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Experiments on mixing in wakes in shallow water

Published online by Cambridge University Press:  09 September 2016

Gioacchino Cafiero  and
Andrew W. Woods
Gioacchino Cafiero
Affiliation:
BP Institute, University of Cambridge, Cambridge CB3 0EZ, UK Department of Industrial Engineering, Experimental Thermo-Fluid dynamics division, University of Naples, Via Claudio 21, 80125, Italy
Andrew W. Woods*
Affiliation:
BP Institute, University of Cambridge, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: andy@bpi.cam.ac.uk
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Abstract

We report on a series of laboratory experiments in which we investigate the mixing in a wake produced downstream of an obstacle in a uniform flow. The fluid is confined within a channel of finite width, and the water depth is small compared with the channel width. The mixing appears to be dominated by dispersion caused by the circulation of the eddies that are shed alternately from each side of the obstacle. However, due to bottom friction, these eddies gradually dissipate downstream. In turn, the intensity of the cross-stream mixing of the tracer decays in the downstream direction, limiting the cross-stream extent of the tracer. We present a time-averaged picture of the experiments which illustrates the deviation of the time-averaged flow in the wake relative to the uniform flow upstream. We then develop a time-averaged model for the flow, using mixing length theory to account for the cross-channel momentum transfer as an eddy viscosity $\unicode[STIX]{x1D706}_{1}ud$, where $2ud$ is the cross-channel integral of the perturbation in the along-channel speed associated with the wake. We also include a frictional stress to account for the bottom friction. The model predicts a similar pattern of variation of the along-channel velocity in both the along- and cross-channel directions to our experimental data. By matching the cross-channel data with the model, we find that the constant $\unicode[STIX]{x1D706}_{1}$ has value 0.2. We also analyse our experimental data to develop a time-dependent picture of the mixing of a stream of dye released into the wake. Using the model for the evolution of the flow, we develop a model for the time-averaged mixing, again based on mixing length theory. The model predicts a similar spatial distribution for the tracer in both the cross-stream and along-stream directions to that seen in our experimental data. By quantitative comparison of the model with the data, we find that the best fit of the empirical eddy diffusivity, $\unicode[STIX]{x1D706}_{2}ud$, with the data occurs with $\unicode[STIX]{x1D706}_{2}=0.22$. We discuss implications of our results for modelling cross-stream mixing in shallow turbulent flow.

JFM classification

Mixing: Mixing Mixing: Turbulent mixing Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 804 , 10 October 2016 , pp. 351 - 369
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Copyright
© 2016 Cambridge University Press 

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