CU06997 Fluid Dynamics

Behaviour of real fluids / Hydraulic Radius

Hydraulic Radius page 117 and page 125

3.1 Real and ideal fluids (page 64,65)

3.2 Viscous flow (page 66-68)

3.3 The stability of laminar flows and the onset of turbulence (p 68-71)

3.5 The Boundary layer (page76-78, )

3.6 Implications of the boundary layer concept (page 84-88)

1

Hydraulic radius

The hydraulic radius is a measure of a river-channel flow or

pipe flow efficiency. Water speed along the channel or pipe

depends on its cross-sectional shape (among other factors),

and the hydraulic radius is a characterisation of the channel

or pipe that intends to capture such efficiency. It is defined as

the ratio of the channel's cross-sectional area to its perimeter:

[m] PerimeterWetted

[m2]Area Wetted

P

AR

R = Hydraulic Radius [m]

A = Wetted Area [m2]

P = Wetted Perimeter [m] 2

R = Hydraulische straal [m]

A = Natte oppervlakte [m2]

P = Natte omtrek [m]

Hydraulic radius P

AR

2

Examples R

3 m

2 m

2 m

P

AR

River w=150 m , h = 3 m

2

Examples R

3 m

2 m

2 m

m 6,02233

23

P

AR

m 86,0223

23

P

AR

River w=150 m , h = 3 m

h m 9,215033

3150

P

AR

2

R and circle’s

2

R and circle’s

m 4

4/1 2 D

D

D

P

AR

m 4)(5,0

)4/1(5,0 2 D

D

D

P

AR

2

Hydraulic Diameter

D = 4 ∗ R

R = Hydraulic Radius [m]

D = Hydraulic Diameter [m]

2

R=D / 4

Partially filled pipes

QD= Discharge full pipe

VD= Velocity full pipe

Qd= Discharge partially full pipe

Vd= Velocity partially full pipe

3

3

h

Partially full pipes

𝐴𝑝 ℎ =1

4∙ 𝐷2 ∙ arccos(1 −

2 ∙ ℎ

𝐷) − (

𝐷

2− ℎ) ∙ ℎ ∙ 𝐷 − ℎ22

𝑅𝑝 ℎ =

14 ∙ 𝐷2 ∙ arccos 1 −

2 ∙ ℎ𝐷

ℎ ∙ 𝐷 − ℎ22 −𝐷

2− ℎ

𝐴𝑝 = Wetted Area partially filled pipe [m2]

𝑅𝑝 = Hydraulic radius partially filled pipe [m]

h = water level partially filled pipe [m]

D = Diameter pipe [m]

3

h

Ideal and Real flow

Ideal flow:

It was in viscid

It was incompressible

It had no surface tension

It always formed a continuum

Real flow

It has viscosity

4

Ideal and Real flow Shear force [N s/m2]

4

Viscosity

Viscosity is a measure of resistance

of a fluid which is being deformed

by either shear stress. In everyday

terms (and for fluids only), viscosity

is "thickness". Thus, water is "thin",

having a lower viscosity, while

honey is "thick" having a higher

viscosity. Viscosity describes a

fluid's internal resistance to flow and

may be thought of as a measure of

fluid frictionn

Velocity

distribution

4

Kinematic viscosity

𝜐 =𝜇

𝜌

𝜇 = Absolute viscosity [kg/ms]

𝜐 = Kinematic viscosity [m2/s] water, 20°C= 1,00 ∙ 10−6

𝜌 = Density of liquid [kg/m3]

4

Laminar and turbulent flow

5

Laminar and turbulent flow

Laminar (layers) Turbulent

Colored water Colored water

Tube of glass Tube of glass

5

Reynolds number, basic formula (p 70)

𝑅𝑒 =𝑢. 𝑙

𝜈

𝑢 = velocity [m/s]

𝜐 = Kinematic viscosity [m2/s]

water, 20°C= 1,00 ∙ 10−6

𝑙 = Length fluid / surface [m]

𝑅𝑒 = Reynolds Number [1]

5

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

5

Boundary layer

Simple definition: The fluid layer where the

boundary has influence on the velocity of the fluid

Ux = velocity free stream, no influence of boundary

6

Boundary layer

6

Boundary layer

6