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Lecture 3

Airplane wing. Lecture 3. Rear wing. Bernoulli ’ s equation. Rain barrel. Tornado damage. If the fluid is incompressible, the volume should remain constant:. dV. dV. Work by pressure during its motion:. Work by pressure.

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Lecture 3

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  1. Airplane wing Lecture 3 Rear wing Bernoulli’s equation. Rain barrel Tornado damage

  2. If the fluid is incompressible, the volume should remain constant: dV dV Work by pressure during its motion: Work by pressure As an element of fluid moves during a short interval dt, the ends move distances ds1 and ds2. A2 v2 ρ2 A1 v1 ρ1 ds1 ds2

  3. Change in kinetic energy: Change in potential energy: y: height of each element relative to some initial level (eg: floor) A2 v2 ρ2 A1 v1 ρ1 ds1 ds2 Kinetic and gravitational potential energy

  4. Bernoulli’s equation Putting everything together: NB: Bernoulli’s equation is only valid for incompressible, non-viscous fluids with a steady laminar flow!

  5. Static vs flowing fluid Pressure at point A (hA below surface): Or gauge pressure: Now we drill a small hole at depth hA. hA Point A is now open to the atmosphere! A x Cylindrical container full of water. hA x A

  6. Container with hole Bernoulli at points A and B (on the surface): (Eqn 1) B x Continuity at points A and B: A x (Eqn 2) Assume the radius of the container is R = 15 cm, the radius of the hole is r = 1 cm and hA = 10 cm. How fast does water come out of the hole? R = 15 cm hA = 10 cm yB yA

  7. DEMO: Container with holes For once, let’s plug in some numbers before the end: Therefore, This is equivalent to taking vB ~ 0 (the container surface moves very slowly because the hole is small ―compared to the container’s base)

  8. h flow ●A ● B 2 equations for vA, vB Measuring fluid speed: the Venturi meter A horizontal pipe of radius RA carrying water has narrow throat of radius RB. Two small vertical pipes at points A and B show a difference in level of h. What is the speed of water in the pipe? Venturi effect: High speed, low pressure Low speed, high pressure

  9. DEMO: Tube with changing diameter

  10. But… It can be used if the speed of the gas is not too large (compared to the speed of sound in that gas). i.e., if the changes in density are small along the streamline Partially illegal Bernoulli Gases are NOT incompressible Bernoulli’s equation cannot be used

  11. Example: Why do planes fly? High speed, low pressure DEMO: Paper sucked by blower. DEMO: Beach ball trapped in air. Low speed, high pressure Net force up (“Lift”)

  12. Aerodynamic grip Race cars use the same effect in opposite direction to increase their grip to the road (important to increase maximum static friction to be able to take curves fast) Tight space under the car ➝ fast moving air ➝ low pressure Higher pressure Lower pressure Net force down

  13. Upward force on a 10 m x 10 m roof: Weight of a 10 m x 10 m roof (0.1 m thick and using density of water –wood is lighter than water but all metal parts are denser): Tornadoes and hurricanes Strong winds ➝ Low pressures vout = 250 mph (112 m/s) vin = 0 The roof is pushed off by the air inside !

  14. The suicide door Air pressure decreases due to air moving along a surface. The high speed wind will also push objects when the wind hits a surface perpendicularly! Modern car doors are never hinged on the rear side anymore. If you open this door while the car is moving fast, the pressure difference between the inside and the outside will push the door wide open in a violent movement. In modern cars, the air hits the open door and closes it again.

  15. Curveballs Speed of air layer close to ball is reduced (relative to ball) Speed of air layer close to ball is increased (relative to ball) Boomerangs are based on the same principle (Magnus effect)

  16. Beyond Bernoulli In the presence of viscosity, pressure may decrease without an increase in speed. Example: Punctured hose (with steady flow). Speed must remain constant along hose due to continuity equation. Lower jet. Real fluid (with viscosity) Friction accounts for the decrease in pressure. Ideal fluid (no viscosity)

  17. The syphon The trick to empty a clogged sink: A x Thin hose → vA ~ 0 h PA = PB = Patm x B

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