Fluid dynamics is the study of how fluids (gases or liquids) flow. Because water is such a common fluid, fluid dynamics is often called hydrodynamics.

When fluid flows through a pipe, the flow or discharge (J) is the mass {J} (or volume {Q}) of fluid that passes a given point per unit of time.

Mathematically:

Q = Av

Describing discharge as mass per unit time is actually more correct, but if the pipe is full of an incompressible (constant ) fluid then either description is fine.

i.e.

J =m/t = V/t = Al/t = Av

Since is usually constant, discharge in terms of volume is…

Q = Av

If an incompressible fluid fills a pipe and flows through it, the discharge stays constant even if the diameter of the pipe changes.

Mathematically:

Q = A1v1 = A2v2 = constant

We can study fluid flow patterns with wind tunnels:

There are many types.

Some are wicked cool!

Some contain wicked cool things!

After designing models based on computed calculations of flow characteristics, the predictions can be checked with a flow test. http://www.esa.int/esaCP/ESA9DBG18ZC_index_0.html

There are two main types of fluid flow:

and

Laminar flow (AKA streamline flow) occurs when the particles of the fluid follow smooth, noncrossing paths.

Note that during laminar flow, neighboring layers of the fluid slide by each other smoothly.

Note that this is a shearing process.

To study this process, two plates are separated by a thin layer of liquid.

A force is applied to the top plate to make it move.

The rapidity of the shearing motion is characterized by the shear rate of the two plates and the fluid between them.

Shear rate = speed of top plate

distance between platesShear rate = v/L = s / t

The viscosity of a fluid is the shear stress required to produce a unit shear rate.

= viscosity = shear stress/shear rate.

= (F/A) / (v/L)

= viscosity = shear stress/shear rate.

= (F/A) / (v/L) = (FL) / (vA)

FYI: = the lower Greek letter eta

F vA / L for any given fluid. So the larger the value, the greater the force resisting the attempted shear under a given set of conditions. (i.e. The fluid is stickier.)

For liquids, the viscosity results from attractive forces between the molecules. For gases, the viscosity results from collisions between the molecules.

The SI unit for is N.s/m2, or Pa.s. This is called the poiseuille (Pl). Other units are the cgs unit the poise (P), for which 1 P = 0.1 Pl, and prefix versions of each.

The greater the viscosity in a fluid, the greater the heat generated as it is sheared under a given set of conditions.

Because of viscosity, it takes a pressure difference at the ends of a horizontal pipe to have laminar fluid flow through it at a steady rate. A French scientist named Poiseuille studied this in the 1800s and developed the formula that bears his name:

Q = (r

4(P1-P2)) / (8L)

Q = (r

4(P1-P2)) / (8L)

All My Loving

Pi r fourth delta P

Over 8 eta L

Shows how fast V’s flowing…

That’s Q

When fluid flows beyond a certain speed, the laminar flow breaks down into turbulent flow.

Turbulent flow is characterized by small whirlpools called eddies, which consume an enormous amount of energy. This increases the drag on the object in the fluid flow far above the drag created by viscosity during streamline (laminar) flow. For liquids in a pipe, this translates to a need for a much higher (and less predictable) ____________ to maintain the flow.

The Reynold’s Number (NR) is a dimensionless experimental number that gives an indication of the velocity at which turbulence will occur in a fluid.

Mathematically:

NR = vD/

Fluid flow will usually be laminar if NR does not exceed about 2000 for fluid flowing through a pipe, or about 10 for obstacles.

In the 1700s, a mathematician named Daniel Bernoulli studied the pressure associated with moving fluids and came to a startling conclusion:

http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Bernoulli_Daniel.html

Bernoulli’s Principle basically states that… As a fluid’s velocity increases, its internal pressure decreases!

Bernoulli’s Principle applies to a variety of phenomena.

Mathematically:

P1 + ½ v12 + gh1 = P2 + ½ v2

2 + gh2

Bernoulli’s Equation is really a restatement of the Law of Conservation of Energy: The total energy of a closed system remains constant. (This is true unless there is a ____________ change.)

P1 + ½ v12 + gh1 = P2 + ½ v2

2 + gh2

Since work is done whenever a force is applied through a distance,

work is done whenever pressure forces a volume of fluid to move as well.

P1 + ½ v12 + gh1 = P2 + ½ v2

2 + gh2

Note: W = (F/A) x A x lA A

Also, since work must be done to accelerate an object, faster moving objects have more kinetic energy.

