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ENVIRONMENTAL FLUID MECHANICS

Turbulence

Benoit Cushman-RoisinThayer School of Engineering

Dartmouth College

Hand sketch of turbulent flow by Leornado da Vinci (1452-1519)

Homogeneous and Isotropic Turbulence

Turbulent flow field seen as a population of eddies of various sizes and orbital speeds

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Distribution of turbulent vortices in homogeneous & isotropic turbulence

Assumption has been made that all turbulent eddies of same diameter d share the same orbital velocity u.

There exists therefore a function .

o

( )u do

Concept of the energy cascade

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Kolmogorov Eddy Cascade Theory:

Essentially a dimensional argument

o

The driver of turbulence is the energy passing through the cascade, which is energy per time and per mass (units: W/kg = m2/s3).

The only way to make the velocity (units m/s) be a function of d (units m) is by the following power law:

with A being a universal dimensionless constant (with experimental value close to 1).

1/3( )u d A d

( )u do

The observed –5/3 energy spectrum decay agrees with the Kolmogorov theory. C = 1.5

Plotted horizontally is the wavenumber k of the observed fluctuations (from a Fourier decomposition), rendered dimensionless.

Plotted vertically is the energy E per mass per wavenumber, suitably made dimensionless.

Turbulent energy spectrum for homogeneous and isotropic turbulence.

Dimensions are:k → m-1

E → J/kg per k = m3/s2

–5/3 power law

2/3 5/3( )E k C k

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Shear Turbulence

Typical velocity distributions across realistic channels.Velocities are largest away from the bottom and sides.

Turbulent flow along a wall

Side viewWall is the bottom boundary.Flow is from left to right.

Top viewHydrogen bubbles are generated at line on left, and flow proceeds to the right.

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http://www.cfm.brown.edu/people/cait/near.wall.html

Hairpin vortex in shear turbulence in proximity of a wall

Evidence of logarithmic velocity profile

Define friction velocity from bottom stress:

*bu

Dimensional analysis requires:

*

*

( )u z u zf

u

Observations reveal that the function f is the logarithm:

*

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( )2.44ln 5.2

u zu z

u

for *11u z

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Turbulent flow along a rough boundary

Velocity profile is still logarithmic but no longer dependent on the fluid’s viscosity.

Instead, a new parameter, the roughness height zo, enters the expression.

* 0

( )2.44ln

u z z

u z

Roughness heights

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Drag Coefficient

The drag coefficient CD is defined as the coefficient of proportionality between bottom stress b and square of velocity:

2uCDb

in which is the height-averaged velocity.

We can calculate the drag coefficient for the logarithmic profile along a rough wall if we also know the height h of the fluid:

u

1ln)(

1 0 ln)(

0

*

00

*

z

hudzzu

huhz

z

zuzu

h

from which follows a relation between the turbulent velocity and average velocity:

1)/ln( 0*

zh

uu

The drag coefficient follows:

2

0

2

*2

2*

2 1)/ln(

zhu

u

u

u

uC b

D

For = 0.41 and h/z0 in the range 50 – 1000, CD varies between 0.005 and 0.020.

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Eddy Viscosity & Mixing Length

In an attempt to apply the formulation of wall turbulence to more general turbulent flows, the concepts of eddy viscosity and mixing length are introduced.

The eddy viscosity is the apparent viscosity of the turbulent fluid:In analogy with molecular viscosity = (du/dz),

we write: b = T (du/dz)

For wall turbulence, we have:

z

u

dz

du

z

zuzu

ub

*

0

*

2*

ln)(

*** )(

)/(

uuzzu

dzdu

m

bT

in which T is called the eddy viscosity.

The eddy viscosity can thus be interpreted as the product of a mixing length lm with the turbulent velocity.

For wall turbulence, the mixing length is lm = z and the turbulent velocity is obtained from u*2 = b/.

The question is:

What should the mixing length and turbulent velocity be in more general turbulent flows?

Thought: Turbulence is the result of instabilities, and instabilities are caused by shear in the flow.

So, connect with the flow shear:

dz

duu

dz

duuu

dz

dummTb **

2* )(

dz

duu mmT

2*

Generalization to more general, 3D turbulent flows:

1) Define strain tensor as

i

j

j

iij x

u

x

uS

2

1and overall strain S from

3

1

3

1

22 2i j

ijSS

This strain S can be viewed as the generalization of the shear du/dz and thus:

SmT2

2) Better: Define the rate-of-rotation tensor

i

j

j

iij x

u

x

u

2

1

and overall shear from

3

1

3

1

22 2i j

ij

from which follows the alternative formulation 2mT

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And what about the mixing length?

For wall turbulence, lm = z, which suggests a geometric relation.

One possibility is: lm = times distance to closest wall.

For channel flow (with two walls, one on each side, say at y = 0 and y = W), one often takes:

W

yWym

)(