CU06997 Fluid Dynamics

Open channel flow 1 5.1 Flow with a free surface (page 122)

5.2 Flow classification (page 122, 123)

5.3 Channels and their properties (page 123-125)

5.4 Velocity distributions (page 126,127)

5.5 Laminar and turbulent flow (page 127-129)

5.6 Uniform flow (page 129 -138)

1

Flow with a free surface

1

Classification of flows, see part 2

1. Steady uniform flow example: pipe with constant D and Q

example: channel with constant A and Q

2. Steady non-uniform flow example: pipe with different D and constant Q

example: channel with different A and constant Q

3. Unsteady uniform flow example: channel with constant A and different Q

4. Unsteady non-uniform flow example; channel with different A and Q

2

Types of flow

2

Geometric properties

3

Velocity distributions

3

Velocity distributions

𝑉𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =𝑄

𝐴=

𝑉1𝐴1 + 𝑉2𝐴2 + 𝑉3𝐴3

𝐴1 + 𝐴2 + 𝐴3

𝑄𝑡𝑜𝑡𝑎𝑎𝑙 = 𝑄1 + 𝑄3 + 𝑄3=𝑉1𝐴1 + 𝑉2𝐴2 + 𝑉3𝐴3

3

Reynolds number, see part 3

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. 𝐷

𝜈

𝑅𝑒 =𝑉. 4𝑅

𝜈

In this course we only look at turbulent flow 3

Open channel, with bed slope >0

2211 AuAuQ

Head loss

Reference line

𝑦1 + 𝑧1 +𝑢1

2

2𝑔= 𝑦2 + 𝑧2 +

𝑢22

2𝑔+ ∆𝐻1−2

4

u1

Reference [m]

Surfacelevel y +z [m]

Total Head H [m]

P1

z1

y1

u12/2g

P2 z2

y2

u22/2g

u2

21

2

222

2

111

22 H

g

uzy

g

uzy

Head loss [m] ΔH

Velocity Head [m]

Open channel, with bed slope <= 0

4

Length

Chezy formula

Chezy formula describes the mean velocity of uniform, turbulent flow

ΔH

𝑉 = 𝐶 ∙ 𝑅 ∙ 𝑆𝑓

𝑉 = Mean Fluid Velocity [m/s]

R = Hydraulic Radius [m]

𝑆𝑓 = Hydraulic gradient [1]

𝐶 =8𝑔

𝜆 Chezy coefficient [m1/2/s]

𝑆𝑓 =ΔH

𝐿

5

Chezy coefficient

/s][m 12

log18 1/2

k

RC

In this course we assume a hydraulic rough boundary

Boundary hydraulic rough

kS = surface roughness [m]

5

Surface roughness kS [m] Equivalent Sand Roughness,

Material (ft) (mm)

Copper, brass 1x10-4 - 3x10-3 3.05x10-2 - 0.9

Wrought iron,

steel 1.5x10-4 - 8x10-3 4.6x10-2 - 2.4

Asphalt-lined

cast iron 4x10-4 - 7x10-3 0.1 - 2.1

Galvanized iron 3.3x10-4 - 1.5x10-

2 0.102 - 4.6

Cast iron 8x10-4 - 1.8x10-2 0.2 - 5.5

Concrete 10-3 - 10-2 0.3 - 3.0

Uncoated Cast

Iron 7.4x10-4 0.226

Coated Cast Iron 3.3x10-4 0.102

Coated Spun

Iron 1.8x10-4 5.6x10-2

Cement 1.3x10-3 - 4x10-3 0.4 - 1.2s

Wrought Iron 1.7x10-4 5x10-2

Uncoated Steel 9.2x10-5 2.8x10-2

Coated Steel 1.8x10-4 5.8x10-2

Wood Stave 6x10-4 - 3x10-3 0.2 - 0.9

PVC 5x10-6 1.5x10-3

Compiled from Lamont (1981), Moody (1944), and

Mays (1999)

5

Manning’s formula

𝑉 =𝑅

23 ∙ 𝑆𝑓

12

𝑛 𝑄 =

1

𝑛∙

𝐴53

𝑃23

∙ 𝑆𝑓

12

𝑉 = Mean Fluid Velocity [m/s]

R = Hydraulic Radius [m]

𝑆𝑓 = Slope Total head [1]

𝐴 = Wetted Area [m2]

𝑃 = Wetter Perimeter [m]

𝑛 = Mannings roughness coefficient [s/m1/3]

n

RC

61

Manning’s formula describes the

mean velocity of uniform,

turbulent flow

6

Manning's roughness coefficient

6

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]

7

Flowing water and energy

][2

2

1111 m

g

uyzH

u1

Reference /datum [m]

Surface level [m]

Total head H [m]

P1

z1

y1

u12/2g Velocity head [m]

y = Pressure head [m]

z = Potential head [m]

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]

8

Equilibrium / normal depth

Discharge, cross-section, energy

gradient and friction are constant

yyb

yb

P

AR

2

.

Side view

Cross-section

𝑉 = 𝐶 ∙ 𝑅 ∙ 𝑆𝑜

𝑞 = 𝑉 ∙ 𝐴 = 𝐶 𝑦 ∙ 𝑆𝑜2 ∙ 𝑦 ∙ 𝑏

𝑆0 = 𝑆𝑓

yn

yn

𝑦𝑛 =𝑞2

𝑏2 ∙ 𝐶2 ∙ 𝑆0

3

9

Equilibrium / normal depth

𝑦𝑛 =𝑞2

𝑏2 ∙ 𝐶2 ∙ 𝑆0

3

yn = normal depth [m]

q = discharge [m3/s]

b = width [m]

𝑆0 = bed slope [1]

𝑆𝑓 = Hydraulic gradient caused by friction [1]

𝐶 =8𝑔

𝜆 Chezy coefficient [m1/2/s]

𝑆0 = 𝑆𝑓

9

Equilibrium / normal depth

Dredged area

𝑦𝑛 =𝑞2

𝑏2 ∙ 𝐶2 ∙ 𝑆0

3

yn

yn

yn

yn

9