fluid dynamics - applications

8/19/2019 Fluid Dynamics - Applications

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1http://www.physics.usyd.edu.au/teach_res/jp/fluids/wfluids.htm

web notes: Fluidslect5.pdf

flow4.pdf flight.pdf

Lecture 5

Dr Julia Bryant

Fluid dynamics - Applications

of Bernoulli’s principleFluid statics• What is a fluid?Density

Pressure • Fluid pressure and depthPascal’s principle

• BuoyancyArchimedes’ principle

Fluid dynamics • Reynolds number

• Equation of continuity • Bernoulli’s principle

• Viscosity and turbulent flow

• Poiseuille’s equation


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Bernoulli’s Equationfor any point along a flow tube or streamline

p + 1 v 2 + g y = constant2

Between any two points along a flow tube or streamline p1 + 1 " v 1

2 + " g y 1 = p2 + 1 " v 22 + " g y 2

2 2 Dimensions

p [Pa] = [N.m-2] = [N.m.m-3] = [J.m-3]

1 " v 2 [kg.m-3.m2.s-2] = [kg.m-1.s-2] = [N.m.m-3] = [J.m-3]2

" g h [kg.m-3 m.s-2. m] = [kg.m.s-2.m.m-3] = [N.m.m-3] = [J.m-3]

Each term has the dimensions of energy / volume or energy density.

1 " v 2

KE of bulk motion of fluid2 " g h GPE for location of fluid

p pressure energy density arising from internal forces within

moving fluid (similar to energy stored in a spring)


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Solving Bernoulli’s Equation

1. Establish points 1 and 2 along the flow.

2. Define the coordinate system - where is

y=0?

3. List your known and unknown variables.4. Solve for the unknown variables, possibly

using the continuity equation.


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(1) Surface of liquid

(2) Just outside hole

v 2 = ? m.s-1

What is the speed with which liquid flows from a hole at

the bottom of a tank? Where does it hit the ground?

R?

Fluid is flowing from thesurface ==> Bernoulli’s applies


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(1) Surface of liquid

(2) Just outside hole

v 2 = ? m.s-1

y 2 = 0

v 1 ~ 0 m.s-1 (large tank)

y = h

p1 = patm

p2 = patm

Firstly, what is the speed with which liquid flows from a hole at

the bottom of a tank?

y 1

y 2

h = (y 1 - y2 )


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Assume liquid behaves as an ideal fluid, Bernoulli's equation

can be applied

p1 + 1 " v

1

2 + " g y 1 = p

2 + 1 " v

2

2 + " g y 22 2

A small hole is at level (2) and the water level at (1) drops

slowly (if tank is large) # v 1 = 0

p1 = patm p2 = patm

" g y 1 = 1 "v 22 + " g y 2

2

v 22 = 2 g (y 1 – y 2) = 2 g h h = (y 1 - y2 )

v 2 = $ (2 g h) Torricelli formula (1608 – 1647)

This is the same velocity as a particle falling freely through a

height h

y 1

y 2

h

y = 0

p 1

p 2


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Recall d=1 a t2

2 t= # (2d/a)

So, time taken to fall d= y 1 - h ist= # {2(y

1

- h)/g}

Distance,R = v2 t

= $ (2 g h) . # {2(y 1 - h)/g} =2 # {h(y

1- h)}

v 2 = $ (2 g h)

Secondly, where does it hit the ground?

y 1

y 2

h

R?


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0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20

h

R

R =2 # {h(y 1 - h)}


h=4

R?

DEMO

h=8.5

h=14

y 1=20


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R =2 # {h(y 1 - h)}


Y 1=16

R?

DEMO

4.5

10

0

5

10

15

20

0 2 4 6 8 10 12 14 16

h

R


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Venturi Meter

A Venturi meter is used to measure the speed of the flow

in a pipe.

What is the speed v1 of flow in section 1 of the system?

DEMO


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Venturi Meter

Bernoulli’s equation only applies along the same

streamline. Therefore we must choose points 1 and 2

along the pipe, but not in the vertical tubes.


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Venturi Meter

Q: What do we know? y 1 = y 2 no height difference

We will need to use:

• Bernoulli$s equation ( Assume liquid behaves as an ideal fluid)

p1 + 1 " v 12 + " g y 1 = p2 + 1 " v 2

2 + " g y 22 2

• Continuity equation A1 v 1 = A2 v 2

• hydrostatic equation for the vertical pipes p 1 - p 2 = "m g h


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From Bernoulli:

p 1 + 1 " v 12 + " g y 1 = p 2 + 1 " v 2

2 + " g y 2 2 2

y 1

= y 2

p 1 – p 2 = 1 " (v 22 - v 1

2) From the continuity equation v 2 = v 1 (A1 / A2) 2

p 1 p 2

d

v 1 = 2 gh

{(A1 / A2)2 - 1}

So, p 1 – p 2 = 1 " ((A1 / A2)2 v 1

2 - v 12)

2

p 1 – p 2 = 1 " v 12 ((A1 / A2)

2 - 1) A1 > A2 so p 1 – p 2>0 p 1 > p 2

2 From hydrostatics p 1 = p 0 + " g (d+h)

p 2 = p 0 + " g d p 1 - p 2 = " g h

"g h = 1 " v 12 {(A1 / A2)

2 - 1} 2


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DEMO

or

Bernoulli’s Bar


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How to impress your mother in a pub!From Bernoulli:

p 1 + 1 " v 12 + " g y 1 = p 2 + 1 " v 2

2 + " g y 2

2 2y 1 = y 2 = 0

p 1 – p 2 = 1 " (v 22 - v 1

2)

2

Continuity equation A2 v 2 = A1 v 1% A2 v 1 p 2


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How does a

siphon work?


