Joseph M. Powers

Simulations of pulsating one-dimensional detonations with true fifth order accuracy

A novel, highly accurate numerical scheme based on shock-fitting coupled with fifth order spatial and temporal discretizations is applied to a classical unsteady detonation problem to generate solutions with unprecedented accuracy. The one-dimensional reactive Euler equations for a calorically perfect mixture of ideal gases whose reaction is described by single-step irreversible Arrhenius kinetics are solved in a series of calculations in which the activation energy is varied. In contrast with nearly all known simulations of this problem, which converge at a rate no greater than first order as the spatial and temporal grid is refined, the present method is shown to converge at a rate consistent with the fifth order accuracy of the spatial and temporal discretization schemes. This high accuracy enables more precise verification of known results and prediction of heretofore unknown phenomena. To five significant figures, the scheme faithfully recovers the stability boundary, growth rates, and wave-numbers predicted by an independent linear stability theory in the stable and weakly unstable regime. As the activation energy is increased, a series of period-doubling events are predicted, and the system undergoes a transition to chaos. Consistent with general theories of non-linear dynamics, the bifurcation points are seen to converge at a rate for which the Feigenbaum constant is 4.66+/-0.09, in close agreement with the true value of 4.669201.... As activation energy is increased further, domains are identified in which the system undergoes a transition from a chaotic state back to one whose limit cycles are characterized by a small number of non-linear oscillatory modes. This result is consistent with behavior of other non-linear dynamical systems, but not typically considered in detonation dynamics. The period and average detonation velocity are calculated for a variety of asymptotically stable limit cycles. The average velocity for such pulsating detonations is found to be slightly greater than the Chapman-Jouguet velocity.

On slow manifolds of chemically reactive systems

This work addresses the construction of slow manifolds for chemically reactive flows. This construction relies on the same decomposition of a local eigensystem that is used in formation of what are known as Intrinsic Low Dimensional Manifolds (ILDMs). We first clarify the accuracy of the standard ILDM approximation to the set of ordinary differential equations which model spatially homogeneous reactive systems. It is shown that the ILDM is actually only an approximation of the more fundamental Slow Invariant Manifold (SIM) for the same system. Subsequently, we give an improved extension of the standard ILDM method to systems where reaction couples with convection and diffusion. Reduced model equations are obtained by equilibrating the fast dynamics of a closely coupled reaction/convection/diffusion system and resolving only the slow dynamics of the same system in order to reduce computational costs, while maintaining a desired level of accuracy. The improvement is realized through formulation of an ellipt...

Accurate Spatial Resolution Estimates for Reactive Supersonic Flow with Detailed Chemistry

Ar obust method is developed and used to provide rational estimates of reaction zone thicknesses in onedimensional steady gas-phase detonations in mixtures of inviscid ideal reacting gases whose chemistry is described by detailed kinetics of the interactions of N molecular species constituted from L atomic elements. The conservation principles are cast as a set of algebraic relations giving pressure, temperature, density, velocity, and L species mass fractions as functions of the remaining N‐L species mass fractions. These are used to recast the N‐L species evolution equations as a self-contained system of nonlinear ordinary differential equations of the form dYi/dx = fi (Y1 ,..., YN‐L). These equations are numerically integrated from a shock to an equilibrium end state. The eigenvalues of the Jacobian of fi are calculated at every point in space, and their reciprocals give local estimates of all length scales. Application of the method to the standard problem of a stoichiometric Chapman‐Jouguet hydrogen‐air detonation in a mixture with ambient pressure of 1 atm and temperature of 298 K reveals that the finest length scale is on the order of 10 −5 cm; this is orders of magnitude smaller than both the induction zone length, 10 −2 cm, and the overall reaction zone length, 10 0 cm. To achieve numerical stability and convergence of the solution at a rate consistent with the order of accuracy of the numerical method as the spatial grid is refined, it is shown that one must employ a grid with a finer spatial discretization than the smallest physical length scale. It is shown that published results of detonation structures predicted by models with detailed kinetics are typically underresolved by one to five orders of magnitude.

Viscous detonation in H2-O2-Ar using intrinsic low-dimensional manifolds and wavelet adaptive multilevel representation

A standard ignition delay problem for a mixture of hydrogen-oxygen-argon in a shock tube is extended to the viscous regime and solved using the method of intrinsic low-dimensional manifolds (ILDM) coupled with a wavelet adaptive multilevel representation (WAMR) spatial discretization technique. An operator-splitting method is used to describe the reactions as a system of ordinary differential equations at each spatial point. The ILDM method is used to eliminate the stiffness associated with the chemistry by decoupling processes which evolve on fast and slow time scales. The fast time scale processes are systematically equilibrated, thereby reducing the dimension of the phase space required to describe the reactive system. The WAMR technique captures the detailed spatial structures automatically with a small number of basis functions thereby further reducing the number of variables required to describe the system. A maximum of only 300 collocation points and 15 scale levels yields results with striking resolution of fine-scale viscous and induction zones. Additionally, the resolution of physical diffusion processes minimizes the effects of potentially reaction-inducing artificial entropy layers associated with numerical diffusion.

