Saffman-Taylor streamer discharges

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Presentation transcript:

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

Talk overview Minimal PDE model for streamers Characteristics of single streamers Interacting streamers : periodic array of streamers in 2D Numerical solutions + characteristics FMB explicit solution fits very well the numerical fronts Conclusions

Minimal PDE model for streamers In dimensionless unit, the minimal PDE model reads s : Electron density r : Ion density Solved in the background of a homogeneous field Negative streamer Non attaching gas like nitrogen Normal condition → Electron impact ionization : Townsend’s Approximation → Poisson equation Electric potential Electric field Cf talks of Alajandro Luque and Chao Li

Characteristics of single streamers (Valid also for interacting streamers) Evolution of some initial condition in the E-field produced by two electrodes: Solutions after the avalanche phase looks like (3D + cylindrical symmetry) z (mm) r (mm) Net charge ( ) z (mm) r (mm) Electric Field (kV/cm) Thin charge layer E-field enhancement E-field screening [Montijn, Ebert, Hundsdorfer, J. Comp. Phys. 2006, Phys. Rev. E 2006, J. Phys. D 2006 Luque, Ebert, Montijn, Hundsdorfer, Appl. Phys. Lett. 2007]

Net charge and E-field evolution for single streamers On branching as a Laplacian instability: [Arrayas, Ebert, Hundsdorfer, Phys. Rev. Lett. 2002, Ebert et al., Plasma Sour. Sci. Techn. 2006]

Interacting streamers as periodic array : previous work G V Naidis, J. Phys. D 29, 779 (1996) 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

Interacting streamers : periodic array of streamers in 2D Anode Direction of propagation L Cathode Neumann boundary conditions for: Potential and Densities = Symmetry axis

Characteristics of Interacting streamers If : Streamers do not branch 256

Net charge and E-field evolution for interacting streamers

Characteristics of Interacting streamers Uniformly translating streamers Results are robust against changes in the initial condition

FMB mathematical setup Single streamers x [E. D. Lozansky and O. B. Firsov, J. Phys. D 6, 976 (1973)] Interacting streamers: periodic array of streamers in 2D Ideally conducting streamer Free moving boundary for Hele-Shaw flow Saffman-Taylor solution y L

Hele-Shaw Flow Radial Symmetry Hole Colored Water → Glycerol

Hele-Shaw Flow Radial Symmetry Hole Glycerol Colored Water →

→ Hele-Shaw Flow Radial Symmetry Hole Colored Water Glycerol Channel configuration

Saffman-Taylor solution Family of possible solutions

Selection: Boundary condition The boundary condition doesn’t allow any selection mechanism [B. Meulenbroek, U. Ebert and L. Schäfer, PRL 95 (2005) F. Brau, A. Luque, B. Meulenbroek, U. Ebert and L. Schäfer, (2007)]

Saffman-Taylor solution For small surface tension, only the finger with is selected

Saffman-Taylor solution Experiment Theory

Comparison: PDE vs FMB

Comparison: PDE vs FMB No free parameter Saffman-Taylor finger The tip of the Saffman – Taylor finger coincide with the maximum of the net charge of the PDE solution No free parameter Saffman-Taylor finger

Comparison: PDE vs FMB Maximum of the net charge along y axis for each value of x

Comparison: PDE vs FMB Saffman-Taylor finger

Various evolutions of the streamer fronts + comparison with Saffman-Taylor Front From the left: Saffman-Taylor finger

Conclusions Interacting streamers viewed as a periodic array of streamers in 2D present remarkable features: If the streamer spacing is small enough for a given background electric field, streamers do not branch. When streamers do not branch they reach a steady state which is an attractor of the dynamics. This steady state is well approximated by a solution from hydrodynamics: the Saffman-Taylor finger.

Predictions If you are in the right part of the phase diagram: In contrast to single streamers, branchings should be mostly suppressed After some transient evolution, the velocity should reach a constant value This value is in 2D. In 3D, we expect that the following linear relation should hold In physical units:

Branching

Electric field along the streamer axis