[Paper Review] The Structure of Radiative Shock Waves. I. The method of global iterations

Yu. A. Fadeyev, Denis Gillet|arXiv (Cornell University)|Feb 6, 1998

Gas Dynamics and Kinetic Theory4 references4 citations

TL;DR

This paper presents a global iteration method for self-consistently solving the coupled equations of fluid dynamics, radiation transfer, and atomic kinetics in steady, plane-parallel radiative shock waves in partially ionized hydrogen gas. The method converges to a stable solution that accurately captures the strong feedback between the shock flow and its radiation field, demonstrating robustness for shocks with upstream velocities from 15 to 60 km/s and Mach numbers between 2.3 and 9.3.

ABSTRACT

The structure of steady plane-parallel radiative shock waves propagating through the hydrogen gas undergoing partial ionization and excitation of bound atomic states is investigated in terms of the self-consistent solution of the equations of fluid dynamics, radiation transfer and atomic kinetics. The shock wave model is represented by a flat finite slab with no incoming radiation from external sources at both its boundaries. The self-consistent solution is obtained using the global iteration procedure each step of which involves (1) integration of the fluid dynamics and rate equations for the preshock and postshock regions, consecutively, both solutions being fitted by the Rankine-Hugoniot relations at the discontinuous jump; (2) solution of the radiation transfer equation for the whole slab. The global iteration procedure is shown to converge to the stable solution which allows for the strong coupling of the gas flow and the radiation field produced by this flow. Application of the method is demonstrated for the shock waves with upstream velocities from 15 km/s to 60 km/s (that is with upstream Mach numbers from 2.3 to 9.3) and the hydrogen gas of unperturbed temperature T=3000K and density rho = 1e-10 gm/cm^3.

Motivation & Objective

• To develop a self-consistent numerical method for modeling radiative shock waves where fluid dynamics, radiation transfer, and atomic kinetics are fully coupled.
• To address the challenge of strong feedback between the shock-generated radiation field and the gas structure in partially ionized hydrogen.
• To enable accurate computation of shock structure across a wide range of upstream velocities (15–60 km/s) and Mach numbers (2.3–9.3).
• To validate the convergence and stability of the global iteration procedure in simulating complex radiative shock systems.
• To provide a foundation for future studies of radiative shocks in astrophysical environments such as H II regions and supernova remnants.

Proposed method

• The method employs a global iteration procedure that alternates between solving fluid dynamics and rate equations in the pre- and postshock regions, with solutions matched via Rankine-Hugoniot jump conditions.
• At each iteration step, the radiation transfer equation is solved over the entire finite slab domain, accounting for self-consistent emission and absorption.
• The fluid and atomic kinetics equations are solved consecutively in the pre- and postshock zones, ensuring continuity across the shock discontinuity.
• The radiation field computed in one iteration is used as input for the next, progressively refining the coupling between radiation and gas structure.
• The method uses a finite slab geometry with no external radiation at the boundaries, simulating isolated shock systems.
• Convergence is achieved when changes in the radiation field and gas structure between iterations fall below a specified tolerance, ensuring a stable self-consistent solution.

Experimental results

Research questions

• RQ1 How can the self-consistent coupling between fluid dynamics, radiation transfer, and atomic kinetics be achieved in radiative shock waves?
• RQ2 What numerical method ensures stable and convergent solutions for the full system of equations in plane-parallel radiative shocks?
• RQ3 How does the radiation field influence the structure and thermodynamics of shocks with Mach numbers between 2.3 and 9.3?
• RQ4 What role does partial ionization and excitation of hydrogen play in shaping the shock structure and radiation output?
• RQ5 Can the global iteration method reliably model shocks across a wide range of upstream velocities (15–60 km/s) without external radiation sources?

Key findings

• The global iteration method successfully converges to a stable, self-consistent solution for radiative shock waves in partially ionized hydrogen gas.
• The method accurately captures the strong feedback between the shock’s radiation field and the gas structure, particularly in high-Mach-number shocks.
• For upstream velocities of 15–60 km/s and Mach numbers from 2.3 to 9.3, the method produces physically consistent shock profiles with proper energy redistribution.
• The radiation field is dominated by hydrogen line emission and continuum processes, especially in the postshock region, with significant contributions from bound-bound and bound-free transitions.
• The solution remains stable and convergent even under strong coupling, demonstrating the robustness of the global iteration approach.
• The computed shock structure shows significant deviations from non-radiative models, especially in the postshock cooling layer, due to efficient radiative energy loss.