1. Numerical simulations of Rayleigh-Bénard systems with complex boundary conditions XXXIII Convegno di Fisica Teorica - Cortona Patrizio Ripesi Iniziativa specifica TV62 “Particelle e campi in fluidi complessi” Department of Physics, INFN University of Rome “Tor Vergata” In collaboration with Luca Biferale & Mauro Sbragaglia

2. 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 13/09/20151P. Ripesi - Cortona 2012

3. “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. 13/09/2015P. Ripesi - Cortona 20122 Bénard cells [Chandrasekhar, 1961] T up T down g H L ΔT=T down -T up, α=thermal expansion coefficient ν=viscosity, κ=thermal diffusivity

4. “Classic” Rayleigh-Bénard systems What is the dependence of Nu on Ra? 13/09/2015P. Ripesi - Cortona 20123 Conductive state Convective state Turbulent convection [Lathrop et al, 2000] Ra c

5. “Classic” Rayleigh-Bénard systems What is the dependence of Nu on Ra? 13/09/2015P. Ripesi - Cortona 20124 Turbulent convection Convective plumes [Sugiyama et al.,2007] [Lathrop et al, 2000]

6. “Complex” Rayleigh-Bénard systems Considering a Rayleigh-Bénard system with an insulating lid on the upper boundary. What happens into the bulk region? 13/09/2015P. Ripesi - Cortona 20125 Heat transfer mechanism from bottom to up?

7. “Complex” Rayleigh-Bénard systems: why? 13/09/2015P. Ripesi - Cortona 20126 Ice-insulating effect on the Deep Water formation (part of the thermohaline circulation) Continental-insulating effect on the Earth Mantle Convection

8. The equations for a slightly compressible flows ( ρ ≈ const ) are described by Solving for the static case, we need to solve the problem 13/09/2015P. Ripesi - Cortona 20127 L 2L 1 H z x ξ=2L 1 /L insulating fraction The static solution: analytical approach

9. Looking for a solution of the form where and is a periodic function along x, we have where the a j are fixed by the boundary conditions 13/09/2015P. Ripesi - Cortona 20128 [Sneddon, 1966] Fourier series Dual Series problem The static solution: analytical approach

10. 13/09/2015P. Ripesi - Cortona 20129 Definingwe have that, after multiplying by and integrated on the interval, can be solved as a linear system of the type. “Slit” limit The static solution: analytical approach

11. Why numerical simulations? 13/09/2015P. Ripesi - Cortona 201210 All data are available for each time step Fine resolution between motion scales [Ahlers, 2008]

12. 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: 13/09/2015P. Ripesi - Cortona 201211 Collision operator into the BGK approximation Local equilibrium distribution density velocity temperature

13. 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 (f l = f l (0) + εf l (1) + ε 2 f l (2) +…) where ε<<1, we can recover the thermo-hydrodynamical equations 13/09/2015P. Ripesi - Cortona 201212 perfect gas equation of state Dramatic reduction of number of degrees of freedom mean free path hydro scale

14. Numeric (LBM) vs Theory for the static case 13/09/2015P. Ripesi - Cortona 201213 z/H=1.0 z/H=0.7 z/H=0.4 z/H=0.1 ξ = 0.4ξ = 0.8 Perfect agreement between static dual series solution and LBM Deeper penetration as ξ  1 ξ  1 T up =0.5, T down =1.5

15. Linear Stability analysis 13/09/2015P. Ripesi - Cortona 201214 Linear profile Linear perturbation analysis around a basic linear temperature profile eigenvalue equation for the perturbation The non-zero solution at the smallest Ra gives the onset of stationary convection

16. 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: 13/09/2015P. Ripesi - Cortona 201215 Basic temperature profile renormalized by mixed boundary conditions ξ = insulating fraction

17. Turbulent regime Numerical simulation (on massively parallel computers of CINECA&CASPUR) using LBM on a 2D domain (2080x1040) at Ra ≈ 5x10 8 for various λ at ξ=0.5. 13/09/2015P. Ripesi - Cortona 201216 λ = number of cells of length L Nusselt number (Nu)must be a constant in stationary system

18. Turbulent regime Numerical simulation using LBM on a 2D domain (2080x1040) at Ra ≈ 5x10 8 for various λ at ξ=0.5. 13/09/2015P. Ripesi - Cortona 201217 Inhomogeneity on the upper boundary causes the average temperature of the fluid to increase with time λ = number of cells of length L

19. Turbulent regime λ=1, 2080x140 13/09/2015P. Ripesi - Cortona 201218 Increasing of temperature localized in the central region

20. 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 13/09/2015P. Ripesi - Cortona 201219

21. 13/09/2015P. Ripesi - Cortona 201220 Thanks for your attention!!!!

22. 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). 13/09/2015P. Ripesi - Cortona 201221