Perturbative corrections for the scaling of heat transport in a Hele-Shaw geometry and its application to geological vertical fractures

JOURNAL OF FLUID MECHANICS(2019)

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摘要
In this work, we investigate numerically the perturbative effects of cell aperture in heat transport and thermal dissipation rate for a vertical Hele-Shaw geometry, which is used as an analogue representation of a planar vertical fracture at the laboratory scale. To model the problem, we derive a two-dimensional set of equations valid for this geometry. For Hele-Shaw cells heated from below and above, with periodic boundary conditions in the horizontal direction, the model gives new nonlinear scalings for both the time-averaged Nusselt number < Nu >(tau) and dimensionless mean thermal dissipation rate < v >(tau) in the high-Rayleigh regime. We demonstrate that < Nu >(tau) and < v >(tau) depend upon the cell anisotropy ratio epsilon, which measures the ratio between the cell gap and height. We show that < Nu >(tau) values in the high-Rayleigh regime decrease when epsilon grows, supporting the field observations at the fracture scale. When epsilon <<1, our results are in agreement with the scalings found using the Darcy model. The numerical results satisfy the theoretical relation < Nu >(tau) = Ra < v >(tau), which is obtained from the model. This latter relation is valid for all values of Rayleigh number considered. The perturbative effects of cell aperture are observed only in the exponents of the scalings < Nu >(tau) similar to Ra gamma((epsilon)) and < v >(tau) similar to Ra gamma(epsilon)-1.
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关键词
convection in porous media,Hele-Shaw flows,mixing and dispersion
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