Numerical simulations of pulsating detonations: I. Nonlinear stability of steady detonations

Very-long-time numerical simulations of an idealized pulsating detonation with one irreversible reaction having an Arrhenius form are performed using a hierarchical adaptive second-order Godunov-type scheme. The initial data are given by the steady solution and the truncation error produces the perturbation to trigger the instability. The detonation is allowed to run for thousands of half-reaction times of the underlying steady wave to ensure that the final amplitudes and periods of the nonlinear oscillations are achieved. Thorough resolution studies are performed for various representative regimes of the instability. It is shown that to obtain quantitatively good solutions over 50 numerical grid points in the half-reaction length of the steady detonation are required, while to obtain a converged solution over 100 points are required, even near the stability boundary. This is much higher resolution than has generally been used in previous papers in either one or two dimensions. Resolutions of less than approximately 20 points per half-reaction length give very poor predictions of the periods and amplitudes near the stability boundary or entirely spurious solutions for more unstable detonations. The evolution of the converged solutions as the activation energy increases, and the detonation becomes more unstable, is also investigated.     

Numerical simulations of hydrogen detonations with detailed chemical kinetics

Time-dependent, multidimensional simulations of unstable propagating detonations were performed using a detailed thermochemical reaction model for a stoichiometric argon-diluted hydrogen–oxygen mixture at low pressures and a hydrogen–air mixture at atmospheric pressure. Detonation cells computed for the low-pressure, dilute H₂–O₂–Ar systems were regular in shape, and their sizes compared reasonably well with experimental observations. The computed H₂–air cells at atmospheric conditions were qualitatively different from those observed in experiments, and their widths range from less than 1 mm to nearly 5 mm with multilevel hierarchal structures. The effective activation energy of the H₂–air mixture, based on constant-volume ignition delay times computed using the detailed thermochemical model, varies between 5 and 40 over the range of post-shock temperatures and pressures in the simulations and is, on average, significantly larger than expected based on the regularity of experimental cellular patterns. Analysis of the simulations suggests that vibrational relaxation of the gas molecules, a process which is ignored when calibrating detailed chemical reaction models, occurs on time scales similar to the ignition delay times for the detonations and may be a source of discrepancy between numerical and experimental results.     

On the dynamics of pulsating detonations

To understand the circumstances in which a pulsatingdetonation wave may prefer a low-frequency mode ofoscillation, and the implications this has formathematical modelling, the dynamics of a detonation waveare studied when the underlying linear stability spectrumconsists of at most two unstable modes. One mode α₁ has a period much larger than the time-scaleof particle passage through the half-reaction length inthe steady one-dimensional detonation; the second α₂ has a smaller period than α₁. Thequestion addressed in this paper concerns the long-timebehaviour of the pulsating detonation in the presence ofthe unstable modes α₁ and α₂. Westudy two general scenarios related to a crossing of theneutral stability boundaries traced out by α₁ and α₂. In all cases where the mode α₁ is unstable, the pulsating detonation emerges with alow-frequency, large-amplitude oscillation, regardlessof the relative growth rates of α₁ and α₂. Only in one case, where thehigher-frequency mode α₂ is alone unstable, isthe long-time nonlinear pulsation of high frequency. Inthis case, the amplitude of the oscillation issignificantly smaller than that observed in thelow-frequency oscillations. In all cases, the finalperiod of the nonlinear oscillation is closely related tothat of the relevant underlying linear mode.     

Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations

To understand the nonlinear dynamical behaviour of a one-dimensional pulsating detonation, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analysed using basic nonlinear dynamics and chaos theory. To illustrate the transition pattern from a simple harmonic limit-cycle to a more complex irregular oscillation, a bifurcation diagram is constructed from the computational results. Evidence suggests that the route to higher instability modes may follow closely the Feigenbaum scenario of a period-doubling cascade observed in many generic nonlinear systems. Analysis of the one-dimensional pulsating detonation shows that the Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be in reasonable agreement with the universal value of d = 4.669. Using the concept of the largest Lyapunov exponent, the existence of chaos in a one-dimensional unsteady detonation is demonstrated.     

