CU06997 Fluid dynamics

Froude number (page 148)

5.9 Critical depth meters (page 155 – 158)

1

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

Critical Depth

V

Reference /datum [m]

Water depth y [m]

y

V2/2g Velocity head [m]

y

B

g

VyH

2

2

yBVQv

22

2

2 yBg

QyH 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

22

2

2 yBg

QyH 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

0.00

1.00

2.00

3.00

4.00

5.00

6.000.3

00

0.5

90

0.8

80

1.1

70

1.4

60

1.7

50

2.0

40

2.3

30

2.6

20

2.9

10

3.2

00

3.4

90

3.7

80

4.0

70

H (

tota

l h

ead

) (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

cyH23

min

22

2

2 yBg

QyH 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

Critical Depth and Critical Velocity

cyH23

min

Sub-critical flow Supercritical flow

𝑦𝑐 =𝑄2

𝑔 ∙ 𝐵2

3

𝑉𝑐 = 𝑔 ∙ 𝑦𝑐2

3

h = y in this graph

Froude number

𝑦𝑐 =𝑄2

𝑔 ∙ 𝐵2

3

𝑉𝑐 = 𝑔 ∙ 𝑦𝑐

2 𝐹𝑟 =

𝑉

𝑔𝑦𝑐2

=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

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

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

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

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

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 =

𝑅23 ∙ 𝑆𝑐

12

𝑛

4

Critical bed slope channel /river

4

Hydraulic jump [watersprong]

When supercritical flow [schietend] changes to subcritical

flow [stromend] a hydraulic jump will occur

5

21

2

2

3

21

vvv

vvwa

Hydraulic jump, energy loss

5