The finite volume method solution of the radiative transfer equation for photon transport in non-thermal gas discharges: application to the calculation of photoionization in streamer discharges

This paper presents the development of a direct accurate numerical method to solve the monochromatic radiative transfer equation (RTE) based on a finite volume method (FVM) and its application to the simulation of streamer propagation. The validity of the developed model is demonstrated by performing direct comparisons with results obtained using the classic integral model. Comparisons with approximate solutions of the RTE (Eddington and SP3 models) are also carried out. Specific validation comparisons are presented for an artificial source of radiation with a Gaussian shape. The reported results demonstrate that whatever the value of the absorption coefficient, the results obtained using the direct FVM are in excellent agreement with the reference integral model with a significantly reduced computation time. When the absorption coefficient is high enough, the Eddington and SP3 methods are as accurate and become faster than the FVM. However, when the absorption coefficient decreases, approximate methods become less accurate and more computationally expensive than the FVM. Then the direct finite volume and the SP3 models have been applied to the calculation of photoionization in a double-headed streamer at ground pressure. For high values of the absorption coefficient, positive and negative streamers calculated using the SP3 model and the FVM for the photoionization source term are in excellent agreement. As the value of the absorption coefficient decreases, discrepancies between the results obtained with the finite volume and the SP3 models increase, and these differences increase as the streamers advance. For low values of the absorption coefficient, the use of the SP3 model overestimates the electron density and underestimates the photoionization source term in both streamers in comparison with the FVM. As a consequence, for low values of the absorption coefficient, positive and negative streamers calculated using the SP3 model for the photoionization source term propagate more slowly than those calculated using the FVM.

[1]  W. Hundsdorfer,et al.  Branching of negative streamers in free flight. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Ningyu Liu,et al.  Effects of photoionization on propagation and branching of positive and negative streamers in sprites , 2004 .

[3]  L. Loeb.  THE MECHANISM OF SPARK DISCHARGE IN AIR AT ATMOSPHERIC PRESSURE. , 1929, Science.

[4]  USE OF DISCRETE-ORDINATES METHODS FOR SOLUTION OF PHOTON PROBLEMS , 1966 .

[5]  George E. Georghiou,et al.  Secondary emission effects on streamer branching in transient non-uniform short-gap discharges , 2003 .

[6]  G. Raithby,et al.  Prediction of radiative transfer in cylindrical enclosures with the finite volume method , 1992 .

[7]  J. Paillol,et al.  The use of an improved Eddington approximation to facilitate the calculation of photoionization in streamer discharges , 2006 .

[8]  N. Babaeva,et al.  Dynamics of positive and negative streamers in air in weak uniform electric fields , 1997 .

[9]  R. Goody.  Principles of atmospheric physics and chemistry , 1995 .

[10]  S. Starikovskaia,et al.  Role of photoionization processes in propagation of cathode-directed streamer , 2001 .

[11]  F. Bastien,et al.  The determination of basic quantities during glow-to-arc transition in a positive point-to-plane discharge , 1979 .

[12]  H. Tagashira,et al.  Computer simulation of a nitrogen discharge at high overvoltages , 1976 .

[13]  Tzeng,et al.  Development of an electron avalanche and its transition into streamers. , 1988, Physical review. A, General physics.

[14]  L. Loeb,et al.  The Mechanism of Spark Discharge in Air at Atmospheric Pressure. I , 1940 .

[15]  P. Williams,et al.  Two‐dimensional studies of streamers in gases , 1987 .

[16]  J. Lowke,et al.  Streamer propagation in air , 1997 .

[17]  Andreas F. Molisch,et al.  Radiation Trapping in Atomic Vapours , 1999 .

[18]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[19]  G. C. Pomraning.  The Equations of Radiation Hydrodynamics , 2005 .

[20]  Axel Klar,et al.  Simplified P N approximations to the equations of radiative heat transfer and applications , 2002 .

[21]  A. Kulikovsky.  The role of photoionization in positive streamer dynamics , 2000 .

[22]  K. D. Lathrop.  USE OF DISCRETE ORDINATES METHODS FOR SOLUTION OF PHOTON TRANSPORT PROBLEMS , 1966 .

[23]  A. Bourdon,et al.  Efficient models for photoionization produced by non-thermal gas discharges in air based on radiative transfer and the Helmholtz equations , 2007 .

[24]  E. Lewis,et al.  Computational Methods of Neutron Transport , 1993 .

[25]  G. Naidis.  On photoionization produced by discharges in air , 2006 .

[26]  Wu,et al.  Formation and propagation of streamers in N2 and N2-SF6 mixtures. , 1988, Physical review. A, General physics.

[27]  S. Pancheshnyi.  Role of electronegative gas admixtures in streamer start, propagation and branching phenomena , 2005 .

[28]  W. Hundsdorfer,et al.  Photoionization in negative streamers: Fast computations and two propagation modes , 2006, physics/0609247.

[29]  A. Bourdon,et al.  Application of photoionization models based on radiative transfer and the Helmholtz equations to studies of streamers in weak electric fields , 2007 .

[30]  A. Kulikovsky.  Positive streamer between parallel plate electrodes in atmospheric pressure air , 1997 .

[31]  Andrew Pollard,et al.  SPATIAL DIFFERENCING SCHEMES OF THE DISCRETE-ORDINATES METHOD , 1996 .