The appearance of turbulence at the bottom of propagating surface waves

Direct numerical simulations of momentum and continuity equations are performed to investigate the transition process from the laminar regime to turbulence within the boundary layer generated at the bottom of a propagating surface wave characterized by an amplitude 𝑎 ∗ and a length 𝐿∗ such that the ratio 𝑎 ∗∕𝐿∗ assumes small but finite values and the velocity close to the bottom is the sum of an oscillatory component and a steady one. Perturbations of the laminar flow are found to appear for values of the Reynolds number 𝑅𝑒𝛿 = 𝑈 ∗ 0 𝛿 ∗ 𝜈 ∗ slightly smaller than the critical value which leads to the instability of the laminar regime in a Stokes boundary layer and falls around 100 (𝑈 ∗ 0 being the amplitude of the velocity oscillations close to the bottom, 𝛿 ∗ the Stokes viscous length and 𝜈 ∗ the kinematic viscosity of the fluid). Close to the critical conditions, perturbations of the Stokes flow appear when the velocity induced by the wave close to the bottom reverses from the onshore direction to the offshore direction but the flow tends to relaminarize during the other phases of the wave cycle. When the Reynolds number is increased of a relatively small amount, turbulence is generated also after the passage of the wave trough, when the velocity induced by the wave close to the bottom reverses from the offshore to the onshore direction. The obtained results suggest that, in the investigated range of the parameters, turbulence is present throughout the wave cycle only when the Reynolds number is larger than a value that depends on the ratio between the water depth and the length of the propagating wave but it falls between 500 and 700.