A rigorous bound on the vertical transport of heat in Rayleigh-Bénard convection at infinite Prandtl number with mixed thermal boundary conditions

Whitehead, Jared P.; Wittenberg, Ralf W.

doi:10.1063/1.4896223

Cited by 19 publications

(20 citation statements)

References 40 publications

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“…For channels and shear flows, upper bounds on wall drag coefficients have been derived rigorously from the incompressible Navier-Stokes equations (Busse 1969;Howard 1972;Constantin & Doering 1994, 1995Kerswell 1997;Nicodemus et al 1998;Hoffmann & Vitanov 1999;Kerswell 2002;Plasting & Kerswell 2003;Seis 2015). For buoyancy-driven thermal convection between planar boundaries, the authors and others have proven lower bounds on mean temperature (Lu et al 2004;Whitehead & Doering 2011a, 2012Goluskin 2015a,b) and upper bounds on heat transport (Howard 1963;Constantin & Doering 1996;Kerswell 1997Kerswell , 2001Otero et al 2002;Plasting & Ierley 2005;Wittenberg 2010;Otto & Seis 2011;Whitehead & Doering 2011b, 2012Wen et al 2013;Wang & Whitehead 2103;Whitehead & Wittenberg 2014;Choffrut et al 2016).…”

Section: Introductionmentioning

confidence: 99%

Bounds for convection between rough boundaries

Goluskin¹,

Doering²

2016

J. Fluid Mech.

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We consider Rayleigh-B\'enard convection in a layer of fluid between rough no-slip boundaries where the top and bottom boundary heights are functions of the horizontal coordinates with square-integrable gradients. We use the background method to derive an upper bound on mean heat flux across the layer for all admissible boundary geometries. This flux, normalized by the temperature difference between the boundaries, can grow with the Rayleigh number ($Ra$) no faster than ${\cal O}(Ra^{1/2})$ as $Ra \rightarrow \infty$. Our analysis yields a family of similar bounds, depending on how various estimates are tuned, but every version depends explicitly on the boundary geometry. In one version the coefficient of the ${\cal O}(Ra^{1/2})$ leading term is $0.242 + 2.925\Vert\nabla h\Vert^2$, where $\Vert\nabla h\Vert^2$ is the mean squared magnitude of the boundary height gradients. Application to a particular geometry is illustrated for sinusoidal boundaries.Comment: 15 pages; v2: result strengthened so that the gradients of the functions defining the boundaries need only be square-integrable as opposed to uniformly bounde

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Section: Introductionmentioning

confidence: 99%

Bounds for convection between rough boundaries

Goluskin¹,

Doering²

2016

J. Fluid Mech.

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“…We do this by combining the careful construction of an asymmetric background temperature field, inspired by the optimal profiles from Fantuzzi et al (2018), with new estimates for the coupling between temperature and vertical velocity. These differ fundamentally from the estimates that apply to infinite-Pr Rayleigh-Bénard convection (Doering et al 2006;Whitehead & Doering 2011;Whitehead & Wittenberg 2014) due to the different boundary conditions (BCs) for the velocity field.…”

Section: Introductionmentioning

confidence: 71%

New bounds on the vertical heat transport for Bénard–Marangoni convection at infinite Prandtl number

2019

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We prove a new rigorous upper bound on the vertical heat transport for Bénard-Marangoni convection of a two-or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number Ma 1 the Nusselt number Nu is bounded asymptotically by Nu const. × Ma 2/7 (ln Ma) −1/7 . Key to our proof are a background temperature field with a hyperbolic profile near the fluid's surface and new estimates for the coupling between temperature and vertical velocity.

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“…Since the work of Doering and Constantin, this method has been applied to a wide variety of problems in fluid dynamics. Examples include upper bounds on the rate of energy dissipation in surface-velocity-driven flows (Doering & Constantin 1992, 1994Marchioro 1994;Wang 1997;Plasting & Kerswell 2003), pressure-driven flows (Constantin & Doering 1995) and surface-stress-driven flows (Tang, Caulfield & Young 2004;Hagstrom & Doering 2014); upper bounds on the heat transfer in different configurations of Rayleigh-Bénard convection (Doering & Constantin 1996Otero et al 2002;Plasting & Ierley 2005;Wittenberg 2010;Whitehead & Doering 2011;Whitehead & Wittenberg 2014;Goluskin 2015;Goluskin & Doering 2016;Fantuzzi 2018) and Bénard-Marangoni convection (Hagstrom & Doering 2010;Fantuzzi, Pershin & Wynn 2018;Fantuzzi, Nobili & Wynn 2020); and upper bounds on buoyancy flux in stably stratified shear flows (Caulfield & Kerswell 2001;Caulfield 2005).…”

Section: Introductionmentioning

confidence: 99%