A Moving Boundary Model Motivated by Electric Breakdown Chiu-Yen Kao

Transcription

1 A Moving Boundary Model Motivated by Electric Breakdown Chiu-Yen Kao Department of Mathematics, The Ohio State University; Department of Mathematics and Computer Science, Claremont McKenna College Workshop on Mathematical Models of Electrolytes with Application to Molecular Biology, Jan 5-7, 2012, Taipei, Taiwan

2 Acknowledgement Collaborators: Ute Ebert, and Saleh Tanveer (CWI Amsterdam & Faculty of Applied Physics, Technische Universiteit Eindhoven, The Netherlands and Department of Mathematics, The Ohio State State University)

3 Outline 1. Introduction to electric breakdown 2. The dimensionless mathematical model density model for streamers moving boundary model for streamers 3. The conformal mapping 4. The theoretical and numerical results linearized evolution nonlinear evolution

4 Lightening Discharge

5 Electrostatic Discharges Left Column: Positive discharges in artificial air (N2 : O2 = 80 : 20) at room temperature. A voltage pulse of 35 kv and of 130 ns duration is applied to the upper needle electrode, the distance to the plate electrode below is 16 cm. Only the upper 5 cm of the gap are shown. Right Column: a zoom into the formation of positive streamers. As the pressure in (d)-(f) is 200mbar, i.e. 1/5 of the pressure in (a)-(c).

6 Density Models for Streamers [Ebert et al, 1996]: Electronic Density: Density of positive ions: Dimensionless diffusion coefficient::: Townsend s ionization coefficients: is a positive function that vanishes for vanishing field. A useful approximation is:

7 Density Models for Streamers The physical units in nitrogen are approximately n e = σ /cm 3 (N/N 0 ) 2 for the electron number density, n + = ρ /cm 3 (N/N 0 ) 2 for the ion number density, Ephys Edimensionless 200 kv/cm(n/ N 0 ) for the electric field, and dimensionless times and lengths are multiples of 3ps(N/N 0 ) and 2.3µm(N/N 0 ). The diffusion constant is D = 0.1. Here N is the gas particle number density and N 0 is the same at standard temperature and pressure, i.e. the dimensional parameters directly show the scaling relations between similar discharges at different gas densities N/N 0.

8 Numerical Calculation of Density Model The earliest stage is the initial avalanche (lower row), the next the propagating streamer (middle row), the latest the branching streamer

9 Basic Oberservation The active space charge layer is very thin. Its thickness is much smaller than its radius of curvature. Inside the streamer the electric field is very small. For sufficiently large field the streamer moves with a velocity v proportional to the electric field at the tip. Further more the region outside the streamer contains no charges, and far from the streamer the electric field becomes constant.

10 A Dimensionless Mathematical Model Normal Velocity: of the interface is given by the drift velocity of the electrons, which is proportional to the local electrostatic field Governing Equation: electrostatic Interfacial Condition: undercooling condition where Far Field Condition: of the interface is given by the drift velocity of the electrons, which is proportional to the local electrostatic field

11 Examples of Conformal Mapping a conformal map is a function which preserves angles

12 Review of Conformal Mapping

13 Analytic Function

14 The Conformal Mapping

15 Equations after mapping A simple solution corresponding to a steadily translating circle is given

16 Discussion 1. Linear Evolution: perturbations are advected to the back of the circle, where they decay. 2. Nonlinear Evolution: the circle is the asymptotic attractor for small perturbations, but larger perturbations may lead to branching.

17 Linear Perturbation of the Circle We here consider linear perturbations of the circle, making the ansatz leading order equation for small η, on elimination of χ leads to where where is re-scaled time.

18 Linear Perturbation of the Circle [Meulenbroek et. al.,2005] [Ute et. al.,2007] 1. ω = 1 is stable: all the complex ω-plane, except for ω = 1, is mapped into a neighbourhood of ζ = ω = 1 is unstable.

19 Linear Perturbation of the Circle Any perturbation not centred precisely at ω = 1 is advected towards ω = 1, where it vanishes asymptotically. As τ, only a shift of the circle is left.

20 Linear Perturbation of the Circle For ε 1 the same automorphism describes the advection of small disturbances from the front to the back of the bubble. Outside an arbitrarily small, but fixed neighborhood of ω= 1,the behavior of β(ω,τ) for τ on the circle ω=e α is given by where (λ j,β j ) are the eigenvalues and the corresponding eigenfunctions which have a singularity at ω = 1. The spectrum has been proven to be discrete, and except for the trivial translation mode λ 0 = 0, β 0 (ω) = 1, all eigenvalues are in the left half complex plane.

21 Numerical Approach for Linear Evolution Linear evolution: based on the recurrence relation for the coefficients b k (τ) in the Taylor expansion It yields

22 Numerical Approach for Linear Evolution With the b k (0) given by the initia Runge Kutta time stepping condition, the b k (τ) can be determined recursively by the method. We choose the cut-off N = 2000 in the simulation. Adaptive time steps are chosen which ensure that the difference between 4-th order and 5-th order Runge Kutta methods is within In the following we present results for δβ(ω,τ) = β(ω,τ) β(0,τ). The subtraction eliminates the overall shift of the evolving body. We choose the initial condition β(ω, 0) = ω 5. In physical space, Re[ωδβ] is the component of the perturbation normal to the unperturbed but shifted circle at angle α.

23 Numerical Result

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25 Fourier Methods on Boundary The shape of the interface is given by We restrict ourselves to interfaces symmetric with respect to the real axis, so that We use Fourier representation

26 Symmetry of coefficients For a given shape of the interface the potential is determined

27 Update formula for Coefficients

28 Translating Circle (a) ε = 0.01 (b) ε = 0.5. We can see that the circle translate to the right with the speed 2/(1+ε).

29 Comparison Initial condition: β(ω,0) = 0.03ω 5. (a) and (b) show the nonlinear and the linear evolution respectively for ε = 1/10. (c) shows the nonlinear evolution for ε=0.(n =512, t = )

30 Nonlinear Results: I f(ω,0) = 1 2ω + 0.5ω 2 (N = 512, t = ). The solutions are shown at t = 0.2n.

31 Nonlinear Results: II f(ω,0)= 1/ω +0.75ω 0.2ω 3 (N =512, t =0.0005).The solutions are shown at t =0.01nfor (a),(b),(c) and at t =0.05n for (d).

32 Nonlinear Results: III Ice-cone: f (ω, 0) = 1/ω 0.1ω 2 (N = 512, t = ). The solutions are shown at t = 0.1n.

33 Accuracy Test: III Ice-cone: f (ω, 0) = 1/ω 0.1ω 2

34 Area Conservation Test: III Ice-cone: f (ω, 0) = 1/ω 0.1ω 2

35 Future Directions 1. from the microscopic scale of collisions of electrons with neutral molecules to a hierarchy of macroscopic scales ranging from thin space charge layers within each streamer finger up to the streamer tree with possibly thousands of branches. 2. Combined undercooling and surface tension relaxation 3. Nonlinear evolution analysis 4. Three dimensional computational approach

36 The End