Experiments on turbulent dispersion

P Tabeling, M C Jullien, P Castiglione

ENS, 24 rue Lhomond, 75231 Paris (France)

Outline

• 1 - Dispersion in a smooth field (Batchelor regime)

• 2 - Dispersion in a rough field (the inverse cascade)

Important theoretical results have been obtained in the fifties, sixties, (KOC theory, Batchelor regime,.. )

In the last ten years, theory has made important progress for the case of rough velocity fields essentially after the Kraichnan model (1968) was rigorously solved (in 1995 by two groups)

In the meantime, the case of smooth velocity fields, called the Batchelor regime, has been solved analytically.

Experiments on turbulent dispersion have been performed since 1950, leading to important observations such as scalar spectra, scalar fronts,...

- However, up to recent years no detailed :- Investigation of lagrangian properties, pair or multipoint statistics- Reliable measurement of high order statistics

In the last years, much progress has been done

IB

Magnet

Principle of the experiment

2 fluid layers, salt and Clear water

U

The experimental set-up

15 cm

5- 8 mm

Is the flow we produce this way two-dimensional ?

- Stratification accurately suppresses the vertical component (measured as less than 3 percents of the horizontal component)

- The velocity profile across the layer is parabolic at all times and quickly returns to this state if perturbed (the time constant has been measured to be on the order of 0.2s)

- Under these circumstances, the equations governing the flow are 2D Navier Stokes equations + a linear friction term

- Systematic comparison with 2D DNS brings evidence the system behaves as a two-dimensional system

Part 1 : DISPERSION IN A SMOOTH VELOCITY FIELD

FORCING USED FOR A SMOOTH VELOCITY FIELD

A typical (instantaneous) velocity field

Velocity profile for two components, along a line

U smooth- U can be expanded in Taylor series everywhere (almost)- The statistical statement is :

(called structure function of order 2)This situation gives rise to the Batchelor regime

U roughStructure functions behave as

<(U(x+r) −U(x))2 >~r2

<(U(x+r) −U(x))2 >~ra with a<2

A way to know whether a velocity field is smooth or rough, is to inspect the energy spectrum E(k)

E(k) ~k−β

If < 3 then the field is roughIf > 3 then the field is smooth

This is equivalent to examining the velocity structure function,

<u(x+r) −u(x) >~ra

For which the boundary between rough and smooth is a=1

CHARACTERISTICS OF THE VELOCITY FIELD(GIVING RISE TO BATCHELOR REGIME)

10-4

10-3

10-2

10-1

100

101

1 10k (cm -1 )

-3

kF k

l

k

Energy Spectrum 2D Energy spectrum

RELEASING THE TRACER

Drop of a mixture of fluoresceindelicately released on the free surface

Evolution of a drop after it has been released

CHARACTERIZING THE BATCHELOR REGIME

There exists a range of time in whichstatistical properties are stationary

Turbulence deals with dissipation : something is injected at large scales and ‘ burned ’ at small scales; in between there is a self similar range of scales called « cascade »

The rule holds for tracers : the dissipation is

χ=D< ∇C∫2dxdy>

In a steady state, is a constant

DISSIPATION AS A FUNCTION OF TIME

0

1

2

3

4

5

6

0 5 10 15 20 25 30 35 40t (s)

TWO WORDS ON SPECTRA...

The spectrum Ec(k) is related to the Fourier decomposition of the field

Its physical meaning can be viewed through the relation

<(C(x,y)−<C>)2 >= EC(k)dk0

∞

∫

They are a bit old-fashioned but still very useful

0.01

0.1

1

10

100

1000

1 10

Eθ

( )k

(k cm-1)

110100

1 10

*k Eθ( )k

SPECTRUM OF THE CONCENTRATION FIELD

2D Spectrum

Does the k -1 spectrum contain much information ?

C=1

r

C=0

< ΔCr( )p

>=<C(x+r)−C(x)( )p>~r0

E(k) ~k−1

GOING FURTHER…. HIGHER ORDER MOMENTS

In turbulence, the statistics is not determined by the second order moment only (even if, from the practical viewpoint, this may be often sufficient)

Higher moments are worth being considered, to test theories, and to better characterize the phenomenon.

A central quantity :Probability distribution function (PDF) of the increments

r

C1C2

ΔCr =C1 −C2

The pdf of Cr is called :P(Cr)

Taking the increment across a distance r amounts to apply a pass-band filter, centered on r.

r

Cr

10-5

0.0001

0.001

0.01

0.1

1

-10 -5 0 5 10

C

PDF for r = 0.9 cm

10-6

10-5

0.0001

0.001

0.01

0.1

1

-6 -4 -2 0 2 4 6

C

PDF for r = 11 cm

Two pdfs, at small and large scale

10-5

0.0001

0.001

0.01

0.1

1

-4 0 4 8 12Θ/σ

10-50.00010.0010.010.1

-15-10-5051015

PDF OF THE INCREMENTS OF CONCENTRATIONIN THE SELF SIMILAR RANGE

Structure functions

Sp =<(C(x+r)−C(x))p >=<(ΔCr)p >

= (ΔCr)pp(ΔCr)d(ΔCr )∫

The structure function of order p is the pth-moment of the pdf of the increment

10

100

1000

104

105

106

0.01 0.1 1 10

Sp(r)

r (cm)

p=4

p=2

p=6

p=8

0.1 1S2(r)

r (cm)

STRUCTURE FUNCTIONS OF THE CONCENTRATION INCREMENTS

TO UNDERSTAND = SHOW THE OBSERVATIONS CAN BE INFERRED

FROM THE DIFFUSION ADVECTION EQUATIONS

The answer is essentially YES, after the work byChertkov, Falkovitch, Kolokolov, Lebedev, Phys Rev E54,5609 (1995)

- k-1 Spectrum- Exponential tails for the pdfs- Logarithmic like behaviour for the structure functions

DO WE UNDERSTAND THESE OBSERVATIONS ?

