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