Chapter 4 Verification
5.1 Background Research
Numerical studies of irregular detonations in the past have been challenging as de- tailed chemical kinetics alone dramatically increase the required computational ex- pense. Many multi-dimensional simulations to date have included simple one-step chemistry, but progress was made in the late 90s when researchers began to used re- duced detailed chemical kinetic models for the hydrogen-oxygen system as by Oran et al. (1), Inaba and Matsuo (77), and Eckett (49). Simulations in three-dimensions with detailed kinetics in hydrogen-oxygen-argon were first made in 2003 by Deiterding (40) and in 2004 a simplified ethylene-air model by Khoklov (89).
5.1.1 Recent Research on Unsteady Effects of Detonations
Only over the past 10 years have the effects of diffusion (mass, viscous, heat) in detonations been investigated. Diffusive/viscous processs are important in irregular detonations, but only because of their interactions with the highly unstable shock fronts. Because of their complexity and small size, these instabilities have been diffi- cult to observe and quantify in both experiments and simulation. Some recent works which tackle these issues are discussed below.
Lee and Radulescu (99) describes the general contemporary theories for detona- tion propagation and structure. They explain the differences of three-dimensional stable and unstable detonations with the basic historical ZND theory and highlight its limitations. Between transverse wave collisions, the velocity of the leading shock generally fluctuates between 1.6 and 0.7 times the average velocity. They discuss how the highly variable flow is still related to an overall mean wave that moves close to the CJ speed.
Experiments of unstable (irregular) detonations have been carried out by multiple research groups. In Austin et al. (7, 6) experiments were conducted for various mix- tures diluted with Ar (more stable) or N2 (less stable). The regularity or irregularity (stability or instability) of the detonations was investigated with schlieren images and chemiluminescene of OH. Of their mixtures tested, d etonations were found to be most unstable for propane-air. This seems to be the most unstable hydrocarbon studied experimentally in detonations to date. Halouaet al. (62) investigated experimentally unstable gaseous detonation of stoichiometric propane/oxygen mixtures, diluted or not with argon or helium. They identified four modes of unsteady propagation: stable detonation, stuttering mode, galloping mode, and fast flame. A comprehensive study was also undertaken by Pintgenet al.(128). Experiments, 2D Euler simulations, and
nonreactive 3 shock theory were compared for H2/O2/Ar diluted mixtures. Reasons were surmised for the possible existence of unburnt pockets behind the incident shock wave. These unburnt structures known as keystones show up in experiments when the fluorescence of OH is measured. OH is one of the radicals present in hydrocarbon-air reactive zones. Radulescu et al. (135) also experimentally investigated the gas igni- tion mechanism in typical irregular detonations. They have very clear high-resolution experimental photos of the unstable detonations. For systems with high activation energies and a sensitive dependence on temperature fluctuations, large portions of gas escape shock-induced ignition. The ignition of the remaining gas relies on turbulent mixing between burned and unburned gases.
Numerical studies on the unsteady effects of detonations have been conducted by many researchers. Noteable ones are discussed below in chronological order. Gamezo et al. (55,56) used a one-step reaction with different activation energies to study the appearance and nature of unreacted gas pockets downstream of the front, and the oscillation of the center-line velocity. Unstable detonations of ethylene-O2were exam- ined by Khokhlov et al. (89). Their model consisted of the reactive Euler equations with Arrhenius kinetics, comparing the OH concentration to that found in experi- ments. They found the solutions to be sensitive to the adiabatic index (specific heat ratio) and the molecular weight. Higher specific heat ratios lead to higher post shock temperatures and more stability. Radulescuet al. (134) analyzes the cellular reaction zone structure of unstable methane-oxygen detonations, which are characterized by large hydrodynamic fluctuations and unreacted pockets with a fine structure. The quantitative comparison between experiment and numerics also permits identification of the current limitations of numerical simulations in capturing these effects. The flow fields were obtained from numerical solutions of the Favre-averaged Euler equations in time and space. They added artificial diffusion to the Riemann solver to suppress the entropy oscillations. The simulations revealed two important length scales, the first being associated with the chemical exothermicity and the second (the hydrody- namic thickness) with the slower dissipation of the hydrodynamic fluctuations, which govern the location of the average sonic surface. In the paper of Massa et al. (116), they state that three principal phenomena occur in the evolution of the triple-point shear layer: diffusion, the Kelvin-Helmholtz instability, and the auto-ignition of the shocked unreacted stream. They find that the chemical energy release at the transver- sal front is the dominant energetic contribution for high activation energy mixtures.
