saffman-taylor streamer discharges fabian brau, cwi amsterdam fabian brau, cwi amsterdam alejandro...

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Saffman-Taylor Saffman-Taylor streamer discharges streamer discharges Fabian Brau Fabian Brau , , CWI Amsterdam CWI Amsterdam Alejandro Luque Alejandro Luque , , CWI Amsterdam CWI Amsterdam Ute Ebert Ute Ebert , , CWI Amsterdam CWI Amsterdam Eindhoven University of Eindhoven University of Technology Technology Streamers, sprites, leaders, lightning: from micro- to macroscales Leiden, 11 October 2007

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Page 1: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Saffman-Taylor Saffman-Taylor

streamer dischargesstreamer discharges Fabian BrauFabian Brau,, CWI Amsterdam CWI Amsterdam

Alejandro LuqueAlejandro Luque, , CWI AmsterdamCWI Amsterdam

Ute EbertUte Ebert, , CWI AmsterdamCWI Amsterdam

Eindhoven University of TechnologyEindhoven University of Technology

Streamers, sprites, leaders, lightning:

from micro- to macroscales

Leiden, 11 October 2007

Page 2: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Talk overviewTalk overview

Minimal PDE model for streamersMinimal PDE model for streamers Characteristics of single streamersCharacteristics of single streamers Interacting streamers : periodic Interacting streamers : periodic

array of streamers in 2Darray of streamers in 2D Numerical solutions + characteristicsNumerical solutions + characteristics FMB explicit solution fits very well the FMB explicit solution fits very well the

numerical frontsnumerical fronts

ConclusionsConclusions

Page 3: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Minimal PDE model for streamersMinimal PDE model for streamersIn In dimensionlessdimensionless unit, the minimal PDE unit, the minimal PDE

model readsmodel reads

→ Electron impact ionization :

Townsend’s Approximation→ Poisson equation

Electric

potential

Electric field

: Electron density

: Ion density

Solved in the background of a homogeneous field

Negative streamer

Non attaching gas like nitrogen

Normal conditionCf talks of Alajandro Luque and Chao Li

Page 4: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Characteristics of Characteristics of singlesingle streamers streamers(Valid also for (Valid also for interactinginteracting streamers) streamers)

Evolution of some initial condition in the Evolution of some initial condition in the E-field produced by two electrodesE-field produced by two electrodes::Solutions after the avalanche phase looks like (Solutions after the avalanche phase looks like (3D + cylindrical symmetry))

z (m

m)

r (mm)

Net charge ( )

-3cm

z (m

m)

r (mm)

Electric Field (kV/cm)Thin charge

layerE-field enhancement

E-field screening

[Montijn, Ebert, Hundsdorfer, J. Comp. Phys. 2006, Phys. Rev. E 2006, J. Phys. D 2006Luque, Ebert, Montijn, Hundsdorfer, Appl. Phys. Lett. 2007]

Page 5: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Net charge and E-field evolutionNet charge and E-field evolutionfor for singlesingle streamers streamers

Net Charge

E-field

On branching as a Laplacian instability:[Arrayas, Ebert, Hundsdorfer, Phys. Rev. Lett. 2002, Ebert et al., Plasma Sour. Sci.

Techn. 2006]

Page 6: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

InteractingInteracting streamers as streamers as periodic array : previous workperiodic array : previous work

Streamers with fixed radius

Essentially 1D

Charge density along a line

Study how the interaction affects the charge density, the electric field and the velocity

G V Naidis, J. Phys. D 29, 779 (1996)

Page 7: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

L

InteractingInteracting streamers : streamers : periodic array of streamers periodic array of streamers

in 2Din 2D

Neumann boundary conditions for: Potential and Densities

= Symmetry axis

Anode

Cathode

Direction of

propagation

Page 8: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Characteristics of Characteristics of InteractingInteracting streamersstreamers

If : Streamers do not branchIf : Streamers do not branchmax( )L L E¥<

256

0.06 cm

100 kV/cm

L

;

;

Page 9: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Net charge and E-field evolutionNet charge and E-field evolutionfor for interactinginteracting streamers streamers

Net Charge

E-field

256

0.5

L

=

=

Page 10: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Characteristics of Characteristics of InteractingInteracting streamersstreamers

Uniformly translating streamersUniformly translating streamers

Results are robust against changes in the Results are robust against changes in the initial conditioninitial condition

3310 ns-´

32.310 mm-´

Page 11: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

FMB mathematical setupFMB mathematical setup

Ideally conducting streamer

Single streamers

Interacting streamers:

periodic array of streamers in 2D

y

x

pf Û

Free moving boundary for

Hele-Shaw flow

Saffman-Taylor solution

L

[E. D. Lozansky and O. B. Firsov, J. Phys. D 6, 976 (1973)]

