1

Shear

Turbulence

Turbulence in Fluids

Benoit Cushman-RoisinThayer School of Engineering

Dartmouth College

Ubiquity of shear Turbulence

Water speed across channels and wind speeds over the earth’s surface exhibit a variation of velocity away from the boundary.Shear is present whenever there is flow along a boundary.

2

viscositykinematic elocity,friction v where

)(

*

*

*

wu

zuf

u

zu

In general,

boundary fromelevation

density

viscositydynamic

stress wall with ),,,()(

z

μ

zFzu wb

Dimensional analysis reduces this to:

zuf

zu *

*u

)(

Question:

What is the shape of the universal function                                   ?

To answer this, we turn to observations.

3

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.

(From

Kline et al., 1967)

Visualization by Hydrogen bubbles

Top view

(From Kline et al., 1967) (From Singer & Joslin, 1995 )

Numerical simulation

Hairpin vortices

4

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

Hairpin vortices in numerical simulations

http://www.csi.tu-darmstadt.de/

Logarithmic profile

41.0h wit

2.5ln)(

2.5ln44.2)(

***

*

*

uzuu

zu

zu

u

zu

(Source: Pope, 2000, Figure 7.7, page 275)

5

Very near the wall, flow is not turbulent but laminar: 

layerBuffer 305

sublayer Viscous 50

:separation lTraditiona

11

:atintersect

profiles log andLinear

forces)other oflack for verticalin the(constant

**

*

*

*

uz

u

uz

zu

zuzu

dz

ud

b

b

(Pope, 2000, page 276)

6

zu

y *

zu

y *

zu

y *

(Pope, 2000, page 282)

Note thatimplying production is locally balanced by dissipation

1P

6.04.0 1

'''2

1 ____2

____2

____2

wvuk

(Pope, 2000, page 540)

k

ww

vv

uu

_______

_____

______

''

''

''

7

Skin-Friction Coefficient(Pope, 200, Section 7.1.5, pages 278 →)

The skin friction coefficient cf is defined as

in which U is the vertically average velocity

with h being the depth of the fluid (water depth in a channel).

The log profile provides:

2

2*

2

2*

2

22

21 U

u

U

u

Uc wall

f

dzzuh

Uh

0

)(1

*0* 2.5ln1

02.5ln uν

hu

κ

udz(z)u

h Uh z u

ν

zu

κ

u(z)u **

h**

2

*2

*2*

2*

2

2* 2.5ln

12

2.5ln1

22

hu

huu

u

U

uc f

Express this in terms of the Reynolds number:

2.5ln

1Re **

huhuhU

Laminar velocity profile

Turbulent log-law velocity profile

Pope, 2000, Figure 7.10, page 279

Expressing the skin-friction coefficient in terms of the Reynolds number cannot be done algebraically in an explicit way, but a graph can be made of the curve.

8

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 z0, enters the expression.

0

* ln)(z

zuzu

Roughness heights

9

Drag Coefficient

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

2UCDb

in which U is the vertically average 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:

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.

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 = eddy (du/dz)

For wall turbulence, we have:

z

u

dz

ud

z

zuzu

ub

*

0

*

2*

ln)(

*** )(

)/(

uuzzu

dzud

m

beddy

in which eddy is 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?

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

dz

udu

dz

uduu

dz

udmmTb **

2* )(

dz

udu 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/dzand thus: Smeddy

2

2) Slightly 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 2meddy

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

)(

11

River flow is 3D and unsteady (turbulent).

But: length of river >> width & depth

As a result, the downstream velocity, aligned with the channel, dominates the flow.

The 1D assumption may be made, withh(x,t) = water depthu(x,t) = water velocity

with x = downstream distance and t = time.

The presence of two flow variables [h(x,t) and u(x,t)] necessitates two physical statements.

These are:

1. Conservation of mass (what goes in, goes out)

2. Momentum budget (with 3 forces: pressure, gravity and friction)

Additional assumption: incompressibilitydensity = constant = 1000 kg/m3 (freshwater)

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Budget for a stretch dx of river

S = Slope = sin P = Wetted perimeter

A = Area W = Width h = max depth

Conservation of mass:

Amount stored in stretch dx = what goes in – what goes out.

