The electric arc discharge in a liquid medium is used in several applications such as the sterilization of the liquid by UV radiation, the fracturing of rocks by shock wave, the circuit breakers in oil bath or the forming of mechanical parts. Thus, describing the physics of the arc in a liquid and in particular its interaction with a liquid interface is an important issue to better characterize this type of configuration. However, such a challenging task requires to couple high-fidelity solver for compressible two-phase flows with liquid phase change and a plasma solver to describe the plasma and its interaction with the bubble. To study this type of medium, we use a compressible formulation of the fluid equations. For this purpose, a pressure based solver has been developed for the computation of the energy conservation equation. Moreover, our numerical model uses the immersed boundary method to simulate the solid electrodes. The numerical model is briefly described in this paper and the first results of the electric arc discharge in steam water are presented. To our knowledge this pressure based model has never been used to describe plasmas and electric arc discharge

Coseru, Sergiu

Freton, Pierre

Gonzalez, Jean-Jacques

Tanguy, Sebastien

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

doi:10.14311/ppt.2023.3.159Plasma Physics and Technology 10(3):159–162, 2023 © Department of Physics, FEE CTU in Prague, 2023A PRESSURE BASED COMPRESSIBLE SOLVER FOR ELECTRICARC-PLASMA SIMULATIONS. Coserua,∗, S. Tanguyb, P. Fretona,b, JJ. Gonzaleza,ba IMFT, UMR CNRS 5502, Université de Toulouse, 2 Allée du Professeur Camille Soula, 31400 Toulouse, Franceb LAPLACE, UMR CNRS 5213, Université Paul Sabatier Toulouse 3, 118 route de Narbonne, 31062 ToulouseCedex France∗ sergiu.coseru@laplace.univ-tlse.frAbstract. The electric arc discharge in a liquid medium is used in several applications such as thesterilization of the liquid by UV radiation, the fracturing of rocks by shock wave, the circuit breakersin oil bath or the forming of mechanical parts. Thus, describing the physics of the arc in a liquidand in particular its interaction with a liquid interface is an important issue to better characterizethis type of configuration. However, such a challenging task requires to couple high-fidelity solver forcompressible two-phase flows with liquid phase change and a plasma solver to describe the plasma andits interaction with the bubble. To study this type of medium, we use a compressible formulation of thefluid equations. For this purpose, a pressure based solver has been developed for the computation ofthe energy conservation equation. Moreover, our numerical model uses the immersed boundary methodto simulate the solid electrodes. The numerical model is briefly described in this paper and the firstresults of the electric arc discharge in steam water are presented. To our knowledge this pressure basedmodel has never been used to describe plasmas and electric arc discharge.Keywords: electric arc, plasma, compressible, pressure-based.1. IntroductionNumerical Simulations is a powerful tool which canbe complementary to experiments for a better under-standing of the plasma behavior in water. The electricarc in liquid water creates a centimeter-sized bubbleof plasma and acoustic waves. The simulation basedon commercial CFD solvers, as Fluent, can hardly sim-ulate the interaction between, the acoustic waves, theplasma and two-phase flows. When dealing with simu-lation of compressible flows one have to be careful forthe numerical stability at different Mach numbers [1].We can achieve an asymptotically preserving solutionthrough compressible semi-implicit solvers based onpressure [2]. The compressible entropic two-phase flowsolver, developed in [3] seem adapted for the prob-lem at hand. In particular, the formulation of theenergy conservation equation based on the pressurevariable is suitable to describe acoustic waves, whichare described implicitly, while being well suited todescribe two-phase flows. The objective is verify theaptitude of these solvers to correctly simulate plasmaflows. This solver is integrated in the home made codeDIVA [4] in which specific plasma solvers have alsobeen integrated. Moreover, this compressible solver iscoupled to an immersed boundary method to accountfor complex geometry, as the solid surface of the elec-trodes [5]. We describe in this paper some preliminaryresults obtained with our solver on single-phase flowswith acoustic waves in presence of an electrical arcbetween two electrodes in a water gas medium. Fu-ture developments on the extension of our solver tocompressible two-phase flows with liquid-vapor phasechange are also discussed.2. Hypotheses□ 2 Dimensional and axisymmetric hypothesis.□ Local Thermodynamical Equilibrium (LTE) forplasma□ Laminar flow□ No sheath description□ No numerical resolution inside the solid material.