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Bounds on heat transport for convection driven by internal heating

Published online by Cambridge University Press:  26 May 2021

Ali Arslan Open the ORCID record for Ali Arslan [Opens in a new window] ,
Giovanni Fantuzzi Open the ORCID record for Giovanni Fantuzzi [Opens in a new window] ,
John Craske Open the ORCID record for John Craske [Opens in a new window]  and
Andrew Wynn Open the ORCID record for Andrew Wynn [Opens in a new window]
Ali Arslan*
Affiliation:
Department of Aeronautics, Imperial College London, LondonSW7 2AZ, UK
Giovanni Fantuzzi
Affiliation:
Department of Aeronautics, Imperial College London, LondonSW7 2AZ, UK
John Craske
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, LondonSW7 2AZ, UK
Andrew Wynn
Affiliation:
Department of Aeronautics, Imperial College London, LondonSW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

The mean vertical convective heat transport $\langle wT \rangle$ between isothermal plates driven by uniform internal heating is investigated by means of rigorous bounds. These are obtained as a function of the Rayleigh number R by constructing feasible solutions to a convex variational problem, derived using a formulation of the classical background method in terms of a quadratic auxiliary function. When the fluid's temperature relative to the boundaries is allowed to be positive or negative, numerical solution of the variational problem shows that best previous bound $\langle wT \rangle \leqslant 1/2$ (Goluskin & Spiegel, Phys. Lett. A, vol. 377, issue 1–2, 2012, pp. 83–92) can only be improved up to finite R. Indeed, we demonstrate analytically that $\langle wT \rangle \leqslant 2^{-21/5} {\textit {R}}^{1/5}$ and therefore prove that $\langle wT\rangle < 1/2$ for ${\textit {R}} < 65\,536$. However, if the minimum principle for temperature is invoked, which asserts that internal temperature is at least as large as the temperature of the isothermal boundaries, then numerically optimised bounds are strictly smaller than $1/2$ until at least ${\textit {R}}=3.4\times 10^{5}$. While the computational results suggest that the best bound on $\langle wT\rangle$ approaches $1/2$ asymptotically from below as ${\textit {R}}\rightarrow \infty$, we prove that typical analytical constructions cannot be used to prove this conjecture.

JFM classification

Mathematical Foundations: Variational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 919 , 25 July 2021 , A15
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Bercovici, D. 2011 Mantle convection. In Encyclopedia of Solid Earth Geophysics (ed. H.K. Gupta). Springer.CrossRefGoogle Scholar
Bouillaut, V., Lepot, S., Aumaître, S. & Gallet, B. 2019 Transition to the ultimate regime in a radiatively driven convection experiment. J. Fluid Mech. 861, R5.CrossRefGoogle Scholar
Boyd, S. & Vandenberghe, L. 2004 Convex Optimization. Cambridge University Press.CrossRefGoogle Scholar
Chernyshenko, S.I. 2017 Relationship between the methods of bounding time averages. arXiv:1704.02475.Google Scholar
Chernyshenko, S.I., Goulart, P.J., Huang, D. & Papachristodoulou, A. 2014 Polynomial sum of squares in fluid dynamics: a review with a look ahead. Phil. Trans. R. Soc. Lond. A 372 (2020), 20130350.Google ScholarPubMed
Choffrut, A., Nobili, C. & Otto, F. 2016 Upper bounds on Nusselt number at finite Prandtl number. J. Differ. Equ. 260 (4), 3860–3880.CrossRefGoogle Scholar
