1. A self-consistent Lattice Boltzmann Model for the compressible Rayleigh-Bénard problem Andrea Scagliarini Department of Physics, University of Rome “Tor Vergata” and INFN IS TV62 “Particles and Fields in Turbulence”

2. In collaboration with: Roberto Benzi (Rome) Luca Biferale (Rome) Hudong Chen (Boston) Xiaowen Shan (Boston) Mauro Sbragaglia (Rome) Sauro Succi (Rome) Federico Toschi (Eindhoven)

3. Outline Why studying the compressible Rayleigh-Bénard convection. Kinetic theory and the Lattice Boltzmann (LB) Equation. Failure of standard LB models with thermal flows. LB Equation with consistent energy/temperature dynamics (definition of effective equilibria in terms of locally “shifted” thermohydrodynamic fields). Rayleigh-Bénard convection in a perfect gas: non Oberbeck- Boussinesq (NOB) and compressibility effects. Perspectives (massive simulations on dedicated architectures; determination of strong NOB effects on the heat transfer in the highly turbulent regime; the route to non-ideality).

4. Rayleigh-Bénard (RB) convection: what? A Rayleigh-Bénard system is a layer of fluid under gravity heated from below (H. Bénard, 1901). The dynamic behaviour is determined by the geometry, the physical properties of the fluid and the temperature difference between top and bottom. Instance of out of equilibrium driven system, showing pattern formation and self-organization Bénard cells

5. Rayleigh-Bénard (RB) convection: where? Thermal convection plays a crucial role in the heat transfer mechanism… …in the ocean (causing oceanic currents), earth mantle (thus being in the determinant in the terrestrial magnetic dipole reversal). in the atmosphere, in stars,

6. HYDRODYNAMIC NAVIER-STOKES KINETIC BOLTZMANN EQUATION MICROSCOPIC MOLECULAR DYNAMICS Chapman-Enskog perturbative expansion ? (particle-particle interactions) (Particles p.d.f.) (continuum description) The 10 orders of magnitude hierarchy

7. Brief overview of (continuum) kinetic theory The central quantity in kinetic theory is the probability density function whose evolution is described by the Boltzmann Equation The moments (in the velocities) of the pdf correspond to the hydrodynamic fields: density velocity temperature collision operator “local” equilibrium ( )

8. Lattice BGK Boltzmann Equation Approximations: 1)Linear collision operator (BGK approximation); 2)Discretization of physical space; 3)Discretization of velocity space (the very strong one!) (!!!) The system is NOT, in general, incompressible! perfect gas equation of state Chapman- Enskog expansion

9. Standard LBGK (hydrodynamic limit) The Chapman-Enskog expansion of the Lattice Boltzmann equation in the BGK approximation leads to the Navier-Stokes equations with the ideal gas equation of state (???) where What is, then, the temperature? “Many authors have introduced different quantities called temperatures which do not coincide with the true temperature in the sense of thermodynamics and statistical mechanics…” (M. Ernst, 1991) variance of the velocity probability density function

10. Projection procedure onto Hermite basis (how to treat an added body-force) Boltzmann BGK equation with external/internal force Expansion of the distribution function on the basis of the Hermite polynomials in veocity space! Who is this guy on the lattice ??? Implementation of the body-force term (Shan et al., J. Fluid. Mech. 550, 413 (2006))

11. Discretizing the velocity space The moments of the distribution function are uniquely determined by the coefficients of the Hermite expansion The moments of the distribution function are exactly preserved up to the Nth order if one truncates the higher order terms in the series expansion The integrals are evaluated via a Gauss-Hermite quadrature where w a and  a are respectively the weights and abscissae of the quadrature

12. “Effective” equilibria The body-force term so formulated can be absorbed into local equilibria, thus obtaining a BGK equation without forcing but with effective equilibria: Mandatory on the lattice!! Examples: External body-force (e.g.: gravity): Rayleigh-Bénard systems/Rayleigh-Taylor instabilities Internal force from pseudo-potentials (Shan-Chen model): multiphase fluids (Shan et al., J. Fluid. Mech. 550, 413 (2006)) (Sbragaglia et al., submitted to J. Fluid. Mech.(2009))

13. Case study: Compressible Rayleigh-Bénard convection Most of studies on the RB convection are tipically carried out for incompressible flows under the Oberbeck-Boussinesq (OB) approximation (that is assuming that the transport coefficients are all constant and that the temperature dependence of the density is linearized in the buoyancy force) Ideal gas equation of state! Important effects due to the compressibility. Strongly NOB convection. QUESTIONS: How to determine these effects? How to control them (if possible)?

14. Transition to convection The dimensionless controlling parameter is the Rayleigh number Conductive Convective Turbulent state state convection The response of the system to the increase of the Rayleigh number can be checked by measuring the Nusselt number it is the dimensionless heat flux!

15. NOB and compressibility effects ( * Nie et al., PRE 77, 035701 (2008)) ( ** Spiegel, Ap. J. 141, 1068 (1965) and Gough et al., Ap. J. 206, 536 (1976)) The theoretical prediction results from a linear stability analysis of the hydrodynamic equations **. Data from numerical simulations based on a 2D LB algorithm with 37 lattice speeds *. The departure from the OB regime can be controlled by the so called depth parameter

16. What else? Achieving a clear understaning of what are precisely the effects of having an intrinsically compressible system on the heat transfer.

17. Performing massive simulations (on large sized domains) in order to study the Rayleigh-Taylor instability A denser fluid is, in some way, accelerated against a lighter one… (for more details, see next talk!!) A strinking example is provided by an astrophysical phenomenon: the explosion of a Supernova. Simulations will be carried out on a large cell-based parallel cluster with APE-like topology. (In collaboration with the INFN section of Ferrara) “Crab Nebula”

18. Studying convection in non-ideal systems, which may undergo phase transitions… …the discovery of hot (boiling) water!!!

19. Thanks for your attention!!!