A turbulent patch arising from a breaking internal wave

Abstract

Results of direct numerical simulations of the development of a breaking internal gravity wave are presented. The wave was forced by the imposition of an appropriate bottom boundary shape (a two-dimensional cosine hill) within a density-stratified domain having a uniform upstream velocity and density gradient. The focus is on turbulence generation and maintenance within the turbulent patch generated by the wave breaking. Pathlines, density contours, temporal and spatial spectra, and second moments of the velocity and density fluctuations and turbulent kinetic energy balance terms obtained from the data averaged over the span in the mixed zone are all used in the analysis of the flow. Typical Reynolds numbers, based on the vertical scale of the breaking region and the upstream velocity, were around 6000 and the fully resolved computations yielded sufficient resolution to capture the fine-scale transition processes as well as the subsequent fully developed turbulence. It is shown that globally, within the turbulent patch, there is an approximate balance in the production, dissipation and transport processes for turbulent kinetic energy, so that the patch remains quasi-steady over a significant time. Although it is far from being axially homogeneous, with turbulence generation occurring largely near the upstream bottom part of the patch where the mean velocity shear is particularly large, it has features not dissimilar to those of a classical turbulent wake.

Study of instability and turbulence development scenarios in a flow with stable stratification and two-dimensional obstacle

“…The results of DNS at fixed Prandtl number Pr = 1 and lower Reynolds numbers (Re < 4000) than in [1,2] demonstrate that, for instance, for Re = 2000 ( Fig. 1-4) multiple peaks in spanwise spectra (corresponding to the wavelength 0.5H < λ y ≤ 5H of spanwise periodical instability) occur during perturbation growth at times 20 < t < 32.…”

“…All details of the governing equations non-dimensionalized by inflow velocity U and obstacle height H, and their numerical realization are given in [1,2].…”

“…The study is carried out to continue numerical investigations of instability and turbulence development mechanisms started in [1,2] where the DNS data for Re = 4000 and Pr = 1 were presented, with the grid resolution sufficient to capture the fine-scale transition features [1] and the subsequent turbulence behavior [2]. Such a phenomenon has also been explored in laboratory experiments in water tanks with towed body [3,4,5].…”

“…The secondary instability of the density field generated after internal lee wave overturning reveals a range of spanwise modes. The smallest mode with the wavelength λ y ~ 0.5H for Re = 4000 and Pr = 1 [1,2] corresponds to perturbations of Rayleigh-Taylor instability (RTI) type, growing and resulting in convective mushroom-like structures, with associated Kelvin-Helmholtz instability rolls. At late times of transition to turbulence the smallerscale vortices transform into the bigger vortex structures, and another dominant mode (λ y ~ 2.5H) arises when the large-scale toroidal vortex structures can be visualized [1].…”

“…At late times of transition to turbulence the smallerscale vortices transform into the bigger vortex structures, and another dominant mode (λ y ~ 2.5H) arises when the large-scale toroidal vortex structures can be visualized [1]. This larger-scale mode can be associated with the most unstable perturbation of the initial laminar two-dimensional vortex pair in the place of lee wave overturning [1,2] and has also been observed in both numerical and physical experiments [3,4,6]. Visualization of large structures is complicated by the presence of a cascade of finer-scale eddies, leading to the formation of the inertial "-5/3" range and the dissipative range of spanwise spectra with steeper slopes.…”