Accurate Spatial Resolution Estimates for Reactive Supersonic Flow with Detailed Chemistry

Abstract

A robust 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 dY i /dx = f i (Y 1 ,. .. , Y N-L). These equations are numerically integrated from a shock to an equilibrium end state. The eigenvalues of the Jacobian of f i 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.

Verified and validated calculation of unsteady dynamics of viscous hydrogen–air detonations

“…The relevant length is the induction zone, that is, the region between the lead shock and the point where significant chemical reaction commences. Good estimates for a similar mixture are given by Powers & Paolucci (2005). The reaction kinetics are such that the induction zone length ind 10 −2 cm.…”

“…The detailed kinetics mechanism employed is drawn from Miller et al (1982) and is used by Powers & Paolucci (2005). It contains 9 species, 3 elements and 19 reversible reactions where nitrogen is treated as a non-reacting species and is shown in table 1.…”

“…In detonations propagating into mixtures at atmospheric pressure, there are reaction dynamics and gradients in flow variables evolving on scales near a micrometre or smaller (Powers & Paolucci 2005). For a steady, travelling, inviscid, Chapman-Jouguet (CJ) hydrogen-air detonation propagating into ambient atmospheric conditions, the reaction length scales range from 10 −5 to 10 0 cm at equilibrium, and near the detonation front this breadth of scales can be even wider.…”

Detonation in hydrogen–nitrous oxide–diluent mixtures: An experimental and numerical study

“…A more realistic simulation of fine structure of the detonation flow field and reaction zone requires using a viscous model, Navier-Stokes equations, rather than the inviscid Euler model [62]. Powers and Paolucci [60] concluded from 1D simulation results that a submicron mesh would be needed to spatially resolve the smallest gradients for a detonation in a H 2 -air mixture at P 1 =101 kPa and T 1 =298 K. Because the computational time increases with the number of species, Ni, as 2 N i , such a high resolution is extraordinarily difficult to achieve for 2D simulation when including realistic chemistry [63]. Moreover, it is unclear if this level of resolution is helpful for two-dimensional simulation as spatial graidents and diffusive fluxes are most important wihtin regions of high shear, which require either sub-grid scale models ot three-dimensional simulations.…”

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

“…However, the techniques described here can, in principle, be adapted to models which better represent actual physical systems with detailed kinetics and complex state equations [2]. Nevertheless, the widely disparate length scales present in such systems pose such a formidable modeling challenge that it is unlikely that a fully resolved unsteady detonation for a system with detailed kinetics exists as of yet [3].…”