Hovercraft. Nikolay Sibiryakov

Transcription

1 1 Hovercraft Nikolay Sibiryakov

2 The problem 2 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.

3 3 First observations

4 Floating over the table 4

5 Force balance 5 F R mg

6 The reactive force 6 S F R = ps p d = 13,8 mm S = 1,5 cm 2 F R = 0,15 N mg = 0,28 N

7 Viscous friction 7 p p atm p F visc. fric. p atm

8 Force balance over the table 8 F R mg F H F H

9 Floating under the ceiling 9

10 Bernoulli s principle 10 p v v out v out p atm p p atm

11 Force balance under the celling 11 FB FB mg F R

12 Final force balance 12 FH FH F R mg F B F B mg F B F B F H F H F R

13 Outline of the report 13 Pressure under the disk Hovering experiments Theoretical model Hovering time vs. craft s weight how the relevant parameters influence the time of the 'low-friction' state

14 14 Air pressure under the disk

15 Parameters of the hovercraft 15 D = 25 cm V = 8 dm 3 balloon + disk m = 28,3 g R = 6 cm S = 110 cm 2

16 Plug-in nozzles mm 4.3 mm 2.6 mm 1.9 mm

17 Setup for pressure measurement 17 Side barriers Holes 3.5 mm diameter Relative pressure sensor

18 Pressure distribution under the disk 18 Relative pressure (Pa) Bernoulli viscous viscous Nozzle diameter 1,9 mm 2,6 mm 4,2 mm 9,0 mm 13,8 mm Distance from center (mm)

19 19 Hovering experiments how the relevant parameters influence the time of the 'low-friction' state

20 Time vs. nozzle cross-section Time (s) Free flow Hovering 1 0,1 1,0 10,0 100,0 1000,0 Nozzle cross-section (mm 2 )

21 21 Narrow nozzles Viscosity dominates

22 What do we already know? Time (s) Free flow Hovering Small nozzle diameter 1 0,1 1,0 10,0 100,0 1000,0 Nozzle cross-section (mm 2 ) Almost free outflow Relative pressure (Pa) Nozzle diameter 1,9 mm 2,6 mm 4,2 mm 9,0 mm 13,8 mm Pressure drops Distance along from center (mm) the radius

23 Viscous regime: Hele-Shaw cell 23 Q Continuity condition: Darcy s law: v( r) Q v( r) 2 r 2 3 dp 12 dr 6 Q p( r) ln R r

24 Narrow nozzle (d = 1.9 mm) 24 p Pa ln r mm 6 Q R p( r) ln 3 r

25 Hovering time with narrow nozzle 25 p Flow under the disk: mg p* 2ln 2 R R a p* Flow through the nozzle: p* p u 2 Velocity in the nozzle: u 2 p mg R 1 2ln p R 2 a

26 Hovering time with narrow nozzle 26 0 V mg R 1 2ln S p p R a 2 1/2 We should know it!

27 Measurement of relative pressure 27 Rulers Compressor Relative pressure sensor

28 Pressure vs. volume 28 Relative pressure (kpa) 5,0 4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0 1% of atmospheric pressure Volume (dm 3 )

29 Hovering time with narrow nozzle 29 mg R 0 1 2ln 2 p R a 1/2 For our parameters of the balloon and the disk 0 1 0,02 ln R a 1,08 If R/a = 60, 0 +

30 30 Wide nozzles Viscosity + Bernoulli

31 What do we already know? Time (s) Free flow Hovering Big nozzle diameters 1 0,1 1,0 10,0 100,0 1000,0 Nozzle cross-section (mm 2 ) Outflow differs from the free outflow Relative pressure (Pa) Distance from center (mm) Nozzle diameter 1,9 mm 2,6 mm 4,2 mm 9,0 mm 13,8 mm Pressure under the central area is less than atmospheric

32 Wide gap: Bernoulli s regime 32 Continuity condition: Bernoulli s principle: Q v( r) 2 r p v 2 2 const p( r) Relative Pressure is negative 2 1 Q R r 4

33 Combined regime 33 6 Q R 1 Q 1 1 p( r) ln 3 r R 2 r 2 2 Viscous Bernoulli

34 Parabolic velocity profile 34 6 Q R 27 Q 1 1 p( r) ln 3 r R 2 r 2 2 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.

35 Force balance mg QR 27 Q R F ln R a Viscous Bernoulli Both terms are large compared with the weight, so they are approximately equal to each other Q 2 70 R 9 ln R / a

36 Outflow from the nozzle under the disk p p 0 p p 0 As outflow from the nozzle into the atmosphere Q 2 a v v 0 0 p Cylindrical entry

37 Hovering time with wide nozzle 37 Q 2 70 R Q 2 a v 9 ln R / a 0 9 ln R / a V 70 2 a R 2 v 0

38 Theory-experiment comparison Free flow Hovering Time (s) ,1 1,0 10,0 100,0 1000,0 Nozzle cross-section (mm 2 )

39 39 Hovering time vs. weight

40 The hovering time 40 mg 1 2ln 0 R 2 p R a 9 ln R / a V a R v0 The hovering time almost does not depend on the weight of the vessel until this weight is not very large.

41 Experiment 41

42 Time vs. weight (nozzle 13.8 mm) Time (s) ,0 0,5 1,0 1,5 Weight (N)

43 43 Summary

44 Conclusions 44

45 References 45 Jackson J.D., Symmons G.R. (1965) An investigation of a laminar flow between two parallel disks. Appl. Sci. Res. 15, Armengoll J., Calbó J., Pujol T., Roura P. (2011) Bernoulli correction to viscous losses: Radial flow between two parallel discs. Am. J. Phys. 76, Izarra Ch., Izarra G. (2014) Stokes equation in a toy CD hovercraft. Eur. J. Phys. 32,

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