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C

B

A

D

y A

y B

y C

Assume that the

liquid behaves as an

ideal fluid, theequation of continuity

and Bernoulli's

equation can be

used.

y D = 0

p A = patm = pD


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C

B

A

D

y A

y B

y C

y D = 0

What do we know?

p A = patm = pD

v A =0 approximately

What do we need to find?

vD = ?

yC = ?

Focus on falling water

not rising water

patm - pC % 0 patm % " g y C


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pC + 1 " v C2 + " g y C = pD + 1 " v D

2 + " g y D2 2

From equation of continuity v C = v D pC = pD + " g (y D - y C) = patm + " g (y D - y C)

The pressure at point C can not be negative

pC % 0 and y D = 0

pC = patm - " g y C % 0y C & patm / (" g )

For a water siphon

patm ~ 105 Pa

g ~ 10 m.s-1

" ~ 103

kg.m-3

y C & 105 / {(10)(103)} m

y C 10 m

Consider points C and D and apply Bernoulli's principle.

C

B

A

D

y A

y B

y C

y D = 0


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C

B

A

D

y A

y B

y C

How fast does

the liquidcome out?


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p A + 1 " v A2 + " g y A = pD + 1 " v D

2 + " g y D2 2

v D2 = 2 ( p A – pD) / " + v A

2 + 2 g (y A - y D)

p A – pD = 0 y D = 0 assume v A2


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FLUID FLOW

MOTION OF OBJECTS IN FLUIDS

How can a plane fly?

Why does a cricket ball swing or a baseball curve?

web notes: flow4.pdf flight.pdf


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Lift F L

drag F D

Resultant F R

Motion of object through fluid

Fluid moving around stationary object

FORCES ACTING ON OBJECT MOVING THROUGH FLUID

Forward thrust

by engine


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C

D

B A

Uniform motion of an object through an ideal fluid

(' = 0)

• Consider a cylinder.• The fluid will slide freely over the

surface.

• Pressure near A and B are equal and

greater than undisturbed flow(streamlines further apart).

• Pressure near C and D are equal andlower then undisturbed flow

# Resultant force = zero

• No drag or lift.• The pattern is symmetrical


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When real fluids flow they have a certain

internal friction called viscosity. It exists in

both liquids and gases and is essentially a

frictional force between different layers of

fluid as they move past one another.

In liquids the viscosity is due to the cohesive

forces between the molecules whilst in

gases the viscosity is due to collisions

between the molecules.

“VISCOSITY IS DIFFERENT TO DENSITY”


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Drag force

In a viscous fluid, a thin layer of fluid sticks to thesurface of an object and the resulting friction leads to a

drag force on the object.

The flow is no longer complete around the object, and

the flow lines break away from the surface resulting ineddies behind the object. The pressure in the eddies is

lowered and the pressure difference gives pressure drag

force.


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low pressure region

high pressure region

rotational KE of eddies # heating effect # increasein internal energy #

temperature increases

Drag force due to pressure difference

Drag force is

opposite to thedirection of motion

motion of air

motion of object


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high pressure region

Drag force due

to pressure difference

v

v

flow speed (high) v air + v

# reduced pressure

flow speed (low) v air - v # increased pressure

v air (v ball)

Boundary layer – airsticks to ball(viscosity) – airdragged around withball

MAGNUS EFFECT

motion of air

motion of object

What happens if the object is spinning?


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Golf ball with backspin (rotating CW) with air stream going from

left to right. Note that the air stream is deflected downward with a

downward force. The reaction force on the ball is upward. This

gives the longer hang time and hence distance carried.

The trajectory of a

golf ball is notparabolic


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Professional golf drive

Initial speed v 0 ~ 70 m.s-1

Angle ~ 6°

Spin ( ~ 3500 rpm

Range ~ 100 m (no Magnus effect)

Range ~ 300 m (Magnus effect)


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Stagnation line

Higher v, lower pressure

Lift increases

with angle ofattack, untilstall.

0°

5°

10°

How can a plane fly?


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Direction plane is moving w.r.t. the air

Direction air is moving w.r.t. plane

low pressure drag

!

attack angle

lift

downwash

huge vortices

momentum transfer

low

pressure

high

pressure

DEMO


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Why do some liquidssplash more?


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Why do cars need different oils in hot

and cold countries?

Why do engines run more freely as it

heats up?

Have you noticed that skin lotions are

easier to pour in summer than winter?

Why is honey sticky?


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When real fluids flow they have a certain

internal friction called viscosity. It exists in

both liquids and gases and is essentially africtional force between different layers of

fluid as they move past one another.

In liquids the viscosity is due to the cohesive

forces between the molecules whilst in

gases the viscosity is due to collisions

between the molecules.

“VISCOSITY IS DIFFERENT TO DENSITY”


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stationary wall

L A viscous fluid

plate

X

Z

A useful model

Assumptions


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stationary wall

L

Viscous fluid will flow

plate moves with speed v

X

Z

A useful model

Q: Direction?

Q: Highest speeds?Q: Lowest speeds?


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stationary wall

low speed

high speed

plate moves with speed v

X

Z

v x = 0

v x = v

A useful model:

Newtonian fluids

water, most gases

linearvelocity

gradient

d

L v x


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stationary wall

plate exerts force F

velocitygradient

is proportional toshearstress

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

A useful model:

Newtonian fluidsover area A

fluid dynamics - applications

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