□ Only the water gas phase is considered in the mod-eling3. TheoryWe present in this section the physical model forcompressible flows which has been coupled to theImmersed Boundary Method developed in [5]. Theformulation of the energy equation is pressure based,and equations of state accounting for a gas and plasmastate enable to compute the temperature field fromthe density and pressure fields obtained by solvingconservation equations. The model uses the equa-tions of conservation of mass, momentum and energydescribed here:The mass equation for the change of density ρ:∂ρ∂t+ ∇ · (ρu⃗) = 0. (1)The momentum equation to compute the velocityfield u⃗:159S. Coseru, T. Sebastien, F. Pierre, G. Jean-Jacques Plasma Physics and TechnologyDu⃗Dt+ ∇pρ= 1ρ∇ · τ + g⃗. (2)The volume forces coefficient is g⃗ in this particularcase it is the gravitational acceleration. The viscositytensor is τ and the viscosity coefficient is µ:τ = µ(∇u⃗ + ∇u⃗T ) − 23µ∇ · u⃗I. (3)The pressure based energy equation for the pressurep, proposed in [3], can be expressed as:DpDt= c2 DρDt+ ∂p∂s∣∣∣∣ρDsDt. (4)Here s is the entropy DDt is the material derivative, cis the speed of sound. The left hand side representingthe evolution of pressure. On the right hand sidethe first term is associated to acoustic waves, andthe second one is associated to heat transfer. Theterm responsible for the acoustic propagation c2 DρDt issolved implicitly. Instead of the time step dt = dxu+c ,numerical stability is ensured with dt = dxu whereu is the speed of the fluid. This is an advantage inthe plasma medium in which the sound speed canreach very large values, such as 10 km/s which wouldinvolve very small values of the time step with classicalcompressible solver. This pressure based solver allowsstability gain of the solver while being able to describeacoustic waves. To solve these equations, we useclassical second order finite volume schemes for thespatial discretization, except the convection terms (−→u ·∇)(−→u ) which are solved using a fifth-order WENO-Z(weighted essentially non-oscillatory) method [6].To close our system of equations, an equation ofstate is required. Although, the present work is fo-cused on electrical arc in a single-phase flow, cubicequations of state have been implemented since theyallow to model the thermodynamics state, both in agas phase and a liquid phase, as shown in [3]. This willbe an asset for coming works on electrical arc in a liq-uid pool. For higher temperatures when dissociationof the molecules happen, data tables are implementedin our solver for a more accurate description of thedensity, temperature and pressure in the plasma state.These data tables are obtained using a solver basedon minimization of energy.An electric current with a linear profile is imposedon half of the radius of the upper electrode and thelower electrode is at a reference potential taken to bezero. The boundary conditions on the electric fieldare imposed on the electrodes using the ImmersedBoundary Method and the lateral conditions imposean electrical insulation n⃗ · ∇V = 0 with n⃗ being thenormal vector, here V is the electric potential. Solvingthe following equation gives the electric potential fieldwhich is coupled to the conductivity coefficient σ∇ · (σ∇V ) = 0. (5)It is noteworthy that the voltage drop deduced bythe model only represents the column contributionas the sheathes description is not included in ourmodel. From the electric potential field computation,the electric current density field J⃗ is computed inthe fluid and the corresponding heat release due toJoule effect J⃗J⃗σ measured inWm3 is included as a sourceterm in the pressure equation DsDt . The magneticfield induced by the electric current is also takeninto account. This magnetic field is calculated usinga simplification, assuming that the electric currentis parallel to the symmetry axis of the cylindricalelectrodes. This allows to write simply the magneticfield as:B⃗ = µ0R∫ R0jz r dr, (6)where B⃗(R) is the magnetic field, R is the radiuscoordinate of the point in axisymmetric coordinate,and jz is the electric current component parallel tothe axis of symmetry. The effect of this magnetic fieldwill be to deflect the charged fluid flow between theelectrodes. At the same time a constant current isassumed inside the volume of the electrodes meaningjz(r) is a constant, from which the magnetic fieldcreated around the electrodes is deduced.In the plasma phase the thermodynamical prop-erties of plasma, depending on the local tempera-ture and pressure, are extracted from tables. Thesound speed is computed with a simple perfect gaslaw c =√γ pρ =√cpcvpρ , γ being the heat capacityratio, which is consistent with experimental data.Radiation is included using the net emission coef-ficient ϵ. The term 4πϵ is added to the change ofentropy term DsDt .4. Results4.1. AcousticsA first benchmark is proposed in order to illustratethe capability of our compressible solver to describethe interaction between an acoustic wave and a solidcomplex geometry imposed with an Immersed Bound-ary method. It consists in a long tube of air throughwhich an acoustic wave of an arbitrary pressure ampli-tude propagates. Here the amplitude of the pressureis taken at 100 Pa . The section of the tube is re-duced at the middle of the domain, which generatesa reflected acoustic wave and a transmitted acousticwave. This behavior is highlighted in Figure 1, wherewe can visualize the propagation of the acoustic waveat different times: t = 0 ms, t = 0.75 ms, t = 1.9 ms,and the formation of a reflected wave after the impactof the incident wave with the section reduction. Atheoretical prediction of the amplitudes of both wavescan be obtained from a linear acoustic theory. Thereflection and transmission coefficients, which dependon the section ratio, have been plotted in Figure 2160vol. 10 no. 3/2023 A pressure based compressible solver for electric arc-plasma simulationFigure 1. Acoustic wave propagation in a tube. Thewave is propagated from right to left with a changeof section in the