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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