Jerome J. Erpenbeck

Stability of Idealized One‐Reaction Detonations

The hydrodynamic stability of steady, one‐dimensional detonations in an ideal‐gas medium, undergoing the irreversible, unimolecular reaction A → B, having Arrhenius rate constant, is obtained in numerical form by application of the general theory of detonation stability. Stability results, as well as the pressure and progress variable profiles for the steady detonations, are presented for a (constant) heat capacity ratio γ of 1.2, a heat of reaction Q, relative to thermal energy in the cold reactants, of 50, activation energies Q‡ (in the same units) of 10 and 50, and at several detonation velocities, D. The neutral stability curves, consisting of the values of the disturbance wavelength transverse to the steady flow for which the steady detonation of given velocity changes stability (as functions of detonation velocity), are investigated. One‐dimensional disturbances are found to be unstable at low detonation velocities (i.e., near the CJ point), for the larger activation energy (but not the smaller) but...

STABILITY OF STEADY-STATE EQUILIBRIUM DETONATIONS

The hydrodynamic stability of one‐dimensional, steady, equilibrium detonation waves is investigated through solution of the initial‐value problem for the detonation equations, linearized in perturbations from the steady‐state solution. A criterion for stability is then found depending on the sign of the real parts of the zeros of a certain function of the steady flow. Determination of stability for a specific case through this criterion would involve extensive, but entirely feasible, numerical computations. The hydrodynamic stability of states of chemical equilibrium is also proved, this problem being pertinent to equilibrium detonations from both physical and mathematical considerations.

Molecular dynamics calculations of the hard-sphere equation of state

The equation of state of the hard-sphere fluid is studied by a Monte Carlomolecular dynamics method for volumes ranging from 25V0 to 1.6V0, whereV0 is the close-packed volume, and for system sizes from 108 to 4000 particles. TheN dependence of the equation of state is compared to the theoretical dependence given by Salsburg for theNPT ensemble, after correction for the ensemble difference, in order to obtain estimates for the thermodynamic limit. The observed values of the pressure are compared with both the [3/2] and the [2/3] Padé approximants to the virial series, using Kratkys value for the fifth virial coefficientB5 and choosingB6 andB7, to obtain a least-squares fit. The resulting values ofB6 andB7 lie within the uncertainties of the Ree-Hoover-Kratky Monte Carlo estimates for these virial coefficients. The values ofB8,B9, andB10 predicted by our optimal [3/2] approximant are also reported. Finally, the Monte Carlo-molecular dynamics equation of state is compared with a number of analytic expressions for the hard-sphere equation of state.

Stability of Step Shocks

The hydrodynamic stability of a steady, plane, step shock through a fluid medium with arbitrary equation of state is investigated through consideration of the initial‐value problem for the time‐dependent hydrodynamic equations, linearized in perturbations from the steady flow. If the stability function, Fs = 1 + κ ‐ κ2(v1 ‐ v) pS/T (with κ the Mach number, v the specific volume, T the absolute temperature, pS the entropy derivative of pressure at constant volume, and subscript 1 referring to the preshock state) is negative, disturbances grow exponentially with time and the shock is unstable. The character of the shock Hugoniot with respect to stability is discussed.

Molecular dynamics calculations of shear viscosity time-correlation functions for hard spheres

The time-correlation function for shear viscosity is evaluated for hard spheres at volumes of 1.6 and 3 times the close-packed volume by a Monte Carlomolecular dynamics technique. At both densities, the kinetic part of the timecorrelation function is consistent, within its rather large statistical uncertainty, with the long-timet−3/2 tail predicted by the mode-coupling theory. However, at the higher density, the time-correlation function is dominated by the cross and potential terms out to 25 mean free times, whereas the mode-coupling theory predicts that these are asymptotically negligible compared to the kinetic part. The total time-correlation function decays roughly asαt−3/2, withα much larger than the mode-coupling value, similar to the recent observations by Evans in his nonequilibrium simulations of argon and methane. The exact value of the exponent is, however, not very precisely determined. By analogy with the case of the velocity autocorrelation function, for which results are also presented at these densities, it is argued that it is quite possible that at high density the asymptotic behavior is not established until times substantially longer than those attainable in the present work. At the lower density, the cross and potential terms are of the same magnitude as the kinetic part, and all are consistent with the mode-coupling predictions within the relatively large statistical uncertainties.

