© crown copyright met office turbulent dispersion: key insights of g.i.taylor and l.f.richardson...
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© Crown copyright Met Office
Turbulent dispersion: Key insights of G.I.Taylor and L.F.Richardson and developments stemming from themDave Thomson, 17th November 2010
Royal Meteorological Society Meeting: Turbulence - a 'resolved' problem?
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Key papers
• Two papers by Taylor and Richardson from the 1920’s form the foundation for much of what we know about turbulent dispersion:
• Taylor (1921) “Diffusion by continuous movements”
• Richardson (1926) “Atmospheric diffusion shown on a distance-neighbour graph”
• Contrasting styles!
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Contents
• Why focus on turbulent dispersion?
• Setting the scene for Taylor’s and Richardson’s work
• Taylor’s (1921) paper and ideas arising
• Richardson’s (1926) paper and ideas arising
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Why focus on turbulent dispersion?
• “The only reason we are interested in turbulence is because of its dispersive properties” – Philip Chatwin
• A (deliberatively provocative) exaggeration of course but one with some underlying truth, especially if include dispersion of momentum as well as heat & material
• Dispersion and mixing is fundamental to turbulence
• Whenever one tries to characterise what turbulence is, the ability to disperse and mix material is always a key characteristic
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Setting the scene
• Reynolds (1883, 1894)
• Statistical description, Reynolds stresses/fluxes
• Turbulence requires viscosity << velocity x length scale
• Boussinesq (1877, 1899)
• Turbulent flux ≈ “eddy-diffusivity” × gradient
• Einstein (1905) – Brownian motion
• Independent random jumps leads to flux proportional to concentration gradient
• Langevin (1908)
• Refined view of Brownian motion with continuously changing velocity
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Taylor (1921)
Instead of assuming “flux = eddy-diffusivity × gradient”, express dispersion in terms of how fluid elements move:
Hence
Note is the Lagrangian correlation function of the velocity
t
dttwtz0
')'()(
t t
dtdttwtwtz0 0
2 ''')''()'()(
)'''()''()'( ttRtwtw
tztz 2)(
t
)(tz
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Taylor (1921)
Expect correlation to decay in time, on time-scale say
This leads to
Effective eddy diffusivity is different for material of different ages
t zz
tw
KttLw 22
t L<<
t L>> )( 2LwK
tL
)()()( tRtTwtw
L
2ww
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Importance of allowing for finite time scale of velocity changes
• Two examples:
• Dispersion in convective boundary layer:
• Upwind diffusion in light winds:
• Both cases very hard to simulate with an Eulerian “fixed frame of reference” approach, but can be treated easily and naturally with a Lagrangian approach.
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Convective boundary layer
See:Deardorff & Willis – water tank resultsde Baas, Nieuwstadt & van Dop – Lagrangian model simulations
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Upwind diffusion in light winds
Material of different ages at same location makes Eulerian treatment difficult
Wind
Source
Puffs growing following Taylor (1921)
Puffs growing following a diffusion equation
– too much upwind spread for same plume growth downwind
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A Lagrangian model (“NAME”)
‘Particles’ tracked through the flow following resolved flow and modelled turbulence Turbulence represented by adding a random component of motion following ideas derived from Taylor (1921)
(Chernobyl simulation)
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“Taylor” diffusion (Taylor 1953-4)
Along-flow dispersion in a pipe (channel/river/boundary-layer) due to the variation in the mean flow:
u
time scale for mixing across pipe
Correlation time for along-flow velocity = time to mix across pipe
Hence along-flow diffusivity =
Slow mixing across pipe implies fast along pipe diffusion
In meteorology can be important in stable boundary layers
2u
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Richardson (1926)
n
nn tkttx 5cos2/1)(
“Does the wind possess a velocity? This question, at first sight foolish, improves on acquaintance … it is not obvious that Δx/Δt attains a limit as Δt → 0”
He gives an example:
In fact Richardson is wrong here – Δx/Δt can’t fluctuate wildly or there would be infinite turbulent energy
However “Does the wind possess an acceleration?” is a useful question
In the limit of small viscosity Δu/Δt blows up: position velocity acceleration
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Richardson (1926)
“The so-called constant K (the eddy-diffusivity) varies in a ratio from 2 to a billion (cm2 s-1)”
t z
• Wind tunnel • Atmosphere
Time- (or ensemble-) average picture
Growth relative to puff centroid
As cloud gets bigger everlarger eddies come into play
What was the “mean flow” becomes an “eddy”
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Richardson (1926)
Consider all pairs of “particles” in the dispersing cloud:
Distribution of pair separations seems to obey diffusion equation with separation dependent diffusivity K ~ ar4/3
r.m.s. pair separation ~ (at)3/2 for t >> r02/3/a (r0 = initial r)
distribution of pair separation r :– strongly peaked
a ~ ε1/3
Pair separation r
Mean square separation of all pairs = 2 × mean square spread of cluster
r
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Statistical structure of concentration fluctuations
• Fluctuations important for toxic/flammable/reactive materials and odours
• Depends on peaked nature of pair separation distribution
• Also depends on the way triads, tetrads etc of particles separate
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Richardson’s law:r.m.s. pair separation ~ ε1/2t3/2 for t >> r0
2/3/ε1/3
• In complex flow with chaotic trajectories: expect r ~ r0 exp(t/T)
– can make r small if r0 small enough
• For Richardson’s law, can’t reduce r by making r0 small
– trajectories cease to be deterministic (in the limit of small viscosity)
• This is the analogue for trajectories of butterfly effect for whole weather system
• Understanding this non-determinism properly requires understanding of the zero viscosity limit
– hence turbulence is not a ‘resolved’ (or resolvable) problem
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