0

constant

0

0,0

AUx

At

AUx

At

dtdx

AUAUAdxdxxx

dttt

This equation is attributed to Leonardo da Vinci (1452-1519).

where A(h) is a known function of the water depth h (channel profile given)

Then

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Case of a rectangular channel with constant width:

A = Wh, with W = constant

P = h + W + h

00

hUx

ht

AUx

At

Momentum Budget:

bottom along force frictional Braking

force nalgravitatio Downslope

aheadforcepressure Braking

rearinforcepressure Pushing

atexitingMomentum

atenteringMomentumstretchinsideMomentum

dxx

xdt

d

14

)0( levelat width channel )(

)( pressure (gage) chydrostati )(

)()( force Pressure0

hzzzw

zhgzp

dzzwzppdAFh

p

x

hgA

x

h

dh

dF

x

F

gAdzzwg

dzzgwzwzhgdh

dF

dzzwzhgF

pp

h

h

hzp

h

p

)(

)()]()([

)()(

0

0

0

For later:

perimeter wetted and

stress bottom with

)(

)(

area bottom stress bottom force Frictional

)(

sin)( force nalGravitatio

2

2

P

UC

PdxUC

Pdx

gASdx

gSAdx

mg

Db

D

b

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Putting it altogether:

dxPUC

gASdx

FF

AUAUdt

AUdxAUdx

D

dxxpxp

dxxx

tdttt

2

at at

at

2

at

2at a

||

][][

or, in differential form:

22 UPCgASx

FAU

xAU

t Dp

and after some simplifications and use of volume conservation:

2UA

PCgS

x

hg

x

UU

t

UD

Momentum in and out

Pressure force, rear and front

Downslope gravity

Bottom friction

inertia gravityfriction

For convenience, we define the hydraulic radius:

so that the momentum equation becomes:

For a broad flat channel, which is a good approximation for most rivers:

R

UCgS

x

hg

x

UU

t

UD

2

h

UCgS

x

hg

x

UU

t

Uh

W

Wh

hW

WhR D

2

2

This equation is attributed to Adhemar de Saint-Venant (1797-1886).

Together, this momentum equation and the mass-conservation equation form a 2 x 2 nonlinear system for the flow variables h and u.

perimeter wetted

area sectional-cross

P

AR

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1. Uniform frictional flow:

0 and 0

xt

Only the momentum equation remains and it becomes:

DD C

ghSU

h

UCgS 0

2

The balance is between the forward force of gravity and the retarding force of bottom friction.

The formula is due to Antoine de Chézy (1718-1798).

Manning’s formula

River data show that the drag coefficient CD is not a constant but depends on depth.

If we use the logarithmic velocity profile of wall turbulence, we obtain:

20

2

]1)/[ln(

zhCD

and the Chézy formula becomes

1ln

02 z

hghS

C

ghSU

D

Using abundant data, Robert Manning (1816-1897) determined that a power of h was adequate, with the 2/3 power giving the best fit, and he wrote:

2/13/21SR

nU

in which the coefficient n is now called the Manning Coefficient.

Note that this expression is not dimensionally correct. So, care must be taken to use metric units.

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Examples of Manning Coefficients:

http://wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htm

Clark Fork at St. Regis, Montanan = 0.028

Clark Fork above Missoula, Montanan = 0.030

Middle Fork Flathead Rivernear Essex, Montana

n = 0.041

South Fork Clearwater Rivernear Grangeville, Idaho

n = 0.051

Rock Creek near Darby, Montana

n = 0.075

Smooth cement canaln = 0.012

Numerical values of the Manning Coefficient:

CHANNEL TYPE n

Artificial channels finished cement 0.012

unfinished cement 0.014

brick work 0.015

rubble masonry 0.025

smooth dirt 0.022

gravel 0.025

with weeds 0.030

cobbles 0.035

Natural channels mountain streams 0.045

clean and straight 0.030

clean and winding 0.040

with weeds and stones 0.045

most rivers 0.035

with deep pools 0.040

irregular sides 0.045

dense side growth 0.080

Flood plains farmland 0.035

small brushes 0.125

with trees 0.150

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The enigma of Roman water engineers

Segovia, Spain

Roman engineers had no conception of time at the scale of the minute and second. So, they had no concept of water velocity and dealt only with water depths.

So, how were they able to build properly designed aqueducts and sewage draining passages?

The Romans were lucky because velocity is directly related to water depth, and water depth could then be used as the only variable.

It also helped that water depth happens to be the more sensitive function of discharge among the two variables.

Answer: