oscillatory boundary layer (OBL) flows over a smooth surface are studied using laser Doppler velocimetry in a large experimental oscillatory flow tunnel. The experiments cover a range of Reynolds numbers in the transitional regime (Reδ=254−1315). Motivated by inconsistencies in the literature, the focus is to shed light regarding the phase shift Δϕ...

## Contexts in source publication

**Context 1**

... owing to the fact that only instantaneous values are presented in such works, no solid conclusion can be reached regarding the ensemble-average bed friction behaviour and the phase difference of its maximum value with respect to the maximum free stream velocity. Once again, no analysis is presented explaining the presence of a phase lag in the data set, but instead a phase difference diagram showing phase lead values is included (Appendix, p. 203, figure 21) by the authors. The bed shear stress measurements of also include phase-lag observations for Re δ of 762. ...

**Context 2**

... numerical studies have also shown the presence of a phase lag at the intermittent turbulent regime (Spalart & Baldwin 1989;Vittori & Verzicco 1998;Costamagna et al. 2003;Bettencourt & Dias 2018). Figure 2 in the paper by Spalart & Baldwin (1989) shows that for Re δ = 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 Re δ = 800. This means that there is a threshold value for Re δ for which the phase difference between bed shear stress and free stream velocity maximum shifts to negative values. ...

## Citations

... Several OBL studies (e.g. Spalart & Baldwin 1987;Scandura 2007;Scandura et al. 2016; van der A, Scandura & O'Donoghue 2018;Fytanidis, García & Fischer 2021;Mier, Fytanidis & García 2021) considered the simplified case of flow over a smooth wall. For practical OBLs that occur in the coastal zone, the seabed is rough, composed of sediment particles. ...

The hydrodynamics of turbulent oscillatory flow over a gravel-based irregular rough wall is investigated using laser-Doppler anemometry measurements of velocities in a large oscillatory flow tunnel and direct numerical simulation (DNS) of the Navier–Stokes equations. The same periodic irregular roughness was used for both experiments and DNS. Four flow shapes are investigated: sinusoidal, skewed, asymmetric and combined skewed–asymmetric. The experiments were conducted for target Reynolds numbers (based on the Stokes length and standard deviation of free-stream velocity) of $R_{\delta,\sigma }=800$ and $R_{\delta,\sigma }=1549$ ; DNS was conducted for flows with target $R_{\delta,\sigma }=800$ . Boundary layer thickness, bottom phase lead and friction factor are in good agreement with previous studies. For the first time, evidence of Prandtl's secondary flows of the second kind in oscillatory flow is presented. Turbulence structure is visualised using isosurfaces of $\lambda _{2}$ (Jeong & Hussain J. Fluid Mech. , vol. 285, 1995, pp. 69–94), revealing densely packed structures that grow stronger and weaker in correspondence with the free-stream velocity. Reynolds and dispersive stresses peak just below the highest roughness crest, with dispersive stress vanishing a short distance above the roughness. Bursts of turbulence kinetic energy and wake kinetic energy are generated each flow half-cycle, with variable behaviour depending on flow shape. Non-Gaussian turbulence statistics are observed that originate near the wall, becoming increasingly non-Gaussian far from the wall. Probability density functions of turbulence statistics can be closely approximated by a fourth-order Gram–Charlier distribution at most phases and elevations, though when statistics deviate more strongly from Gaussian, streamwise and wall-normal (spanwise) statistics are better described by a Pearson type IV (VII) distribution.

... In Mier, Fytanidis & García (2021), the experimental observations of mean flow structure and bed-shear stress/free-stream velocity maximum phase difference were presented for the case of intermittently turbulent oscillatory boundary-layer (OBL) flows over smooth walls. A revision of the bed-shear stress/free-stream velocity maxima phase difference diagram was proposed and a threshold value of Re δ = 763 was identified as a critical Re δ value for which phase lag starts being observed (Re δ = U o δ/ν, where δ = [2ν/ω] 1/2 is the Stokes length, U o is the maximum free-stream velocity of the oscillation, ν is the kinematic viscosity and ω = 2π/T is the angular velocity based on the period of the oscillation T). ...

