Stratified flows with vertical layering of density: theoretical and experimental study of the time evolution of flow configurations and their stability

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  • March 21, 2019
Creator
  • Moore, Matthew N. J.
    • Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
  • A vertically moving boundary in a stratified fluid can create and maintain a horizontal density gradient, or vertical layering of density. We study an idealized two dimensional problem in which a wall moves upwards with constant speed and maintains a viscously entrained boundary layer of heavy fluid. Additionally, we study an axisymmetric analogue of this problem in which the moving wall is replaced by a moving fiber. We construct exact solutions under the assumptions of steady-state shear flow for both the two dimensional and axisymmetric problems, in the cases in which the domain is either semi-infinite or bounded horizontally. Most attention is focused on the situation in which the density profile has a sharp transition. In the semi-infinite domain, it is found that a relationship between the size of the entrained layer and the towing speed is required to hold for a steady, shear solution to exist. The condition is found to be a result of the over-restrictive assumptions of steady flow in a semi-infinite domain and no such condition is required in the bounded domain. In the bounded domain, a two-parameter family of shear solutions is constructed after the physically-based assumption of vanishing flux is made. We conduct experiments that successfully create the axisymmetric shear flows from an initially stable stratification. In order to determine the time evolution of the flow observed in the experiments, a lubrication model is developed and is shown to be in excellent agreement with observations. Additionally, we determine the full time evolution of the flow in the case of no stratification, and this solution is asymptotic to the experimental system for short times. We perform stability analysis on the family of exact shear solutions in both two dimensions and the axisymmetric geometry, using asymptotic and numerical methods. The stability properties of the flow depend strongly on the size of the entrained layer. A critical layer size is found, below which the flow configuration is stable and beyond which the flow configuration is unstable. This bifurcation is independent of the Reynolds number of the flow and the Reynolds number only affects the magnitude of the amplification or damping of disturbances. It is found that unstable layer sizes are possible to achieve from the initial value problem of stable stratification. Layer sizes which are predicted to be unstable are observed in the experiment, however the amplification of disturbances is not observed because the rate of amplification is too small. Experimental measurements show excellent agreement with predictions from the time dependent lubrication model over a large range of times, as well as good agreement with the homogeneous model for short times.
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  • In Copyright
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  • "... in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics."
Advisor
  • Camassa, Roberto
Degree granting institution
  • University of North Carolina at Chapel Hill
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Place of publication
  • Chapel Hill, NC
Access right
  • Open access
Date uploaded
  • March 18, 2013

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