The shear-layer instability appears to play no role in the formation of localized ex- plosions. They find hot spots occurring in the high and medium activation energy
mixtures near the sonic-transition locus of the initially supersonic unreacted stream.
Here, molecular diffusion heating is responsible for ignition supporting the localized and multidimensional nature of these irregular explosions.
5.1.2 Recent Research on Diffusive, Reactive Navier-Stokes
As discussed in Powers (131), viscosity, be it from physical or numerical errors, can have a large influence on detonation and reactive flow solutions in genernal. Singh et al. (146) considered both viscous and inviscid models and qualitatively measured the influences of both physical and numerical viscosity on two-dimensional detona- tion solutions. Here, the physical viscosity in the Navier-Stokes model was adjusted so that the viscous layers were roughly one-tenth the length of the induction length.
Both the Euler and Navier-Stokes models were subjected to a grid-refinement study.
In the Euler calculations, intrinsic numerical viscosity, which depends on the size of the grid and the details of the particular numerical method chosen, always played a role in the solution downstream of the shock. In the Navier-Stokes calculations at coarse resolutions, the same artificial viscosity dominates the physical viscosity, and the structures depend on the grid resolution. As the grid is refined for the Euler calcu- lations, the artificial viscosity decreases, and fewer downstream instabilities, such as the Kelvin-Helmholtz instability, are suppressed. At coarse resolutions in the Navier- Stokes model predicts similar results as the Euler model. In this case, the inherent numerical viscosity of the method dominates the physical viscosity. However, as the grid is refined in the Navier-Stokes calculations, the physical viscosity dominates, and no finer-scale structures are apparent. We see these same effects in §6.2.
The first works including diffusion processses for a detailed chemistry and trans- port model began with a one-dimensional analysis. Singhet al. (147) used a wavelet based method to efficiently solve the one-dimensional reactive Navier-Stokes equa- tions. In a paper by Arienti and Shepherd (5), simplified zero- and one-dimensional models were used to estimate the role of diffusion in detonations. From the triple points, there exist mixing layers of hot products and relatively colder unreacted reac- tants. The rate of mixing affects the time for which the colder flow ignites, and has more affect on the overall flow field if the detonation is highly unstable (in other words, sensitive or having a high activation energy). The zero-dimensional model assumes instantaneous mixing and the one-dimensional model assumes a one-dimensional lam- inarily diffusing flame. A study by Singh et al. (147) considered a one-dimensional Navier-Stokes model for a detailed H2-O2-Ar mixture and thus resolved shocks of
finite thickness. The viscous layer overlapped some of the finest reaction zone lengths but is distinct from the better understood induction zone. In another paper (145), they also examine the combustion modes possible behind shock waves through solu- tions of the one-dimensional, steady reactive Navier-Stokes (diffusive but nonviscous) equations with a detailed chemical reaction mechanism for stoichiometric methane-air mixtures.
Many researchers have simulated diffusion for two- and three-dimensional deto- nations, yet, almost all have neglected multi-component chemistry and/or have also neglected to resolve the diffusive scales. One work that stands out is that of the NRL group (88), which has modeled DDT (detonation to deflagration transition) in 2D and 3D. In their simulations they have used two-component chemistry. They found DDT to be very sensitive to the specific heat ratio.
The most recent and complex simulations to date that are relevant to this thesis are that of Massaet al. (116). In their two-dimensional simulations, also discussed in
§1.1.6, they neglected the shock waves (which leaves out the main source of detonation instability) and simulated with detailed chemistry the shear layer behind detonation triple points. They investigated the role of vortical structures associated with Kelvin- Helmholtz instability in the formation of localized ignition.