Capturing photoionization shadows in streamer simulations using the discrete ordinates method

Numerical simulations of streamer propagation involving photoionization are presented, utilizing an ANSYS Fluent implementation that employs unstructured meshes and automatic mesh refinement. Two approximate methods for radiative transfer, used to handle computation of the photoionization source terms, are compared: the Eddington approximation and the discrete ordinates (DOs) method. The former is commonly employed in streamer simulations, while the latter is well-established in other branches of computational physics, such as radiative heat transfer. A 2D test case with two distinct regions, where streamer propagation can be triggered thanks to the protruded electrodes, is introduced. The two regions are partially separated by an opaque solid insulator barrier to study the effects of photoionization shadows on streamer inception and propagation. The primary positive streamer is initiated by placing a neutral plasma patch close to one of the electrode protrusions, while the secondary positive streamer, in the other region of the computational domain, is initiated by photoionization originating from the primary streamer zone. The Eddington approximation results in an excessively high photoionization source in the secondary streamer inception zone, as it fails to capture the shadowing effects of the opaque dielectric barrier. Consequently, this leads to a fast secondary streamer inception process, followed by rapid streamer propagation. On the other hand, the DOs method accurately captures the shadow, leading to a delayed secondary streamer inception. It is also shown that both methods exhibit very similar results when the dielectric barrier is transparent and the shadow is absent. This work demonstrates that using the DOs method for streamer simulations offers considerable advantages over the Eddington approximation, especially in cases involving more complex geometries where shadows need to be captured for accurate streamer inception and dynamics.