Numerical solution of under-resolved detonations

A new fractional-step method is proposed for the numerical solution of high speed reacting flows, where the chemical time scales are often much smaller than the fluid dynamical time scales. When the problem is stiff, because of insufficient spatial/temporal resolution, a well-known spurious numerical phenomenon occurs in standard finite volume schemes: the incorrect calculation of the speed of propagation of discontinuities. The new method is first illustrated considering a one-dimensional scalar hyperbolic advection/reaction equation with stiff source term, which may be considered as a model problem to under-resolved detonations. During the reaction step, the proposed scheme replaces the cell average representation with a two-value reconstruction, which allows us to locate the discontinuity position inside the cell during the computation of the source term. This results in the correct propagation of discontinuities even in the stiff case. The method is proved to be second-order accurate for smooth solutions of scalar equations and is applied successfully to the solution of the one-dimensional reactive Euler equations for Chapman–Jouguet detonations.     

Nonlinear dynamics and chaos regularization of one-dimensional pulsating detonations with small sinusoidal density perturbations

In this work, we explore the effect of initial density variation in the combustible mixture on the nonlinear dynamics of one-dimensional gaseous detonation propagation. Studies of nonlinear dynamical behavior of one-dimensional pulsating detonation are frequently based upon the reactive Euler simulations with one-step Arrhenius chemistry. In regions of the control parameters space, i.e., activation energy E_a, the 1-D detonation dynamics are shown to exhibit chaotic behavior at values of 28.5 and 30.0. Using small sinusoidal initial density perturbations, this investigation shows the emergence of various nonlinear temporal patterns as a function of the perturbation wavelength. It demonstrates that the cooperative behavior between the intrinsic instability and imposed small perturbation can lead to regularization of chaotic oscillations in one-dimensional gaseous pulsating detonation. Hence, by means of a small perturbation, an otherwise chaotic motion is rendered more stable and predictable. This result thus has implications for how intrinsically unstable detonation dynamics can be controlled.     

Numerical resolution of pulsating detonation waves

The canonical problem of the one-dimensional, pulsating, overdriven detonation wave has been studied for over 30 years, not only for its phenomenological relation to the evolution of multidimensional detonation instabilities, but also to provide a robust, reactive, high-speed flowfield with which to test numerical schemes. The present study examines this flowfield using high-order, essentially non-oscillatory schemes, systematically varying the level of resolution of the reaction zone, the size and retention of information in the computational domain, the initial conditions, and the order of the scheme. It is found that there can be profound differences in peak pressures as well as in the period of oscillation, not only for cases in which the reaction front is under-resolved, but for cases in which the computation is corrupted due to a too-small computational domain. Methods for estimating the required size of the computational domain to reduce costs while avoiding erroneous solutions are proposed and tested.     

A review of direct numerical simulations of astrophysical detonations and their implications

Multi-dimensional direct numerical simulations (DNS) of astrophysical detonations in degenerate matter have revealed that the nuclear burning is typically characterized by cellular structure caused by transverse instabilities in the detonation front. Type Ia supernova modelers often use onedimensional DNS of detonations as inputs or constraints for their whole star simulations.While these one-dimensional studies are useful tools, the true nature of the detonation is multi-dimensional. The multi-dimensional structure of the burning influences the speed, stability, and the composition of the detonation and its burning products, and therefore, could have an impact on the spectra of Type Ia supernovae. Considerable effort has been expended modeling Type Ia supernovae at densities above 1×10⁷ g·cm^-3 where the complexities of turbulent burning dominate the flame propagation. However, most full star models turn the nuclear burning schemes off when the density falls below 1×10⁷ g·cm^-3 and distributed burning begins. The deflagration to detonation transition (DDT) is believed to occur at just these densities and consequently they are the densities important for studying the properties of the subsequent detonation. This work will review the status of DNS studies of detonations and their possible implications for Type Ia supernova models. It will cover the development of Detonation theory from the first simple Chapman–Jouguet (CJ) detonation models to the current models based on the time-dependent, compressible, reactive flow Euler equations of fluid dynamics.     

