impact: void collapse and jet formation

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Impact: Void collapse and jet formation Detlef Lohse Devaraj van der Meer Raymond Bergmann Gabriel Caballero Phys. Rev. Lett. 93, 198003 (2004) Nature 432, 689 (2004) Phys. Rev. Lett. 96, 154505 (2006) Phys. Rev. Lett. 99, 018001 (2007)

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Page 1: Impact: Void collapse and jet formation

Impact:Void collapse and jet formation

Detlef LohseDevaraj van der MeerRaymond Bergmann

Gabriel Caballero • Phys. Rev. Lett. 93, 198003 (2004)• Nature 432, 689 (2004)• Phys. Rev. Lett. 96, 154505 (2006)• Phys. Rev. Lett. 99, 018001 (2007)

Page 2: Impact: Void collapse and jet formation

Astroid impact on earth

Page 3: Impact: Void collapse and jet formation

Craters …on the moon

Moltke Tycho central peak

Page 4: Impact: Void collapse and jet formation

Craters

Mars explorer, January 2004

…on Mars

Arizona

…on earth

Page 5: Impact: Void collapse and jet formation

Speculation on craterformation

Source:Jan Smit,Amsterdam,Dept. Geology

Page 6: Impact: Void collapse and jet formation

What’s really going on?

Page 7: Impact: Void collapse and jet formation

Downscaled experiments: Impact of steel ball on fine sand

Page 8: Impact: Void collapse and jet formation

Problem: reproducibility

Page 9: Impact: Void collapse and jet formation

Controlled experiments

Ball dropped on decompactified, very fine sand

Page 10: Impact: Void collapse and jet formation

Ball at release point Maximum jet height

Jet height > Release height !

Page 11: Impact: Void collapse and jet formation

Jet height vs release height

Siggi Thoroddsen, and Amy Shen, Phys. Fluids 13, 4 (2001):

Page 12: Impact: Void collapse and jet formation

Impact of ball on decompactified sand

Page 13: Impact: Void collapse and jet formation

Impact of ball on decompactified sand

3 events: • Impact creates splash

• A jet is formed

• Granular eruption

Page 14: Impact: Void collapse and jet formation

Planetary impact:

V = 20 km/s d = 1m-150 km Y ≈ 1 kbar

HS experiments:

V = 0.1-1 km/s d = 0.2-2 cm Y ≈ 50 bar

Our experiments:

V = 0-10 m/s d = 1-5 cm Y < 20 Pa

Impact: planetary vs. lab

Page 15: Impact: Void collapse and jet formation

How to look into the sand?

1. Analogy to (opaque) liquid2. “2D” experiments (falling cylinder)3. Discrete particle simulations

Page 16: Impact: Void collapse and jet formation

1. Ball or drop impact on water

Page 17: Impact: Void collapse and jet formation

Air entrainment through impact

11 cm

Page 18: Impact: Void collapse and jet formation

Detlef LohsePhys. Today 56, No. 2, p. 36 (2003)

Page 19: Impact: Void collapse and jet formation

Mechanism

1. Void formation2. Void collapse due to hydrostatic pressure3. Jet formations at singularity point4. Bubble formation

Page 20: Impact: Void collapse and jet formation

Quantitative analysis of void collapse in liquid

Phys. Rev. Lett. 96, 154505 (2006)

Page 21: Impact: Void collapse and jet formation

Pulled disk impacting on a liquid

linear motor

1m

0.5m

Page 22: Impact: Void collapse and jet formation
Page 23: Impact: Void collapse and jet formation

vimpact = 1.0 m/s

Rdisk = 0.03 m

Pulled disk through a liquid

Fr = v2impact /Rdisk g = 3.4

Page 24: Impact: Void collapse and jet formation

Void profiles as function of time

Page 25: Impact: Void collapse and jet formation

Dimensional analysis

Relevant parameters: •  disk radius Rdisk

•  mean velocity V •  gravity g

Irrelevant parameters: •  surface tension (We)

•  viscosity (Re)

