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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: whitehead@mathematics.byu.edu
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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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References

REFERENCES

Ahlers, G. & Behringer, R. P. 1978 Evolution of turbulence from Rayleigh–Bénard instability. Phys. Rev. Lett. 40 (11), 712–716.CrossRefGoogle Scholar
Bradshaw, P. 1969 The analogy between streamline curvature and buoyancy in turbulent shear flow. J. Fluid Mech. 36, 177–191.CrossRefGoogle Scholar
Brauckmann, H. J., Eckhardt, B. & Schumacher, J. 2017 Heat transport in Rayleigh–Bénard convection and angular momentum transport in Taylor–Couette flow: a comparative study. Phil. Trans. A 375 (2089), 20160079.CrossRefGoogle ScholarPubMed
Brauckmann, H. J., Salewski, M. & Eckhardt, B. 2016 Momentum transport in Taylor–Couette flow with vanishing curvature. J. Fluid Mech. 790, 419–452.CrossRefGoogle Scholar
Busse, F. H. 1969 Bounds on transport of mass and momentum by turbulent flow between parallel plates. Z. Angew. Math. Phys. 20 (1), 1–14.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamics and Hydro-Magnetic Stability. Clarendon Press.Google Scholar
Chossat, P. & Iooss, G. 2012 The Couette–Taylor Problem. Springer.Google Scholar
Constantin, P. 1994 Geometric statistics in turbulence. SIAM Rev. 36 (1), 73–98.CrossRefGoogle Scholar
Doering, C. R. & Constantin, P. 1992 Energy dissipation in shear driven turbulence. Phys. Rev. Lett. 69 (11), 1648–1651.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49, 4087–4099.CrossRefGoogle ScholarPubMed
Doering, C. R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53 (6), 5957–5981.CrossRefGoogle ScholarPubMed
Doering, C. R., Otto, F. & Reznikoff, M. G. 2006 Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh–Bénard convection. J. Fluid Mech. 560, 229–241.CrossRefGoogle Scholar
Dubrulle, B., Dauchot, O., Daviaud, F., Longaretti, P.-Y., Richard, D. & Zahn, J.-P. 2005 Stability and turbulent transport in Taylor–Couette flow from analysis of experimental data. Phys. Fluids 17 (9), 095103.CrossRefGoogle Scholar
Dubrulle, B. & Hersant, F. 2002 Momentum transport and torque scaling in Taylor–Couette flow from an analogy with turbulent convection. Eur. Phys. J. B 26, 379–386.CrossRefGoogle Scholar
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 a Fluxes and energy dissipation in thermal convection and shear flows. Europhys. Lett. 78 (2), 7.CrossRefGoogle Scholar
Eckhardt, B., Grossmann, S. & Lohse, D. 2007 b Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders. J. Fluid Mech. 581, 221–250.CrossRefGoogle Scholar
Ezeta, R., Sacco, F., Bakhuis, D., Huisman, S. G., Ostilla-Mónico, R., Verzicco, R., Sun, C. & Lohse, D. 2020 Double maxima of angular momentum transport in $\eta = 0.91$ TC turbulence. arXiv:2006.03528.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2000 Transition from the Couette–Taylor system to the plane Couette system. Phys. Rev. E 61 (6 Pt B), 7227–7230.CrossRefGoogle ScholarPubMed
Gollub, J. P. & Swinney, H. L. 1975 Onset of turbulence in a rotating fluid. Phys. Rev. Lett. 35 (14), 927–930.CrossRefGoogle Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17 (3), 405–432.CrossRefGoogle Scholar
Howard, L. N. 1972 Bounds on flow quantities. Annu. Rev. Fluid Mech. 4, 473–494.CrossRefGoogle Scholar
Jeffreys, H. 1928 Some cases of instability in fluid motion. Proc. R. Soc. Lond. A 118, 195–208.Google Scholar
Lenaers, P., Li, Q., Brethouwer, G., Schlatter, P. & Örlü, R. 2012 Rare backflow and extreme wall-normal velocity fluctuations in near-wall turbulence. Phys. Fluids A 24 (3), 035110.CrossRefGoogle Scholar
Manneville, P. 2010 Instabilities, Chaos and Turbulence. Imperial College Press.CrossRefGoogle Scholar
Nagata, M. 1986 Bifurcations in Couette flow between almost corotating cylinders. J. Fluid Mech. 169, 229–250.CrossRefGoogle Scholar
Nagata, M. 2013 A note on the mirror-symmetric coherent structure in plane Couette flow. J. Fluid Mech. 727, R1.CrossRefGoogle Scholar
Nickerson, E. C. 1969 Upper bounds on torque in cylindrical Couette flow. J. Fluid Mech. 38 (4), 807–815.CrossRefGoogle Scholar
Otto, F. & Seis, C. 2011 Rayleigh–Bénard convection: improved bounds on the Nusselt number. J. Math. Phys. 52, 083702.CrossRefGoogle Scholar
Plasting, S. C. & Kerswell, R. R. 2003 Improved upper bound on the energy dissipation rate in Plane Couette flow: the full solutions to Busse's problem and the Constantin–Doering–Hopf problem with one-dimensional background fields. J. Fluid Mech. 477, 363–379.CrossRefGoogle Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. J. Sci. 32 (192), 529–546.CrossRefGoogle Scholar
Salewski, M. & Eckhardt, B. 2015 Turbulent states in plane Couette flow with rotation. Phys. Fluids A 27 (4), 045109.CrossRefGoogle Scholar
Spiegel, E. A. 1963 A generalization of the mixing-length theory of thermal convection. Astrophys. J. 138, 216–225.CrossRefGoogle Scholar
Veronis, G. 1970 The analogy between rotating and stratified fluids. Annu. Rev. Fluid Mech. 2, 37–66.CrossRefGoogle Scholar
Whitehead, J. P. & Doering, C. R. 2011 The ultimate regime of two-dimensional Rayleigh–Bénard convection with stress-free boundaries. Phys. Rev. Lett. 106, 244501.CrossRefGoogle Scholar
Whitehead, J. P. & Doering, C. R. 2012 Rigid rigorous bounds on heat transport in a slippery container. J. Fluid Mech. 707, 241–259.CrossRefGoogle Scholar