Cu06997 lecture 9_open channel
3 likes2,837 views
This document summarizes key concepts in fluid dynamics related to open channel flow: 1) It describes different types of open channel flow including steady uniform/non-uniform flow and unsteady uniform/non-uniform flow. 2) Common equations for mean flow velocity are presented, including the Chezy and Manning's formulas which relate velocity to hydraulic radius and slope. 3) The concepts of hydraulic radius, roughness coefficients, shear stress, specific energy, and equilibrium/normal depth are defined and their representative equations shown.
![Reynolds number, see part 3 𝑅𝑒 = 𝑉. 𝐷
𝜈
𝜇= Absolute viscosity [m2/s] 𝑉. 4𝑅
𝜐= Kinematic viscosity [kg/ms] 𝑅𝑒 =
water, 20°C= 1,00 ∙ 10−6 𝜈
𝜌 = Density of liquid [kg/m3]
𝑉 = Velocity [m/s]
D = Hydraulic diameter [m]
R= Hydraulic Radius = D/4 [m]
𝑅𝑒 = Reynolds Number [1]
𝑹𝒆 > 𝟒𝟎𝟎𝟎 Turbulent flow
𝑹𝒆 < 𝟐𝟎𝟎𝟎 Laminar flow
3 In this course we only look at turbulent flow](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-8-320.jpg)
![Open channel, with bed slope <= 0
2 2
u u
y1 z1 1
y2 z2 H 1 2
2
2g 2g
Head loss [m]
u12/2g ΔH
Total Head H [m]
y1 u22/2g Velocity Head [m]
P1
u1 Surfacelevel y +z [m]
z1 y2
P2
u2 z2
4 Reference [m]](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-10-320.jpg)
![Chezy formula 𝑉= 𝐶∙ 𝑅 ∙ 𝑆𝑓
Chezy formula describes the mean velocity of uniform, turbulent flow
𝑉= Mean Fluid Velocity [m/s]
R= Hydraulic Radius [m]
𝑆𝑓 = Hydraulic gradient [1]
8𝑔
𝐶= Chezy coefficient [m1/2/s]
𝜆
ΔH
𝑆𝑓 =
𝐿
ΔH
5 Length](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-11-320.jpg)
![Chezy coefficient
In this course we assume a hydraulic rough boundary
Boundary hydraulic rough 12 R
C 18 log [m1/2/s]
k
kS = surface roughness [m]
5](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-12-320.jpg)
![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](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-13-320.jpg)
![Manning’s formula describes the
Manning’s formula mean velocity of uniform,
turbulent flow
2 1 5
1
𝑅3 ∙ 𝑆2 1 𝐴3
𝑆2
1
𝑉=
𝑓 𝑄= ∙ ∙ R 6
𝑛 2 𝑓 C
𝑛 𝑃3 n
𝑉= 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](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-14-320.jpg)
![Mean boundary shear stress
𝜏0 = 𝜌 ∙ 𝑔 ∙ 𝑅 ∙ 𝑆0
τ0 = shear stress at solid boundary [N/m2]
R= Hydraulic Radius [m]
𝑆0 = Slope of channel bed [1]
7](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-16-320.jpg)
![Flowing water and energy
2
u
H1 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]](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-17-320.jpg)
![Specific Energy
𝑉2
𝐸𝑠 = 𝑦 +
2𝑔
𝑉= Mean Fluid Velocity [m/s]
p
y= = Pressure Head / water depth [m]
ρ∙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]](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-18-320.jpg)
![Equilibrium / normal depth
𝑆0 = 𝑆 𝑓
3 𝑞2
𝑦𝑛 =
𝑏 2 ∙ 𝐶 2 ∙ 𝑆0
