Curved quasi-steady detonations: Asymptotic analysis and detailed chemical kinetics

We consider the problem of slightly-curved, quasi-steady diverging detonation waves. For sufficiently small curvature, the reaction zone structure equations can be formulated as a two point boundary value problem for solutions containing a sonic point. Analytical solutions to this boundary value problem are obtained for the case of a one-step Arrhenius reaction in the limit of high activation energy. The analytic solution results in a nonlinear relationship between detonation velocity and curvature. For extremely small curvature, this relationship is consistent with previous linear analyses, whereas in the nonlinear regime it predicts a critical maximum curvature, beyond which no quasi-steady solutions with a sonic point can be found. We also analyse the curved detonation structure for the gaseous fuel-oxidizer combination of H_2 and O_2 with various diluents. Using a standard shooting method we generate numerical solutions of the two-point boundary value problem based on realistic thermochemistry and a detailed chemical reaction mechanism. Similar to the large activation energy results, the numerical solutions reveal a nonlinear detonation speed-curvature relation and a critical maximum curvature for the existence of quasi-steady solutions. The relation of this critical curvature to the critical scales of multi-dimensional detonation is discussed.