Detonation Stability for Disturbances of Small Transverse Wavelength

The stability of one‐dimensional, steady detonations to periodic disturbances transverse to the flow is examined in the limit of small wavelength, 2π/e → 0. The asymptotic criterion for stability is found to depend largely on the steady‐state profile of c0 2η (where c0 is the frozen sound speed, η is the sonic parameter 1 − u 2/c0 2, and u is the mass velocity relative to the von Neumann shock) as a function of distance from the shock. Detonations for which c0 2η decreases monotonically are found to be stable in the e → ∞ limit but stability in cases in which this quantity increases either monotonically or up to a maximum is determined through simple integral functions of the steady‐flow variables. In contrast to the labor involved with application of the general theory of detonation stability, the asymptotic result can be applied straightforwardly to any detonation, irrespective of the equation of state and the complexity of the chemical kinetics. The results for an idealized, one‐reaction, A → B, system with an Arrhenius rate constant are detailed.

Nonlinear Theory of Unstable Two‐Dimensional Detonation

A two‐dimensional theory is presented for the structure of detonations for systems whose steady‐state solutions are hydrodynamically unstable. Periodic boundary conditions are imposed in the direction transverse to the steady flow. The analysis, which depends on the assumption that the steady detonation is only “slightly” unstable, permits the time‐dependent, two‐dimensional flow to be described by a system of two, complex, autonomous, ordinary differential equations in the time. Both traveling‐wave and standing‐wave solutions are found to be possible, and their significance is discussed. The theory is applied to an ideal‐gas, one‐reaction system and solutions of both types are found to exist and to be stable for a variety of transverse periods within the unstable regime. With increasing heat of reaction, the magnitude of the perturbations within these waves increases rapidly.

Theory of detonation stability

The theory of the hydrodynamic stability of steady one-dimensional detonation waves, having finite rates of chemical reactions and neglecting other transport properties, is reviewed. The discussion includes (1) the general theory and its application to certain idealized detonations, (2) the stability of the square-wave detonation model, and (3) the application of acousticray tracing to steady detonations and its relation to stability. It is concluded that at present only the considerations discussed in (1) have been fruitful in differentiating stable from unstable detonations. The problem of the application of this theory to an arbitrary system is discussed. The square-wave calculations appear to indicate that such detonations are generally unstable, a conclusion consistent with the trend observed in (1) for large activation energy, but the validity of this conclusion remains in doubt. The ray analysis of (3) appears to lend added physical insight into the short-wavelength asymptotic analysis of (1), but does not otherwise add new information with respect to stability.

The mutual diffusion constant of binary, isotopic hard‐sphere mixtures: Molecular dynamics calculations using the Green–Kubo and steady‐state methods

The mutual diffusion constant of a binary mixture of equal diameter hard spheres is estimated using the method of molecular dynamics. The mixture considered is equimolar, with a species mass ratio of ten to one in a volume that is three times the close‐packed volume. Two molecular dynamics methods are used: the standard Green–Kubo technique based on the evaluation of equilibrium velocity correlation functions, and a nonequilibrium method that generates a steady diffusive flow along a composition gradient by imposing special boundary conditions on two opposing faces of the cubic volume. We find that both methods yield, within about 3%, the same value of the diffusion coefficient.

Note on the fractal dimension of hard sphere trajectories

Using Monte Carlo molecular dynamics, a new, careful study is made of the approach of the trajectory of a typical particle in a hard sphere fluid to that of a Brownian particle, discussed before by Powles and Quirke and Rapaport. The apparent fractal dimension of the trajectory, as a function of reduced length scale,Δ(η), characterizes the transition from mechanical to Brownian motion and differs markedly from 2 in all present computer simulations.