Page 12: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Hele-Shaw FlowHele-Shaw FlowHoleHole

GlycerolGlycerol

Colored Colored WaterWater→

Radial Symmetry

Page 13: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Hele-Shaw FlowHele-Shaw FlowHoleHole

GlycerolGlycerol

Colored Colored WaterWater→

Radial Symmetry

Page 14: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Hele-Shaw FlowHele-Shaw FlowHoleHole

GlycerolGlycerol

Colored Colored WaterWater→

Radial Symmetry

Channel configuration

Page 15: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Saffman-Taylor solutionSaffman-Taylor solution

( )( )(1 ) 1 2( , ) ln 1 cos

2 2

: channel width

finger width0 1:

channel width

L yx y t vt

L

L

Ev

l

l

pp l

l

¥

- é ù= + +ê úû

<

=

ë

<

Family of possible solutions

Page 16: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Selection: Boundary Selection: Boundary conditioncondition

The boundary condition doesn’t allow any selection mechanismThe boundary condition doesn’t allow any selection mechanism0ff+ -- =

surface tension

local curvature0

ˆ width space charge layer

ss

n

ffff

ff fe

k

e

k+ -+ -

+ - +

=ì ìï ïïï - = íïï ï =- = Þ ïí îïïï - = ×Ñ =ïîr

In the limit of small or

A selection is possible

s e

[B. Meulenbroek, U. Ebert and L. Schäfer, PRL 95 (2005)

F. Brau, A. Luque, B. Meulenbroek, U. Ebert and L. Schäfer, (2007)]

Page 17: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Saffman-Taylor solutionSaffman-Taylor solution

For small surface tension, only the finger with is selected

1/ 2l =

Page 18: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Saffman-Taylor solutionSaffman-Taylor solution

Experiment

Theory

Page 19: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Comparison: PDE vs FMBComparison: PDE vs FMB256

0.5

L

=

=

Page 20: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Comparison: PDE vs FMBComparison: PDE vs FMB256

0.5

L

=

=

Saffman-Taylor finger 1/ 2l =

The tip of the Saffman – Taylor finger coincide with the maximum of the net charge of the PDE solution

No free parameter

Page 21: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Comparison: PDE vs FMBComparison: PDE vs FMB256

0.5

L

=

=

Maximum of the net charge along y axis

for each value of x

Page 22: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Comparison: PDE vs FMBComparison: PDE vs FMB

Saffman-Taylor finger

256

0.5

L

=

=

1/ 2l =

Page 23: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Various evolutions of the streamer Various evolutions of the streamer fronts + comparison with Saffman-fronts + comparison with Saffman-

Taylor FrontTaylor FrontFrom the left:

0.5, 256

0.5, 256

wider initial cond

Fig. 1:

itio

Fig. 2:

Fig. 3:

n

0.6

376

E L

E L

E

L

¥

¥

¥

= =

= =

=

=

Saffman-Taylor finger 1/ 2l =

Page 24: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

ConclusionsConclusions

If the streamer spacing is small enough for a given If the streamer spacing is small enough for a given background electric field, streamers do not branch.background electric field, streamers do not branch.

When streamers do not branch they reach a steady When streamers do not branch they reach a steady state which is an attractor of the dynamics.state which is an attractor of the dynamics.

This steady state is well approximated by a solution This steady state is well approximated by a solution from hydrodynamics: the Saffman-Taylor finger.from hydrodynamics: the Saffman-Taylor finger.

Interacting streamers viewed as a periodic array Interacting streamers viewed as a periodic array of streamers in 2D present remarkable features:of streamers in 2D present remarkable features:

Page 25: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

PredictionsPredictions

In contrast to single streamers, branchings In contrast to single streamers, branchings should be mostly suppressedshould be mostly suppressed

After some transient evolution, the velocity After some transient evolution, the velocity should reach a constant valueshould reach a constant value

This value is in 2D. In 3D, we This value is in 2D. In 3D, we expect that the following linear relation should expect that the following linear relation should holdhold

In physical units: In physical units:

2v E E+¥» »

with 3v cE c¥» »

[cm/ s] 380 [V/ cm]v c E¥»

If you are in the right part of the phase diagram:

Page 26: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

BranchingBranching

376

0.7

L

=

=

Page 27: Saffman-Taylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro Luque, CWI Amsterdam Alejandro Luque, CWI Amsterdam

Electric field along the Electric field along the streamer axisstreamer axis

-1200kV cm´