Constantin, P. & Doering, C.R. 1995 Variational bounds on energy dissipation in incompressible flows. II. Channel flow. Phys. Rev. E 51 (4), 3192.CrossRefGoogle ScholarPubMed
Constantin, P. & Doering, C.R. 1996 Heat transfer in convective turbulence. Nonlinearity 9 (4), 1049–1060.CrossRefGoogle Scholar
Constantin, P. & Doering, C.R. 1999 Infinite Prandtl number convection. J. Stat. Phys. 94 (1–2), 159–172.CrossRefGoogle Scholar
Debler, W.R. 1959 The onset of laminar natural convection in a fluid with homogenously distributed heat sources. PhD thesis, University of Michigan.Google Scholar
Doering, C.R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49 (5), 4087.CrossRefGoogle ScholarPubMed
Doering, C.R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53 (6), 5957.CrossRefGoogle ScholarPubMed
Doering, C.R. & Constantin, P. 1998 Bounds for heat transport in a porous layer. J. Fluid Mech. 376, 263–296.CrossRefGoogle Scholar
Doering, C.R. & Constantin, P. 2001 On upper bounds for infinite Prandtl number convection with or without rotation. J. Math. Phys. 42 (2), 784–795.CrossRefGoogle Scholar
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
Emara, A.A. & Kulacki, F.A. 1980 A numerical investigation of thermal convection in a heat-generating fluid layer. Trans. ASME J. Heat Transfer 102 (3), 531–537.CrossRefGoogle Scholar
Fantuzzi, G., Goluskin, D., Huang, D. & Chernyshenko, S.I. 2016 Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization. SIAM J. Appl. Dyn. Syst. 15 (4), 1962–1988.CrossRefGoogle Scholar
Fantuzzi, G., Pershin, A. & Wynn, A. 2018 Bounds on heat transfer for Bénard–Marangoni convection at infinite Prandtl number. J. Fluid Mech. 837, 562–596.CrossRefGoogle Scholar
Fantuzzi, G. & Wynn, A. 2015 Construction of an optimal background profile for the Kuramoto–Sivashinsky equation using semidefinite programming. Phys. Lett. A 379 (1–2), 23–32.CrossRefGoogle Scholar
Fantuzzi, G. & Wynn, A. 2016 Optimal bounds with semidefinite programming: an application to stress-driven shear flows. Phys. Rev. E 93 (4), 043308.CrossRefGoogle ScholarPubMed
Fantuzzi, G., Wynn, A., Goulart, P.J. & Papachristodoulou, A. 2017 Optimization with affine homogeneous quadratic integral inequality constraints. IEEE Trans. Autom. Control 62 (12), 6221– 6236.CrossRefGoogle Scholar
Foias, C., Manley, O. & Temam, R. 1987 Attractors for the Bénard problem: existence and physical bounds on their fractal dimension. Nonlinear Anal. Theory Meth. Applics. 11 (8), 939–967.CrossRefGoogle Scholar
Fujisawa, K., Fukuda, M., Kobayashi, K., Kojima, M., Nakata, K., Nakata, M. & Yamashita, M. 2008 SDPA (SemiDefinite Programming Algorithm) and SDPA-GMP User's Manual – Version 7.1.1. Tech. Rep. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan.Google Scholar
Fujisawa, K., Kim, S., Kojima, M., Okamoto, Y. & Yamashita, M. 2009 User's manual for SparseCoLO: conversion methods for SPARSE COnic-form linear optimization problems. Research Report B-453 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan.Google Scholar
Fukuda, M., Kojima, M., Murota, K. & Nakata, K. 2001 Exploiting sparsity in semidefinite programming via matrix completion I: general framework. SIAM J. Optim. 11 (3), 647–674.CrossRefGoogle Scholar
Goluskin, D. 2015 Internally heated convection beneath a poor conductor. J. Fluid Mech. 771, 36–56.CrossRefGoogle Scholar
