Relationship between airflow at the critical height and momentum transfer to the traveling waves

Direct numerical simulations of the airflow over periodic traveling waves were performed to investigate the applicability of two mechanisms related to momentum transfer from the turbulent airflow to the traveling waves, i.e., the critical-layer mechanism and the nonseparated sheltering. We found three regimes classified by the wave age. In the first regime at low wave ages ( 2 ⩽ c ∕ u * ⩽ 4 ) , the critical height is located in the thin inner region where nonseparated sheltering works, and the perturbations of the shear stress in the inner region directly affect the streamwise phase of pressure on the wave surface. In the second regime at intermediate wave ages ( 4 < c ∕ u * ⩽ 12 ) , the critical layer is located in the thick inner region and has a strong link to momentum transfer across the interfacial wave. In particular, the phase of wave-induced pressure on the wave surface ∣ ϕ p ̃ ∣ bot is determined by the phase of wave-induced vertical velocity at the critical height ∣ ϕ w ̃ ∣ z = z c as ∣ ϕ p ̃ ∣ bot ≈ ∣ ϕ w ̃ ∣ z = z c + π ∕ 2 . In this regime, the growth rate of the interfacial wave is predicted well by the critical-layer mechanism. In the third regime at high wave ages ( c ∕ u * ⩾ 16 ) , the critical height is in the outer region far above the inner region, and the assumption of the critical-layer mechanism is better satisfied. However, momentum transfer across the interface in this regime is determined mainly by the local flow in the inner region and it is only weakly affected by the flow at the critical height. Then, the nonseparated sheltering works also at this high wave age.