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

41

2

44

2

11

22 H

g

uh

g

uh

4411 AuAuQ

Pressure

Head

Total

Head

Head loss

1

Reynolds number:

p 93 (pipe), p 127 (open channel)

𝜇 = Absolute viscosity [m2/s]

𝜐 = Kinematic viscosity [kg/ms]

water, 20°C= 1,00 ∙ 10−6

𝜌 = Density of liquid [kg/m3]

𝑉 = Velocity [m/s]

D = Hydraulic diameter [m]

R = Hydraulic Radius = D/4 [m]

𝑅𝑒 = Reynolds Number [1]

𝑹𝒆 > 𝟒𝟎𝟎𝟎 Turbulent flow

𝑹𝒆 < 𝟐𝟎𝟎𝟎 Laminar flow

𝑅𝑒 =𝑉. 4𝑅

𝜈

𝑅𝑒 =𝜌 ∙ 𝑉 ∙ 𝐷

𝜇=

𝑉 ∙ 𝐷

𝜈

1

Laminar flow, frictional head loss

[Energieverlies tgv wrijving]

ℎ𝑓 =32 ∙ 𝜇 ∙ 𝐿 ∙ 𝑉

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]

𝑔 = earths gravity [m/s2] 2

Total Head

Pressure Head

Laminar flow, wall shear stress

[Schuifspanning]

𝜏0=

4 ∙ 𝜇 ∙ 𝑉

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

• Friction loss (wrijvingsverlies)

• Local loss (lokaal verlies)

[m] 2g

uΔΗ

2

• ΔH = Head loss or Energy loss [m]

• u2/2g = Velocity head [m]

• ξ (ksie) = Loss coëfficiënt [1]

3

Darcy-Weisbach

2g

u

2g

u

4ΔΗ

22

f R

L

• ΔH = Head loss by friction [m]

• u2/2g = Velocity head [m]

• L = Length [m]

• λ = (lamda) = Friction coëfficiënt[1]

• ξ (ksie) = Loss coëfficiënt [1]

• R = hydraulic radius [m]

R

Lf

4

Total Head

Pressure Head

3

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 !!!

3

• During exams Fluid Dynamics, the λ will be given

Colebrook-White transition formula

1

𝜆= −2 ∙ 𝑙𝑜𝑔

𝑘𝑠

3,70 ∙ 𝐷+

2,51

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

𝑉 = −2 2𝑔 ∙ 𝐷 ∙ 𝑆𝑓∙ 𝑙𝑜𝑔𝑘𝑠

3,70𝐷+

2,51υ

D 2𝑔∙𝐷∙𝑆𝑓

with 𝑆𝑓 =ℎ𝑓

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

[m] 2g

uΔΗ

2

l

4

Head loss Sudden Pipe Enlargement

2g

VVΔΗ

2

21

l

∆𝐻𝑙= (1 −

𝐴1

𝐴2)2∙

𝑉12

2𝑔

4

Head loss Sudden Pipe Enlargement

∆𝐻𝑙 =(𝑉1 − 𝑉2)2

2𝑔 ∆𝐻𝑙= (1 −

𝐴1

𝐴2)2∙

𝑉12

2𝑔 𝜉𝑙 = (1 −

𝐴1

𝐴2)2

∆𝐻𝑙 = 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

∆𝐻𝑙= (𝐴1

𝐴3− 1)2∙

𝑉22

2𝑔 and 𝐴3 ≅ 0,6 ∙ 𝐴2 ∆𝐻𝑙= 0,44 ∙

𝑉22

2𝑔

∆𝐻𝑙 = Head Loss due to sudden pipe contraction [m]

𝑉2 = Mean Fluid Velocity after sudden pipe contraction [m/s]

𝑔 = earths gravity [m/s2]

4

Local head loss coefficients

∆𝐻𝑙 = 𝑘𝑙 ∙𝑢2

2𝑔

𝑘𝑙 = 𝜉𝑙

4