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What happens when Ra grows large: Turbulent convection Statistical approach

What happens when Ra grows large: Turbulent convection Statistical approach Dynamics of individual (coherent) plumes. Turbulent convection ( s =0.71, a =2 p , Ra=10 7 ). Turbulent convection ( s =0.71, a =2 p , Ra=10 7 ). Turbulent convection ( s =0.71, a =2 p , Ra=10 7 ).

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What happens when Ra grows large: Turbulent convection Statistical approach

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  1. What happens when Ra grows large: Turbulent convection Statistical approach Dynamics of individual (coherent) plumes

  2. Turbulent convection (s =0.71, a=2p, Ra=107)

  3. Turbulent convection (s =0.71, a=2p, Ra=107)

  4. Turbulent convection (s =0.71, a=2p, Ra=107)

  5. Turbulent convection (s =0.71, a=2p, Ra=107)

  6. Turbulent convection: Statistical properties and transition from soft to hard turbulence Scaling of the heat transport: Nu vs Ra Nu =Qtotal / Qconduction= 1 + < wq >

  7. Turbulent convection

  8. Turbulent convection

  9. Turbulent convection

  10. Turbulent convective plumes

  11. Clustering of plumes and formation of large-scale order Large-scale wind and generation of mean shear (rectification process, k=0): Krishnamurti and Howard (1981) Howard and Krishnamurti (1986) Massaguer, Spiegel and Zahn (1992) Large-scale wind leads to plume clustering: Heslot et al (1987) Kadanoff (2001) Instability of the long-wave modes in turbulent convection: Elperin et al (2003) Numerical simulation with periodic b.c.: No k=0 mode Hartlep, Tilgner and Busse (2003) Parodi, von Hardenberg, Passoni, Provenzale, Spiegel, PRL (2004), PLA (2008)

  12. Coarsening of the plume pattern:

  13. Coarsening of the plume pattern:

  14. Coarsening of the plume pattern:

  15. Coarsening of the plume pattern:

  16. The coarsening is due toclustering of convective plumes:

  17. Turbulent RB convection undergoesa process of energy transferfrom the scales of the linear instabilityto the largest scales (box size).At later times,the system becomes statistically stationary.

  18. It is not a mean shear ( k = 0 )but rather a circulation at the largest scales (k=1)The large-scale structures areclusters of individual plumes

  19. The large-scale wind in closed containeris probably due to the same process,with the energy piling up at the k=1/2 mode It is not the large-scale circulation that generates plume clustering but viceversa

  20. Is there an upper scale where the clustering is arrested ?How does the clustering depend on theRayleigh number ?

  21. Is there an upper scale where the clustering is arrested ?

  22. Is there an upper scale where the clustering is arrested ?

  23. Dependence on the Rayleigh number

  24. What causes the clustering ? Option 1: attraction of same-sign plumes Option 2: the interaction of the lower and upper boundary layers by the agency of plumes Other view:The fixed-flux instability of a coarse-grained field( with Reff << R )

  25. In the fully turbulent regime, the system recovers a “statistical” roll pattern

  26. Convection has still a lot to teach: Effect of rotation Transport Predictability Simplified models of moist convection

  27. A summary of NS and fully-developed turbulence (incompressible, homogeneous flow) Non dimensional version

  28. A summary of NS and fully-developed turbulence (incompressible, homogeneous flow) In the limit : formation of boundary layers

  29. Far from boundary layers: inertial range (Kolmogorov 1941, or K41) log E(k) Direct energy cascade from large to small scales k-5/3 E log k

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