1

The Problem

of

Turbulent

Dispersion

Turbulence in Fluids

Benoit Cushman-RoisinThayer School of Engineering

Dartmouth College

OUTLINE

The nature of turbulent dispersion

Problems with the eddy-diffusivity approach

Construction of an alternative model

Checking the new model

2

Idea of diffusion:

Flux from left to right:

Flux from right to left:

Net flux:

1cuq

2cuq

)( 21 ccuqqq

Limit of small x:

dx

dcD

x

xxcxcxu

ccuq

xx

x

)()(lim)(lim

)]([lim

00

210

where )(lim0

xuDx

This is Fick’s first law.

3

Adolf Eugen Fick (1829-1901)German physiologistinventor of contact lensesafter whom Fick’s Law of diffusion is named.

Adolf Fick’s original diffusion experiment

4

For molecular diffusion:

D = u’ with

u’ = thermal velocity of molecules(Brownian motion)

= mean free path (average distance between consecutive collisions)

Adolf Fick (1855)Robert Brown (1827)Albert Einstein (1905)

)(lim0

xuDx

In theory:

And for turbulent diffusion?

Simply imagine that turbulence acts like molecular agitation but in a more vigorous way.

Dmolecular → Deddy

with Deddy >> Dmolecular

Concept of eddy diffusivitydue to Joseph Boussinesq (1877)

5

and write:

D = u* with

u* = turbulent velocity

= turbulent eddy diameter(mixing length)

Concept of mixing length due to Ludwig Prandtl (1925)

)(lim0

xuDx

To construct an eddy diffusivity?

Start again from

One-dimensional system:

Budget:2

2

x

cD

t

c

x

q

t

c

Solution for a localized and instantaneous release:

where M is the mass released at t = 0 and at x = 0.

This is called Fick’s second law.

tD

x

tD

Mtxc

4exp

4),(

2

6

Dt

x

Dt

Mtxc

4exp

4),(

2

2

2

max 2exp

x

c

c

with

DtDttD

Mtc 2 ,2,

4)( 2

max

or

Diffusionexperimentby injection of SF6 in theHudson River

… fitting 2

proportional to the first power of time

(Clark et al., Envl. Sci. Tech., 30, 1527-1532, 1996)

7

cm/s11.1

m/day 958

t

A parabola fits better

… and the power,raised from 1 to 2, may not even be enough!

Turbulent jet

x

~ x

… looks more like a cone than a paraboloid

8

9

Smokestack plumes tend to grow linearly with downwind distance.

Smokestack dispersion in the horizontal, cross-wind direction

10

Flow over a flat plate:

Laminar flow,

Turbulent flow over smooth plate:

Turbulent flow over rough plate:

xxx x

)(Re

91.4

Ux

x Re

085.01085.0

)(Re

079.0 xxx x

xx

x

z

dx

d o

)(constant

log89.17

365.2

2.012.0

)(Re

37.0 xxx x

:10Re10 75 x

:Re107x

:10Re 5x

Conclusion:

There is systematic evidence that turbulent diffusion creates spreading at a rate proportional to time (or distance in the presence of entraining fluid motion.)

This is contrary to the prediction of a diffusivity model, for which spreading proceeds proportionally to the square root of time (or distance).

11

Actually, there are two bothersome elements about eddy diffusivity:

1. The limit is spurious because there is a whole range of values for u’ and x.

2. The patch size increases more like t (or x) than like t1/2 (or x1/2).

)(lim0

xux

Further ammunition:

Lewis Fry Richardson and Henry Stommel intheir famous “parsnip” paper”, J. Meteorol., 1948

“The variation of K depends on a geometrical quantity , and Fick’s equation is also geometrical in so far as it contains ∂2/∂x2. For this reason it is difficult to regard the variation of K as an outer circumstance detached from Fick’s equation. There appears to be a fault in the equation itself.”

(The subject of this quote is diffusion in the atmosphere and ocean, K is the diffusivity, and is the cluster size.)

