AE 1350
Lecture Notes #6
We have studied...
• Meaning of Incompressible Flow
• How and why speed of the flow affects
compressibility
• Streamlines and stream tubes
• Continuity Equation
Topics to be studied
• Conservation of Momentum
– Euler’s equation
• Conservation of Energy
– Bernoulli’s equation (even though Euler
thought of it first.)
• Practical Applications of Bernoulli’s
Equation
Continuity
Mass Flow Rate In = Mass Flow Rate Out
r1 V1 A1 = r2 V2 A2
Station 1
Density r1
Velocity V1
Area A1
Station 2
Density r2
Velocity V2
Area A2
Momentum Equation
Based on Newton’s Second Law:
Rate of Change of Momentum of a particle= Forces acting on it
Consider an infinitesimally small slice of stream tube in space.
Rate of change of momentum of the fluid particles within this
stream tube must be due to forces acting on it.
Momentum Equation (Contd..)
Mass Flow Rate in = Mass Flow rate out
r VA = (r+dr)(V+dV)(A+dA)
Density r
velocity V
Area =A
Momentum rate in=
Mass flow rate times velocity
= rV2A
Density r+dr
velocity V+dV
Area =A+dA
Momentum Rate out=
Mass flow rate times velocity
= r VA (V+dV)
Momentum Equation (Contd..)
Density r
velocity V
Area =A
Momentum rate in=
Mass flow rate times velocity
= rV2A
Density r+dr
velocity V+dV
Area =A+dA
Momentum Rate out=
Mass flow rate times velocity
= r VA (V+dV)
Rate of change of momentum within this element =
Momentum rate out - Momentum rate in
= r VA (V+dV) - rV2A = r VA dV
Momentum Equation (Contd..)
Density r
velocity V
Area =A
Density r+dr
velocity V+dV
Area =A+dA
Rate of change of momentum as fluid particles
flow through this element= r VA dV
By Newton’s law, this momentum change must be caused by
forces acting on this stream tube.
Forces acting on the Stream tube
Horizontal Force = Pressure times area of the ring=(p+dp/2)dA
(p+dp)(A+dA)
Pressure
times
area=pA
Area of this ring = dA
Net force = pA + (p+dp/2)dA-(p+dp)(A+dA)=- Adp - dp • dA/2-Adp
Product of two
small numbers
Momentum Equation
From the previous slides,
Rate of change of momentum when fluid particles flow
through the stream tube = rAVdV
Forces acting on the stream tube = -Adp
We have neglected all other forces - viscous, gravity, electrical
and magnetic forces.
Equating the two factors, we get:
rVdV+dp=0
This equation is called the Euler’s Equation
Bernoulli’s Equation
Euler equation:
rVdV + dp = 0
For incompressible flows, this equation may be integrated:
r  VdV +  dp  0
Or ,
1
rV 2 + p  Const
2
Bernoulli’s
Equation
Kinetic Energy + Pressure Energy = Constant
Applications of Bernoulli’s
Equation
• See examples 4.1 through 4.3 in the text.
• We will do more worked out examples at
our next lecture.
• Important Applications include:
– Pitot Tube
– Venturi Meter
– Flow over airfoils
Pitot tubes are used on aircraft as a speedometer.
The Venturi Meter
It is used to measure Flow rates. Gas companies, Water works,
and aircraft fuel monitors all use this device.
How does the Venturi Meter work?
r1V1 A1  r 2V2 A2
Incompressible _ Flow : r1  r 2  r
Thus,
 A12  2 p2  p1 
V 1  2  
r
 A2 
Solve for V1 :
2
1
V1 A2

V2 A1
Bernoulli : p1 +
1
1
rV12  rV22  p2  p1
2
2
V22  2 p2  p1 
2
V1 1  2  
r
 V1 
1
1
rV12  p2 + rV2 2
2
2
V1 
2 p2  p1 
 A12 
r 1  2 
 A2 
Compute flow rate :
Flow _ rate  rV1 A1
That’s all folks!!!