© Crown copyright Met Office Turbulent dispersion: Key insights of G.I.Taylor and L.F.Richardson and developments stemming from them Dave Thomson, 17 th.

© 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?


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!


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


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


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


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


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


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.


Convective boundary layer

See:Deardorff & Willis – water tank resultsde Baas, Nieuwstadt & van Dop – Lagrangian model simulations


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


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)


“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


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


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”


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


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


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

© Crown copyright Met Office Turbulent dispersion: Key insights of G.I.Taylor and L.F.Richardson and developments stemming from them Dave Thomson, 17 th.

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