Page 26: Impact: Void collapse and jet formation

hs

Rdisk⇠ Fr1/2

ds ⇠ hs

Dimensional analysis

Closure time ts ~ Rdisk1/2 / g1/2

Depth at closure time hs ~ V ts

������

Page 27: Impact: Void collapse and jet formation

Experimental & numerical scaling law

hs/R =1.0 Fr1/2

Page 28: Impact: Void collapse and jet formation

Boundary Integral simulation

Potentialtheory

Fr = 7.8

Page 29: Impact: Void collapse and jet formation

Boundary Integral simulation

Page 30: Impact: Void collapse and jet formation

Comparison BI simulation with experiment

t=-48ms t=35ms t=57ms

Page 31: Impact: Void collapse and jet formation

Comparison BI simulation with experiment

t=88ms t=115ms t=131ms

Page 32: Impact: Void collapse and jet formation

Simplified potential flow analysis: 2D Rayleigh-Plesset equtation

Page 33: Impact: Void collapse and jet formation

expnum

Rs/R

disk

t – tsexp

num

theory ~ (ts– t)1/2

log(

Rs/R

disk

)

log(t – ts)

sgdhhh)hh(h =++∞

2212 log

( ) 21/s

disk

ttRh

−∝

( ) 022

22 ==+ h

dtd)hh(h

Rayleigh-type singularity t --> ts

At the end, ln to –inf neglect the rest, simplifies, great agreement

Page 34: Impact: Void collapse and jet formation

vimpact ≈ 0.5 m/s

Rdisk = 0.03 m

Slow impact:

Toroidal bubble!

Fr = 0.8

Page 35: Impact: Void collapse and jet formation

vimpact ≈ 3 m/s

Rdisk = 0.01 m Fr = 100

Fast impact:

Surface seal

Page 36: Impact: Void collapse and jet formation

What exactly happens at collapse?

singularity

Page 37: Impact: Void collapse and jet formation

Focusing of energy à jets

Page 38: Impact: Void collapse and jet formation

Vimpact = 1.0 m/sHdisk = 0.03 m

Very close to pinch-off

12800 fps

45 mm

Zoom in, to increase 12.8 fps, capillary waves, instrability

Page 39: Impact: Void collapse and jet formation

Vimpact = 1.0 m/sHdisk = 0.03 m

Even closer to pinch-off

48000 fps

6 mm

1. Capillary instability?2. Instability through fast air-flow?

Instability clearer, 48 fps, air rushing out, Kelvin Helmholtz, frequency bubble cloud +/- 10 kHz, 1 mm bubble radius, pure lyinertial collapse of the neck

Page 40: Impact: Void collapse and jet formation

Outwards airflow in the end

Air flow reverses!

Flow visualization with smoke particles

Page 41: Impact: Void collapse and jet formation
Page 42: Impact: Void collapse and jet formation

Back to granular matter:

Rayleigh-Plesset type model for collapse of sand void

Page 43: Impact: Void collapse and jet formation

Cavity collapse

Sand pressure

Initial conditions

.

Page 44: Impact: Void collapse and jet formation

Rayleigh-type dynamics of cavity collapse

Euler equation in cylindrical coordinates

2D slice at depth z

Equation for 2D collapsing void

Continuity equation and boundary conditions r v(r) = R(t) R(t)

.

Page 45: Impact: Void collapse and jet formation

Rayleigh model at high impact velocity

bubble formation !

Page 46: Impact: Void collapse and jet formation

Experiments vs. hydrodynamic theory

T = -21ms T = 37ms T = 78ms

Page 47: Impact: Void collapse and jet formation

T = 100ms T = 116ms T = 191ms

Experiments vs. hydrodynamic theory

Page 48: Impact: Void collapse and jet formation

How to look into the sand?

1. Analogy to water2. “2D” experiments (falling cylinder)3. Discrete particle simulations

Page 49: Impact: Void collapse and jet formation

2D experimental setup

Page 50: Impact: Void collapse and jet formation

2D experiment: high impact velocity

Just as in water:1. void formation2. void collapse3. two jets (sheets in 2D)4. bubble formation

Page 51: Impact: Void collapse and jet formation

3. Discrete particle simulations

• soft sphere code• N = 1000000• ds = 0.5 mm• db = 15 mm• quasi 2D (8 grains thick)• pre-fluidized

Page 52: Impact: Void collapse and jet formation

Discrete particle simulation

Page 53: Impact: Void collapse and jet formation

3D discrete particle simulation

Page 54: Impact: Void collapse and jet formation

Does sandbed support weight?

prepared sand

D. Lohse, R.Rauhe, D. van der Meer, R. Bergmann, Nature 432, 689 (2004)

Page 55: Impact: Void collapse and jet formation

Sandbed does not support weight

Page 56: Impact: Void collapse and jet formation

“Dry quick sand”

Packing density only 41% !No!

Page 57: Impact: Void collapse and jet formation

Myth from Lawrence of Arabia…

Page 58: Impact: Void collapse and jet formation

final depth ~ mass

Page 59: Impact: Void collapse and jet formation

Jet height vs mass: threshold behavior

Page 60: Impact: Void collapse and jet formation

A force model to explain the observations

Page 61: Impact: Void collapse and jet formation

Model: Coulomb friction

Coulomb friction

Force balance

Final depth

Solution

P

?