12

HEURISTIC APPROACH to obtain a model that

predicts spreading at the rate of

instead of

t

t

Dt

tDDt

tt

2 2

22

In other words, the diffusivity D needs to grow proportionally to the length scale.

Dispersion in the upper mixed layer of the ocean

D ~ size

D ~ sizeholds across 4 decades.

13

Dispersion in the upper mixed layer of the ocean

D ~ size

Note:Many argue that

D ~ ℓ4/3

instead ofD ~ ℓ1

but there is no reason why systems at vastly different scales should all possess the same coefficient of proportionality between D and the power of ℓ.

From the perspective of Fourier decomposition, there is a spectrum of wavenumber values, k (= 2π / wavelength).

The preceding statement then translates into the need to have the diffusivity D being inversely proportional to the wavenumber k characterizing the patch or plume.

k

uD

kDD

k

*1scalelength

1 scalelength

where u* is a turbulent velocity scale

14

ckudt

cd

k

uD

ckDdt

cd

x

cD

t

c

ˆˆ

ˆˆ

**

22

2

d

xcxcu

t

ccku

dt

cd

2

**

)()(ˆ

ˆ

Fourier transform the budget equation:

Now, take the back Fourier transform:

then make D depend on k:

HEURISTIC APPROACH:

dtxctxcu

t

c

2* ),(),(

“Turbulent diffusion is a non local effect…, and a description of the diffusion of some kind of integral equation is more to be expected.”

We obtain an integral (non-local) equation:

George K. Batchelor and A. A. Townsend, 1956, In Surveys in Mechanics, edited by G. K. Batchelor and R. M. Davies, Cambridge University Press, page 360.

instead of the differential (local) equation: 2

2

x

cD

t

c

15

d

xcxcu

t

c

2* )()(

Solution of

corresponding to an instantaneous (t = 0) and localized (x = 0) release of mass M is

2*

2*

)(),(

tux

tuMtxc

22

2

max

),(

xc

txc

or

with tutu

Mc *

*max

We obtain indeed what we want:A patch size that grows like time !

and

… and without ever invoking a spurious limit )(lim

0xuD

x

We thus seem to have overcome our two problems.

The approach, however, was force-feeding the answer, and it would be far more satisfactory to have a more rigorous derivation of the integral term.

We should do better, and fortunately we can…

16

0

x

cu

t

c

',' uuccc

x

cu

t

c

''

3. Average the equation:

Closure problem ! Sorry, Osborne, we shall not average and then try to solve the equation but rather solve the equation first and then average.

Osborne Reynolds (1894)

The traditional approach in turbulence is to perform the so-called Reynolds decomposition:

1. Write equation:

2. Decompose variables into mean and fluctuation:

FORMAL MODEL:

0

x

cu

t

c

↑

fluctuatingturbulent velocity

Solution over a short time interval t during which u can be considered constant is:

),(),( ttuxcttxc

Then, ensemble average over the turbulent fluctuations:

duufttuxcttxc )(),(),(

(simple advance with flow)

17

duufttuxcttxc )(),(),(

duuf )(with being the probability that the turbulent fluid flow has an instantaneous velocity within the interval .],[ duuu

Thus,

duuft

txcttuxc

t

txcttxc)(

),(),(),(),(

A few algebraic manipulations:

1. The probability density function f(u) must be normalized

duuftxcttuxctxcttxc

duuf

)(),(),(),(),(

1)(

duufttuxcttxc )(),(),(

2. Divide both sides by t:

18

3. Switch from velocity u to displacement = ut :

where g(,t) = probability of jump of length in time interval t

dtg

t

txctxc

t

txcttxc),(

),(),(),(),(

Properties of probability distribution function ),( tg

1. Normalization:

2. Zero mean:

3. Divisibility:

1),(

dtg

0),(

dtg

),(),()2,( dtgtgtg

jump in 1st Δt

additionaljump in2nd Δt

because time interval t is arbitrary and limit t → 0 should not be singular

19

Dimensional analysis requires that the function g have the dimensions of 1/. Thus, the parameter t must somehow be absorbed in a dimensionless construct.

