Entrainment and mixing in stratified shear flows

Abstract

The results of a laboratory experiment designed to study turbulent entrainment at sheared density interfaces are described. A stratified shear layer, across which a velocity difference Δ U and buoyancy difference Δ b is imposed, separates a lighter upper turbulent layer of depth D from a quiescent, deep lower layer which is either homogeneous (two-layer case) or linearly stratified with a buoyancy frequency N (linearly stratified case). In the parameter ranges investigated the flow is mainly determined by two parameters: the bulk Richardson number Ri B = Δ bD /Δ U 2 and the frequency ratio f N = ND =Δ U . When Ri B > 1.5, there is a growing significance of buoyancy effects upon the entrainment process; it is observed that interfacial instabilities locally mix heavy and light fluid layers, and thus facilitate the less energetic mixed-layer turbulent eddies in scouring the interface and lifting partially mixed fluid. The nature of the instability is dependent on Ri B , or a related parameter, the local gradient Richardson number Ri g = N 2 L / (∂ u /∂ z ) 2 , where N L is the local buoyancy frequency, u is the local streamwise velocity and z is the vertical coordinate. The transition from the Kelvin–Helmholtz (K-H) instability dominated regime to a second shear instability, namely growing Hölmböe waves, occurs through a transitional regime 3.2 < Ri B < 5.8. The K-H activity completely subsided beyond Ri B ∼ 5 or Ri g ∼ 1. The transition period 3.2 < Ri B < 5 was characterized by the presence of both K-H billows and wave-like features, interacting with each other while breaking and causing intense mixing. The flux Richardson number Ri f or the mixing efficiency peaked during this transition period, with a maximum of Ri f ∼ 0.4 at Ri B ∼ 5 or Ri g ∼ 1. The interface at 5 < Ri B < 5.8 was dominated by ‘asymmetric’ interfacial waves, which gradually transitioned to (symmetric) Hölmböe waves at Ri B > 5:8. Laser-induced fluorescence measurements of both the interfacial buoyancy flux and the entrainment rate showed a large disparity (as large as 50%) between the two-layer and the linearly stratified cases in the range 1.5 < Ri B < 5. In particular, the buoyancy flux (and the entrainment rate) was higher when internal waves were not permitted to propagate into the deep layer, in which case more energy was available for interfacial mixing. When the lower layer was linearly stratified, the internal waves appeared to be excited by an ‘interfacial swelling’ phenomenon, characterized by the recurrence of groups or packets of K-H billows, their degeneration into turbulence and subsequent mixing, interfacial thickening and scouring of the thickened interface by turbulent eddies. Estimation of the turbulent kinetic energy (TKE) budget in the interfacial zone for the two-layer case based on the parameter α, where α = (− B + ε)/ P , indicated an approximate balance (α ∼ 1) between the shear production P , buoyancy flux B and the dissipation rate ε, except in the range Ri B < 5 where K-H driven mixing was active.