Goluskin, D. 2016 Internally Heated Convection and Rayleigh–Bénard Convection. Springer.CrossRefGoogle Scholar
Goluskin, D. & Doering, C.R. 2016 Bounds for convection between rough boundaries. J. Fluid Mech. 804, 370–386.CrossRefGoogle Scholar
Goluskin, D. & Fantuzzi, G. 2019 Bounds on mean energy in the Kuramoto–Sivashinsky equation computed using semidefinite programming. Nonlinearity 32 (5), 1705–1730.CrossRefGoogle Scholar
Goluskin, D. & van der Poel, E.P. 2016 Penetrative internally heated convection in two and three dimensions. J. Fluid Mech. 791, R6.CrossRefGoogle Scholar
Goluskin, D. & Spiegel, E.A. 2012 Convection driven by internal heating. Phys. Lett. A 377 (1–2), 83–92.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Howard, L.N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10 (4), 509–512.CrossRefGoogle Scholar
Jahn, M. & Reineke, H.-H. 1974 Free convection heat transfer with internal heat sources, calculations and measurements. In Proceedings of the 5th International Heat Transfer Conference, Tokyo, pp. 74–78. Begel House Inc.CrossRefGoogle Scholar
Kakac, S., Aung, W.M. & Viskanta, R. 1985 Natural Convection: Fundamentals and Applications. Hemisphere Publishing Corp., p. 1191.Google Scholar
Kulacki, F.A. & Goldstein, R.J. 1972 Thermal convection in a horizontal fluid layer with uniform volumetric energy sources. J. Fluid Mech. 55 (2), 271–287.CrossRefGoogle Scholar
Lee, S.D., Lee, J.K. & Suh, K.Y. 2007 Boundary condition dependent natural convection in a rectangular pool with internal heat sources. Trans. ASME J. Heat Transfer 129 (5), 679–682.CrossRefGoogle Scholar
Lu, L., Doering, C.R. & Busse, F.H. 2004 Bounds on convection driven by internal heating. J. Math. Phys. 45 (7), 2967–2986.CrossRefGoogle Scholar
Malkus, W.V.R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225 (1161), 196–212.Google Scholar
Mayinger, F., Jahn, M., Reineke, H. & Steibnberner, V. 1976 Examination of thermohydraulic processes and heat transfer in a core melt. Tech. Rep. BMFT RS 48/1. Institut für Verfahrenstechnik der TU, Hanover Germany.Google Scholar
Miles, J.W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10 (4), 496–508.CrossRefGoogle Scholar
Nemirovski, A. 2006 Advances in convex optimization: conic programming. In International Congress of Mathematicians, vol. 1, pp. 413–444.Google Scholar
Otero, J., Dontcheva, L.A., Johnston, H., Worthing, R.A., Kurganov, A., Petrova, G. & Doering, C.R. 2004 High-Rayleigh-number convection in a fluid-saturated porous layer. J. Fluid Mech. 500, 263–281.CrossRefGoogle Scholar
Otero, J., Wittenberg, R.W., Worthing, R.A. & Doering, C.R. 2002 Bounds on Rayleigh–Bénard convection with an imposed heat flux. J. Fluid Mech. 473, 191–199.CrossRefGoogle Scholar
Otto, F. & Seis, C. 2011 Rayleigh–Bénard convection: improved bounds on the Nusselt number. J. Math. Phys. 52 (8), 083702.CrossRefGoogle Scholar
Peckover, R.S. & Hutchinson, I.H. 1974 Convective rolls driven by internal heat sources. Phys. Fluids 17 (7), 1369–1371.CrossRefGoogle Scholar
Plasting, S.C. & Kerswell, R.R. 2003 Improved upper bound on the energy dissipation rate in plane Couette flow: the full solution to Busse's problem and the Constantin-Doering-Hopf problem with one-dimensional background field. J. Fluid Mech. 477, 363–379.CrossRefGoogle Scholar
Priestley, C.H.B. 1954 Vertical heat transfer from impressed temperature fluctuations. Austral. J. Phys. 7 (1), 202–209.CrossRefGoogle Scholar
