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

4. Types of flow
2

5. Geometric properties
3

6. Velocity distributions
3

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

15. Manning's roughness coefficient
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