Mean flow structure and velocity–bed shear stress maxima phase difference in smooth wall, transitionally turbulent oscillatory boundary layers: direct numerical simulations

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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 (Mieret 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 oscillationT). 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.