Cu06997 lecture 9_open channel

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Cu06997 lecture 9_open channel

  1. 1. CU06997 Fluid Dynamics Open channel flow 1 5.1 Flow with a free surface (page 122) 5.2 Flow classification (page 122, 123) 5.3 Channels and their properties (page 123-125) 5.4 Velocity distributions (page 126,127) 5.5 Laminar and turbulent flow (page 127-129) 5.6 Uniform flow (page 129 -138)1
  2. 2. Flow with a free surface1
  3. 3. Classification of flows, see part 2 1. Steady uniform flow example: pipe with constant D and Q example: channel with constant A and Q 2. Steady non-uniform flow example: pipe with different D and constant Q example: channel with different A and constant Q 3. Unsteady uniform flow example: channel with constant A and different Q 4. Unsteady non-uniform flow example; channel with different A and Q2
  4. 4. Types of flow2
  5. 5. Geometric properties3
  6. 6. Velocity distributions3
  7. 7. Velocity distributions 𝑄 𝑉1 𝐴1 + 𝑉2 𝐴2 + 𝑉3 𝐴3 π‘‰π‘Žπ‘£π‘’π‘Ÿπ‘Žπ‘”π‘’ = = 𝐴 𝐴1 + 𝐴2 + 𝐴3 𝑄 π‘‘π‘œπ‘‘π‘Žπ‘Žπ‘™ = 𝑄1 + 𝑄3 + 𝑄3 =𝑉1 𝐴1 + 𝑉2 𝐴2 + 𝑉3 𝐴33
  8. 8. Reynolds number, see part 3 𝑅𝑒 = 𝑉. 𝐷 𝜈 πœ‡= Absolute viscosity [m2/s] 𝑉. 4𝑅 𝜐= Kinematic viscosity [kg/ms] 𝑅𝑒 = water, 20Β°C= 1,00 βˆ™ 10βˆ’6 𝜈𝜌 = Density of liquid [kg/m3]𝑉 = Velocity [m/s]D = Hydraulic diameter [m]R= Hydraulic Radius = D/4 [m]𝑅𝑒 = Reynolds Number [1] 𝑹𝒆 > πŸ’πŸŽπŸŽπŸŽ Turbulent flow 𝑹𝒆 < 𝟐𝟎𝟎𝟎 Laminar flow3 In this course we only look at turbulent flow
  9. 9. Open channel, with bed slope >0 2 2 𝑒1 𝑒2𝑦1 + 𝑧1 + = 𝑦2 + 𝑧2 + + βˆ†π»1βˆ’2 2𝑔 2𝑔 Q ο€½ u1 οƒ— A1 ο€½ u2 οƒ— A2 Head loss Reference line4
  10. 10. Open channel, with bed slope <= 0 2 2 u u y1  z1  1 ο€½ y2  z2   H 1ο€­ 2 2 2g 2g Head loss [m] u12/2g Ξ”H Total Head H [m] y1 u22/2g Velocity Head [m] P1 u1 Surfacelevel y +z [m] z1 y2 P2 u2 z24 Reference [m]
  11. 11. Chezy formula 𝑉= πΆβˆ™ 𝑅 βˆ™ 𝑆𝑓Chezy formula describes the mean velocity of uniform, turbulent flow 𝑉= Mean Fluid Velocity [m/s] R= Hydraulic Radius [m] 𝑆𝑓 = Hydraulic gradient [1] 8𝑔 𝐢= Chezy coefficient [m1/2/s] πœ† Ξ”H 𝑆𝑓 = 𝐿 Ξ”H5 Length
  12. 12. Chezy coefficient In this course we assume a hydraulic rough boundary Boundary hydraulic rough 12 R C ο€½ 18 log [m1/2/s] k kS = surface roughness [m]5
