cu06997 lecture 5_reynolds_and_r
TRANSCRIPT
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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)
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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]
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Hydraulic radius P
AR
2
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Examples R
3 m
2 m
2 m
P
AR
River w=150 m , h = 3 m
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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
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R and circle’s
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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
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Hydraulic Diameter
D = 4 ∗ R
R = Hydraulic Radius [m]
D = Hydraulic Diameter [m]
2
R=D / 4
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Partially filled pipes
QD= Discharge full pipe
VD= Velocity full pipe
Qd= Discharge partially full pipe
Vd= Velocity partially full pipe
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h
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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]
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h
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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
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Ideal and Real flow Shear force [N s/m2]
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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
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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]
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Laminar and turbulent flow
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Laminar and turbulent flow
Laminar (layers) Turbulent
Colored water Colored water
Tube of glass Tube of glass
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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]
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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
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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
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Boundary layer
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Boundary layer
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