Hovercraft

Nikolay Sibiryakov

1

The problem

A simple model hovercraft can be built using a CD and a balloon filled with air attached via a tube. Exiting air can lift the device making it float over a surface with low friction. Investigate how the relevant parameters influence the time of the 'low-friction' state.

2

First observationsFirst observations

3

Floating over the table 4

Force balance

mgRF

5

The reactive force

p

FR = pS

Sd = 13,8 mmS = 1,5 cm2

FR = 0,15 N

mg = 0,28 N

6

Viscous friction

atmp p

atmpp. .visc fricF

7

Force balance over the table

mgRF

HFHF

8

Floating under the ceiling 9

Bernoulli’s principle

atmpoutvoutv vp

atmp p

10

Force balance under the celling

mg

RF

BF BF

11

Final force balance

mgmg

RF

RF

BF

BFBF

BFHF HF

HFHF

12

Outline of the report

Pressure under the disk

Hovering experiments

Theoretical model

…how the relevant parameters influence the time of the 'low-friction' state…

Hovering time vs. craft’s weight

13

Air pressure Air pressure under the diskunder the disk

14

Parameters of the hovercraft

balloon + diskm = 28,3 g

R = 6 cmS = 110 cm2

D = 25 cmV = 8 dm3

15

Plug-in nozzles

1.9 mm2.6 mm4.3 mm9.0 mm

16

Setup for pressure measurement

Side barriers

Relative pressure sensor

Holes3.5 mm diameter

17

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 10 20 30 40 50 60 70

Distance from center (mm)

Rel

ativ

e pr

essu

re (P

a)

1,9 mm2,6 mm4,2 mm9,0 mm13,8 mm

Nozzle diameter

Pressure distribution under the disk

viscous

viscous

Bernoulli

18

HoveringHoveringexperimentsexperiments

“…how the relevant parameters influence the time of the 'low-friction' state…”

19

1

10

100

1000

0,1 1,0 10,0 100,0 1000,0Nozzle cross-section (mm2)

Tim

e (s

)

Free flowHovering

Time vs. nozzle cross-section 20

Narrow nozzles Narrow nozzles Viscosity dominatesViscosity dominates

21

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 10 20 30 40 50 60 70

Distance from center (mm)

Rel

ativ

e pr

essu

re (P

a)

1,9 mm2,6 mm4,2 mm9,0 mm13,8 mm

Nozzle diameter

1

10

100

1000

0,1 1,0 10,0 100,0 1000,0Nozzle cross-section (mm2)

Tim

e (s

)

Free flowHovering

What do we already know?

Almost free outflow

Pressure drops along the radius

Small nozzle diameter

22

Viscous regime: Hele-Shaw cell

( )2

Qv rr

Continuity condition:

Darcy’s law:2

( )12

dpv rdr

3

6( ) lnQ Rp rr

Q

23

Narrow nozzle (d = 1.9 mm)

6051 lnp Pa

r mm

3

6( ) lnQ Rp rr

24

Hovering time with narrow nozzle

2*p p u

p

p*Flow through the nozzle:

2* 2lnmg RpR a

Flow under the disk:

Velocity in the nozzle:

221 2lnp mg Ru

p R a

25

Hovering time with narrow nozzle

1/2

21 2lnV mg RS p p R a

0

We should know it!

26

Measurement of relative pressure

Rulers

Relative pressure sensor

Compressor

27

0,00,51,01,52,02,53,03,54,04,55,0

0 1 2 3 4 5 6 7 8 9

Volume (dm3)

Rel

ativ

e pr

essu

re (k

Pa)

Pressure vs. volume

1% of atmospheric pressure

28

Hovering time with narrow nozzle

For our parameters of the balloon and the disk

0 1 0,02 ln Ra

If R/a = 60, 01,08

+

1/2

0 21 2lnmg Rp R a

29

Wide nozzlesWide nozzlesViscosity + BernoulliViscosity + Bernoulli

30

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 10 20 30 40 50 60 70

Distance from center (mm)

Rel

ativ

e pr

essu

re (P

a)

1,9 mm2,6 mm4,2 mm9,0 mm13,8 mm

Nozzle diameter

1

10

100

1000

0,1 1,0 10,0 100,0 1000,0Nozzle cross-section (mm2)

Tim

e (s

)

Free flowHovering

What do we already know?

Outflow differs from the free outflow

Pressure under the central area is less than atmospheric

Big nozzle diameters

31

Wide gap: Bernoulli’s regime

( )2

Qv rr

Continuity condition:

2

const2vp

Bernoulli’s principle:

2

2 2 2 2

1 1 1( )4

Qp rR r

Relative Pressure is negative

32

Combined regime

2

3 2 2 2 2

6 1 1 1( ) ln4

Q R Qp rr R r

Viscous Bernoulli

33

Parabolic velocity profile

2

3 2 2 2 2

6 27 1 1( ) ln140

Q R Qp rr R r

viscous Bernoulli with a parabolic profile

Armengoll J., Calbó J., Pujol T., Roura P. (2011) “Bernoulli correction to viscous losses: Radial flow between two parallel discs”.

34

Force balance

2 2

3 23 27 ln

70RQR Q Rmg F

a

Both terms are large compared with the weight, so they are approximately

equal to each other

270

9 ln /RQ

R a

Viscous Bernoulli

Outflow from the nozzle under the disk

p

p0

As outflow from the nozzle into the atmosphere

02Q a v 0pv

p0p

Cylindrical entry

Hovering time with wide nozzle

270

9 ln /RQ

R a

02Q a v

2

0

ln /970 2

R aV

a R v

37

1

10

100

1000

0,1 1,0 10,0 100,0 1000,0Nozzle cross-section (mm2)

Tim

e (s

)

Free flowHovering

Theory-experiment comparison

+

38

Hovering timeHovering timevs.vs.

weightweight

39

The hovering time

0

21 2lnmg Rp R a

2

0

ln /970 2

R aV

a R v

The hovering time almost does not depend on the weight of the vessel until this weight is not

very large.

40

Experiment 41

0

2

4

6

8

10

12

0,0 0,5 1,0 1,5

Weight (N)

Tim

e (s

)

Time vs. weight (nozzle 13.8 mm)

+

42

SummarySummary

43

Conclusions 44

References

• Jackson J.D., Symmons G.R. (1965) “An investigation of a laminar flow between two parallel disks”. Appl. Sci. Res. 15, 59–75.

• Armengoll J., Calbó J., Pujol T., Roura P. (2011) “Bernoulli correction to viscous losses: Radial flow between two parallel discs”. Am. J. Phys. 76, 730–737.

• Izarra Ch., Izarra G. (2014) “Stokes equation in a toy CD hovercraft”. Eur. J. Phys. 32, 89–99.

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