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
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
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