Numerical simulations of Rayleigh-Bénard systems with complex

boundary conditions

XXXIII Convegno di Fisica Teorica - Cortona

Patrizio RipesiIniziativa specifica TV62 “Particelle e campi in fluidi complessi”Department of Physics, INFN University of Rome “Tor Vergata”

In collaboration with Luca Biferale & Mauro Sbragaglia

Outline

• Complex Rayleigh-Bénard convection: why ?

• Transition to steady convection (theory and numerics)

• Kinetic theory and Lattice Boltzmann model (LBM)

• Turbulent regimes with mixed boundary conditions

• Conclusions and perspectives

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“Classic” Rayleigh-Bénard systems

A Rayleigh-Bénard system is a layer of fluid subject to an external gravity field placed between two plates, heated from below and cooled from above. The dynamic behavior is determined by the geometry, the temperature difference and the physical properties of the fluid.

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Bénard cells

[Chandrasekhar, 1961]

Tup

Tdown

gH

L

ΔT=Tdown-Tup , α=thermal expansion coefficient ν=viscosity, κ=thermal diffusivity

“Classic” Rayleigh-Bénard systems

What is the dependence of Nu on Ra?

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Conductive state

Convective state

Turbulent convection[Lathrop et al, 2000]

Rac

“Classic” Rayleigh-Bénard systems

What is the dependence of Nu on Ra?

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Turbulent convectionConvective plumes [Sugiyama et al.,2007]

[Lathrop et al, 2000]

“Complex” Rayleigh-Bénard systems

Considering a Rayleigh-Bénard system with an insulating lid on the upper boundary. What happens into the bulk region?

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Heat transfer mechanism from bottom to up?

“Complex” Rayleigh-Bénard systems: why?

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Ice-insulating effect on the Deep Water formation (part of the

thermohaline circulation)

Continental-insulating effect on the Earth Mantle Convection

The equations for a slightly compressible flows ( ρ ≈ const ) are described by

Solving for the static case, we need to solve the problem

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L

2L1

H z

x

ξ=2L1/L insulating fraction

The static solution: analytical approach

Looking for a solution of the form

where and is a periodic function along x, we have

where the aj are fixed by the boundary conditions

19/04/23 P. Ripesi - Cortona 2012 9[Sneddon, 1966]

Fourier series

Dual Series problem

The static solution: analytical approach

Why numerical simulations?

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• All data are available for each time step• Fine resolution between motion scales

[Ahlers, 2008]

Numerics: a little bit of Kinetic theory…

The main feature of the Kinetic theory is the formulation of an equation (called the Boltzmann equation) which describe the evolution for the single particle distribution function (pdf) f(ξ,x,t)

The momenta (in velocity spaces) of the pdf give to the hydrodynamical fields:

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Collision operator into the BGK approximation

Local equilibrium distribution

density

velocity

temperature

The Lattice Boltzmann Model

Discretized BGK Boltzmann equation

From this equation, it can be shown that by using a Chapman-Enskog expansion of the distribution function (fl = fl

(0) + εfl (1)+ ε2fl

(2)+…) where ε<<1, we can recover the

thermo-hydrodynamical equations

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perfect gas equation of state

Dramatic reduction of number of degrees of freedom

mean free path

hydro scale

Numeric (LBM) vs Theory for the static case

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z/H=1.0

z/H=0.7

z/H=0.4

z/H=0.1

ξ = 0.4 ξ = 0.8

• Perfect agreement between static dual series solution and LBM

• Deeper penetration as ξ 1

ξ 1

Tup=0.5, Tdown=1.5

Transition to the convection in the limit L<<H

Limiting temperature profile for the case L<<H:

Linear stability analysis with a renormalized basic temperature profile provides a new estimate for the critical Rayleigh number:

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Basic temperature profile renormalized by mixed

boundary conditions

ξ = insulating fraction

Turbulent regime

Numerical simulation (on massively parallel computers of CINECA&CASPUR) using LBM on a 2D domain (2080x1040) at Ra ≈ 5x108 for various λ at ξ=0.5.

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λ = number of cells of length L

Nusselt number (Nu)must be a

constant in stationary system

Turbulent regime

Numerical simulation using LBM on a 2D domain (2080x1040) at Ra ≈ 5x108 for various λ at ξ=0.5.

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Inhomogeneity on the upper boundary causes the average temperature of the

fluid to increase with time

λ = number of cells of length L

Turbulent regime

λ=1, 2080x140

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Increasing of temperature localized in the central region

Conclusions & Perspectives

• Insulating lid on cold boundary can alter the classic RB convection, leading to an increasing of the bulk temperature of the fluid depending on size (ξ) and wave-number (λ) of the lids

• How the global heat transfer (Nu) is affected by changing ξ and λ for different Ra ? Ongoing work

• 3D numerical simulations of a case of geophysical interest ( like ice and Deep-Water formation) Planned

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Thanks for your attention!!!!

References

•Ahlers G., Grossmann S., Lohse D. “Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection”. Rev. Mod. Phys., 81, 503-537, (2008).

•Chandrasekhar S.“Hydrodynamic and Hydromagnetic Stability”. Dover Pub., (1961).

•Duffy DG. “Mixed Boundary value problems”. Chapman & Hall/CRC, (2008).

•Ripesi P., Biferale L., Sbragaglia M. “High resolution numerical study of turbulent Rayleigh-Bénard convection with non-homogeneous boundary conditions, using a Lattice Boltzmann Method”. in preparation.

•Sneddon I. “Mixed boundary value problems in potential theory”. North-Holland Pub. Co., (1966).

•Sugiyama K., Calzavarini E., Grossmann S., Lohse D. “Non-Oberbeck-Boussinesq effects in two-dimensional Rayleigh-Bénard convection in glycerol”. Europhys. Lett., 80,(2007).

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