Turbulent dynamics of sinusoidal oscillatory flow over a wavy bottom

Abstract

A direct numerical simulation study is conducted to investigate sinusoidal oscillatory flow over a two-dimensional wavy wall. The height and wavelength of the bottom profile, and the period and amplitude of the free-stream oscillation, are selected to mimic a wave-driven boundary layer over vortex ripples on a sandy seabed. Two cases with different Reynolds numbers$(Re)$are considered, and the higher-$Re$case achieves a fully developed turbulent state with a wide separation between the energy-containing and dissipative scales. The oscillatory flow is characterized by coherent columnar vortices, which are the main transport agents of turbulent kinetic energy and enstrophy. Two classes of coherent vortices are observed: (i) a primary vortex formed at the lee side of the ripple by flow separation at the crest; (ii) a secondary vortex formed beneath the primary vortex by vortex-induced separation. When the free-stream velocity weakens, these vortices form a counter-rotating vortex dipole and eject themselves over the crest with their mutual induction. Turbulence production peaks twice in a half-cycle; during the formation of the primary vortex and during the ejection of the vortex dipole. The intensity of the former peak remains low in the lower-$Re$case, as the vortex dipole follows a higher altitude trajectory limiting its interactions with the bottom, and leaving minimal residual turbulence around the ripples for the subsequent half-cycle. Flow snapshots and spectral analysis reveal two dominant three-dimensional features: (i) an energetic vortex mode with a preferred spanwise wavelength close to the ripple wavelength; (ii) streamwise vortical structures in near-wall regions with a relatively shorter spanwise spacing influenced by viscous effects. The vortex mode becomes strong when the cores of the vortices are strained to an elliptical form while moving towards the crest. Following the detachment of the vortices from the ripple, the vortex mode in the higher-$Re$case breaks down the spanwise coherence of the columnar vortices and decomposes them into intermittent patches of turbulent vortex clusters. The distribution of wall shear stress over the ripple is also analysed in detail. The peak values are observed near the ripple crest around the ejection of the vortex dipole and the maximum free-stream velocity. In the former, both the vortex mode and streamwise vortices have strong footprints on the wall, yielding a bimodal wall-shear-stress spectrum with two distinctive peaks. In the second high-stress regime, decaying coherent vortices impose strong inhomogeneity on the wall shear stress as their wall-attached parts sweep the ripples. These spanwise variations in the wall shear provide insights into the instability of two-dimensional sand ripples.