Euler's and Bernoulli's Equations

Overview

On November 7, 1940, the Tacoma Narrows Bridge began to twist and undulate up and down under relatively moderate (42 mph) cross-winds. A few hours later, the undulations and twisting increased out of control and the bridge finally broke apart in one of the most spectacular engineering failures ever recorded on film. According to Sir Alfred Pugsley, a prominent structural engineer of the mid-1960s, "had the designers of the Tacoma Narrows Bridge known more aerodynamics, the collapse might have been averted". His comment was very appropriate, as the behavior of this bridge was very similar to the response of airplane wings subjected to uncontrolled turbulence. A bridge, just like an airplane wing, can under certain conditions exhibit aeroelastic instabilities, as the pressure forces from the wind bend it one way, while its steel structure (aluminum for an airplane) is pulling in the opposite direction.

This example illustrates the importance of being able to predict pressure loads on structures, whether stationary or moving through a fluid. In the case of an airplane for example, it is the pressure forces generated from the relative motion of the beast through the air, which translate into lift and drag.

When the flow involves a liquid, there is an additional reason for being able to predict the pressure in certain critical parts of the flow (ex., pump impellers, the crest of hydrofoils or marine propellers, spillways of dams and even bends in pipelines). The fluid may boil while flowing under normal temperatures, if the pressure drops below its vapor pressure value at the particular temperature. This phenomenon is known as cavitation and can cause significant damage to any hardware in contact with the cavitating fluid.

Fluid pressure is also important to know in many everyday applications. For example:

Our blood pressure, an important indicator of our health, is directly related to the flow of blood in our veins and arteries.
Atmospheric pressure readings, an important indicator of the weather, are related to the vortex strength in cyclonic storms.

Learning Objectives

	By the end of this unit, you should be able to:
	1	Derive Euler's equation of motion for a fluid.
	2	Derive Bernoulli's equation and list the assumptions made in the derivation.
	3	Apply Bernoulli's equation in a variety of problems including flow velocity measurements and pressure calculations.
	4	Predict cavitation in enclosed pipes or hydraulic machines.
	5	Describe the differences between ideal (fully attached) and real (separated) flow over a circular cylinder.

Discussion Questions

1	What is the basic principle behind Euler's equation?
2	What is the basic principle behind Bernoulli's equation?
3	Under what conditions are we allowed to use Bernoulli's equation in a flow problem?
4	Write the Bernoulli equation so that each term represents: a. energy per unit mass b. energy per unit weight c. energy per unit volume
5	What kind of energy does each term in the Bernoulli equation represent?
6	Why is the Bernoulli equation so important?
7	Consider the flow field around a circular cylinder. How does flow separation affect the pressure on the surface of the cylinder?
8	What is cavitation ?. Think of and discuss at least 2 engineering examples of cavitation.
9	Use the Bernoulli principle to explain how airplane wings and hydrofoils generate lift.
10	How did the Tacoma Narrows bridge collapse?
11	When you wash your car using your garden hose, why do you put your finger at the mouth of the hose when you try to clean dirty spots?

Applications

1	Rotating tank of water.
2	Water jet flowing out of a nozzle.
3	Hydrofoil operating under cavitating conditions.

Workout Problems

Take-Home Problem (team effort, 20 points)

A soccer ball is pressurized to 20 kPa-g at the start of a game. It develops a small leak from a hole with an area of 0.006 mm². Would the ball feel noticeably softer at the end of the 1st half of the game (45 min)?

Links

1.	Bernoulli's equation
2.	Pitot Tube
3.	Lift