Page 62: Impact: Void collapse and jet formation

Depth vs time

experiments

+ model

Page 63: Impact: Void collapse and jet formation

Large-Fr impact on sand

Surface seal, just as in

water

Page 64: Impact: Void collapse and jet formation

Oblique impact

Page 65: Impact: Void collapse and jet formation

Oblique impact

45°v = 3.7m/s

Page 66: Impact: Void collapse and jet formation

Oblique impact

45°v = 3.7m/s

Page 67: Impact: Void collapse and jet formation

Conclusions

Hydrodynamic description seems to work at least semiquantitatively (for soft sand)

Series of events in both liquid and sand:1. void formation2. void collapse3. two jets 4. bubble formation

D. Lohse et al., Phys. Rev. Lett. 93, 198003 (2004)

Page 68: Impact: Void collapse and jet formation

Granular void collapse analyzed by…

•Experiment•Analogy to liquid•Boundary Integral simulations•Dimensional analysis•Discrete particle simulations•Simple continuum Rayleigh type model

Page 69: Impact: Void collapse and jet formation

Breakdown of hydrodynamic description

… at large enough compactification of sand when strong enough force chains will have built up.

But how?

- sudden breakdown?

- continuous breakdown?

Page 70: Impact: Void collapse and jet formation

Is this the full story?

Page 71: Impact: Void collapse and jet formation

Large-Fr impact on sand

Surface seal

Fr=100

Page 72: Impact: Void collapse and jet formation

Analyse effect of ambient air

Air

Pneumatic release mechanism

Vacuumpump

Page 73: Impact: Void collapse and jet formation

Effect of ambient pressure on…

• … splash• … jet• …penetration depth

Page 74: Impact: Void collapse and jet formation

Splash depends on

ambient pressure

Ejectie 9 mbar calibratie

Page 75: Impact: Void collapse and jet formation

Jet much less pronounced

under reduced pressure!

1000 mbar25 mbarsee also Royer et al.,Nature Phys. 1, 164 (2005)

Page 76: Impact: Void collapse and jet formation

D = 2.5cm ; Fr = 32 ; t = 159ms

25 50 100 150 200 300 400 600 800 1000

Pressure (mbar)

Effect of ambient air pressure

Page 77: Impact: Void collapse and jet formation

Jet height vs ambient pressure: saturation effects: two regimes

Page 78: Impact: Void collapse and jet formation

Ball trajectory in sand

Page 79: Impact: Void collapse and jet formation

Final depth of intruder vs p

Page 80: Impact: Void collapse and jet formation

Final depth described by force balance model

1000 mbar

400 mbar

25 mbar

Page 81: Impact: Void collapse and jet formation

Coulomb friction coefficient depends on ambient pressure

Page 82: Impact: Void collapse and jet formation

Final depth correlated with jet heightTwo regimes:

Regime 1 (low p):Jet height in-creases linearly with final depth

Regime 2 (high p):Jet height inde-pendent of final depth

How to explain thesetwo regimes ?

High pressure (1 bar)

Low pressure (25 mbar)

Page 83: Impact: Void collapse and jet formation

Closure time

Page 84: Impact: Void collapse and jet formation

Closure time: nearly constant

Page 85: Impact: Void collapse and jet formation

Trajectories: when closure?Impact velocity

Closure time1 bar

400 mbar

100 mbar

25 mbar

High pressures:Identical trajectories until closure time

Low pressures:Trajectories deviate substantially

→ same jet height (regime 2)

→ final depth determines jet height (regime 1)

Page 86: Impact: Void collapse and jet formation

Final question:

What causes the sphere to penetrate less at lower pressures (i.e., the friction reduction)?

The sand bed is fluidized by the air flow around the impacting ball (Resand grains ≈ 5)!

Page 87: Impact: Void collapse and jet formation

Impact of ball on decompactified sand

Page 88: Impact: Void collapse and jet formation

Height of sand bed vs time at impact

Ambient air leads to

expansion of granular bed at impact: extra fluidization

Page 89: Impact: Void collapse and jet formation

Conclusions II

- Ambient air pressure strongly influences the penetration depth of the ball and thus the jet height

- Ambient air pressure hardly affects the collapse of the cavity

- Autofluidization effect

Gabriel Caballero et al., Phys. Rev. Lett. 99, 018001 (2007)

- Two regimes: high p: trajectories unchanged up to closurelow p: trajectories deviate: jet height <-> depth

Page 90: Impact: Void collapse and jet formation

Collaborators: • Raymond Bergmann• Gabriel Caballero• Martin van der Hoef• Hans Kuipers• Devaraj van der Meer• Rene Mikkelsen• Andrea Prosperetti• Remco Rauhe• Marijn Sandtke• Mark Stijnman• Michel Versluis• Ko van der Weele• Christiaan ZeilstraFinancial support from FOM

Page 91: Impact: Void collapse and jet formation
Page 92: Impact: Void collapse and jet formation

Scaling for position of singularity

hs/R =0.69 Fr1/3

ttouch(z)=tget(z) + tcollapse(z)

Minimize:

hs(Fr) ~ Fr 1/3

Different from scaling law in water!

