This document discusses concepts in fluid dynamics including specific energy, critical depth, the Froude number, subcritical and supercritical flow, critical velocity, and hydraulic jumps. It provides equations for calculating critical depth, Froude number, critical velocity, and critical bed slope in open channels. Diagrams show the relationship between total head, water depth, and Froude number below and above critical depth.

1. CU06997 Fluid dynamics
Froude number (page 148)
5.9 Critical depth meters (page 155 – 158)
1

2. Specific Energy
V
Channel bed as datum [m]
Surface level [m]
Total head H or Specific energy Es [m]
y
V2/2g Velocity head [m]
y = Pressure head [m]
= water depth [m]
𝐸𝑠 = 𝑦 +
𝑉2
2𝑔
𝑉 = Mean Fluid Velocity [m/s]
y =
p
ρ∙g
= Pressure Head / water depth [m]
1

3. Critical Depth
V
Reference /datum [m]
Water depth y [m]
y
V2/2g Velocity head [m]
y
B
g
V
yH
2
2
 yBVQv 
22
2
2 yBg
Q
yH v


H
Suppose Q and B are given, what could by the value of H and y
Total head H or Specific energy Es [m]
2
P1 P1

4. 22
2
2 yBg
Q
yH v


𝐻 = y +
𝑄2
2𝑔 ∙ 𝐵2
∙
1
𝑦2
Example
B= 2 m, Q = 6 m3/s
y
B
H
𝐻 = y + 0.45 ∙
1
𝑦2
2

5. 0.00
1.00
2.00
3.00
4.00
5.00
6.00 0.300
0.590
0.880
1.170
1.460
1.750
2.040
2.330
2.620
2.910
3.200
3.490
3.780
4.070
H(totalhead)(m)
y (water depth) (m)
Sub-critical or Supercritical flow
Stromend of schietend water
Total head
H=3/2*h
Supercritical flow
Schietend water
Sub-critical flow
Stromend water
Example
B= 2 m, Q = 6 m3/s
2
cyH 2
3
min 

6. 22
2
2 yBg
Q
yH v


𝐻 = y +
𝑄2
2𝑔 ∙ 𝐵2
∙
1
𝑦2
Differentiation [Differentiëren]
dH/dy = 0 gives
y
B
H
2
𝑦𝑐 =
𝑄2
𝑔 ∙ 𝐵2
3
Represents lowest point graph.
Means point with the lowest
H for a given Q and B

7. Critical Depth and Critical Velocity
cyH 2
3
min 
Sub-critical flow Supercritical flow
𝑦𝑐 =
𝑄2
𝑔 ∙ 𝐵2
3
𝑉𝑐 = 𝑔 ∙ 𝑦𝑐
2
3
h = y in this graph

8. Froude number
𝑦𝑐 =
𝑄2
𝑔 ∙ 𝐵2
3
𝑉𝑐 = 𝑔 ∙ 𝑦𝑐
2
𝐹𝑟 =
𝑉
𝑔𝑦𝑐
2
=
𝑉
𝑉𝑐
yc = critical depth [m]
Q = discharge [m3/s]
B = width [m]
Vc = critical velocity [m/s]
V = actual velocity [m/s]
Fr = Froude number [-]
Subcritical flow [stromend] Fr < 1 V < Vc
Supercritical flow [schietend] Fr > 1 V > Vc
3

9. Froude number
Fr>1
• Supercritical flow [schietend water]
• Water velocity > wave velocity
• Disturbances travel downstream
• Upstream water levels are unaffected by
downstream control
Fr<1
• Subcritical flow [stromend water]
• Water velocity < wave velocity
• Disturbances travel upstream and downstream
• Upstream water levels are affected by
downstream control
3

10. Froude number<1 Subcritical
[stromend]
Consequences for strategy to calculate water levels
What happens downstream affect the upstream water level
So most of the time you start downstream and go upstream
3

11. Question 3de
50 m
Ø300 PVC
Ø500 beton
Ø250 PVC
Pump=20 l/s
P4 P3 P2
GL +6.00 m
Rain=66 l/s
Waste=10 l/s
Rain=225 l/s
Waste=10 l/s
+5,5 m
Q=66 l/s
v=0,93 m/s
I=1:244
Q=291 l/s
v=1,48 m/s
I=1:166
Q=0 l/s
v=0 m/s
I=0
P1
In example m = 1,83

12. Froude number>1 Supercritical
[schietend]
Consequences for strategy to calculate water levels
What happens downstream does not affect the upstream
water level
So most of the time you start upstream and go downstream
3

13. Critical bed slope channel /river
Q and B (width channel) are given
Step 1 Calculate yc
Step 2 Calculate R and Vc
Step 3 Calculate Sc using Chezy or Manning
𝑦𝑐 =
𝑄2
𝑔 ∙ 𝐵2
3
𝑉𝑐 = 𝐶 ∙ 𝑅 ∙ 𝑆𝑐
𝑉𝑐 =
𝑅
2
3 ∙ 𝑆𝑐
1
2
𝑛
4

14. Critical bed slope channel /river
4

15. Hydraulic jump [watersprong]
When supercritical flow [schietend] changes to subcritical
flow [stromend] a hydraulic jump will occur
5

16.  
 21
2
2
3
21
vvv
vv
wa



Hydraulic jump, energy loss
5