middle. After the incident wave im-pacts at the section reduction, it splits into 2 waves:transmitted wave going to the left and the reflectedwave going to the right. The figure times are: for(a),t = 0 ms, for (b) t = 0.75 ms, for (c) t = 1.9 mswhere x-axis represents different section ratios. Thisdemonstrates a good agreement between the theoryand our numerical predictions.Figure 2. Comparisons between numerical simulationand linear acoustic theory on the transmission andreflection coefficients. The point represent numericalsimulation for different section ratios of tubes and thecontinuous graph represents the theoretical solution.4.2. PlasmaWe present in this subsection, some preliminary resultsobtained with the overall solver which couples a solverfor compressible flow with an immersed boundarymethod and a plasma solver to describe the electric arc.The simulation is performed on an axisymmetric grid.Figure 3. The temperature field between the blackelectrodes taken 90 microseconds after the start of thesimulation.The mesh is regular with a 40 micrometers cell sizewhich corresponds to a mesh containing 256×512 cells.The gas used is water vapour at an initial pressureslightly below atmospheric pressure 95800 Pa. Thecomputational domain is a cylinder of 1 cm radius andFigure 4. Pressure wave traveling away from the elec-trodes 6 microseconds after the start of the simulation161S. Coseru, T. Sebastien, F. Pierre, G. Jean-Jacques Plasma Physics and TechnologyFigure 5. The velocity streamlines 20 microsecondsafter the start of the simulation. The streamlines aresuperimposed on the temperature field.2 cm height, and the electrodes are 3 mm apart andhave a radius of 1.5 mm. Neumann boundary condi-tions are imposed on the temperature n⃗ · ∇T = 0 onthe electrodes. The initial conditions are an uniformtemperature everywhere except between the electrodeswhere a hot canal of 7000 K is initialized to allow theplasma formation at the beginning of the simulation.We impose an electric current with a total intensity atan arbitrary value of 100 A constant in time. In Fig-ure 3, we can visualize, the temperature field and theexpansion of the plasma with a maximum temperaturearound 20000 K in the vicinity of the electrode. Thesolver has shown its ability to describe the pressurewave in the test case in Figure 4. The arc ignition isnot represented in our model as the plasma is initial-ized with a hot canal (7000 K) so the shock wave inFigure 4 is only due to initial temperature differencebetween the hot canal and the surrounding cold gas.The velocity streamlines are presented in Figure 5.Velocity between both electrodes has a typical mag-nitude of 40 m.s−1. The electric potential and theelectric current streamlines are presented in Figure 6.We can observe the current density distribution in theplasma column. We can also observe that the totalvoltage obtained is around 7 Volts. This value corre-sponds to the column voltage as sheath description isnot taken into account.5. ConclusionIn this paper, we have briefly described the numer-ical solver that we are currently developing for theinteraction between an electric arc discharging in aliquid pool and a plasma bubble. It combines a solverfor compressible flow with an Immersed Boundarymethod to account for complex geometry in our com-putational domain and a plasma solver for electricarcs. Our solver enables to capture the generation ofacoustic waves and their interactions with the plasmaFigure 6. Electric potential field between the electrodesrepresented with the color-map. The streamlines rep-resent the passage of electric current.medium. As the present work only considers a singlephase flow, future works will focus on its coupling witha two-phase flow solver accounting for liquid-vaporphase change, in order to predict the formation andthe growth of a plasma bubble in a liquid pool.References[1] F. Xiao, R. Akoh, and S. Ii. Unified formulation forcompressible and incompressible flows by usingmulti-integrated moments ii: Multi-dimensional versionfor compressible and incompressible flows. Journal ofComputational Physics, 213(1):31–56, 2006.doi:10.1016/j.jcp.2005.08.002.[2] N. Kwatra, J. Su, J. T. Grétarsson, and R. Fedkiw. Amethod for avoiding the acoustic time step restriction incompressible flow. Journal of Computational Physics,228(11):4146–4161, 2009.doi:10.1016/j.jcp.2009.02.027.[3] A. Urbano, M. Bibal, and S. Tanguy. A semi implicitcompressible solver for two-phase flows of real fluids.Journal of Computational Physics, 456:111034, 2022.doi:10.1016/j.jcp.2022.111034.[4] A. Dalmon, M. Lepilliez, S. Tanguy, et al. Directnumerical simulation of a bubble motion in a sphericaltank under external forces and microgravity conditions.Journal of Fluid Mechanics, 849:467–497, 2018.doi:10.1017/jfm.2018.389.[5] M. Lepilliez, E. R. Popescu, F. Gibou, and S. Tanguy.On two-phase flow solvers in irregular domains withcontact line. Journal of Computational Physics, 321:1217–1251, 2016. doi:10.1016/j.jcp.2016.06.013.[6] R. Borges, M. Carmona, B. Costa, and W. S. Don. Animproved weighted essentially non-oscillatory scheme forhyperbolic conservation laws. Journal of ComputationalPhysics, 227(6):3191–3211, 2008.doi:10.1016/j.jcp.2007.11.038.162

A Pressure Based Compressible Solver for Electric Arc-plasma Simulation

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A Pressure Based Compressible Solver for Electric Arc-plasma Simulation

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