Cu06997 lecture 10_froude

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 E
s [m]
y
V
2
/2g Velocity head [m]
y = Pressure head [m]
= water depth [m]
�
??????=??????+
??????
2
2??????

??????= 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
V
2
/2g
Velocity head [m]
y
B g
V
yH
2
2
 yBVQ
v
 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 E
s [m]
2
P1 P1

4. 22
2
2 yBg
Q
yH
v

 ??????=y+
�
2
2??????∙�
2
∙
1
??????
2

Example
B= 2 m, Q = 6 m3/s
y
B
H
??????=y+0.45∙
1
??????
2

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 (total head) (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

 ??????=y+
�
2
2??????∙�
2
∙
1
??????
2

Differentiation [Differentiëren]
dH/dy = 0 gives
y
B
H
2
??????
??????=
�
2
??????∙�
2
3

Represents lowest point graph.
Means point with the lowest
H for a given Q and B

7. Critical Depth and Critical Velocity c
yH
2
3
min

Sub-critical flow Supercritical flow
??????
??????=
�
2
??????∙�
2
3

??????
??????=??????∙??????
??????
2

3
h = y in this graph

8. Froude number
??????
??????=
�
2
??????∙�
2
3

??????
??????=??????∙??????
??????
2

�??????=
??????
????????????
??????
2
=
??????
??????
??????

y
c = critical depth [m]
Q = discharge [m
3
/s]
B = width [m]
V
c = critical velocity [m/s]
V = actual velocity [m/s]
Fr = Froude number [-]

Subcritical flow [stromend] Fr < 1 V < V
c
Supercritical flow [schietend] Fr > 1 V > V
c
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 = 1,8 3

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 y
c
Step 2 Calculate R and V
c
Step 3 Calculate S
c using Chezy or Manning
??????
??????=
&#3627408452;
2
??????∙&#3627408437;
2
3

??????
??????=&#3627408438;∙&#3627408453;∙&#3627408454;
??????
??????
??????=
&#3627408453;
2
3∙&#3627408454;
??????
1
2
??????

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