[Paper Review] Stability of detonations in the ZND limit

Kevin Zumbrun|ArXiv.org|Jun 15, 2009

Combustion and Detonation Processes39 references3 citations

TL;DR

This paper establishes that the stability of strong viscous detonation waves in the small-viscosity (ZND) limit is equivalent to the combined stability of the limiting ZND detonation profile and the associated viscous Neumann shock. It proves that nonstable eigenvalues of the viscous detonation converge to those of the ZND profile at low frequencies and to scaled eigenvalues of the Neumann shock at high frequencies, providing a rigorous foundation for interpreting numerical instability results in detonation theory.

ABSTRACT

Confirming a conjecture of Lyng--Raoofi--Texier--Zumbrun, we show that stability of strong detonation waves in the ZND, or small-viscosity, limit is equivalent to stability of the limiting ZND detonation together with stability of the viscous profile associated with the component Neumann shock. More, on bounded frequencies the nonstable eigenvalues of the viscous detonation wave converge to those of the limiting ZND detonation, while on frequencies of order one over viscosity, they converge to one over viscosity times thos of the associated viscous Neumann shock. This yields immediately a number of examples of instability and Hopf bifurcation of reacting Navier--Stokes detonations through the extensive numerical studies of ZND stability in the detonation literature.

Motivation & Objective

• To resolve a conjecture by Lyng–Raoofi–Texier–Zumbrun on the stability of viscous detonations in the ZND limit.
• To establish a precise connection between the spectral stability of viscous detonation waves and the stability of their limiting ZND and Neumann shock components.
• To characterize the asymptotic behavior of unstable eigenvalues in the viscous detonation problem as viscosity tends to zero.
• To provide a theoretical framework that explains and predicts instability and Hopf bifurcations observed in numerical studies of reacting Navier–Stokes detonations.

Proposed method

• Analyzes the Evans–Lopatinski determinant for the ZND model and the Evans determinant for the reactive Navier–Stokes (rNS) system to study spectral stability.
• Uses a decomposition into 'fast' (Neumann shock) and 'slow' (reaction zone) coordinate regions to analyze eigenvalue behavior across different frequency regimes.
• Applies asymptotic ODE theory, including conjugation, convergence, and tracking lemmas, to handle the singular perturbation structure as viscosity ε → 0.
• Employs Duhamel’s principle and fixed-point arguments in a perturbative framework to construct and bound transition matrices between stable and unstable subspaces.
• Introduces a transformation to constant-coefficient form via fast-slow coordinate scaling, enabling application of standard exponential dichotomy theory.
• Uses the implicit function theorem to establish regularity of solutions with respect to parameters in the perturbation analysis.

Experimental results

Research questions

• RQ1 Is the stability of viscous detonation waves in the ZND limit equivalent to the combined stability of the limiting ZND profile and the viscous Neumann shock?
• RQ2 How do the unstable eigenvalues of the viscous detonation wave behave as viscosity ε approaches zero?
• RQ3 Do the nonstable eigenvalues of the viscous detonation converge to those of the ZND profile at low frequencies and to scaled eigenvalues of the Neumann shock at high frequencies?
• RQ4 Can the spectral instability patterns of viscous detonations be rigorously linked to numerical observations in the detonation literature?

Key findings

• Stability of viscous detonation waves in the ZND limit is equivalent to the joint stability of the limiting ZND detonation and the viscous Neumann shock profile.
• On bounded frequencies, the nonstable eigenvalues of the viscous detonation converge to those of the limiting ZND detonation as ε → 0.
• On frequencies of order 1/ε, the nonstable eigenvalues converge to 1/ε times the eigenvalues of the viscous Neumann shock profile.
• The convergence of eigenvalues is uniform and quantitatively controlled via asymptotic ODE theory and fixed-point arguments in the singular perturbation regime.
• The results provide a rigorous justification for interpreting numerical instability patterns in reacting Navier–Stokes systems through the lens of ZND and Neumann shock stability.
• The analysis confirms the conjecture of Lyng–Raoofi–Texier–Zumbrun and extends the theoretical understanding of detonation stability in the small-viscosity limit.