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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
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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)
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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
_______
_____
______
''
''
''
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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.
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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 z0, enters the expression.
0
* ln)(z
zuzu
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 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
)(
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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: