Uniform Channel Flow Basic Concepts Hydromechanics VVR090
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1 Uniform Channel Flow Basic Concepts Hydromechanics VVR090 ppt by Magnus Larson; revised by Rolf L Feb 2014
2 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
3 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
4 Definition Sketch for Uniform Flow
5 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
6 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
7 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
8 VVR Feb (43)
9 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
10 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
11 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
12 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.
13 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
14 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
15 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 )
16 Example of Roughness Heights (k s )
17 Definition of Reynolds Number Definitions of Reynolds number: Re u4r Re u * * uk * o s grs o
18 Criteria for Turbulent Flow Types 0 Re 4 smooth * 4 Re 100 transition * 100 Re rough *
19 Pipe Flow Friction Factors Hydraulically smooth flow: f Re Re 100,000 1 f Re f 2.0log Re 100, Hydraulically rough flow: 1 12R 2.0log f ks
20 Colebrook s formula applicable for the transition region: 1 k log 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
21 Selecting a Suitable Roughness
22 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
23 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 Cut in rock In fine gravel In coarse gravel 0.028
24 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.
25 Influence of Vegetation
26 Influence of Cross-Section Size and Shape, and Irregulariy
27 Influence of Obstruction and Channel Alignment
28 Example of Manning s n from Chow (1959) (illustrative pictures in the following)
29 Manning s Roughness n
30 Manning s Roughness n
31 Manning s Roughness n
32 Manning s Roughness n
33 Example 5.1 Given a trapezoidal channel with a bottom width of 3 m, side slopes of 1.5:1, a longitudinal slope of , and a resistance coefficient of n = 0.013, determine the normal discharge if the normal depth of flow is 2.6 m.
34 Example 5.2 Given a trapezoidal channel with a bottom width of 3 m, side slopes of 1.5:1, a longitudinal slope of , and a resistance coefficient of n = 0.13, find the normal depth of flow for a discharge of 7.1 m 3 /s.
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Experiment- To determine the coefficient of impact for vanes. Experiment To determine the coefficient of discharge of an orifice meter.
SUBJECT: FLUID MECHANICS VIVA QUESTIONS (M.E 4 th SEM) Experiment- To determine the coefficient of impact for vanes. Q1. Explain impulse momentum principal. Ans1. Momentum equation is based on Newton s
GEOL 652. Poudre River Fieldtrip
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Lecture 10: River Channels
GEOG415 Lecture 10: River Channels 10-1 Importance of channel characteristics Prediction of flow was the sole purpose of hydrology, and still is a very important aspect of hydrology. - Water balance gives
What factors affect the angle of a slope?
Climate Rock type and Structure What factors affect the angle of a slope? Aspect Fast mass movements Slides: Slides are movements along the SLIP PLANE, i.e. a line of weakness in the rock or soil structure.
Open-channel hydraulics
Open-channel hydraulics STEADY FLOW IN OPEN CHANNELS constant discharge, other geometric and flow characteristics depended only on position Uniform flow Non-uniform flow S; y; v const. i i 0 i E y 1 y
When water (fluid) flows in a pipe, for example from point A to point B, pressure drop will occur due to the energy losses (major and minor losses).
PRESSURE DROP AND OSSES IN PIPE When water (luid) lows in a pipe, or example rom point A to point B, pressure drop will occur due to the energy losses (major and minor losses). A B Bernoulli equation: