Problem 05 Levitation
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1 Problem 05 Levitation reporter: Denise Sacramento Christovam
2 Problem 05 Levitation A light ball (e.g. a Ping-Pong ball) can be supported on an upward airstream. The airstream can be tilted yet still support the ball. Investigate the effect and optimize the system to produce the maximum angle of tilt that results in a stable ball position. Reporter: Denise Christovam 2
3 Contents Introduction Flow characteristics Aerodynamic forces Threshold angle Oscillations Optimization Experiments Flow format Flow velocity Drag crisis Threshold angle Ball Variation Oscillations Conclusion Analysis Optimization Reporter: Denise Christovam 3
4 Flow Characteristics Reynolds number Molecular behavior Radius of the ball Velocity Fluid s density Dynamic viscosity Re < 2000: laminar flow 2000 < Re < 2400: transition flow Re > 2400: turbulent flow Typically over turbulent flow Reporter: Denise Christovam 4
5 Velocity v' (m/s) Velocity (v) Team of Brazil Flow Characteristics Flow velocity* Bernoulli s Principle Central Velocity (v') Distance d (cm) Edge Velocity Distance (r) Fixed v at 12.5 m/s and d at m. *BECKER, Aaron, SANDHEINRICH, Robert, and BRETL,Timothy. Automated Manipulation of Spherical Objects in Three Dimensions Using A Gimbaled Air Jet. Reporter: Denise Christovam 5
6 Aerodynamic Forces Drag: Re > 1000: Surface Forces (lift): Roughness Format Flow C L 0.5 Reporter: Denise Christovam 6
7 Aerodynamic Forces Magnus-driven force: rotational lift 90 : idealized angle (impossible to reach theoretical) Magnus effect is relevant for rough spheres It can be neglected for highly smooth spheres e.g. Golf Ball Reporter: Denise Christovam 7
8 Aerodynamic Forces Kutta-Joukowski Lift Theorem for a Cyllinder Lift r= radius of the ball M: Magnus force Γ: strength of rotation r : radius of the cyllinder Reporter: Denise Christovam 8
9 Aerodynamic Forces M D L D W θ 1 θ 1 W Rough Sphere Smooth Sphere Reporter: Denise Christovam 9
10 Threshold Angle θ Fluid s side pressure: stable perpendicular position (r medium =0) 2 : rotation in the orthogonal plane to the levitation z z θ 2 θ 2 r d θ y θ y x θ 2 will be considered 0⁰, as the object of study is the inclination towards the ground. x Reporter: Denise Christovam 10
11 Threshold Angle Ball s weight Smooth z L/M D Drag coefficient Air density Ball s radius Fluid s central velocity W Lift coefficient Angular velocity θ y Rough Reporter: Denise Christovam 11
12 Oscillations Mainly due to small pressure differences inside the jet and turbulence Type: Perpendicular (ordinate) Perpendicular (abscissa) Perpendicular: slightly similar to simple harmonic oscillations Reporter: Denise Christovam 12
13 Optimization Initial velocity of the jet Minimum height: maximum angle Minimum d initial: d The greater the pressure difference between center and sides, the greater the stability. Therefore, initial jet velocity must be high. Radius and mass of the ball Small mass (smaller weight) Great radius (greater lift and drag) Reporter: Denise Christovam 13
14 Optimization Nozzle s diameter Fluid cone s formation; Big nozzle: continuity law Small nozzle: viscous fluid in a pipe (lost of pressure leads to lost in lift) Jet nozzle must be of the same magnitude of the ball s radius. Ball surface Upward force balancing weight; Magnus-driven force + lift; Rough ball Oscillations More oscillations, the ball is more likely to fall Flow Must be laminar before reaching the ball (directed force), only then the predicted angle is valid Reporter: Denise Christovam 14
15 Material 1. Light balls 2. Anemometer (±0.1 m/s) 3. Millimeter Scale (±1 mm) 4. Protractor (±1 ) 5. Hair dryer Camera (120 FPS) Talc Scales (± g) Reporter: Denise Christovam 15
16 Experimental Description Experiment 1: show the cone s formation that helps in stability. Experiment 2: show that the global equation for flow velocity is applicable. Experiment 3: verify if our experimental conditions match with the drag crisis. Experiment 4: threshold angle. Experiment 5: variation of balls. Experiment 6: oscillations analysis Reporter: Denise Christovam 16
17 Experiment 1: Flow Format Flow Format Velocity Drag crisis Threshold Angle Ball Variation Oscillations Reporter: Denise Christovam 17
