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)
1

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

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

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]

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

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

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]

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
3

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?
3

Moody diagram
3

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

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

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

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

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
4

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]

Local head loss coefficients
𝑢2 𝑘 𝑙 = 𝜉𝑙
∆𝐻 𝑙 = 𝑘 𝑙 ∙
2𝑔
4