J. Fluid Mech. zyxwvutsrqp (1996), uol. 309, pp. zyxwvutsrqp 225-215 zyxwvut Copyright zyxwvutsr 0 1996 Cambridge University Press 225 On the dynamics of multi-dimensional detonation By JIN YAO AND D. SCOTT STEWART? Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL 61801, USA (Received 25 January zyxwvu 1995 and in revised form 22 September 1995) We present an asymptotic theory for the dynamics of detonation when the radius of curvature of the detonation shock is large compared to the one-dimensional, steady, Chapman-Jouguet (CJ) detonation reaction-zone thickness. The analysis considers additional time-dependence in the slowly varying reaction zone to that considered in previous works. The detonation is assumed to have a sonic point in the reaction- zone structure behind the shock, and is referred to as an eigenvalue detonation. A new, iterative method is used to calculate the eigenvalue relation, which ultimately is expressed as an intrinsic, partial differential equation (PDE) for the motion of the shock surface. Two cases are considered for an ideal equation of state. The first corresponds to a model of a condensed-phase explosive, with modest reaction rate sensitivity, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D,, the first normal time derivative of the normal shock velocity, zyxwvu D,,, and the shock curvature, IC. The second case corresponds to a gaseous explosive mixture, with the large reaction rate sensitivity of Arrhenius kinetics, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D,, its first and second normal time derivatives of the normal shock velocity, b,, B,, and the shock curvature, IC, and its first normal time derivative of the curvature, k. For the second case, one obtains a one-dimensional theory of pulsations of plane CJ detonation and a theory that predicts the evolution of self-sustained cellular detonation. Versions of the theory include the limits of near-CJ detonation, and when the normal detonation velocity is significantly below its CJ value. The curvature of the detonation can also be of either sign, corresponding to both diverging and converging geometries. 1. Introduction Previous work, Stewart zyxwvu & Bdzil (1988), Bdzil & Stewart (1989), has developed an asymptotic theory for weakly curved, slowly varying detonation that propagates near the Chapman-Jouguet (CJ) velocity, DcJ, for the explosive, and has found that the normal detonation shock velocity D, is a function of the total shock curvature, IC. We call this relation, the (D,,K)-relation, and it is a partial differential equation (PDE) for the motion of the detonation shock surface. The functional form of the (D,, Ic)-relation follows from an asymptotic argument and is solely determined by the explosive material's equation of state and reaction rate law. In this paper, we extend the asymptotic analysis by considering the additional time- dependence which is required when the normal detonation shock velocity deviates t Author to whom correspondence should be addressed.