18 Experiment 1: Flow Format Flow Format α Velocity Drag crisis Threshold Angle α= 12.3 Ball Variation Oscillations Reporter: Denise Christovam 18
19 Velocity v' (m/s) Velocity v (m/s) Team of Brazil Experiment 2: Flow Velocity Flow Format Velocity Drag crisis Threshold Angle Ball Variation Oscillations Distance d (cm) Central Velocity (v') Distance d (cm) Approximately the same experimental behavior as theoretically predicted (differ by an arbitrary constant) Reporter: Denise Christovam 19
20 Velocity (v) Edge velocity (m/s) Team of Brazil Experiment 2: Flow Velocity Flow Format Velocity Drag crisis Threshold Angle Distance r (m) Same experimental behavior as theoretically predicted Ball Variation Oscillations Distance (r) Reporter: Denise Christovam 20
21 Experiment 3: Drag Crisis Flow Format Experimental Setup: Velocity Drag crisis D = W Threshold Angle Ball Variation Oscillations ρ: 1.2 kg/m³ W s : ² N W r : 0.44 N R s : 19 mm R r : 21.5 mm Reporter: Denise Christovam 21
22 Experiment 3: Drag Crisis Flow Format Velocity Measured d= 27.5 cm, v = 12.5 m/s Measured C Ds 0.20 Measured C Dr 0.23 Drag Crisis Drag Crisis Drag crisis Threshold Angle Ball Variation Oscillations Reporter: Denise Christovam 22
23 Experiment 4: Threshold Angle Flow Format Velocity Drag crisis Threshold Angle Ball Variation Oscillations Reporter: Denise Christovam 23
24 Experiment 4: Threshold Angle Flow Format Velocity Drag crisis Threshold Angle Ball Variation Smooth Ball (in degrees) Average 42.0 Theoretical 43.9 Standard Deviation 3.1 Rough Ball (in degrees) Average 41.5 Theoretical 42.7 Standard Deviation 1.4 Oscillations Error source: turbulence before ball, number of frames (120 FPS) Reporter: Denise Christovam 24
25 Angle (degrees) Team of Brazil Experiment 5 Ball Variation Flow Format Mass variation Velocity Drag crisis Threshold Angle Ball Variation Threshold Angle x Mass Mass (g) R=7.35 cm R=9.95 cm z θ y Oscillations Reporter: Denise Christovam 25
26 Frequency (Hz) Team of Brazil Experiment 5 Ball Variation Flow Format Frequency of rotation Greater mass Velocity Frequency of Rotation x Mass Drag crisis Closer to the nozzle (weight) 8 Threshold Angle 6 4 R=7.35 cm R=9.95 cm Higher velocity Ball Variation Oscillations Mass (g) Higher frequency Reporter: Denise Christovam 26
27 Angle (degrees) Team of Brazil Experiment 5 Ball Variation Flow Format Velocity Frequency of rotation Threshold Angle x Frequency z 47 Drag crisis Threshold Angle Ball Variation Frequency (Hz) R=7.35cm R=9.95 cm θ y Oscillations Reporter: Denise Christovam 27
28 Experiment 7: Oscillations Flow Format Perpendicular oscillations Velocity Drag crisis Threshold Angle Ball Variation Oscillations (120 FPS) Reporter: Denise Christovam 28
29 Position (mm) Position (mm) Team of Brazil Experiment 7: Oscillations Flow Format Velocity Drag crisis Threshold Angle Ball Variation Oscillations 40,0 30,0 20,0 10,0 0,0-10,0 40,0 30,0 20,0 10,0 0,0 Perpendicular (x) Time (s) Perpendicular (y) Time (s) Secondary oscillations due to eventual turbulence Resembles SHO (constant period of oscillation) Reporter: Denise Christovam 29
30 Conclusions The main cause of the levitation are the pressure differences within the fluid, due to velocity variation, which cause aerodynamic forces; The threshold angle in idealized situations is about 90. As it s impossible to reach in any experimental condition, the maximum is dependent on the sphere and jet used, and is 41,5. fluid s velocity/mass/radius surface nozzle aerodynamic forces and lift cone Magnus effect lift cone Reporter: Denise Christovam 30
31 References DOMINIQUE LEGENDRE and JACQUES MAGNAUDET (1998). The lift force on a spherical bubble in a viscous linear shear flow. Journal of Fluid Mechanics, 368, pp DOMINIQUE LEGENDRE and JACQUES MAGNAUDET (1997). A note on the lift force on a spherical bubble or drop in a low-reynolds-number shear flow. Physics of Fluids, v. 9, n 1, pp EGGERS, Jens. 3.5 Flow past a sphere (Univ. of Bristol). Flow past a sphere II: Stoke's lat, the bernoulli equation, turbulence, boundary layers, flow separation BECKER, Aaron, SANDHEINRICH, Robert, and BRETL,Timothy. Automated Manipulation of Spherical Objects in Three Dimensions Using A Gimbaled Air Jet. C.E. Aguiar e G. Rubini. A aerodinâmica da bola de futebol Reporter: Denise Christovam 31
32 Thank you! Reporter: Denise Christovam 32
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