Numerical simulations of generic singularities

Numerical simulations of the approach to the singularity in vacuum spacetimes are presented here. The spacetimes examined have no symmetries and can be regarded as representing the general behavior of singularities. It is found that the singularity is spacelike and that, as it is approached, the spacetime dynamics becomes local and oscillatory.     

Numerical simulations of two-dimensional QED

We describe the computer simulation of two-dimensional QED on a 64 × 64 Euclidean space-time lattice using the Susskind lattice fermion action. The order parameter for chiral symmetry breaking and the low-lying meson masses are calculated for both the model with two continuum flavours, which arises naturally in this formulation, and the model with one continuum flavour obtained by including a nonsymmetric mass term and setting one fermion mass equal to the cut-off. Results are compared with those obtained using the quenched approximation, and with analytic predictions.     

Numerical simulations of the reditron

The reflected-electrons discrimination microwave generator (reditron) is a high-power, narrow-band, and single-mode microwave generator that makes exclusive use of the oscillatory character of the virtual-cathode of a relativistic electron beam. The complex, nonlinear character of the virtual-cathode device necessitates particle-in-cell plasma simulation techniques. Investigations indicate two sources of the radiation: (1) the trapped electrons reflexing between the real and virtual cathodes, and (2) the oscillation of the virtual cathode. In the conventional design, the two mechanisms coexist and interfere with each other destructively, causing degradation of the efficiency of microwave generation. The authors have investigated a configuration with a slotted, thick anode and an external magnetic field, which effectively eliminates the reflexing electrons. Two-dimensional particle-in-cell simulations showed that such a configuration exploits the oscillation of the virtual cathode exclusively, and it generates single-mode, narrow bandwidth, and high-power microwave radiation with a potential efficiency over 10%. It was found that further optimization could be achieved by the use of a density (current) modulated electron beam at appropriate frequencies     

Numerical simulations studies of glasses

In the recent years the use of the molecular dynamics technique became very common in glass studies. The key point for given a predictive capability to the method lies in the interatomic interaction model. The purpose of the present paper is to review the various potentials that have been devised for that purpose, focussing on their validity limits and basis. We show then on simple examples taken from studies of pure silica as well as of alkali-silica glasses, the present capability of the approach.     

Size effects in triplet-triplet annihilation: II. Monte carlo simulations

The dynamics of triplet-triplet annihilation (TTA) is theoretically studied in linear chains and nanoparticles, modeled as 1D, 2D, and 3D regular lattices, as a function of size M, of the rate of excitation migration W, and of the rate of excitation annihilation V in the diffusion-influenced limit (V ≫ W). It is shown that a sum of two exponentials is usually sufficient for fitting experimental phosphorescence and triplet-triplet absorption decays. The first term describes the decay of domains containing initially one triplet, while the second one reflects the disappearance of domains containing initially two triplets. Monte Carlo calculations were carried out to compute the survival probability of an annihilating pair of triplets, yielding expressions for the dependence of the rate constant of TTA on the parameters M, W, and V in one, two, and three dimensions. The text was submitted by the authors in English.     

Numerical investigation of the accuracy of particle image velocimetry technique in gas-phase detonations

We numerically investigate the accuracy of the Particle Image Velocimetry (PIV) technique for the flow characterization in high-speed, compressible regimes, in particular in gas-phase detonations. We carry out synthetic PIV reconstruction of the flow field in a two-dimensional, planar detonation propagating under atmospheric conditions and modelled using single-step Arrhenius kinetics. The flow is uniformly seeded with monodispersed Al₂O₃ particles with sizes 50 and 200 nm, along with initially co-located massless Lagrangian tracer particles. The effect of massive particles on the detonation speed and thermodynamic state of the flow is investigated and is found to be negligible. We further assess the ability of massive particles to sample the flow field and while it is found that 50 nm particles sample the flow field better than the 200 nm ones, they also exhibit significant clustering. By comparing the trajectories of massive particles with those of massless tracers, it is shown that almost all massive particles rapidly diverge from the actual flow pathlines. Finally, we quantify the accuracy of the PIV reconstruction of the velocity field in comparison with the actual velocity field in the numerical simulations. It is shown that while PIV is generally capable of capturing the bulk flow features in the streamwise direction, its accuracy is not sufficient to characterize the transverse velocity component or velocity fluctuations.     

