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By Woo Chang Chung

Bernoulli’s Principle

and Simple Fluid Dynamics



Pressure

Pressure is defined as force per unit area.

Standard unit is Pascal, which is N/m2

For liquid pressure, the medium is considered

as a continuous distribution of matter.For gas pressure, it is calculated as theaverage pressure of molecular collisions onthe container.

Pressure acts perpendicular on the surface.

Pressure is a scalar quantity – pressure hasno particular direction (i.e. acts in every

direction).



Pascal’s Law

Pf = P0 + gh“When there is an increase in pressure at any point in a confined fluid, there isan equal increase at every point in the container.”

In a fluid, all points at the same depth must be at the same pressure.

Consider a fluid in equilibrium.

PA - ρ Ahg – P0 A = 0

P = P0 + ρ gh



Hydraulics

Pressure is equal at the bottom of both containers (because it’sthe same depth!)

P = F2/ A2

= F1/ A1

and since A1

< A2

, F2

> F1

There is a magnification of force, just like a lever, but work staysthe same! (conservation of energy). W = F1* D1 = F2 * D2

∴ D1 > D2

You have to push down the piston on the

left far down to achieve some change in

the height of the piston on the right.



Continuity Equation

A1v1 = A2v2

“What comes in comes out.”Av= V/s (volume flow rate) = constant

A = area

v = velocity



Bernoulli’s Equation

Where p is the pressure, ρ is the density, v is the velocity,

h is elevation, and g is gravitational acceleration



Derivation of Bernoulli’s Equation

Restrictions Incompressible

Non-viscous fluid (i.e. no friction)

Following a streamline motion (no turbulence) Constant density

*There exists an extended form of equation that

takes friction and compressibility into account, butthat is too complicated for our level of study.



Derivation of Bernoulli’s Equation

Consider the change in total energy of the fluid as it moves from the inlet to the

outlet.

Δ Etotal = Wdone on fluid - Wdone by fluid

Δ Etotal = (1/2mv22 + mgh1) – (1/2mv1

2 + mgh2)

Wdone on fluid - Wdone by fluid = (1/2mv22 + mgh1) – (1/2mv12 + mgh2)

P2V2 - P1V1 = (1/2mv22 + mgh1) – (1/2mv1

2 + mgh2)

P2 – P1 = (1/2ρ v12 + ρ gh1) – (1/2ρ v1

2 + ρ gh1)

Etotal =1/2mv2 + mgh

W = F/ A*A*d = PV

P2 +1/2ρ v1

2 + ρ gh1 = P1 +1/2ρ v1

2 + ρ gh1∴



Venturi Tube

A2 < A1 ; V2 > V1 According to Bernoulli’s Law, pressure at A2 is lower.

Choked flow: Because pressure cannot be negative,total flow rate will be limited. This is useful incontrolling fluid velocity.

P2 +1/2ρ v1

2 = P1 +1/2ρ v1

2 ; ΔP = ρ/2*(v22 – v1

2)



• This is an atomizer, which uses the Venturi effect to spray liquid.

• When the air stream from the hose flows over the straw, the

resulting low pressure on the top lifts up the fluid.

Atomizer(Demonstration)

http://faraday.physics.uiowa.edu/Movies/MPEG/2c20.20.mpg



Torricelli and his Orifice

In 1843, Evangelista Torricelli proved that the flow of liquid throughan opening is proportional to the square root of the height of theopening.

Q = A*√(2g(h1-h2)) where Q is flow rate, A is area, h is height

Depending on the contour

and shape of the opening,

different discharge

coefficients can be applied

to the equation

(of course we assume

simpler situation here).



Derivation of Torricelli’s Equation

We use the Bernoulli Equation:

In the original diagram A1 [top] is much larger than A2 [the opening]. Since

A1V1 = A2V2 and A1 >> A2, V1 ≈ 0 Since both the top and the opening are open to atmospheric pressure,

P1 = P2 = 0 (in gauge pressure).

The equation simplifies down to:

ρgh1 =1/2 ρv2

2 + ρgh21/2 ρv2

2 = pg(h1-h2)V2

2 = 2g(h1-h2)

∴ V2 = √(2g(h1-h2))

Q = Av2 = A √(2g(h1-h2))

P2 +1/2ρ v1

2 + ρ gh1 = P1 +1/2ρ v1

2 + ρ gh1



Pitot – Static Tube

Used for aircrafts as

speedometer

Typically 10 inches long

and ½ wide in diameter.

A pressure transducer

measures the difference

between static pressure

and total pressure (bymeasuring the strains

put by net force on its

metal)





Pitot-Static Tube

There are several holes on the outside and a center holein the center. A center hole is connected to one side ofthe transducer while the outside holes are connected tothe other side.

