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Regime transitions and energetics of sustained stratified shear flows

Published online by Cambridge University Press:  22 July 2019

Adrien Lefauve*
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
J. L. Partridge
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
P. F. Linden
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Email address for correspondence:


We describe the long-term dynamics of sustained stratified shear flows in the laboratory. The stratified inclined duct (SID) experiment sets up a two-layer exchange flow in an inclined duct connecting two reservoirs containing salt solutions of different densities. This flow is primarily characterised by two non-dimensional parameters: the tilt angle of the duct with respect to the horizontal, $\unicode[STIX]{x1D703}$ (a few degrees at most), and the Reynolds number $Re$, an input parameter based on the density difference driving the flow. The flow can be sustained with constant forcing over arbitrarily long times and exhibits a wealth of dynamical behaviours representative of geophysically relevant sustained stratified shear flows. Varying $\unicode[STIX]{x1D703}$ and $Re$ leads to four qualitatively different regimes: laminar flow; mostly laminar flow with finite-amplitude, travelling Holmboe waves; spatio-temporally intermittent turbulence with substantial interfacial mixing; and sustained, vigorous interfacial turbulence (Meyer & Linden, J. Fluid Mech., vol. 753, 2014, pp. 242–253). We seek to explain the scaling of the transitions between flow regimes in the two-dimensional plane of input parameters $(\unicode[STIX]{x1D703},Re)$. We improve upon previous studies of this problem by providing a firm physical basis and non-dimensional scaling laws that are mutually consistent and in good agreement with the empirical transition curves we inferred from 360 experiments spanning $\unicode[STIX]{x1D703}\in [-1^{\circ },6^{\circ }]$ and $Re\in [300,5000]$. To do so, we employ state-of-the-art simultaneous volumetric measurements of the density field and the three-component velocity field, and analyse these experimental data using time- and volume-averaged potential and kinetic energy budgets. We show that regime transitions are caused by an increase in the non-dimensional time- and volume-averaged kinetic energy dissipation within the duct, which scales with $\unicode[STIX]{x1D703}Re$ at high enough angles. As the power input scaling with $\unicode[STIX]{x1D703}Re$ is increased above zero, the two-dimensional, parallel-flow dissipation (power output) increases to close the budget through an increase in the magnitude of the exchange flow, incidentally triggering Holmboe waves above a certain threshold in interfacial shear. However, once the hydraulic limit of two-layer exchange flows is reached, two-dimensional dissipation plateaus and three-dimensional dissipation at small scales (turbulence) takes over, at first intermittently, and then steadily, in order to close the budget and follow the $\unicode[STIX]{x1D703}Re$ scaling. This general understanding of regime transitions and energetics in the SID experiment may serve as a basis for the study of more complex sustained stratified shear flows found in the natural environment.

JFM Papers
© 2019 Cambridge University Press 

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