径向渡越时间振荡器的数值模拟

 用2.5维PIC程序对径向渡越时间振荡器进行了数值模拟，给出了产生微波的详细物理图像，得出了输出微波功率与提取口大小、腔的径向间距、场模式之间的关系。模拟得到了峰值功率约500MW，频率5GHz的TEM₁波，起振时间15ns，峰值效率大于30%。     

Numerical simulations of stochastic differential equations

A simple, very accurate algorithm for numerical simulation of stochastic differential equations is described. Its relationship to colored noise is elucidated and exhibited by explicit results. The especially delicate problem of mean first passage times is highlighted and highly accurate agreement between the numerical simulations and analytic results are shown.     

Numerical simulations of time-resolved quantum electronics

Numerical simulation has become a major tool in quantum electronics both for fundamental and applied purposes. While for a long time those simulations focused on stationary properties (e.g. DC currents), the recent experimental trend toward GHz frequencies and beyond has triggered a new interest for handling time-dependent perturbations. As the experimental frequencies get higher, it becomes possible to conceive experiments which are both time-resolved and fast enough to probe the internal quantum dynamics of the system. This paper discusses the technical aspects–mathematical and numerical–associated with the numerical simulations of such a setup in the time domain (i.e. beyond the single-frequency AC limit). After a short review of the state of the art, we develop a theoretical framework for the calculation of time-resolved observables in a general multiterminal system subject to an arbitrary time-dependent perturbation (oscillating electrostatic gates, voltage pulses, time-varying magnetic fields, etc.) The approach is mathematically equivalent to (i) the time-dependent scattering formalism, (ii) the time-resolved non-equilibrium Green’s function (NEGF) formalism and (iii) the partition-free approach. The central object of our theory is a wave function that obeys a simple Schrödinger equation with an additional source term that accounts for the electrons injected from the electrodes. The time-resolved observables (current, density, etc.) and the (inelastic) scattering matrix are simply expressed in terms of this wave function. We use our approach to develop a numerical technique for simulating time-resolved quantum transport. We find that the use of this wave function is advantageous for numerical simulations resulting in a speed up of many orders of magnitude with respect to the direct integration of NEGF equations. Our technique allows one to simulate realistic situations beyond simple models, a subject that was until now beyond the simulation capabilities of available approaches.     

径向渡越时间振荡器的数值模拟

用2.5维PIC程序对径向渡越时间振荡器进行了数值模拟,给出了产生微波的详细物理图像,得出了输出微波功率与提取口大小、腔的径向间距、场模式之间的关系。模拟得到了峰值功率约500MW,频率5GHz的TEM1波,起振时间15ns,峰值效率大于30%。     

Soft X-ray fluorescence from particulate media: Numerical simulations

We introduce a Monte Carlo ray tracing code for soft X-ray fluorescence in particulate media. We use the code to investigate the observation geometry dependent effects on absolute fluorescence line intensities and relative line ratios due to the medium porosity, incident spectrum, and particle size distribution. In particular, we assess the differences between the results given by the simulations and by the analytical fundamental parameters equation of X-ray fluorescence.     

Numerical simulations of random sound waves

We perform one-dimensional numerical simulations of both driven and impulsively generated sound waves propagating through a medium whose mass density admits time-independent, random fluctuations. While the amplitude of both types of wave is always attenuated, driven sound waves can be either retarded or speeded up depending on their wavenumber and amplitude and on the strength of the random field. The speed of a pulse propagating in the random medium is also altered, in agreement with the findings for the driven waves. The concomitant action of nonlinearity and randomness results in wave speeding for wavenumbers which are of the order of the size of an average random density fluctuation, whereas it gives retardation for larger wavenumbers.