SaffmanTaylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro...

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Transcript of SaffmanTaylor streamer discharges Fabian Brau, CWI Amsterdam Fabian Brau, CWI Amsterdam Alejandro...
SaffmanTaylor SaffmanTaylor
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 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
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
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 Efield produced by two electrodesEfield 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
layerEfield enhancement
Efield 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 Efield evolutionNet charge and Efield evolutionfor for singlesingle streamers streamers
Net Charge
Efield
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 of
propagation
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
E¥
;
;
Net charge and Efield evolutionNet charge and Efield evolutionfor for interactinginteracting streamers streamers
Net Charge
Efield
256
0.5
L
E¥
=
=
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
HeleShaw flow
SaffmanTaylor solution
L
[E. D. Lozansky and O. B. Firsov, J. Phys. D 6, 976 (1973)]
HeleShaw FlowHeleShaw FlowHoleHole
GlycerolGlycerol
Colored Colored WaterWater→
Radial Symmetry
HeleShaw FlowHeleShaw FlowHoleHole
GlycerolGlycerol
Colored Colored WaterWater→
Radial Symmetry
HeleShaw FlowHeleShaw FlowHoleHole
GlycerolGlycerol
Colored Colored WaterWater→
Radial Symmetry
Channel configuration
SaffmanTaylor solutionSaffmanTaylor 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
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)]
SaffmanTaylor solutionSaffmanTaylor solution
For small surface tension, only the finger with is selected
1/ 2l =
SaffmanTaylor solutionSaffmanTaylor solution
Experiment
Theory
Comparison: PDE vs FMBComparison: PDE vs FMB256
0.5
L
E¥
=
=
Comparison: PDE vs FMBComparison: PDE vs FMB256
0.5
L
E¥
=
=
SaffmanTaylor 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
0.5
L
E¥
=
=
Maximum of the net charge along y axis
for each value of x
Comparison: PDE vs FMBComparison: PDE vs FMB
SaffmanTaylor finger
256
0.5
L
E¥
=
=
1/ 2l =
Various evolutions of the streamer Various evolutions of the streamer fronts + comparison with Saffmanfronts + 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
¥
¥
¥
= =
= =
=
=
SaffmanTaylor 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 SaffmanTaylor finger.from hydrodynamics: the SaffmanTaylor 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 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:
BranchingBranching
376
0.7
L
E¥
=
=
Electric field along the Electric field along the streamer axisstreamer axis
1200kV cm´