Rosa, R. & Temam, R.M. 2020 Optimal minimax bounds for time and ensemble averages of dissipative infinite-dimensional systems with applications to the incompressible Navier–Stokes equations. arXiv:2010.06730.Google Scholar
Spiegel, E.A. 1963 A generalization of the mixing-length theory of turbulent convection. Astrophys. J. 138, 216–225.CrossRefGoogle Scholar
Straus, J.M. 1976 Penetrative convection in a layer of fluid heated from within. Astrophys. J. 209, 179–189.CrossRefGoogle Scholar
Tilgner, A. 2017 Bounds on poloidal kinetic energy in plane layer convection. Phys. Rev. Fluids 2 (12), 123502.CrossRefGoogle Scholar
Tilgner, A. 2019 Time evolution equation for advective heat transport as a constraint for optimal bounds in Rayleigh–Bénard convection. Phys. Rev. Fluids 4 (1), 014601.CrossRefGoogle Scholar
Tobasco, I., Goluskin, D. & Doering, C.R. 2018 Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems. Phys. Lett. A 382 (6), 382–386.CrossRefGoogle Scholar
Tritton, D.J. 1975 Internally heated convection in the atmosphere of Venus and in the laboratory. Nature 257 (5522), 110–112.CrossRefGoogle Scholar
Trowbridge, A.J., Melosh, H.J., Steckloff, J.K. & Freed, A.M. 2016 Vigorous convection as the explanation for Pluto's polygonal terrain. Nature 534 (7605), 79–81.CrossRefGoogle ScholarPubMed
Tveitereid, M. 1978 Thermal convection in a horizontal fluid layer with internal heat sources. Intl J Heat Mass Transfer 21 (3), 335–339.CrossRefGoogle Scholar
Waki, H., Nakata, M. & Muramatsu, M. 2012 Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization. Comput. Optim. Appl. 53 (3), 823–844.CrossRefGoogle Scholar
Wang, Q., Lohse, D. & Shishkina, O. 2020 Scaling in internally heated convection: a unifying theory. Geophys. Res. Lett. 47, e2020GL091198.Google Scholar
Wen, B., Chini, G.P., Dianati, N. & Doering, C.R. 2013 Computational approaches to aspect-ratio-dependent upper bounds and heat flux in porous medium convection. Phys. Lett. A 377 (41), 2931–2938.CrossRefGoogle Scholar
Wen, B., Chini, G.P., Kerswell, R.R. & Doering, C.R. 2015 Time-stepping approach for solving upper-bound problems: application to two-dimensional Rayleigh–Bénard convection. Phys. Rev. E 92 (4), 043012.CrossRefGoogle ScholarPubMed
Whitehead, J.P. & Doering, C.R. 2011 a Internal heating driven convection at infinite Prandtl number. J. Math. Phys. 52 (9), 093101.CrossRefGoogle Scholar
Whitehead, J.P. & Doering, C.R. 2011 b Ultimate state of two-dimensional Rayleigh–Bénard convection between free-slip fixed-temperature boundaries. Phys. Rev. Lett. 106 (24), 244501.CrossRefGoogle ScholarPubMed
Whitehead, J.P. & Doering, C.R. 2012 Rigid bounds on heat transport by a fluid between slippery boundaries. J. Fluid Mech. 707, 241–259.CrossRefGoogle Scholar
Wörner, M., Schmidt, M. & Grötzbach, G. 1997 Direct numerical simulation of turbulence in an internally heated convective fluid layer and implications for statistical modelling. J. Hydraul Res. 35 (6), 773–797.CrossRefGoogle Scholar
Yamashita, M., Fujisawa, K., Fukuda, M., Kobayashi, K., Nakata, K. & Nakata, M. 2012 Latest developments in the SDPA family for solving large-scale SDPS. In Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 687–713. Springer.CrossRefGoogle Scholar
Yan, X. 2004 On limits to convective heat transport at infinite Prandtl number with or without rotation. J. Math. Phys. 45 (7), 2718–2743.CrossRefGoogle Scholar