what happens when ra grows large: turbulent convection statistical approach
DESCRIPTION
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 ). - PowerPoint PPT PresentationTRANSCRIPT
What happens when Ra grows large:Turbulent convection
Statistical approach
Dynamics of individual (coherent) plumes
Turbulent convection (=0.71, a=2, Ra=107)
Turbulent convection (=0.71, a=2, Ra=107)
Turbulent convection (=0.71, a=2, Ra=107)
Turbulent convection (=0.71, a=2, Ra=107)
Turbulent convection:
Statistical properties andtransition from soft to hard turbulence
Scaling of the heat transport:Nu vs Ra
Nu =Qtotal / Qconduction= 1 + < w >
Turbulent convection
Turbulent convection
Turbulent convection
Turbulent convective plumes
Clustering of plumes andformation 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 inturbulent 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)
Coarsening of the plume pattern:
Coarsening of the plume pattern:
Coarsening of the plume pattern:
Coarsening of the plume pattern:
The coarsening is due toclustering of convective plumes:
Turbulent RB convection undergoesa process of energy transfer
from the scales of the linear instabilityto the largest scales (box size).
At later times,the system becomes statistically stationary.
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
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
Is there an upper scale where the clustering is arrested ?
How does the clustering depend on theRayleigh number ?
Is there an upper scale where the clustering is arrested ?
Is there an upper scale where the clustering is arrested ?
Dependence on the Rayleigh number
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 )
In the fully turbulent regime, the system recovers a “statistical” roll pattern
Convection has still a lot to teach:
Effect of rotation
Transport
Predictability
Simplified models of moist convection
A summary of NS and fully-developed turbulence
(incompressible, homogeneous flow)
€
∇⋅r
u = 0
∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −∇p + ν 0∇
2 r u
Non dimensional version
€
∇⋅r
u = 0
∂
∂t+
r u ⋅∇
⎛
⎝ ⎜
⎞
⎠ ⎟r u = −∇p +
1
Re∇ 2 r
u
Reynolds number : Re =UL
ν 0
A summary of NS and fully-developed turbulence
(incompressible, homogeneous flow)
In the limit :
formation of boundary layers
€
Re → ∞
Far from boundary layers:
inertial range(Kolmogorov 1941,
or K41)
€
ετ
=u3
l= constant → u ≈ l1/ 3
E(k)dk ≈ u2 ≈ l2 / 3
k ≈1/ l
E(k) ≈ k−5 / 3
log k
log E(k)
k-5/3
E
Direct energy cascadefrom large
to small scales