1

Dr. Pierre Y. JulienDept. of Civil and Environmental Engineering

Colorado State University, Fort Collinspierre@engr.colostate.edu

February 2008

Assisted by: Seema C Shah-Fairbank, P.EGraduate Research Assistantsshah@engr.colostate.edu

Turbulent Velocity Profile

2

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

3

Incipient Motion

4

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

5

Saint-Venant Equation

( )

tgV

xgVV

xhSS

EquationVenanttSainFlowUnsteadyxgVV

xhS

xg

VhzS

SlopeEnergyxhS

xhzS

SlopeSurfaceFreexzS

SlopeBed

of

o

o

f

oo

w

oo

∂∂

−∂∂

−∂∂

−=

−∂∂

−∂∂

−=∂

⎟⎠⎞⎜

⎝⎛ ++∂

−=

∂∂

−=∂+∂

−=

∂∂

−=

22

6

7

Logarithmic Velocity Profile

o

x

zz

uv ln1

* κ=

8

Logarithmic Velocity Profile

⎟⎟⎠

⎞⎜⎜⎝

⎛=

skzLnv χ

κ2.30u*

⎟⎟⎠

⎞⎜⎜⎝

⎛=

skzLnv 2.30u*

κ⎟⎠⎞

⎜⎝⎛=

vzLnv ** u05.9u

κ

9

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

10

Resistance to Flow

11

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

12

Example - Shear Velocity

smu

msmghSu f

113.0

0001312.0*9.9*81.9

*

2*

=

==

13

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