Conference Record of the 1998 IEEE International Symposium on Electrical Insulation, 7-10 June 1998, Washington, DC, USA. Vol. 2., pp. 611--614.

## Simulations of Gas-dynamic Flows During Streamers Propagation at Liquid Dielectrics Breakdowns A. L. Kupershtokh, D. A. Medvedev

Lavrentyev Institute of Hydrodynamics SB RAS
Novosibirsk, 630090, RUSSIA

### Abstract

For the case of selfsimilar solution for the problem of streamer channel expansion at the electric breakdown in liquid, computations using the lattice gas (LG) and lattice Boltzmann equation (LBE) methods were carried out. In the latter case, the computation results agree well with theoretical calculations.

Computations of the prebreakdown stage of the propagation of a rectilinear streamer channel were also carried out. Expansion of the conductive channel and formation of compression waves propagating with a speed close to the sound velocity in a dielectric liquid were observed.

The stochastic evolution of the streamer structure was computed with allowance for the gas-dynamic flows. In this case, the growth proceeds due to both the hydrodynamic movement of the conductive phase and the breakdown of new portions of the substance. Transition between two mechanisms of development of the conductive structure was demonstrated.

### Introduction

An electric breakdown of dielectric liquids is known to occur by propagation of a system of streamers (thin plasma channels) from one electrode to another. A large body of experimental data points to the important role of stochastic processes at breakdown of solid and liquid dielectrics. One can cite as an example the statistical lag time after the voltage supply to the gap, the dissymmetry of the streamer system and the irreproducibility of its detailed structure, the indented shape of oscillograms of the prebreakdown current and the luminescence of the growing structure, etc. Certain success has been recently achieved in the modeling of these stochastic effects [1-4].

On the other hand, gas-dynamic flows (flows of compressible liquid) of the surrounding dielectric have drawn little attention so far. However, the formation of compression waves during the streamer propagation and the expansion of streamer channels have been already observed in the first experiments where the high-speed photorecording of the breakdown process has been conducted. Many experimental results point to the important role of these processes not only at the channel stage, but also at the prebreakdown stage of the streamer channel propagation. For instance, data have been obtained [5] which clearly indicate expansion and pulsations of streamer channels. At the propagation stage, divergent shock waves with an approximately conic shape have been observed. Some experiments point to the decisive role of liquid flows (including vortical ones) as the viscosity of dielectric liquids increases. In this case, the fine structure of streamers vanishes and their evolution resembles the expansion and growth of a cavity (bubble) of irregular shape [6]. Similar processes have been observed in certain development regimes of partial breakdown [7].

However, so far there are no model that would allow one to describe the entire complex of experimentally observed phenomena.

### Lattice Gas methods

The lattice gas (LG) method [8] treats a liquid flow as the dynamics of special particles which can move along the links of a fixed lattice and suffer collisions and scattering in nodes. The LG method is in a sense an extremely simplified method of molecular dynamics.

An eight-directional LG model with three velocity levels 0, 1 and $\sqrt{2}$ was used [9]. It allows one to describe the energy release at the streamer tip.

Another modification of this method is the method of immiscible lattice gases (ILG). Here two kinds (colors) of particles exist, one for a dielectric liquid and the second for a conductive phase. Collision rules are chosen to maximize the flux of particles of given color to adjacent nodes with the majority of particles of the same color. This gives rise to surface tension on the interface.

### Method "Lattice Boltzmann Equation"

This method has evolved from the lattice gas method. In this case there is also a regular spatial lattice and a few velocity values are possible at each node. The ensemble-averaged local one-particle distribution functions are taken as variables. The averaging leads to complete elimination of the statistical noise. It allows one to considerably decrease the number of grid points in the computation domain. This is possible, however, only at the cost of giving up integer arithmetics. The LBE method can be viewed as a solution of the kinetic equation for model particles.

The evolution equations are formally the same as for the lattice gas. The collision operator, however, can be of any form. It allows one to provide, for instance, Galilean invariance. Moreover, the LBE method can be easily extended to three dimensions. Presently, the collision operator in the BGK form [10] is widely used. It simply means the relaxation to the local equilibrium configuration which is determined by the density, velocity, and temperature at a node. In the computations we used a two-dimensional model with four velocity levels: 0, 1, $\sqrt{2}$ and 2 (13 possible velocity vectors) [11].

Figure 1. Structure of the selfsimilar solution. Results of simulation with the LBE method. P is the pressure, r is the density, u is the mass velocity, e is the average kinetic energy per one particle ("temperature").

### Flow Structure during Expansion of the Streamer Conductive Channel

For the one-dimensional problem of the conductive channel expansion a selfsimilar solution exists when three conditions are satisfied:
1) The power released in the channel is W = const.
2) The heat conductivity inside the channel is sufficiently high, and one can roughly consider that the energy released is completely transferred to the channel boundary by both the conductive heat flux and radiation.
3) The total heat flux from the channel is completely absorbed in a thin layer of liquid. It causes a transition of the liquid substance to the channel plasma. Under these conditions, the mass velocity of "plasma" inside the channel is zero, and the temperarure, density and pressure are constant both across the channel section and in time.

Figure 1 shows flow structure for this selfsimilar solution. It was obtained in simulation using the LBE method. In cylindrical geometry a selfsimilar solution also exists [12], but for W(t) = kt.

