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Basics of fluid flow
Types of flow
Fluid
Ideal/Real
Compressible/Incompressible
Flow
Steady/Unsteady
Uniform/Non-uniform
Laminar/Turbulent
Pressure/Gravity (free surface)
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Basics of fluid flow (Chapter 4)
Basics of fluid flow, kinematics
Mechanics
Statics
Dynamics
Kinematics
Kinetics
Kinematics: deals with motion apart from considerations
of mass, force or energy
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Basics of fluid flow
Path lines, streamlines and streak lines
Path line: the trajectory that a fluid particle would make as it moves
around with the flow
Streamline: line that shows the flow direction, local velocity vector is
tangent to the streamline at every point along the line at that instant
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Basics of fluid flow
Types of flow
Steady flow: all fluid/flow properties at any point in the flow do not
change with time; however, conditions may be different at different
points.
Uniform flow: at every point in the flow, the velocity (in both magnitude
and direction) is identical at any given instant.
For steady flows:
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Basics of fluid flow
One -, two -, and three- dimensional flows
This is the most general 3-D flow:
The flow is classified as 2-D if: The flow can be viewed as 1-D if:
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Partial derivative
Differentiating a function of more than one variable with respect to a particular
variable, with the other variables kept constant:
the notation ∂f/∂t means the partial derivative of the function f with respect to t
∂f/ ∂t : partial derivative
df/dt : total derivative
For more info:
http://apollo.lsc.vsc.edu/classes/met380/Fingerhuts_notes/driv.pdf
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Basics of fluid flow
Velocity and Acceleration, 4.12
Convective (spatial)
acceleration
at
Local (temporal)
acceleration
an
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Basics of fluid flow
Flow rate and Mean velocity
Flow rate: the rate at which fluid
crosses a known surface
volume flow rate mass flow rate
The volume flow rate passing through
the element of area dA (in yz plane) is
dQ = u(cosθ)dA=udA’
volume flow rate is equal to the magnitude of the mean
velocity multiplied by the flow area at right angles to the
direction of the mean velocity
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Basics of fluid flow
Flow rate and Mean velocity
The volume flow rate passing through
the element of area dA is
dQ = u·dA =udA´
the local time mean velocity, u, will vary across the
section for real fluid
A
AVudAQ
QAVudAmA
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Basics of fluid flow
Reynolds Transport Theorem & Continuity
QVAVA 2211
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Copyright © The McGraw-Hill Companies, Inc.
FIGURE 5-24
Bernoulli’s Equation
(Energy per unit weight)
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Energy in Steady Flow (Chapter 5)
Energies of a Flowing Fluid (Euler’s Equation)
Kinetic Energy
Potential Energy
1/2mV2 V2/2g
Wz z
Pressure Head
p = γh p/γ
Unit: L
(Energy per unit
weight)
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Derivation of the Bernoulli Equation
The forces acting on a fluid
particle along a streamline.
Steady, incompressible flow:
The sum of the kinetic, potential, and
flow energies of a fluid particle is
constant along a streamline during
steady flow when compressibility and
frictional effects are negligible.
Bernoulli
equation
The Bernoulli equation between any
two points on the same streamline:
Steady flow:
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Energy in Steady Flow (Chapter 5)
Bernoulli’s Equation
Unit: L
(Energy per unit weight)
Basic assumptions:
•Inviscid & incompressible fluid
•Steady flow
•Applies along a streamline
• No energy added or removed from the
fluid along the streamline
Piezometric pressure
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Copyright © The McGraw-Hill Companies, Inc.
FIGURE 5-22
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Energy in Steady Flow, Pipe Flow
V
p/γ p/γ
V2/2g
Bring moving water to a halt, and it'll
drive a column of water up to exactly
the height from which water would flow
to gain that velocity.
Pitot Tube
(Measures stagnation
pressure)
Free stream dynamic
pressure
Free stream static
pressure
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Energy in Steady Flow, Free surface flow
V
V2/2g
Bring moving water to a halt, and it'll
drive a column of water up to exactly
the height from which water would flow
to gain that velocity.
Pitot Tube
(Measures stagnation
pressure)
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Example: Bernoulli’s principle, Pitot Tube
http://www.youtube.com/watch?v=dk39ffdWq_E
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Example:
Spraying Water
into the Air
Example: Water Discharge
from a Large Tank
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The hydraulic
grade line (HGL)
and the energy
grade line (EGL)
for free discharge
from a reservoir
through a
horizontal pipe
with a diffuser.
Hydraulic grade line (HGL), P/g + z The line that represents the sum of
the static pressure and the elevation heads.
Energy grade line (EGL), P/g + V2/2g + z The line that represents the
total head of the fluid.
Dynamic head, V2/2g The difference between the heights of EGL and HGL.
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Energy in Steady Flow
Stagnation pressure, ideal fluid (5.4)
1 2
V
V1 = V, p2 is the
stagnation
pressure
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Energy in Steady Flow
General Energy Equation, steady flow, incompressible fluid
For an incompressible fluid with γ = const. and α =1:
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Energy in Steady Flow
General Energy Equation, steady flow, incompressible fluid
For an incompressible fluid with γ = const. and α =1:
If there is no machine between points 1 and 2:
If head loss is neglected:
Real fluid
Ideal fluid
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Energy in Steady Flow
Power considerations in fluid flow, Derivation of Power Equation
Power: P = (Force) x (Velocity) Power: P = Energy / Time
P = FV (F =ΔpA)
P = (ΔpA)V (Δp = γh)
P = (γhA)V (Q = AV)
P = γhQ
P = (Energy/Weight) x (Weight/Time)
P = ΔpQ
head (h) γQ
P = h γQ
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Energy in Steady Flow
Power considerations in fluid flow, Units of Power
P = γhQ
Horsepower = P = γhQ/550
( [Q] = cfs, [h] = ft, [γ] = pcf )
Power in BG units
Kilowatts = P = γhQ/1000
( [Q] = m3/s, [h] = m, [γ] = N/m3 )
Power in SI units
P = γhQ P: power put into flow by a pump,
then h = hpump
P: power lost because of friction,
then h = hL
Pump efficiency, η = (power output) / (power input)
http://www.waterencyclopedia.com/Po-Re/Pumps-Traditional.html