Outside holes are perpendicular to the direction of traveland are pressurized by static pressure (Ps)The center hole is horizontal to the travel and ispressured by total pressure (Ps + ½ ρv2)

The difference in pressure is equal to ½ ρv2.

After finding out the local density of the air by checkingaltitude and temperature, we can solve for velocity andthis is registered.

Pitot tube does not work well in low velocity andsupersonic velocity.



Misinterpretation of Bernoulli

Does lower pressure generates faster velocity? Or is itthe other way around?

According to Newton’s Second Law, acceleration iscaused by force.

So when the fluid accelerates in the direction of the fluid,there must be force, or difference of pressure in thiscase.

Therefore, lower pressure generates faster velocity, notthe other way around.

The deflection of the streaming is the cause for thegeneration of pressure difference.



Streamlines

A streamline is a path traced out by a masslessparticle as it moves with the flow.

Velocity is zero at the surface.

As you move away from the surface, the velocity

uniformly approaches the free stream value (fluidmolecules nearby the surface are dragged due toviscosity).

The layer at which the velocity reaches the freestream value is called boundary layer . It does not

necessarily match the shape of the object – boundary layer can be detached, creatingturbulence (wing stall in aerodynamic terms).



Aerodynamic Lift

Lift is the fort that keeps an aircraftin the air.

In Bernoulli-an view, lift isproduced by the different ofpressure (faster velocity on the top,slower velocity in the bottom)

In Newtonian view, lift is thereaction force that results from thedownward deflection of the air.

Both views are correct, but thecurrent argument arises from themisapplication of either view.

The most accurate explanationwould take into account thesimultaneous conservation ofmass, momentum, and energy of afluid, but that involvesmultivariable calculus.



Misconceptions of Lift

In many popular literature, encyclopedia, and even textbooks,Bernoulli’s Law is used incorrectly to explain the aerodynamic lift.

#1: Equal transit time

- The air on the upper side of the wing travels faster because ithas to travel a longer path and must “catch up” with the air on thelower side.

The error lies in the specification of velocity. Air is not forced to“catch up” with the downside air. Also, this theory predicts slowervelocity than in reality.

#2: “Venturi” Theory

- Upper surface of the airfoil acts like a Venturi nozzle,constricting the flow. Therefore, velocity is higher on the upperside, and the difference in velocity results in difference in

pressure.

The error lies in the simple assumption that an airfoil is a half-

Venturi nozzle. But the other (phantom) half does not exist!

http://www.grc.nasa.gov/WWW/K-12/airplane/wrong3.html

http://www.grc.nasa.gov/WWW/K-12/airplane/wrong3.html



Coanda Effect

A fluid jet traveling tangential to the surface of a streamlined

boundary remains attached to that surface for some distance as

it travels. The deflection of the stream creates pressure

difference.

Henri Coanda, a Romanian scientist, discovered this effect whenflames and smokes from the world’s first-ever jet engine (built by

him) attached to the fuselage as they f lew out.Due to viscosity, adjacent air

molecules are swept and result in

lower pressure.

Then the steam follows

the boundary

This floating ping pong ball owes its levitation

to the Coanda Effect. (DEMO)

http://faraday.physics.uiowa.edu/heat/2C20.30.htm

http://faraday.physics.uiowa.edu/heat/2C20.30.htm



Sources

Atomizer. University of Iowa. 29 May 2008 <http://faraday.physics.uiowa.edu/Movies/MPEG/2c20.20.mpg>.

Ball in Water Stream. University of Iowa. 29 May 2008 <http://faraday.physics.uiowa.edu/Movies/MPEG/2c20.30b.mpg>.

Hoselton, Mitch. Lesson 61 - Derivation of Bernoulli's Equation. 2003. 23 May 2008<http://faculty.trinityvalleyschool.org/hoseltom/lesson%20plans/ Lesson%2061-

Derivation%20of%20Bernoullis%20Equation.pdf>.

"Index of Aerodynamic Slides." Beginner's Guide to Aerodynamics. NASA. 29 May 2008<http://www.grc.nasa.gov/WWW/K-12/airplane/short.html>. Path: Bernoulli's Equation; AirPressure; Pitot-Static Tube - Speedometer; Bernoulli and Newton; Boundary Layer; Definition ofStreamlines .

Misinterpretations of Bernoulli's Equation. Department of Physics, University Frankfurt. 29 May

2008 <http://user.uni-frankfurt.de/~weltner/Mis6/ mis6.html>.

"Torricelli's Equation." Torricelli's Theorem and the Orifice Equation. Wayne State University. 26May 2008 <http://www.eng.wayne.edu/legacy/forms/4/timmkunns.htm>. Weltner, Klaus, andMartin Ingelman-Sundberg.

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