Uniform Channel Flow – Basic Concepts
Hydromechanics VVR090
ppt by Magnus Larson; revised by Rolf L Feb 2014
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
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
Definition Sketch for Uniform Flow
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
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
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
VVR170. 5 Feb 2013. 8 (43)
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
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
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
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.
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
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
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)
Example of Roughness Heights (ks)
Definition of Reynolds Number
Definitions of Reynolds number:
**
*
4Re
Re s
oo
u R
u k
u gRS
*
*
*
0 Re 4 smooth
4 Re 100 transition
100 Re rough
Criteria for Turbulent Flow Types
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
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
Selecting a Suitable Roughness
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
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
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.
Influence of Vegetation
Influence of Cross-Section Size and Shape, and Irregulariy
Influence of Obstruction and Channel Alignment
Example of Manning’s n from Chow (1959)
(illustrative pictures in the following)
0.012
0.014
0.016
Manning’s Roughness n
0.018
0.018
0.020
Manning’s Roughness n
0.020
0.022
0.024
0.024
0.026
0.028
Manning’s Roughness n0.040
0.040
0.045
0.029
0.030
0.035
Manning’s Roughness n
0.110
0.125
0.150
0.050
0.060
0.080
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.
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.