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

Talk overviewTalk overview

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

of streamers in 2Dof 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

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 streamerNon attaching gas like nitrogenNormal condition

Cf talks of Alajandro Luque and Chao Li

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]

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]

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)

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 ofpropagation

Characteristics of Characteristics of InteractingInteracting streamersstreamers

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

256

0.06 cm100 kV/cm

LE¥

;;

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

Net Charge

E-field256

0.5LE¥

==

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-´

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)]

Hele-Shaw FlowHele-Shaw FlowHoleHole

GlycerolGlycerol

Colored Colored WaterWater→

Radial Symmetry

Hele-Shaw FlowHele-Shaw FlowHoleHole

GlycerolGlycerol

Colored Colored WaterWater→

Radial Symmetry

Hele-Shaw FlowHele-Shaw FlowHoleHole

GlycerolGlycerol

Colored Colored WaterWater→

Radial Symmetry

Channel configuration

Saffman-Taylor solutionSaffman-Taylor solution( )( )(1 ) 1 2( , ) ln 1 cos2 2

: channel widthfinger width0 1: channel width

L yx y t vtL

L

Ev

l

l

pp l

l

¥

- é ù= + +ê úû

<

=

ë

<

Family of possible solutions

Selection: Boundary Selection: Boundary conditioncondition

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

surface tensionlocal 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)]

Saffman-Taylor solutionSaffman-Taylor solution

For small surface tension, only the finger with is selected

1/ 2l =

Saffman-Taylor solutionSaffman-Taylor solution

Experiment

Theory

Comparison: PDE vs FMBComparison: PDE vs FMB256

0.5LE¥

==

Comparison: PDE vs FMBComparison: PDE vs FMB256

0.5LE¥

==

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

Comparison: PDE vs FMBComparison: PDE vs FMB256

0.5LE¥

==

Maximum of the net charge along y axis for each value of x

Comparison: PDE vs FMBComparison: PDE vs FMB

Saffman-Taylor finger

2560.5

LE¥

==

1/ 2l =

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, 256wider initial cond

Fig. 1:

itio

Fig. 2:

Fig. 3:n

0.6376

E L

E L

EL

¥

¥

¥

= =

= =

==

Saffman-Taylor finger 1/ 2l =

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:

PredictionsPredictions

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

After some transient evolution, the velocity should After some transient evolution, the velocity should reach a constant valuereach 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:

BranchingBranching376

0.7LE¥

==

Electric field along the Electric field along the streamer axisstreamer axis

-1200kV cm´