Near- and far-field structure of shallow mixing layers between parallel streams

Zhengyang Cheng; George Constantinescu

doi:10.1017/jfm.2020.638

The dynamics of coherent structures forming in a turbulent shallow mixing layer (ML) between two parallel streams advancing in a constant-depth, open channel is investigated using three-dimensional, time-accurate simulations. The large channel length to flow depth ratio ($L_{x}/H = 400\text{--}800$) allows characterization of the spatial evolution of shallow MLs until the mean velocity difference between the two streams becomes less than 3% of the initial value at the end of the splitter plate. Away from the ML origin, the dynamics and coherence of the Kelvin–Helmholtz (KH) billows are affected by the destabilizing effect of the mean shear between the two streams and by the stabilizing effect of the bed friction. A linear decay of the entrainment coefficient α with the bed-friction factor, S, applies only over the region where merging of neighbouring KH billows is still observed (transition to quasi-equilibrium regime). At larger distances from the origin, where the billows are severely stretched in the streamwise direction before being destroyed, the rates of increase of the ML width, δ, and centreline shift, lcy, become very small and α is exponentially decaying with increasing S toward zero (quasi-equilibrium regime). During the initial stages of the quasi-equilibrium regime where the KH vortices are severely stretched, the ML assumes an undulatory shape in horizontal planes. New relationships are proposed to characterize the downstream variation of the non-dimensional ML width and centreline shift over the transition and quasi-equilibrium regimes. During the transition to equilibrium regime, the ML boundary on the fast-stream side remains close to vertical, while that on the slow-stream side becomes strongly tilted. The ML boundary on the slow-stream side becomes again close to vertical once the quasi-equilibrium regime is reached. During the transition to the equilibrium regime, the passage of the KH billows and the generation of streamwise cells of secondary flow generate regions of high instantaneous bed shear stress, such that the region where the erosive capacity of the flow peaks does not correspond to the fast stream. The paper also investigates the effects of flow shallowness and initial velocity ratio between the two streams on the turbulent kinetic energy inside the ML, the depth-averaged lateral momentum fluxes, the passage frequency and size of the KH billows and the wavelength and period of the undulatory motions of the ML during the early stages of the quasi-equilibrium regime.

Near- and far-field structure of shallow mixing layers between parallel streams

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