turbulent velocity profile - walter scott, jr. college of...
TRANSCRIPT
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Dr. Pierre Y. JulienDept. of Civil and Environmental Engineering
Colorado State University, Fort [email protected]
February 2008
Assisted by: Seema C Shah-Fairbank, P.EGraduate Research [email protected]
Turbulent Velocity Profile
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Turbulent Flow Equations
{{
4444 34444 2143421
3214444 34444 21nfluctuatioturbulent
zzzyzx
viscous
zm
gradientpressure
mnalgravitatioz
convective
zz
zy
zx
local
z
yzyyyxym
my
yz
yy
yx
y
xzxyxxxm
mx
xz
xy
xx
x
zvv
yvv
xvv
vzpg
zv
vyv
vxv
vt
v
zvv
yvv
xvv
vypg
zv
vyv
vxv
vt
v
zvv
yvv
xvv
vxpg
zv
vyv
vxv
vt
v
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
+∂
∂+
∂∂
−∇+∂∂
−=∂∂
+∂∂
+∂∂
+∂∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂
∂+
∂
∂+
∂
∂−∇+
∂∂
−=∂
∂+
∂
∂+
∂
∂+
∂
∂
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
+∂
∂+
∂∂
−∇+∂∂
−=∂∂
+∂∂
+∂∂
+∂∂
++++++
++++++
++++++
2
2
2
1
1
1
υρ
υρ
υρ
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Incipient Motion
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Open Channel• Pressure Distribution
• Shear Stress
• Shear Velocity
( )zhgp
gdzdph
zp
−=
−= ∫∫ρ
ρ0
( )fo
fzx
ghS
Szhg
ρτ
ρτ
=
−=
fo ghSu ==ρτ
*
Sw
So
Datum
zo
h
SfV2/2g
g
g sin at o
t zxp
Q
x
z
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Saint-Venant Equation
( )
tgV
xgVV
xhSS
EquationVenanttSainFlowUnsteadyxgVV
xhS
xg
VhzS
SlopeEnergyxhS
xhzS
SlopeSurfaceFreexzS
SlopeBed
of
o
o
f
oo
w
oo
∂∂
−∂∂
−∂∂
−=
−∂∂
−∂∂
−=∂
⎟⎠⎞⎜
⎝⎛ ++∂
−=
∂∂
−=∂+∂
−=
∂∂
−=
22
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Logarithmic Velocity Profile
o
x
zz
uv ln1
* κ=
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Logarithmic Velocity Profile
⎟⎟⎠
⎞⎜⎜⎝
⎛=
skzLnv χ
κ2.30u*
⎟⎟⎠
⎞⎜⎜⎝
⎛=
skzLnv 2.30u*
κ⎟⎠⎞
⎜⎝⎛=
vzLnv ** u05.9u
κ
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Depth Average Velocity• One Point Method
– Measured down from water surface at 60% of the total flow depth
• Two Point Method– Average the velocity at 20 and 80% of the total flow depth
• Three Point Method– Average of the one –point and two-point methods.
• Surface Method– Determine surface velocity using a float and multiply the
velocity by a coefficient to determine the average velocity
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Resistance to Flow
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Example – Rhine River In 1998 a flood was observed on the Rhine River, in the Netherlands. The following is the data that was obtained on the Rhine on November 3rd.
Q = 9,464 cmsS = 13.12 cm/kmh = 9.9 m (from velocity profile)W = 260 m d90 = 12.190 mmd50 = 1.182 mm
The velocity profile is given as follows:
Determine the shear, mean and fall velocities.
Depth Velocity Depth Velocitym m/s m m/s0.3 0.74 2.2 1.630.3 0.81 3.5 1.920.3 0.72 4 1.850.4 0.47 4.1 1.860.5 0.84 5.3 1.990.8 1.22 6 1.980.9 1.34 7.3 2.081.2 1.38 8 2.041.3 1.47 9 1.9
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Example - Shear Velocity
smu
msmghSu f
113.0
0001312.0*9.9*81.9
*
2*
=
==
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Example - Mean Velocity• Method A – at 60%
• Method B – at 20% and 80%
• Method C –Average of Method A and Method B
smv
sm
sm
v
mhzmhz
mh
mean
mean
82.1
2
59.104.2
98.192.7
9.9
%80
%20
=
+=
====
=
smv
mhzmh
mean 86.1
96.39.9
%60
=
=== Depth Velocity
m m/s3.5 1.92
3.96 1.864 1.85
Depth Velocitym m/s1.3 1.47
1.98 1.592.2 1.63
Depth Velocitym m/s7.3 2.08
7.92 2.048 2.04
smv
sm
sm
v
mean
mean
84.1
2
82.186.1
=
+=
CIVE 413 ENVIRONMENTAL RIVER MECHANICS February 19, 2008 Pierre Y. Julien Turbulent Velocity Profiles and Resistance to Flow Problem # 1 (100 %) Field measurements along a vertical profile of the Rhine River are shown below. The navigable channel width covers 260 m. Consider a rectangular section to determine the hydraulic radius. The bed material is typically d10 = 0.4 mm, d50= 1.3 mm and d90 = 10 mm. The measured slope of the Energy Grade Line was 13.12 cm per km on Nov. 3. Show the velocity profile on linear scale, and also provide a semi-log plot with a fitted line to the data to graphically determine the value of kappa. Determine the following parameters in SI: a) flow depth b) hydraulic radius c) ratio of hydraulic radius to flow depth d) shear stress in Pascals e) shear velocity f) von Kármán constant g) mean flow velocity in m/s (3 points) h) Froude number i) Manning n j) laminar sublayer thickness in mm Date Time Positio Depth Concentration u Depthyrmndday Hrminsec from axis m water mg/l m/s m of measurement
981103 120441 0 9.9 494 0.74 9.6 981103 120647 0 9.9 488 0.81 9.6 981103 120856 0 9.9 498 0.72 9.6 981103 121353 0 9.9 398 0.84 9.4 981103 121626 0 9.9 293 1.22 9.1 981103 122217 0 9.9 432 0.47 9.5 981103 122628 0 9.9 132 1.38 8.7 981103 122850 0 9.9 185 1.34 9 981103 123057 0 9.9 134 1.47 8.6 981103 123431 0 9.9 83 1.63 7.7 981103 123707 0 9.9 49 1.92 6.4 981103 123944 0 9.9 43 1.86 5.8 981103 124158 0 9.9 43 1.85 5.9 981103 124423 0 9.9 35 1.99 4.6 981103 124704 0 9.9 33 1.98 3.9 981103 124922 0 9.9 26 2.08 2.6 981103 125144 0 9.9 25 2.04 1.9 981103 125425 0 9.9 23 1.90 0.9