Axially homogeneous Rayleigh-Bénard convection in a cylindrical cell

L.E. Schmidt, E. Calzavarini, Detlef Lohse, F. Toschi, Roberto Verzicco

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Abstract

Previous numerical studies have shown that the ‘ultimate regime of thermal convection’ can be attained in a Rayleigh–Bénard cell when the kinetic and thermal boundary layers are eliminated by replacing both lateral and horizontal walls with periodic boundary conditions (homogeneous Rayleigh–Bénard convection). Then, the heat transfer scales like Nu~Ra$_{1/2}$ and turbulence intensity as Re~Ra$_{1/2}$, where the Rayleigh number Ra indicates the strength of the driving force (for fixed values of Pr, which is the ratio between kinematic viscosity and thermal diffusivity). However, experiments never operate in unbounded domains and it is important to understand how confinement might alter the approach to this ultimate regime. Here we consider homogeneous Rayleigh–Bénard convection in a laterally confined geometry – a small-aspect-ratio vertical cylindrical cell – and show evidence of the ultimate regime as Ra is increased: in spite of the lateral confinement and the resulting kinetic boundary layers, we still find Nu~Re~Ra$_{1/2}$ at . Further, it is shown that the system supports solutions composed of modes of exponentially growing vertical velocity and temperature fields, with Ra as the critical parameter determining the properties of these modes. Counter-intuitively, in the low-Ra regime, or for very narrow cylinders, the numerical simulations are susceptible to these solutions, which can dominate the dynamics and lead to very high and unsteady heat transfer. As Ra is increased, interaction between modes stabilizes the system, evidenced by the increasing homogeneity and reduced fluctuations in the root-mean-square velocity and temperature fields. We also test that physical results become independent of the periodicity length of the cylinder, a purely numerical parameter, as the aspect ratio is increased.
Original languageEnglish
Pages (from-to)52-68
Number of pages17
JournalJournal of fluid mechanics
Volume691
Issue number9
DOIs
Publication statusPublished - 2012

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convection
aspect ratio
Aspect ratio
Boundary layers
Temperature distribution
temperature distribution
velocity distribution
cells
heat transfer
Heat transfer
thermal boundary layer
Kinetics
support systems
Thermal diffusivity
kinetics
Rayleigh number
thermal diffusivity
free convection
diffusivity
homogeneity

Keywords

  • IR-79929
  • METIS-285893

Cite this

Schmidt, L.E. ; Calzavarini, E. ; Lohse, Detlef ; Toschi, F. ; Verzicco, Roberto. / Axially homogeneous Rayleigh-Bénard convection in a cylindrical cell. In: Journal of fluid mechanics. 2012 ; Vol. 691, No. 9. pp. 52-68.
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abstract = "Previous numerical studies have shown that the ‘ultimate regime of thermal convection’ can be attained in a Rayleigh–B{\'e}nard cell when the kinetic and thermal boundary layers are eliminated by replacing both lateral and horizontal walls with periodic boundary conditions (homogeneous Rayleigh–B{\'e}nard convection). Then, the heat transfer scales like Nu~Ra$_{1/2}$ and turbulence intensity as Re~Ra$_{1/2}$, where the Rayleigh number Ra indicates the strength of the driving force (for fixed values of Pr, which is the ratio between kinematic viscosity and thermal diffusivity). However, experiments never operate in unbounded domains and it is important to understand how confinement might alter the approach to this ultimate regime. Here we consider homogeneous Rayleigh–B{\'e}nard convection in a laterally confined geometry – a small-aspect-ratio vertical cylindrical cell – and show evidence of the ultimate regime as Ra is increased: in spite of the lateral confinement and the resulting kinetic boundary layers, we still find Nu~Re~Ra$_{1/2}$ at . Further, it is shown that the system supports solutions composed of modes of exponentially growing vertical velocity and temperature fields, with Ra as the critical parameter determining the properties of these modes. Counter-intuitively, in the low-Ra regime, or for very narrow cylinders, the numerical simulations are susceptible to these solutions, which can dominate the dynamics and lead to very high and unsteady heat transfer. As Ra is increased, interaction between modes stabilizes the system, evidenced by the increasing homogeneity and reduced fluctuations in the root-mean-square velocity and temperature fields. We also test that physical results become independent of the periodicity length of the cylinder, a purely numerical parameter, as the aspect ratio is increased.",
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Axially homogeneous Rayleigh-Bénard convection in a cylindrical cell. / Schmidt, L.E.; Calzavarini, E.; Lohse, Detlef; Toschi, F.; Verzicco, Roberto.

In: Journal of fluid mechanics, Vol. 691, No. 9, 2012, p. 52-68.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Axially homogeneous Rayleigh-Bénard convection in a cylindrical cell

AU - Schmidt, L.E.

AU - Calzavarini, E.

AU - Lohse, Detlef

AU - Toschi, F.

AU - Verzicco, Roberto

PY - 2012

Y1 - 2012

N2 - Previous numerical studies have shown that the ‘ultimate regime of thermal convection’ can be attained in a Rayleigh–Bénard cell when the kinetic and thermal boundary layers are eliminated by replacing both lateral and horizontal walls with periodic boundary conditions (homogeneous Rayleigh–Bénard convection). Then, the heat transfer scales like Nu~Ra$_{1/2}$ and turbulence intensity as Re~Ra$_{1/2}$, where the Rayleigh number Ra indicates the strength of the driving force (for fixed values of Pr, which is the ratio between kinematic viscosity and thermal diffusivity). However, experiments never operate in unbounded domains and it is important to understand how confinement might alter the approach to this ultimate regime. Here we consider homogeneous Rayleigh–Bénard convection in a laterally confined geometry – a small-aspect-ratio vertical cylindrical cell – and show evidence of the ultimate regime as Ra is increased: in spite of the lateral confinement and the resulting kinetic boundary layers, we still find Nu~Re~Ra$_{1/2}$ at . Further, it is shown that the system supports solutions composed of modes of exponentially growing vertical velocity and temperature fields, with Ra as the critical parameter determining the properties of these modes. Counter-intuitively, in the low-Ra regime, or for very narrow cylinders, the numerical simulations are susceptible to these solutions, which can dominate the dynamics and lead to very high and unsteady heat transfer. As Ra is increased, interaction between modes stabilizes the system, evidenced by the increasing homogeneity and reduced fluctuations in the root-mean-square velocity and temperature fields. We also test that physical results become independent of the periodicity length of the cylinder, a purely numerical parameter, as the aspect ratio is increased.

AB - Previous numerical studies have shown that the ‘ultimate regime of thermal convection’ can be attained in a Rayleigh–Bénard cell when the kinetic and thermal boundary layers are eliminated by replacing both lateral and horizontal walls with periodic boundary conditions (homogeneous Rayleigh–Bénard convection). Then, the heat transfer scales like Nu~Ra$_{1/2}$ and turbulence intensity as Re~Ra$_{1/2}$, where the Rayleigh number Ra indicates the strength of the driving force (for fixed values of Pr, which is the ratio between kinematic viscosity and thermal diffusivity). However, experiments never operate in unbounded domains and it is important to understand how confinement might alter the approach to this ultimate regime. Here we consider homogeneous Rayleigh–Bénard convection in a laterally confined geometry – a small-aspect-ratio vertical cylindrical cell – and show evidence of the ultimate regime as Ra is increased: in spite of the lateral confinement and the resulting kinetic boundary layers, we still find Nu~Re~Ra$_{1/2}$ at . Further, it is shown that the system supports solutions composed of modes of exponentially growing vertical velocity and temperature fields, with Ra as the critical parameter determining the properties of these modes. Counter-intuitively, in the low-Ra regime, or for very narrow cylinders, the numerical simulations are susceptible to these solutions, which can dominate the dynamics and lead to very high and unsteady heat transfer. As Ra is increased, interaction between modes stabilizes the system, evidenced by the increasing homogeneity and reduced fluctuations in the root-mean-square velocity and temperature fields. We also test that physical results become independent of the periodicity length of the cylinder, a purely numerical parameter, as the aspect ratio is increased.

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KW - METIS-285893

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DO - 10.1017/jfm.2011.440

M3 - Article

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JO - Journal of fluid mechanics

JF - Journal of fluid mechanics

SN - 0022-1120

IS - 9

ER -