By replacing the m in the equation with Al, we can see that

P1 + ½ v12 + gh1 = P2 + ½ v2

2 + gh2

W = KE = ½ mv2

W = KE = ½ Vv2

Lastly, since work must be done to raise an object, potential energy may be exchanged for kinetic energy.

By replacing the m in the equation with Al, we can see that

P1 + ½ v12 + gh1 = P2 + ½ v2

2 + gh2

W = PE = KE = mgh

W = PE = KE = Vgh

So all the terms in Bernoulli’s Equation are really energy terms associated with a given volume movement.

P1V + ½ Vv12 + Vgh1 = Constant

This becomes:

P1 + ½ v12 + gh1 = Constant / V

Note that Bernoulli’s Equation ignores viscosity and compressibility. Reality is more closely modeled with the Navier-Stokes equation, but that is beyond the scope of this course.

"That we have written an equation does not remove from the flow of fluids its charm or mystery or its surprise." --Richard Feynman [1964]

http://jef.raskincenter.org/published/coanda_effect.html

http://en.wikipedia.org/wiki/Richard_Feynman

Long before Bernoulli entered the world, Torricelli realized that if a fluid were to flow from a w-i-d-e barrel, the fluid velocity would depend on the height of the fluid above the spigot. He determined that the formula was…

v =

Long before Bernoulli entered the world, Torricelli realized that if a fluid were to flow from a w-i-d-e barrel, the fluid velocity would depend on the height of the fluid above the spigot. He determined that the formula was…

v = √2gh

Why would that be? Well, if we modify Bernoulli we can derive this! (Note: essentially we are giving up _______, and gaining _______.)

P1 + ½ v12 + gh1 = P2 + ½ v2

2 + gh2

Since the air pressure doesn’t change much, P1 + ½ v1

2 + gh1 = P2 + ½ v22 + gh2

Since the fluid is considered to be incompressible… P1 + ½ v1

2 + gh1 = P2 + ½ v22 + gh2

Since the top is still and the bottom is… the bottom… P1 + ½ v1

2 + gh1 = P2 + ½ v22 + gh2

By rearrangement

v = √2gh

So Torricelli is an example of Bernoulli.

How about others?

Bernoulli’s Principle explains the dynamic lift of flying birds and planes, venturi tubes (car carburetors, venturi meters, atomizers), air circulation in burrows, curveballs, many musical instruments, and TIAs.

Bernoulli’s Principle helps to explain the dynamic lift that supports birds and airplanes.

Note also that there are MANY ways to look at flight. Despite the differences in approach, they (the correct interpretations) all work.

http://hyperphysics.phy-astr.gsu.edu/hbase/fluids/airfoil.html#c1

http://jef.raskincenter.org/published/coanda_effect.html

"In aerodynamics, theory is what makes the invisible plain. Trying to fly an airplane without theory is like getting into a fistfight with a poltergeist." --David Thornburg [1992].

Dynamic lift occurs when a moving fluid is turned by a solid object.

http://www.av8n.com/irro/lecture_e.html

Notice that the fluid travels faster over this wing, producing a net upward force, or lift.

http://www.av8n.com/irro/lecture_e.html

Let’s study this a bit:

Link to: http://www.grc.nasa.gov/WWW/K-12/airplane/wrong2.html

Question: Would this undercambered wing generate more or less lift than one which had a flat bottom?

Link to: http://www.grc.nasa.gov/WWW/K-12/airplane/wrong2.html

http://jef.raskincenter.org/published/coanda_effect.html

1)The airfoil does NOT need to be curved.

2)Both the upper and lower surfaces affect the turning of the fluid.

Link to: http://www.grc.nasa.gov/WWW/K-12/airplane/wrong2.html

http://en.wikipedia.org/wiki/Airfoil

The Coanda effect works like this… A fluid moving by a straight object moves straight.

If an object is bent into the path of the fluid, the fluid bends to follow the object.

But if an object is bent away from the path of the fluid, the fluid still bends to follow the object!

The Coanda effect was named after the Romanian inventor Henri Coanda, who helped develop some of the first aircraft to utilize the jet engine.

http://en.wikipedia.org/wiki/Coand%C4%83_effect_movies

http://hyperphysics.phy-astr.gsu.edu/hbase/fluids/kutta.html#c1

Curve balls curve because of Bernoulli.

http://hyperphysics.phy-astr.gsu.edu/hbase/fluids/kutta.html#c1

Atomizers work because of Bernoulli.

http://www.physics.lsa.umich.edu/demolab/demo.asp?id=27

Venturi tubes work because of Bernoulli.

Aspirators work because of Bernoulli.

http://hyperphysics.phy-astr.gsu.edu/hbase/fluids/aspirv.html#c1