... This new diagram explains inconsistencies in the literature regarding the instance when the maximum of the bed-shear stress was predicted with respect to the instance of free-stream velocity maximum. In the present work, flows in the same regime as those of Mier et al. (2021) will be examined in an effort to analyse their characteristics. These results are of relevance for environmental fluid mechanics applications, and coastal engineering and morphodynamics (Sleath 1984;Fredsøe & Deigaard 1992;Nielsen 1992;Garcia 2008;Sumer 2014). ...

... OBL flows can be categorized into different regimes (Akhavan, Kamm & Shapiro 1991a;Pedocchi, Cantero & García 2011;Ozdemir, Hsu & Balachandar 2014), namely: (i) the laminar regime (Re δ < Re δ cr1 ), for which analytical solutions exist for the velocity and shear stress profiles (Batchelor 1967); (ii) the disturbed laminar regime (Re δ cr1 < Re δ < Re δ cr2 ), in which small perturbations are superimposed on the laminar profiles without altering the mean characteristics of the flow such as the mean velocity or shear stress profiles (Carstensen, Sumer & Fredsøe 2010); (iii) the intermittently turbulent regime (Re δ cr2 < Re δ < Re δ cr3 ), for which the flow tends to remain in a quasi-laminar state for part of the acceleration phase until turbulent bursts are observed later during the period (starting at the beginning of the deceleration phase and moving closer to the end of the acceleration phase as Re δ increases), altering both the mean flow velocity profiles and the bed-shear stress signature of the flow (Merkli & Thomann 1975;Hino et al. 1983;Akhavan et al. 1991a;Akhavan, Kamm & Shapiro 1991b); and (iv) the fully turbulent regime (Re δ > Re δ cr3 ) in which high turbulence levels are observed during the whole cycle of the oscillation and the logarithmic layer is valid for most of the time during the oscillation cycle, excluding a period close to the flow reversal (Jensen, Sumer & Fredsøe 1989). Different flow regimes have been identified to alter significantly the temporal variation of mean flow characteristics (Hino, Sawamoto & Takasu 1976;Jensen et al. 1989;Akhavan et al. 1991a;Mier et al. 2021) and bed-shear stress Mier et al. 2021) over the period. Even in the early works of Kajiura (1964), Kamphuis (1975) and Sarpkaya (1993), these researchers had recognized the effect of different flow regimes on the friction coefficient f w (defined as f w = 2τ max /2U 2 o , where τ max is the maximum of the ensemble-average bed-shear stress). ...

Direct numerical simulations of oscillatory boundary-layer flows in the transitional regime were performed to explain discrepancies in the literature regarding the phase difference ${\rm \Delta} \phi$ between the bed-shear stress and free-stream velocity maxima. Recent experimental observations in smooth bed oscillatory boundary-layer (OBL) flows, showed a significant change in the widely used ${\rm \Delta} \phi$ diagram (Mier et al. , J. Fluid Mech. , vol. 922, 2021, A29). However, the limitations of the point-wise measurement technique did not allow us to associate this finding with the turbulent kinetic energy budget and to detect the approach to a ‘near-equilibrium’ condition, defined in a narrow sense herein. Direct numerical simulation results suggest that a phase lag occurs as the result of a delayed and incomplete transition of OBL flows to a stage that mimics the fully turbulent regime. Data from the literature were also used to support the presence of the phase lag and propose a new ${\rm \Delta} \phi$ diagram. Simulations performed for ${\textit {Re}}_{\delta }=671$ confirmed the sensitivity in the development of self-sustained turbulence on the background disturbances ( $\textit{Re}_{\delta}=U_{o}\delta/\nu$ , where $\delta=[2\nu/\omega]^{1/2}$ is the Stokes' length, $U_{o}$ is the maximum free stream velocity of the oscillation, $\nu$ is the kinematic viscosity and $\omega=2{\rm \pi}/T$ is the angular velocity based on the period of the oscillation T ). Variations of the mean velocity slope and intersect values for oscillatory flows are also explained in terms of the proximity to near-equilibrium conditions. Relaminarization and transition effects can significantly delay the development of OBL flows, resulting in an incomplete transition. The shape and defect factors are examined as diagnostic parameters for conditions that allow the formation of a logarithmic profile with the universal von Kármán constant and intersect. These findings are of relevance for environmental fluid mechanics and coastal morphodynamics/engineering applications.