  13. 13. Surface roughness kS [m] Equivalent Sand Roughness, Material (ft) (mm) Copper, brass 1x10-4 - 3x10-3 3.05x10-2 - 0.9 Wrought iron, 1.5x10-4 - 8x10-3 4.6x10-2 - 2.4 steel Asphalt-lined 4x10-4 - 7x10-3 0.1 - 2.1 cast iron 3.3x10-4 - 1.5x10- Galvanized iron 2 0.102 - 4.6 Cast iron 8x10-4 - 1.8x10-2 0.2 - 5.5 Concrete 10-3 - 10-2 0.3 - 3.0 Uncoated Cast 7.4x10-4 0.226 Iron Coated Cast Iron 3.3x10-4 0.102 Coated Spun 1.8x10-4 5.6x10-2 Iron Cement 1.3x10-3 - 4x10-3 0.4 - 1.2s Wrought Iron 1.7x10-4 5x10-2 Uncoated Steel 9.2x10-5 2.8x10-2 Coated Steel 1.8x10-4 5.8x10-2 Wood Stave 6x10-4 - 3x10-3 0.2 - 0.9 PVC 5x10-6 1.5x10-3 Compiled from Lamont (1981), Moody (1944), and Mays (1999)5
  14. 14. Manning’s formula describes theManning’s formula mean velocity of uniform, turbulent flow 2 1 5 1 𝑅3 βˆ™ 𝑆2 1 𝐴3 𝑆2 1 𝑉= 𝑓 𝑄= βˆ™ βˆ™ R 6 𝑛 2 𝑓 Cο€½ 𝑛 𝑃3 n𝑉= Mean Fluid Velocity [m/s]R= Hydraulic Radius [m]𝑆𝑓 = Slope Total head [1]𝐴= Wetted Area [m2]𝑃= Wetter Perimeter [m]𝑛= Mannings roughness coefficient [s/m1/3]6
  15. 15. Mannings roughness coefficient6
  16. 16. Mean boundary shear stress 𝜏0 = 𝜌 βˆ™ 𝑔 βˆ™ 𝑅 βˆ™ 𝑆0 Ο„0 = shear stress at solid boundary [N/m2] R= Hydraulic Radius [m] 𝑆0 = Slope of channel bed [1]7
  17. 17. Flowing water and energy 2 u H1 ο€½ z1  y1  1 [m ] 2g Total head H [m] u12/2g Velocity head [m] Surface level [m] y1 y = Pressure head [m] u1 P1 z1 z = Potential head [m] Reference /datum [m]
  18. 18. Specific Energy 𝑉2 𝐸𝑠 = 𝑦 + 2𝑔 𝑉= Mean Fluid Velocity [m/s] p y= = Pressure Head / water depth [m] Οβˆ™g Total head H or Specific energy Es [m] V2/2g Velocity head [m] Surface level [m] V y y = Pressure head [m] = water depth [m]8 Channel bed as datum [m]
  19. 19. Equilibrium / normal depth Discharge, cross-section, energy gradient and friction are constant yn 𝑆0 = 𝑆 𝑓 Side view 𝑉= πΆβˆ™ 𝑅 βˆ™ π‘†π‘œ yn A b. y Rο€½ ο€½ y Cross-section P b  2οƒ— y π‘ž = 𝑉 βˆ™ 𝐴 = 𝐢 2 𝑦 βˆ™ 𝑆 π‘œβˆ™ 𝑦 βˆ™ 𝑏 3 π‘ž2 𝑦𝑛 =9 𝑏 2 βˆ™ 𝐢 2 βˆ™ 𝑆0
  20. 20. Equilibrium / normal depth 𝑆0 = 𝑆 𝑓 3 π‘ž2 𝑦𝑛 = 𝑏 2 βˆ™ 𝐢 2 βˆ™ 𝑆0 yn = normal depth [m] q= discharge [m3/s] b= width [m] 𝑆0 = bed slope [1] 𝑆𝑓 = Hydraulic gradient caused by friction [1] 8𝑔 𝐢= Chezy coefficient [m1/2/s] πœ†9
  21. 21. Equilibrium / normal depthyn yn yn yn Dredged area 3 π‘ž2 𝑦𝑛 = 𝑏 2 βˆ™ 𝐢 2 βˆ™ 𝑆09

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