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Exact relations between Rayleigh–Bénard and rotating plane Couette flow in two dimensions

Published online by Cambridge University Press:  28 September 2020

Bruno Eckhardt Open the ORCID record for Bruno Eckhardt [Opens in a new window] ,
Charles R. Doering Open the ORCID record for Charles R. Doering [Opens in a new window]  and
Jared P. Whitehead Open the ORCID record for Jared P. Whitehead [Opens in a new window]
Bruno Eckhardt
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, D-35032Marburg, Germany
Charles R. Doering
Affiliation:
Center for the Study of Complex Systems, Department of Mathematics and Department of Physics, University of Michigan, Ann Arbor, MI48109, USA
Jared P. Whitehead*
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT84602, USA
*
†Email address for correspondence: [email protected]
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Abstract

Rayleigh–Bénard convection (RBC) and Taylor–Couette flow (TCF) are two paradigmatic fluid dynamical systems frequently discussed together because of their many similarities despite their different geometries and forcing. Often these analogies require approximations, but in the limit of large radii where TCF becomes rotating plane Couette flow (RPC) exact relations can be established. When the flows are restricted to two spatial independent variables, there is an exact specification that maps the three velocity components in RPC to the two velocity components and one temperature field in RBC. Using this, we deduce several relations between both flows: (i) heat and angular momentum transport differ by $(1-R_{\Omega })$, explaining why angular momentum transport is not symmetric around $R_{\Omega }=1/2$ even though the relation between $Ra$, the Rayleigh number, and $R_{\Omega }$, a non-dimensional measure of the rotation, has this symmetry. This relationship leads to a predicted value of $R_{\Omega }$ that maximizes the angular momentum transport that agrees remarkably well with existing numerical simulations of the full three-dimensional system. (ii) One variable in both flows satisfies a maximum principle, i.e. the fields’ extrema occur at the walls. Accordingly, backflow events in shear flow cannot occur in this quasi two-dimensional setting. (iii) For free-slip boundary conditions on the axial and radial velocity components, previous rigorous analysis for RBC implies that the azimuthal momentum transport in RPC is bounded from above by $Re_S^{5/6}$, where $Re_S$ is the shear Reynolds number, with a scaling exponent smaller than the anticipated $Re_S^1$.

Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , R4
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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