2. AE 2350 Lecture Notes #6 April 19, 1999

3. We have studied... Meaning of Incompressible Flow How and why speed of the flow affects compressibility Streamlines and stream tubes Continuity Equation

4. 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 Read Chapter 6 in the text!

5. Continuity Station 1 Density  1 Velocity V 1 Area A 1 Station 2 Density  2 Velocity V 2 Area A 2 Mass Flow Rate In = Mass Flow Rate Out  1 V 1 A 1 =  2 V 2 A 2

6. 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.

7. Momentum Equation (Contd..) Density  velocity V Area =A Density  d  velocity V+dV Area =A+dA Mass Flow Rate in = Mass Flow rate out  VA = (  +d  )(V+dV)(A+dA) Momentum rate in= Mass flow rate times velocity =  V 2 A Momentum Rate out= Mass flow rate times velocity =  VA (V+dV)

8. Momentum Equation (Contd..) Density  velocity V Area =A Density  d  velocity V+dV Area =A+dA Momentum rate in= Mass flow rate times velocity =  V 2 A Momentum Rate out= Mass flow rate times velocity =  VA (V+dV) Rate of change of momentum within this element = Momentum rate out - Momentum rate in =  VA (V+dV) -  V 2 A =  VA dV

9. Momentum Equation (Contd..) Density  velocity V Area =A Density  d  velocity V+dV Area =A+dA Rate of change of momentum as fluid particles flow through this element=  VA dV By Newton’s law, this momentum change must be caused by forces acting on this stream tube.

10. Forces acting on the Stream tube Pressure times area=pA (p+dp)(A+dA) Horizontal Force = Pressure times area of the ring=(p+dp/2)dA 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

11. Momentum Equation From the previous slides, Rate of change of momentum when fluid particles flow through the stream tube =  AVdV 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:  VdV+dp=0 This equation is called the Euler’s Equation

12. Bernoulli’s Equation Euler equation:  VdV + dp = 0 For incompressible flows, this equation may be integrated: Kinetic Energy + Pressure Energy = Constant Bernoulli’s Equation

13. Applications of Bernoulli’s Equation See examples 6.1 through 6.4 in the text. We will do more worked out examples on Wednesday, April 21. Important Applications include: –Pitot Tube –Venturi Meter –Flow over airfoils

14. Pitot tubes are used on aircraft as a speedometer.

15. The Venturi Meter It is used to measure Flow rates. Gas companies, Water works, and aircraft fuel monitors all use this device.

16. How does the Venturi Meter work?

17. That’s all folks!!!