Page 93: Impact: Void collapse and jet formation

Different scaling laws! water

hs/R =1.0 Fr1/2

sand

hs/R =0.69 Fr1/3

Page 94: Impact: Void collapse and jet formation

High velocity impacts

v = 2.0 m/s

v = 3.6 m/s

Page 95: Impact: Void collapse and jet formation

Oblique impact on water

Page 96: Impact: Void collapse and jet formation

Rayleigh model: low impact velocity

Collapse without air entrainment

Page 97: Impact: Void collapse and jet formation
Page 98: Impact: Void collapse and jet formation

Disk depth vs. Fr1/2

I’M NOT AT ALL SURE ABOUT THE EQUALITY. CHECK MCMAHON & GLASHEEN FOR THEIR DEFINITION OF <v>!!!!!!

Again refer to the big feet of the lizard.

Page 99: Impact: Void collapse and jet formation

Rdisk

Dimensional analysis

Closure time ts ~ Rdisk1/2 / g1/2

Depth at closure time hs ~ V ts

ds ~ hs

I’M NOT AT ALL SURE ABOUT THE EQUALITY. CHECK MCMAHON & GLASHEEN FOR THEIR DEFINITION OF <v>!!!!!!

Page 100: Impact: Void collapse and jet formation

Experimental & numerical scaling law

hs/R =1.0 Fr1/2

Page 101: Impact: Void collapse and jet formation

Air entrainment by shaking fluid: The Faraday experiment

Page 102: Impact: Void collapse and jet formation

Parametric instability

115 121

118 125

128

131

134

137

140

143

h1 h2 h3 h4 h5

Page 103: Impact: Void collapse and jet formation

Void profile just

before singularity

Entrained air

How much air is entrained?

Page 104: Impact: Void collapse and jet formation

Air entrainment

Slope 0.80

Strong tools to look at such questions as air entrainment

Page 105: Impact: Void collapse and jet formation

Profile of void just before singularity

Page 106: Impact: Void collapse and jet formation

Differences liquid vs soft sand

Page 107: Impact: Void collapse and jet formation

The sound of impact

Minnaert formula:

fr ≈ 175 Hz

R0 ≈ 2.0 cm

t = 0.13 s

t = 0.14 s

t = 0.15 s

hydrophone

nb gamma = adiabatic exponent

Page 108: Impact: Void collapse and jet formation

Preparation of sand in our experiments

• Grain size = 40µm• Let air bubble through it• Slowly turn off air stream• Resulting packing density: only 41%!

à Model system for sedimented fine sand in the desert after a sand storm

Page 109: Impact: Void collapse and jet formation

Radius of curvature

Include correction, but nevertheless at low froude there’s a significant deviation. The observed anomalous powerlaw of the neck radius must reflect itself in the in time evolution of the void. Define R, R exp increasing with froude

Page 110: Impact: Void collapse and jet formation

Dimensional numbers at singularity

I’M NOT AT ALL SURE ABOUT THE EQUALITY. CHECK MCMAHON & GLASHEEN FOR THEIR DEFINITION OF <v>!!!!!!

Page 111: Impact: Void collapse and jet formation

Intrinsic scales at singularity (for water)

Below this,

I’M NOT AT ALL SURE ABOUT THE EQUALITY. CHECK MCMAHON & GLASHEEN FOR THEIR DEFINITION OF <v>!!!!!!

Page 112: Impact: Void collapse and jet formation

Intrinsic scales at singularity (for glycerol)

Below this,

I’M NOT AT ALL SURE ABOUT THE EQUALITY. CHECK MCMAHON & GLASHEEN FOR THEIR DEFINITION OF <v>!!!!!!

Page 113: Impact: Void collapse and jet formation

Complete 2D-Rayleigh equation

Viscous ~ capillary: Viscous ~ inertia: Inertia ~ capillary: Purely inertially:

I’M NOT AT ALL SURE ABOUT THE EQUALITY. CHECK MCMAHON & GLASHEEN FOR THEIR DEFINITION OF <v>!!!!!!