yn = normal depth [m]
q= discharge [m3/s]
b= width [m]
𝑆0 = bed slope [1]
𝑆𝑓 = Hydraulic gradient caused by friction [1]
8𝑔
𝐶= Chezy coefficient [m1/2/s]
𝜆
9](https://image.slidesharecdn.com/cu06997lecture9openchannel-130410100720-phpapp01/85/Cu06997-lecture-9_open-channel-20-320.jpg)
Cu06997 lecture 9_open channel
- 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. Flow with a free surface 1
- 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 Q 2
- 7. Velocity distributions 𝑄 𝑉1 𝐴1 + 𝑉2 𝐴2 + 𝑉3 𝐴3 𝑉𝑎𝑣𝑒𝑟𝑎𝑔𝑒 = = 𝐴 𝐴1 + 𝐴2 + 𝐴3 𝑄 𝑡𝑜𝑡𝑎𝑎𝑙 = 𝑄1 + 𝑄3 + 𝑄3 =𝑉1 𝐴1 + 𝑉2 𝐴2 + 𝑉3 𝐴3 3
- 8. Reynolds number, see part 3 𝑅𝑒 = 𝑉. 𝐷 𝜈 𝜇= Absolute viscosity [m2/s] 𝑉. 4𝑅 𝜐= Kinematic viscosity [kg/ms] 𝑅𝑒 = water, 20°C= 1,00 ∙ 10−6 𝜈 𝜌 = Density of liquid [kg/m3] 𝑉 = Velocity [m/s] D = Hydraulic diameter [m] R= Hydraulic Radius = D/4 [m] 𝑅𝑒 = Reynolds Number [1] 𝑹𝒆 > 𝟒𝟎𝟎𝟎 Turbulent flow 𝑹𝒆 < 𝟐𝟎𝟎𝟎 Laminar flow 3 In this course we only look at turbulent flow
- 9. Open channel, with bed slope >0 2 2 𝑢1 𝑢2 𝑦1 + 𝑧1 + = 𝑦2 + 𝑧2 + + ∆𝐻1−2 2𝑔 2𝑔 Q u1 A1 u2 A2 Head loss Reference line 4
- 10. Open channel, with bed slope <= 0 2 2 u u y1 z1 1 y2 z2 H 1 2 2 2g 2g Head loss [m] u12/2g ΔH Total Head H [m] y1 u22/2g Velocity Head [m] P1 u1 Surfacelevel y +z [m] z1 y2 P2 u2 z2 4 Reference [m]
- 11. Chezy formula 𝑉= 𝐶∙ 𝑅 ∙ 𝑆𝑓 Chezy formula describes the mean velocity of uniform, turbulent flow 𝑉= Mean Fluid Velocity [m/s] R= Hydraulic Radius [m] 𝑆𝑓 = Hydraulic gradient [1] 8𝑔 𝐶= Chezy coefficient [m1/2/s] 𝜆 ΔH 𝑆𝑓 = 𝐿 ΔH 5 Length
- 12. Chezy coefficient In this course we assume a hydraulic rough boundary Boundary hydraulic rough 12 R C 18 log [m1/2/s] k kS = surface roughness [m] 5
- 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. Manning’s formula describes the Manning’s formula mean velocity of uniform, turbulent flow 2 1 5 1 𝑅3 ∙ 𝑆2 1 𝐴3 𝑆2 1 𝑉= 𝑓 𝑄= ∙ ∙ R 6 𝑛 2 𝑓 C 𝑛 𝑃3 n 𝑉= 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
- 16. Mean boundary shear stress 𝜏0 = 𝜌 ∙ 𝑔 ∙ 𝑅 ∙ 𝑆0 τ0 = shear stress at solid boundary [N/m2] R= Hydraulic Radius [m] 𝑆0 = Slope of channel bed [1] 7
- 17. Flowing water and energy 2 u H1 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. Specific Energy 𝑉2 𝐸𝑠 = 𝑦 + 2𝑔 𝑉= Mean Fluid Velocity [m/s] p y= = Pressure Head / water depth [m] ρ∙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. Equilibrium / normal depth Discharge, cross-section, energy gradient and friction are constant yn 𝑆0 = 𝑆 𝑓 Side view 𝑉= 𝐶∙ 𝑅 ∙ 𝑆𝑜 yn A b. y R y Cross-section P b 2 y 𝑞 = 𝑉 ∙ 𝐴 = 𝐶 2 𝑦 ∙ 𝑆 𝑜∙ 𝑦 ∙ 𝑏 3 𝑞2 𝑦𝑛 = 9 𝑏 2 ∙ 𝐶 2 ∙ 𝑆0
- 20. Equilibrium / normal depth 𝑆0 = 𝑆 𝑓 3 𝑞2 𝑦𝑛 = 𝑏 2 ∙ 𝐶 2 ∙ 𝑆0 yn = normal depth [m] q= discharge [m3/s] b= width [m] 𝑆0 = bed slope [1] 𝑆𝑓 = Hydraulic gradient caused by friction [1] 8𝑔 𝐶= Chezy coefficient [m1/2/s] 𝜆 9
- 21. Equilibrium / normal depth yn yn yn yn Dredged area 3 𝑞2 𝑦𝑛 = 𝑏 2 ∙ 𝐶 2 ∙ 𝑆0 9