Uniform Channel Flow Basic Concepts Hydromechanics VVR090 ppt by Magnus Larson; revised by Rolf L Feb 2014
SYNOPSIS 1. Definition of Uniform Flow 2. Momentum Equation for Uniform Flow 3. Resistance equations 4. Flow Resistance Coefficients 5. Selecting a Manning s roughness 6. 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: S S S f w o S f = slope of energy grade line S w = slope of water surface S o = slope of channel bed
Definition Sketch for Uniform Flow
Depth for uniform flow is denoted Normal depth (y 0 or y n ) If normal depth y 0 < y c (supercritical flow) then slope is steep If normal depth y 0 > y c (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): F Wsin ALsin m Boundary shear force (resisting motion): F R o LP Shear stress proportional to bottom velocity squared: ku o 2
VVR170. 5 Feb 2013. 8 (43)
Momentum Equation for Uniform Flow cont d Steady state conditions: gravity force = shear forces F m F R ALsin 2 ku LP u R k A P 1/2 RS
3. Resistance equations. a) the Chezy Equation The Chezy equation is given by: u C RS C k 1/2 C has the dimensions L 1/2 /T Antoine Chezy
b) The Manning Equation The Manning equation is given by: 1 n 2/3 u R S n has the dimensions T/L 1/3 Compare with the Chezy equation: Robert Manning C 1/6 R n
General Equation for Uniform Flow Most semi-empirical equations for the average velocity of a uniform flow may be written: u x CR S y 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: h L f 2 L u 4R 2g
Flow Resistance Coefficients II Slope of the energy line: 2 hl f u S L 4R 2g Compare with Manning and Chezy equation: n R 1/6 f 8g C 8g f
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 k s ) Transitional region in between these flows (dependence on Re and k s )
Example of Roughness Heights (k s )
Definition of Reynolds Number Definitions of Reynolds number: Re u4r Re u * * uk * o s grs o
Criteria for Turbulent Flow Types 0 Re 4 smooth * 4 Re 100 transition * 100 Re rough *
Pipe Flow Friction Factors Hydraulically smooth flow: f 0.316 0.25 Re Re 100,000 1 f Re f 2.0log Re 100,000 2.51 Hydraulically rough flow: 1 12R 2.0log f ks
Colebrook s formula applicable for the transition region: 1 k 2.5 2.0log s f 12 R Re f Plots of f versus k s /4R and Re (analogous to a Moody diagram). Friction Factor Relative Roughness Re number
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)
Manning s Roughness n 0.012 0.018 0.014 0.018 0.016 0.020
Manning s Roughness n 0.020 0.024 0.022 0.026 0.024 0.028
Manning s Roughness n 0.029 0.040 0.030 0.040 0.035 0.045
Manning s Roughness n 0.050 0.110 0.060 0.125 0.080 0.150
Example 5.1 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 = 0.013, 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 m 3 /s.