[Paper Review] On the self-similarity of diffracting gaseous detonations and the critical channel width problem

Matei I. Radulescu, Rémy Mével+2 more|arXiv (Cornell University)|May 10, 2021

Combustion and Detonation Processes34 references20 citations

TL;DR

This paper proposes a closed-form model predicting the critical channel width for detonation transmission in 2D gaseous detonations by combining Whitham's geometrical shock dynamics with a shock evolution law based on Radulescu's shock change equations. The model uses the critical lateral strain rate from He and Clavin to derive a quantitative criterion for detonation failure, showing excellent agreement with experimental and simulation data for H₂/O₂/Ar mixtures.

ABSTRACT

One strategy for arresting propagating detonation waves in pipes is by imposing a sudden area enlargement, which provides a rapid lateral divergence of the gases in the reaction zone and attenuates the leading shock. For sufficiently small tube diameter, the detonation decays to a deflagration and the shock decays to negligible strengths. This is known as the critical tube diameter problem. In the present study, we provide a closed form model to predict the detonation quenching for 2D channels. Whitham's geometric shock dynamics, coupled with a shock evolution law based on shocks sustained by a constant source obtained by the shock change equations of Radulescu, is shown to capture the lateral shock dynamics response to the failure wave originating at the expansion corner. A criterion for successful detonation transmission to open space is that the lateral strain rate provided by the failure wave not exceed the critical strain rate of steady curved detonations. Using the critical lateral strain rate obtained by He and Clavin, a closed form solution is obtained for the critical channel opening permitting detonation transmission. The predicted critical channel width is found in very good agreement with our recent experiments and simulations of diffracting H$_2$/O$_2$/Ar detonations.

Motivation & Objective

• To develop a predictive model for the critical channel width that permits detonation transmission into open space.
• To address the long-standing lack of a closed-form predictive model for detonation quenching in weakly unstable gaseous detonations.
• To validate the applicability of curvature-based failure criteria in 2D diffraction of weakly unstable detonations.
• To reconcile theoretical predictions with recent experimental and numerical results on H₂/O₂/Ar detonations.
• To establish a benchmark model for future studies of more unstable detonation regimes.

Proposed method

• Applies Whitham's geometrical shock dynamics to model lateral shock wave evolution during detonation diffraction.
• Integrates a shock evolution law based on Radulescu's shock change equations for shocks weakly supported by detonation products.
• Uses the critical lateral strain rate from He and Clavin's work on steady curved detonations as a failure threshold.
• Derives a closed-form expression for the critical channel width by equating the maximum lateral strain rate in the diffraction process to the critical strain rate.
• Validates predictions against high-resolution numerical simulations and recent experiments using 2H₂+O₂+2Ar mixtures.
• Employs adaptive mesh refinement with 7 µm resolution (100–200 grid points per half-reaction length) to ensure numerical convergence.

Experimental results

Research questions

• RQ1 Can a curvature-based criterion accurately predict the critical channel width for detonation transmission in 2D channels?
• RQ2 Does the lateral strain rate distribution in diffracting detonations match predictions from Whitham's geometrical shock dynamics and the WSB extension?
• RQ3 Can the shock dynamics of weakly unstable detonations be accurately modeled without resolving the cellular structure?
• RQ4 Is the critical lateral strain rate from steady curved detonation theory applicable to transient, diffracting detonations?
• RQ5 How well does the proposed closed-form model agree with experimental and high-resolution simulation data for H₂/O₂/Ar detonations?

Key findings

• The proposed model predicts the critical channel width for detonation transmission with excellent agreement to both numerical simulations and recent experiments on H₂/O₂/Ar detonations.
• The model's prediction is in very good quantitative agreement with experimental data, validating the use of the critical lateral strain rate from He and Clavin as a failure criterion.
• Whitham's geometrical shock dynamics, when combined with the shock evolution law from Radulescu, provides an adequate approximation for lateral strain rate distribution during diffraction.
• The study confirms that weakly unstable detonations can be modeled using steady-state curvature-based criteria, supporting Lee’s critical curvature hypothesis.
• The model demonstrates that the anomalous scaling between 2D and 3D diffraction problems can be reconciled when cellular structure is neglected for weakly unstable conditions.
• The high-resolution simulations (7 µm grid spacing, 100–200 points per half-reaction length) confirm numerical convergence and support the reliability of the model’s validation.