lecture11 uniform channel flow rolf 2017 · 2017. 3. 6. · 1. definition of uniform flow uniform...
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Uniform Channel Flow – Basic Concepts
Hydromechanics VVR090
ppt by Magnus Larson; revised by Rolf L Feb 2014
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SYNOPSIS
1. Definition of Uniform Flow2. Momentum Equation for Uniform Flow 3. Resistance equations 4. Flow Resistance Coefficients 5. Selecting a Manning’s roughness6. Examples/Problems
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1. Definition of Uniform Flow
Uniform flow occurs when:
1. The depth, flow area, and velocity at every cross section is constant
2. The energy grade line, water surface, and channel bottom are all parallel:
f w oS S S
Sf = slope of energy grade line
Sw = slope of water surface
So = slope of channel bed
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Definition Sketch for Uniform Flow
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Depth for uniform flow is denoted ”Normal depth ” (y0 or yn)
If normal depth y0 < yc (supercritical flow) then slope is ”steep”If normal depth y0 > yc (subcritical flow) then slope is ”mild”
Profiles
Mild slope
Steep slope
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Conditions that allow uniform flow to develop are rarely satisfied in practice.
However, it is a concept of great significance in understanding and solving most problems in open-channel hydraulics.
Uniform flow occurs in long, straight, prismatic channel where a terminal velocity can be achieved =>
ENERGY balance between head loss due to turbulent flow and reduction in potential energy
FORCE Balance between gravity and boundary shear forces
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2. Momentum Equation for Uniform Flow
Gravity force (driving motion):
sin sinmF W AL
Boundary shear force (resisting motion):
R oF LP
Shear stress proportional to bottom velocity squared:2
o ku
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VVR170. 5 Feb 2013. 8 (43)
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Momentum Equation for Uniform Flow cont’d
Steady state conditions: gravity force = shear forces
2
1/ 2
sin
m RF F
AL ku LP
u RSk
ARP
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3. Resistance equations. a) the Chezy Equation
The Chezy equation is given by:
1/ 2
u C RS
Ck
C has the dimensions L1/2/TAntoine Chezy
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b) The Manning Equation
The Manning equation is given by:
2/31u R Sn
n has the dimensions T/L1/3
Compare with the Chezy equation:
1/ 6RCn
Robert Manning
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General Equation for Uniform Flow
Most semi-empirical equations for the average velocity of a uniform flow may be written:
x yu CR S
Manning equation is the most commonly employed equation in open channel flow (x=2/3, y=1/2).
It will be used for calculations in the present course.
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4. Flow Resistance Coefficients I
Difficult to estimate an appropriate value on the resistance coefficient in the Manning or Chezy equations.
Should depend on:
• Reynolds number
• boundary roughness
• shape of channel cross section
Compare with the Darcy-Weisbach formula for pipe friction:
2
4 2LL uh fR g
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Flow Resistance Coefficients II
Slope of the energy line:
Compare with Manning and Chezy equation:
2
4 2Lh f uS
L R g
1/ 6
8
8
fn Rg
gCf
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Types of Turbulent Flow
Two main types of turbulent flow:
• hydraulically smooth turbulent flow:
Roughness elements covered by viscous sublayer (resistance depends on Reynolds number Re)
• hydraulically rough flow:
Roughness elements penetrates through the viscous sublayer (resistance coefficient depends on roughness height ks)
Transitional region in between these flows (dependence on Re and ks)
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Example of Roughness Heights (ks)
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Definition of Reynolds Number
Definitions of Reynolds number:
**
*
4Re
Re s
oo
u R
u k
u gRS
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*
*
*
0 Re 4 smooth
4 Re 100 transition
100 Re rough
Criteria for Turbulent Flow Types
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Pipe Flow Friction Factors
Hydraulically smooth flow:
0.25
0.316 Re 100,000Re
Re1 2.0log Re 100,0002.51
f
ff
Hydraulically rough flow:
1 122.0logs
Rkf
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Colebrook’s formula applicable for the transition region:
1 2.52.0log12 Re
skRf f
Plots of f versus ks/4R and Re (analogous to a Moody diagram).
Re number
Fric
tion
Fact
or
Relative Roughness
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Selecting a Suitable Roughness
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5. Selecting a Manning’s roughness
Difficult to apply f from pipe flow.
Manning’s n is often determined based on empirical knowledge, including the main factors governing the flow resistance:
• surface roughness
• vegetation
• channel irregularity
• obstruction
• channel alignment
• sedimentation and scouring
• stage and discharge
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Soil Conservation Service (SCS) Method for n
Determine a basic n for a uniform, straight, and regular channel, then modify this value by adding correction factors.
Each factor is considered and evaluated independently.
Channel Characteristics Basic n
In earth 0.020
Cut in rock 0.025
In fine gravel 0.024
In coarse gravel 0.028
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Procedure:
1. Select basic n
2. Modify for vegetation
3. Modify for channel irregularity
4. Modify for obstruction
5. Modify for channel alignment
6. Estimate n from step 1 to 5
A total n is obtained as the sum of the different contributions.
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Influence of Vegetation
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Influence of Cross-Section Size and Shape, and Irregulariy
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Influence of Obstruction and Channel Alignment
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Example of Manning’s n from Chow (1959)
(illustrative pictures in the following)
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0.012
0.014
0.016
Manning’s Roughness n
0.018
0.018
0.020
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Manning’s Roughness n
0.020
0.022
0.024
0.024
0.026
0.028
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Manning’s Roughness n0.040
0.040
0.045
0.029
0.030
0.035
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Manning’s Roughness n
0.110
0.125
0.150
0.050
0.060
0.080
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Example 5.1Given a trapezoidal channel with a bottom width of 3 m, side slopes of 1.5:1, a longitudinal slope of 0.0016, and a resistance coefficient of n = 0.013, determine the normal discharge if the normal depth of flow is 2.6 m.
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Example 5.2
Given a trapezoidal channel with a bottom width of 3 m, side slopes of 1.5:1, a longitudinal slope of 0.0016, and a resistance coefficient of n = 0.13, find the normal depth of flow for a discharge of 7.1 m3/s.