In the transition layer liquid--"plasma", a new conductive phase is formed. This transition occurs through the flux of liquid molecules which enter the channel plasma after their dissociation and partial ionization. At the channel stage of discharge, the magnitude of the flux of molecules into the channel can achieve values much greater than   j ~ 2·1024sec-1·cm-2[13]. At the initial stage of streamer propagation, the temperature inside the conductive channel (~ 3000 K) is rather low [14]; therefore, the density here is not much different from the liquid density. Thus, the interface between the conductive channel and the surrounding liquid cannot be considered as an impenetrable piston. If the thickness of the transition layer is much less than the channel radius, the transition layer can be treated as a quasi-stationary gas-dynamic rupture. In the frame of reference fixed upon the layer, the mass and momentum conservation laws take the form

r1 D = r2 u2 ,

P1 + r1 D2 = P2 + r1 D u2 .            (1)

Here P is the pressure, r is the density, u is the mass velocity of substance, D is the liquid inflow velocity, u2 = V - u2 is the plasma velocity relative to the rupture, V = u1 + D is the observed velocity of channel expansion (see Fig. 1).

From (1) follows a small pressure jump at a rupture, P1 = P2 + r2 u2 (u2 - D). The addition to the pressure in front of the rupture is simply the reactive pressure due to fast liquid "vaporization". In the selfsimilar case, u2 = 0 , and, hence, u2= V , and the pressure difference is D P = P1 - P2 = r2 V u1 , which agrees quantitatively with the value obtained in computations with the LBE method (Fig. 1).

Figure 2. Expansion of a conductive channel and formation of shock wave. e is the average kinetic energy per one particle ("temperature"), r is the average density, p is the pressure. Computation grid 400x10,000.

### Simulation of the Streamer Channel Expansion

In the present work, the LBE method was used to simulate the channel stage of an electric discharge in liquid. The energy release took place in a thin transition layer where e* < e < emax , which corresponds to the energy input W = const . Here e is the average kinetic energy per one particle ("temperature"). Results are presented in Fig. 1. The solution obtained is selfsimilar. The computed values agree well with theoretical calculations.

Simulations were also performed using LG method with energy release (Fig. 2). When e > 0.9 the substance became conductive and energy releases at nodes. The main difference from the selfsimilar case (Fig. 1) is the higher heat conductivity. Therefore, a portion of energy escapes from the channel and "pre-heats" surrounding liquid.

Figure 3. Propagation of the streamer tip. The velocity is 2.5. Time t=140.
Figure 4. Propagation of the streamer tip. The velocity is 0.6. Time t=300.

### Shock Waves in the Streamer Top Propagation

The propagation of the top of a rectilinear conductive channel between two plane electrodes was simulated by the same method. Density distributions averaged over 10 patterns are shown in Figs. 3 and 4. The black color indicates lower densities.

The expansion of the conductive channel and the formation of compression waves propagating with a speed close to the sound velocity in liquid dielectric, C = 1/$\sqrt{2}$ » 0.7, were observed. If the velocity of the streamer top was higher than C, a divergent wave with an approximately conic shape was generated. Such waves have been observed at the prebreakdown stage of the streamer channel propagation in the experiments of [5].

### Streamer Growth

In the present work, a new model was realized in which two-dimensional flow of a dielectric liquid is simulated by the immiscible lattice gas (ILG) method [9] and the transition of the dielectric to a conductive phase is described by the field fluctuation criterion (FFC) [1,2].

A triangular lattice with six directions of particle velocity was used. In addition, up to nine rest particles can be at each node.

The hydrodynamic flow was computed taking into account the electric pressure eE2/ 8p acting on the interface. The Laplace equation Dj = 0 was then solved in the region occupied by the dielectric. The conductive phase was considered equipotential (j = 0). When the growth criterion E > E* + d was satisfied for a node adjacent to the interface, this node and its 6 neighbors became conductive and some energy released (the pressure increased). In a high electric field, the boundary of a conductive phase moves under action of the electric forces. In addition, the conductive region expands because of the higher pressure in it. The streamer grows as a result of both hydrodynamic movement and the breakdown of new portions of the dielectric (transition to the conductive state).

The relative role of these two mechanisms of development of the conductive structure can be varied by changing the ratio between the streamer growth velocity due to breakdown and the hydrodynamic velocity of viscous flow.

The problem was solved in the rectangular region between two horizontal plane electrodes to which an electric potential difference j(t) was applied. Periodic boundary conditions in the X direction were used.

A transition between two mechanisms of development of the conductive structure was demonstrated. Figure 5 shows an example of the growth of a branched structure mainly through the breakdown of new portions of dielectric. The structure development mainly due to the hydrodynamic flow is presented in Fig. 6. The bubble-like conductive region is accelerated by the electrodynamic force. At a later stage, a flow in the form of a plane vortical dipole moving to the upper electrode is clearly observed. The development of conductive structures of this type has been observed in experiments on breakdown of highly viscous dielectrics [6] and in certain regimes of partial breakdown [7].

Figure 5. Example of formation of branched streamer structure. t = 40.

Figure 6. Example of bubble-like streamer. Formation of a vortex. t = 200 (a) and 350 (b).

### Conclusion

The gas-dynamic processes are shown to have a determinative effect on the dynamics of the streamer structure development. One should take them into account in constructing stochastic models of breakdown in dielectric liquids.

### Acknowledgements

This work was supported by Russian Foundation for Basic Research under grant No. 97-02-18416.

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