The document discusses principles of fluid dynamics including the application of conservation laws and the energy equation to fluid flows. It covers topics such as Bernoulli's equation, velocity and pressure measurement techniques like Pitot tubes, flow through orifices, open channels, and pipes with head losses. Formulas are presented for flow rate calculation using concepts like velocity head, pressure head, and total head.

1. CU06997 Fluid Dynamics
Principles of fluid flow
2.5 Application of the conservations laws to fluid flows (page 25-32)
2.6 Application of the energy equation (page 32 -36)
2.8 Velocity and discharge measurement (page 42 – 48)
1

2. Flowing water and energy
2
u
H1  z1  y1  1
[m ]
2g
Total head H [m]
u12/2g Velocity head [m], [snelheidsh..]
Surface level [m]
y1 y = Pressure head [m]
u1 P1 [drukhoogte]
z1 z = Potential head [m]
. [plaatshoogte]
Reference /datum [m]
1

3. Bernoulli’s Equation, no energy losses
2 2
𝑢1 𝑢2
𝑦1 + 𝑧1 + = 𝑦2 + 𝑧2 + = 𝐻 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡[𝑚]
2𝑔 2𝑔
p
y= = Pressure Head[m] [drukhoogte]
ρ∙g
𝑧= Potential Head[m] [plaatshoogte]
u2
=
2g Velocity Head[m] [snelheidshoogte]
2

4. Bernoulli’s law (without energy losses)
2 2
u u
y1  z1  y2  z2   y 3  z3 
2
 constant
3
2g 2g
Total Head [m]
P1 y1 u22/2g u32/2g Velocity head [m]
z1 y2 Surface Level [m]
P2
u1=0 P3 y3
u2>0 z2 u3>u2 z3
Reference[m]
2

5. Pitot
3

6. Pitot
𝑢2
ℎ=
2∙ 𝑔
3

7. Pitot
2
𝑢= 2∙ 𝑔∙ℎ
𝑢= Fluid Velocity [m/s]
𝑔= earths gravity [m/s2]
h= Difference in pressure [m]
3

8. Flowing water and energy
2
u
H1  z1  y1  1
[m ]
2g
Total Head [m]
v12/2g Velocity head [m]
Surface Level [m]
h1
u=0
P1
z1 u1 Prandtl buis
Reference / datum [m]

9. Torricelli
u12 2
u2
y1  z1   y 2  z2 
2g 2g
u1=0 m/s
y2=0 m
y1+z1-z2=x
u2  2 g  x
4

10. Small orifice [Kleine doorlaat]
4

11. Small orifice [Kleine doorlaat]
4

12. Small orifice
2
𝑄 = 𝐶 𝑣 ∙ 𝐶 𝑐 ∙ 𝐴0 ∙ 2𝑔 ∙ ℎ
𝑄= Flow rate [m3/s]
𝐴= Wetted Area [m2]
𝐶𝑣 = velocity coefficient (0,97-0,99) [-]
𝐶𝑐 = contraction coefficient (0,61-0,66) [-]
𝑔= earths gravity [m/s2]
h= Difference in pressure [m]
4

13. Large orifice
3 3
2 2
2 2
𝑄 = ∙ 𝑏 ∙ 2𝑔 ∙ (ℎ2 − ℎ1 )
3
𝑄= Flow rate [m3/s]
𝑏= Width orifice [m2]
𝑔= earths gravity [m/s2]
h1 = Difference in pressure from top [m]
h2 = Difference in pressure from bottom [m]
5

14. Bernoulli’s law, with head loss
2 2
u u
y1  z1  1
 y 2  z2   H1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>u1 z2
6 Reference [m]

15. Pipe with head loss
2 2
v v
h1  1
 h4  4
 H14
2g 2g Q  v1  A1  v4  A4
Head loss
Total
Head
Pressure
Head
6

16. Open channel
u12 2
u2
h1   z1  h2   z2  H12
2g 2g
Q  v1  A1  v2  A2
Head loss
Reference line
6

17. Bernoulli expressed in m (head)
𝐸 𝑡𝑜𝑡𝑎𝑙 = 𝐸 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 + 𝐸 𝑘𝑖𝑛𝑒𝑡𝑖𝑐
= 𝑚 ∙ 𝑔 ∙ 𝑑 + 1 𝑚 ∙ 𝑢2
2
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 [𝐽 = 𝑁𝑚]
𝐸 𝑡𝑜𝑡𝑎𝑙 𝑢2 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
= 𝑑+ = [m]
𝑚∙𝑔 2𝑔 𝑚∙𝑔
𝑝
𝑑= 𝑧+ 𝑦= 𝑧+ [𝑚]
𝜌∙ 𝑔
𝑽𝟐 Total Head
𝑯= 𝒚+ 𝒛+ [m]
7 𝟐∙𝒈

18. 𝑢2
𝐻 = 𝑦+ 𝑧+ [m]
2∙𝑔
You also could express the energy in Pa (N/m2) instead of m.
𝑝= 𝜌∙ 𝑔∙ 𝑦
If you combine 1 en 2 by multiply al parameters with 𝜌 ∙ 𝑔 (they don’t
change)
𝑢2
𝜌∙ 𝑔∙ 𝐻 = 𝜌∙ 𝑔∙ 𝑦+ 𝜌∙ 𝑔∙ 𝑧+ 𝜌∙ 𝑔∙
2∙ 𝑔
𝑢2
𝐻 = 𝑝+ 𝜌∙ 𝑔∙ 𝑧+ 𝜌∙ [𝑃𝑎]
2
7

19. Bernoulli expressed in Pa (pressure)
𝑢2
𝐻 = 𝑝+ 𝜌∙ 𝑔∙ 𝑧+ 𝜌∙ [𝑃𝑎]
2
H energy [Pa]
𝑢2
𝜌∙
2
Surface level
p1 p = pressure [Pa]
u1 P1
z1 𝜌 ∙ 𝑔 ∙ 𝑧 = “potential” [Pa]
7 Reference /datum [m]