Theory of two-phase detonation—Part I: Modeling☆

Abstract A new, one-dimensional, two-phase model appropriate for describing the detonation of granulated solid propellants or explosives is presented. The model satisfies the principle that the mixture mass, momentum, and energy are conserved, is strictly hyperbolic, and is frame indifferent. Conditions are presented for satisfying the second law of thermodynamics. It is shown that this and previous models do not satisfy the second law under all circumstances. It is shown that in the limit of no chemical reaction or gas phase effects that inclusion of compaction work is in violation of the energy conservation principle. It is also shown that a complete two-phase particle combustion model with constitutive functions dependent on particle radius requires an equation specifying the variation of particle radius; such a relation can be given by a number evolution equation. The model equations are solved in a subsequent study which follows as a separate article.

Review of Multiscale Modeling of Detonation

Issues associated with modeling the multiscale nature of detonation are reviewed, and potential applications to detonation-driven propulsion systems are discussed. It is suggested that a failure of most existing computations to simultaneously capture the intrinsic microscales of the underlying continuum model along with engineering macroscales could in part explain existing discrepancies between numerical predictions and experimental observation. Mathematical and computational strategies for addressing general problems in multiscale physics are first examined, followed by focus on their application to detonation modeling. Results are given for a simple detonation with one-step kinetics, which undergoes a period-doubling transition to chaos; as activation energy is increased, such a system exhibits larger scales than are commonly considered. In contrast, for systems with detailed kinetics, scales finer than are commonly considered are revealed to be present in models typically used for detonation propulsion systems. Some modern computational strategies that have been recently applied to more efficiently capture the multiscale physics of detonation are discussed: intrinsic low-dimensional manifolds for rational filtering of fast chemistry modes, and a wavelet adaptive multilevel representation to filter small-amplitude fine-scale spatial modes. An example that shows the common strategy of relying upon numerical viscosity to filter fine-scale physics induces nonphysical structures downstream of a detonation is given.

Approximate Solutions for Oblique Detonations in the Hypersonic Limit

This article describes analytic solutions for hypersonic flow of a premixed reactive ideal gas over a wedge. The flow is characterized by a shock followed by a spatially resolved reaction zone. Explicit solutions are given for the irrotational flowfield behind a straight shock attached to a curved wedge and for the rotational flowfield behind a curved shock attached to a straight wedge. Continuous solution trajectories exist that connect the state just past the shock to the equilibrium end states found from a Rankine-Hugoni ot theory for changes across oblique discontinuities with energy release. The analytic results are made possible by the hypersonic approximation, which implies that a fluid particles kinetic energy is much larger than its thermal and chemical energy. The leading order solution is an inert oblique shock. The effects of heat release are corrected for at the next order. These results can be used to verify numerical results and are necessary for more advanced analytic studies. In addition, the theory has application to devices such as the oblique detonation wave engine, the ram accelerator, hypersonic airframes, or re-entry vehicles.

Two-phase viscous modeling of compaction of granular materials

An inviscid model for deflagration-to-detonation transition in granular energetic materials is extended by addition of explicit intraphase momenta and energy diffusion so as to (1) enable the use of a straightforward numerical scheme, (2) avoid prediction of structures with smaller length scales than the component grains, and (3) have a model prepared to describe long time scale transients that are present in some slow processes which can lead to detonation. The model is shown to be parabolic, frame invariant, and dissipative. Consideration of the characteristics for cases with and without intraphase diffusion indicate what boundary conditions are necessary for a well posed problem. A simple numerical method, based on a method of lines applied to the nonconservative form of the equations, is shown to predict convergence at the proper rate to unique solutions which agree well with known solutions for an unsteady inviscid shock tube and a steady piston-driven viscous shock. A series of simulations of inert ...

Numerical predictions of oblique detonation stability boundaries

Oblique detonation stability was studied by numerically integrating the two-dimensional, one-step reactive Euler equations in a generalized, curvilinear coordinate system. The integration was accomplished using the Roe scheme combined with fractional stepping; nonlinear flux limiting was used to prevent unphysical solution oscillations near discontinuities. The method was verified on one-and two-dimensional flows with exact solutions, and its ability to correctly predict one-dimensional detonation stability boundaries was demonstrated. The behavior of straight oblique detonations attached to curved walls was then considered. Using the exact, steady oblique detonation solution as an initial condition, the numberical simulation predicted both steady and unsteady oblique detonation solutions when a detonation parameter known as the normal overdrive,fn, was varied. For a standard test case with a specific heat ratio of ψ=1.2, a dimensionless activation energy of Θ=50, and dimensionless heat release ofq=50, an oblique detonation with a constant dimensionless component of velocity tangent to the lead shock,vtan=4.795, underwent transition to unstable behavior atfn=1.77. This is slightly higher than the threshold offn=1.73 predicted by one-dimensional theory; thus, the two-dimensionality renders the flow slightly more susceptible to instability.

Reaction zone structure for strong, weak overdriven, and weak underdriven oblique detonations

A simple dynamic systems analysis is used to give examples of strong, weak overdriven, and weak underdriven oblique detonations. Steady oblique detonations consisting of a straight lead shock attached to a solid wedge followed by a resolved reaction zone structure are admitted as solutions to the reactive Euler equations. This is demonstrated for a fluid that is taken to be an inviscid, calorically perfect ideal gas that undergoes a two‐step irreversible reaction with the first step exothermic and the second step endothermic. This model admits solutions for a continuum of shock wave angles for two classes of solutions identified by a Rankine–Hugoniot analysis: strong and weak overdriven waves. The other class, weak underdriven, is admitted for eigenvalue shock‐wave angles. Chapman–Jouguet waves, however, are not admitted. These results contrast those for a corresponding one‐step model that, for detonations with a straight lead shock, only admits strong, weak overdriven, and Chapman–Jouguet solutions.