CONCLUSION : THEORY AGREES WITH EXPERIMENT

A PIECE OF UNDERSTANDING, CONFIRMED BY THE EXPERIMENT, IS OBTAINED

HOWEVER, THE STORY IS NOT FINISHED

0

2

4

6

8

10

0.0 2.0 4.0 6.0 8.0 10

Isolated pair

x (cm)

0.1

1

10

0.0 2.0 4.0 6.0 8.0 10

t(s)

The life of a pair of particles released in the system

How two particles separate ? exponentially, according to the theory

6.5

7

7.5

8

8.5

9

6.5 7.0 7.5 8.0 8.5 9.0

Isolated pair

x (cm)

Blow-up of the previous figure : the first four seconds

0

5

10

15

20

0.0 2.0 4.0 6.0 8.0 10 12

time

0.01

0.1

1

10

100

0.0 2.0 4.0 6.0 8.0 10 12

time

Separation (squared) for 100000 pairs

LINEAR LINEAR LOG-LINEAR

C Jullien (2001)

Part 2 :DISPERSION IN THE INVERSE CASCADE

Reminding...

• We are dealing with a diffusion advection given by :

DCDt

=∂C∂t

+(u∇)C=DΔC

Two cases : u(x,t) smooth u(x,t) rough

ARRANGEMENT USED FOR THE INVERSE CASCADE

A typical instantaneous velocity field

l

2l

4l

Cartoon of the inverse cascade in 2D

vorticity

streamfunction

Energy spectrum for the inverse cascade

0.1

1

10

0.1 1

E ( k , t )

k / 2π (cm-1

)

( a )

Slope -5/3

injectiondissipation

2D spectrum

Evolution of a drop released in the inverse cascade

Evolution of a drop in the Batchelor regime

How do two particles separate ?

6

8

10

12

14

16

6 8 10 12 14 16x (cm)

Averaged squared separation with timein the inverse cascade

Slope 3

0.001

0.01

0.1

1

10

100

0. 1 10 100( )time s

Why the pairs do not simply diffuse ?

li

X(N) = Xii=1

N

∑

Central limit theorem : the squared separation grows as t2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0τ/t

R(t,τ

)/R(t,0)

(b)

Pairs remember about 60% of their past common life

10-5

10-4

10-3

10-2

10-1

100

101

0 2 4 6 8 10r (cm)

(a)

Lagrangian distributions of the separations

t=1s

t=10 s

10-5

10-4

10-3

10-2

10-1

100

101

0 5 10 15 20s=r/σ

( )b

The same, but renormalized using the r.m.s

Pair separation of particles in turbulence

- An old problem…. starting with Richardson in 1926- Tackled by him, Batchelor, Obukhov, Kraichnan,…- Several predictions for pair distributions- Essentially no reliable data for a long time

- Accurate data obtained only a few years ago- Surprisingly close to Kraichnan model  prediction

How do triangles evolve in the turbulent field ?

9

10

11

12

13

14

15

10 11 12 13 14 15 16

P Castiglione (2001)

How to characterize triangles ?

r1

r2 r3

e1 =r1 −r2

2;e2 =

2r3 −r1 −r26

Introduce

Define the area :R=e1

2 +e22

Introduce shape parameters :

w=2e1 ×e2

R2

χ=12Arctg

2e1e2

e12 −e2

2

⎡

⎣ ⎢

⎤

⎦ ⎥

0 /12 /6

w

0

1

Different configurations mapped by w,

Distribution of the shape parameters of 100000 triangles released in the flow

P Castiglione

Coming back to the dispersion of a blob

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30 35

t (s)

A «QUASI-STEADY » STATE CAN BE DEFINED

Spectrum of the concentration field

0.01

0.1

1

10

100

1000

0.1 1 10/2k (cm-1)

Slope -5/3

2D SpectrumSpectrum

Is the k-5/3 spectrum a big surprise ?

It was given in the fifties by Kolmogorov Corrsin Obukhov, giving rise to the KOC theory

E(k) ~k−5/3

It has been observed by a number of investigators, during the last four decades, in 3D

PDF of the concentration increments in the inertial range

r=7cm

r=1cm

10

100

1000

104

105

106

0.1 1 10 100

r (cm)

n=2

n=4

n=6

n=8

n=10

Structure functions of the concentration field

Sn(r)

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14

ξn

n

K 41

Exponents of the structure functions

The exponents tend to saturate

About the saturation

- Remarkable phenomenon, discovered a few years ago- One conjectures it is a universal phenomenon- It is linked to the presence of fronts- It is linked to the clusterization of particles

C=1 C=0

r

<(C(x+r)-C(x))p>=Nr0

DO WE UNDERSTAND THE OBSERVATIONS ?

Kraichnan model provides a framework for interpreting most of the observations, i.e :

- Existence of deviations from KOC theory- Saturation of the exponents- Presence of fronts- Clusterization of triads- Form of the distributions of pairs

CONCLUSION

Progress has been made, leading to new theories whose relevance to the « real world » has been shown

These theories explain a set of properties which constrain the concentration field

High order moments, multipoint statistics is no more a terra incognita, and we might encourage investigators to more systematically consider these quantities so as to better characterize their system.