Direct numerical simulations of transition and turbulence in smooth-walled Stokes boundary layer

Stokes boundary layer (SBL) is a time-periodic canonical flow that has several environmental, industrial, and physiological applications. Understanding the hydrodynamic instability and turbulence in SBL, therefore, will shed more light on the nature of such flows. Unlike its steady counterpart, the flow in a SBL varies both in space and time, which makes hydrodynamic instability and transition from laminar to turbulent state highly complicated. In this study, we utilized direct numerical simulations (DNS) to understand the characteristics of hydrodynamic instability, the transition from laminar to turbulent state, and the characteristics of intermittent turbulence in a smooth SBL for |$Re_\Delta$|$ReΔ$ in the range of 500–1000. Simulation results show that nonlinear growth plays a critical role on the instability at |$Re_\Delta = 500$|$ReΔ=500$ and 600. However, the nonlinear growth does not warrant sustainable transition to turbulence and the outcome is highly dependent on the amplitude and spatial distribution of the initial velocity disturbance in addition to |$Re_\Delta $|$ReΔ$⁠. Simulation results at |$Re_\Delta = 500$|$ReΔ=500$ confirm that instability and subsequent transitional flow will eventually decay. At |$Re_\Delta = 600$|$ReΔ=600$ nonlinear growth recurs at every modulation period but such transition does not evolve into fully developed turbulence at any time in the modulation cycle. At |$Re_\Delta = 700$|$ReΔ=700$⁠, the flow shows features of fully developed turbulence during some modulation periods and the transitional character of |$Re_\Delta = 600$|$ReΔ=600$ at the remaining. Therefore, we conclude that flow in the range of |$Re_\Delta = 600$|$ReΔ=600$–700 is to be classified as self-sustaining transitional flow. For higher Reynolds number the flow indeed exhibits features of fully developed boundary layer turbulence for a portion of the wave period, which is known as the intermittently turbulent regime in the literature.