Therefore, one must interject a dimensional quantity with which a dimensionless ratio can be formed.

Three possibilities come to mind:

1. Molecular diffusivity D →

2. Turbulent velocity fluctuation u* →

3. Rate of energy cascade , à la Kolmogorov → 2/32/1 t

tD2

tu *

Divisibility demands

)(1

),(**

aGtu

tgtu

a

With

')'()'(22

1daaaGaG

aG

22

1)(

Aa

AaG

Solution is (A = arbitrary constant)

Cauchy probability distribution function

Back in terms of displacement :

2*

2*

)(

1),(

tAu

tuAtg

20

dtuA

tuA

t

txctxc

t

c

dtgt

txctxc

t

txcttxc

22*

22*1),(),(

),(),(),(),(),(

Limit t → 0 yields:

d

xcxcuA

t

c

2* )()(

We recover the model that leads to spreading proportional to time.

Use this probability density function in the governing equation:

d

xcxcuA

t

c

2* )()(

Therefore, the turbulent-dispersion equation

corresponds to dispersion accomplished by particle jumps obeying the Cauchy probability distribution function

22*

22*1

),(tuA

tuAdtg

(where the time interval t does not matter because this probability distribution is infinitely divisible).

21

Simulations of Cauchy jumps (a random walk with “heavy tail”)

Three realizations of 5000 steps of a Cauchy jumps in two dimensions. The origin of each simulation is at [0,0], and the x and ycomponents are given by x = r cos() and y = r sin(), where r follows a Cauchy distribution and follows a random distribution.

Now, that we have the jump probability distribution function, let us determine the corresponding velocity distribution.

dtuA

tAu

dtu

Gtu

dtgduuf

22*

22*

**

1

1

),()(

with tu

2*

22*

22*

222*

1)(

1)(

uAu

Auuf

duttuAtu

tAuduuf

which gives

Note : Au* is not the variance but is nonetheless related to the width of the distribution

22

If the governing equation is

')'(

),(),'(),(2

* dxxx

txctxcuA

t

txc

and the mass budget of the substance being dispersed is

x

txq

t

txc

),(),(

what should the flux q(x,t) be ?

Answer is:

"')"'(

)"()'(),(

2* dxdx

xx

xcxcAutxq

x

x

Generalization to two and three dimensions is easily accomplished.

')'(

),(),'(2

*1 dxxx

txctxcuA

t

c

''])'()'[(

),,(),','(2/322

*2 dydxyyxx

tyxctyxcuA

t

c

1D:

2D:

3D: '''])'()'()'[(

),,,(),',','(2222

*3 dzdydxzzyyxx

tzyxctzyxcuA

t

c

23

Interesting application to momentum.

For this, take c = u:

')'(

),(),'(),(0 2

* dzzz

tzutzuAu

t

tzu

in the semi-infinite domain 0 < z < ∞

The steady case is:

and its exact solution is:

')'(

)()'(0

0 2* dz

zz

zuzuuA

oz

zUzu ln)(

The well known logarithmic velocity profile of shear turbulence along a wall !

24

Recovering the turbulent stress

z

zdzdz

zz

zuzuAu0 2

* "')"'(

)"()'(

withoz

zuzu ln)( *

and2

*u , we obtain:

392.03

31

)1(

)ln(

2

1

0

1

0 2

2*2

*

AA

dbdaab

abAuu

41.0for

(Theodore von Kármán, 1934)

In conclusion, we have a new model for turbulent dispersion

')'(

),(),'(2

12

* dxxx

txctxcuA

t

c x

x

where c is the concentration of the substance, u* = turbulent (friction) velocity, and A = dimensionless constant (could go into definition of u*)

This model correctly reproduces turbulent dispersion, with spreading proportional to t, rather than t1/2.

When applied to momentum, this model also reproduces the logarithmic profile of velocity near a wall.

The model is non local. (The value at x depends on values everywhere else.)

25

Needs:

1. Verification in a variety of applications

2. Generalization to include stratification