This document discusses fluid dynamics concepts related to pipe flow, including laminar and turbulent flow. It covers fundamental concepts such as Reynolds number, laminar flow equations for frictional head loss and wall shear stress. Turbulent flow equations are presented for head loss, Darcy-Weisbach, and Colebrook-White transition formula. Local head losses are discussed for sudden pipe enlargement and contraction.

1. CU06997 Fluid Dynamics
Flow in pipes and closed conduits
4.1 Introduction (page 91)
4.2 The historical context (page 91-93)
4.3 Fundamental concepts of pipe flow (page 94-97)
4.4 Laminar flow (page 97-100)
4.5 Turbulent flow (page 100 – 111)
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2. Pipe with head loss
2 2
u u
h1  1
 h4  4
 H14
2g 2g Q  u1  A1  u4  A4
Head loss
Total
Head
Pressure
Head
1

3. Reynolds number:
p 93 (pipe), p 127 (open channel)
𝜌∙ 𝑉∙ 𝐷 𝑉∙ 𝐷
𝑅𝑒 = =
𝜇= Absolute viscosity [m2/s]
𝜇 𝜈
𝜐= Kinematic viscosity [kg/ms]
water, 20°C= 1,00 ∙ 10−6 𝑉. 4𝑅
𝜌 = Density of liquid [kg/m3] 𝑅𝑒 =
𝑉 = Velocity [m/s] 𝜈
D = Hydraulic diameter [m]
R= Hydraulic Radius = D/4 [m]
𝑅𝑒 = Reynolds Number [1]
𝑹𝒆 > 𝟒𝟎𝟎𝟎 Turbulent flow
𝑹𝒆 < 𝟐𝟎𝟎𝟎 Laminar flow
1

4. Laminar flow, frictional head loss
[Energieverlies tgv wrijving]
Total Head
32 ∙ 𝜇 ∙ 𝐿 ∙ 𝑉 Pressure Head
ℎ𝑓 =
𝜌 ∙ 𝑔 ∙ 𝐷2
ℎ𝑓 = frictional head loss ∆H [m]
𝜇= Absolute viscosity [kg/ms]
𝐿= Length between the Head Loss [m]
𝑉= mean velocity [m/s]
D= Hydraulic Diameter [m]
𝜌= Density of liquid [kg/m3]
2 𝑔= earths gravity [m/s2]

5. Laminar flow, wall shear stress
[Schuifspanning]
4∙ 𝜇∙ 𝑉
𝜏0=
𝑅
τ0 = shear stress at solid boundary [N/m2]
𝜇= Absolute viscosity [kg/ms]
𝑉= mean velocity [m/s]
R= Hydraulic Radius [m]
2

6. Head loss /Energy loss [m]
• Turbulent flow u 2
• Friction loss (wrijvingsverlies) ΔΗ    [m]
2g
• Local loss (lokaal verlies)
• ΔH = Head loss or Energy loss [m]
• u2/2g = Velocity head [m]
• ξ (ksie) = Loss coëfficiënt [1]
3

7. 2 2
Darcy-Weisbach L u u
ΔΗ f      
4 R 2g 2g
Total Head
L
f  
Pressure Head
4R
• ΔH = Head loss by friction [m]
• u2/2g = Velocity head [m]
• L = Length [m]
• λ = (lamda) = Friction coëfficiënt[1]
• ξ (ksie) = Loss coëfficiënt [1]
3 • R = hydraulic radius [m]

8. Remarks friction loss Darcy-Weisbach
• λ (boundary roughness) depends on material and
construction. λ often between 0,01 and 0,10
• λ is not a constant, depends on “boundary layer”.
“Smooth” or “Rough”, Most of the time “Smooth”
How to calculate λ !!!
• During exams Fluid Dynamics, the λ will be given
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9. Colebrook-White transition formula
1 𝑘𝑠 2,51
= −2 ∙ 𝑙𝑜𝑔 +
𝜆 3,70 ∙ 𝐷 Re∙ 𝜆
𝜆= Friction coefficient [1]
D= Hydraulic Diameter 4R [m]
kS = surface roughness [m]
(k-waarde)
Difficult to solve
Could use figure 4.5 page 105
Nowadays computers?
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10. Moody diagram
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11. Colebrook-White and Darcy Weisbach
𝑘𝑠 2,51υ
𝑉 = −2 2𝑔 ∙ 𝐷 ∙ 𝑆 𝑓 ∙ 𝑙𝑜𝑔 +
3,70𝐷 D 2𝑔∙𝐷∙𝑆 𝑓
ℎ𝑓
with 𝑆 𝑓 =
𝐿
𝑉= Average velocity [m/s]
D= Hydraulic Diameter (4R) [m]
kS = surface roughness [m]
𝜐= Kinematic viscosity [kg/ms]
Sf = slope of hydraulic gradient [-]
hf = frictional head loss (∆Hf) [m]
𝐿= Length between the Head Loss [m]
3

12. Turbulent flow ,
Mean boundary shear stress
𝜏0 = 𝜌 ∙ 𝑔 ∙ 𝑅 ∙ 𝑆0
τ0 = shear stress at solid boundary [N/m2]
R= Hydraulic Radius [m]
𝑆0 = Slope of channel bed [1]
In sewer minimum shear stress value
(0.5 – 1.5 N/m2)
3

13. Local head losses
2
u
ΔΗ l    [m]
2g
4

14. Head loss Sudden Pipe Enlargement
V1  V2 
2
∆𝐻 𝑙 = (1 −
𝐴1 2 𝑉1
) ∙
2
ΔΗ l  𝐴2 2𝑔
2g
4

15. Head loss Sudden Pipe Enlargement
∆𝐻 𝑙 = (1 −
𝐴1 2 𝑉1
) ∙
2
𝜉 𝑙 = (1 −
𝐴1 2
) (𝑉1 − 𝑉2 )2
𝐴2 2𝑔 𝐴2 ∆𝐻 𝑙 =
2𝑔
∆𝐻 𝑙 = Head Loss due to sudden pipe enlargement [m]
𝜉𝑙 = Loss coefficient due to sudden pipe enlargement [1]
𝐴= Wetted Area [m2]
𝑉= Mean Fluid Velocity [m/s]
𝑔= earths gravity [m/s2]
1= Before enlargement
2= After enlargement
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16. Head loss Sudden Pipe Contraction
4
2
𝐴1 2
𝑉2 𝑉2
∆𝐻 𝑙 = ( − 1)2 ∙
𝐴3 2𝑔
and 𝐴3 ≅ 0,6 ∙ 𝐴2 ∆𝐻 𝑙 = 0,44 ∙
2𝑔
∆𝐻 𝑙 = Head Loss due to sudden pipe contraction [m]
𝑉2 = Mean Fluid Velocity after sudden pipe contraction [m/s]
𝑔 = earths gravity [m/s2]

17. Local head loss coefficients
𝑢2 𝑘 𝑙 = 𝜉𝑙
∆𝐻 𝑙 = 𝑘 𝑙 ∙
2𝑔
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