^{1,a)}, Antonio L. Sánchez

^{1,2}and Forman A. Williams

^{2}

### Abstract

We present the exact small-amplitude linear Laplace-transform theory describing the propagation of an initially planar detonation front through a gaseous mixture with nonuniform density perturbations, complementing earlier normal-mode results for nonuniform velocity perturbations. The investigation considers the fast-reaction limit in which the detonation thickness is much smaller than the size of the density perturbations, so that the detonation can be treated as an infinitesimally thin front with associated jump conditions given by the Rankine-Hugoniot equations. The analytical development gives the exact transient evolution of the detonation front and the associated disturbance patterns generated behind for a single-mode density field, including explicit expressions for the distributions of density, pressure, and velocity. The results are then used in a Fourier analysis of the detonation interaction with two-dimensional and three-dimensional isotropic density fields to provide integral formulas for the kinetic energy, enstrophy, and density amplification. Dependencies of the solution on the heat-release parameter and propagation Mach number are discussed, along with differences and similarities with results of previous analyses for non-reacting shock waves.

This work was supported by the U.S. AFOSR Grant No. FA9550-12-1-0138 and by the Spanish MCINN through the Program CONSOLIDER-Ingenio2010 (Project No. CSD2010-00011).

I. INTRODUCTION

II. FORMULATION OF THE PERTURBATION PROBLEM FOR INTERACTIONS WITH MONOCHROMATIC DENSITY FIELDS

III. SUMMARY OF THE RESULTS OF THE EXACT SOLUTION WITH THE MONOCHROMATIC DENSITY FIELD

IV. EXPANSION IN THE STRONG-DETONATION LIMIT

V. REPRESENTATIVE TIME-DEPENDENT RESULTS FOR MONOCHROMATIC INTERACTIONS

VI. LONG-TIME RESULTS FOR MONOCHROMATIC INTERACTIONS

VII. DETONATION RESPONSE TO ISOTROPIC DENSITY DISTURBANCES

VIII. CONCLUDING REMARKS

### Key Topics

- Mach numbers
- 28.0
- Vortex dynamics
- 26.0
- Shock waves
- 20.0
- Acoustic distortion
- 14.0
- Shock wave effects
- 12.0

## Figures

A planar thin detonation front travels with velocity D through a nonuniform mono-modulated density field. A set of vortical, entropic, and acoustic perturbations is generated downstream.

A planar thin detonation front travels with velocity D through a nonuniform mono-modulated density field. A set of vortical, entropic, and acoustic perturbations is generated downstream.

Detonation front evolution for k x /k y = 0.3357 (left-hand-side plot) and k x /k y = 0.5035 (right-hand-side plot) as obtained with γ = 1.2, M = 10, and q = 2 from (A13) (solid curves) and from (23b) (dashed curves).

The variation with the incident angle θ = tan −1(k y /k x ) of the amplitudes of the permanent solution for γ = 1.2, M = 10, and q = (0, 1, 2, 5, 10), with the grey tone used for each curve progressing from darkest (q = 0) to lightest (q = 10) as the exothermicity increases.

The variation with the incident angle θ = tan −1(k y /k x ) of the amplitudes of the permanent solution for γ = 1.2, M = 10, and q = (0, 1, 2, 5, 10), with the grey tone used for each curve progressing from darkest (q = 0) to lightest (q = 10) as the exothermicity increases.

The variation with the propagation Mach number M of the peak amplitude of the permanent solution at ζ0 = 1 for γ = 1.2 and q = (0, 1, 2, 5, 10), with the grey tone used for each curve progressing from darkest (q = 0) to lightest (q = 10) as the exothermicity increases and with dashed and solid curves employed for M/M cj < 3 and M/M cj > 3, respectively.

The variation with the propagation Mach number M of the peak amplitude of the permanent solution at ζ0 = 1 for γ = 1.2 and q = (0, 1, 2, 5, 10), with the grey tone used for each curve progressing from darkest (q = 0) to lightest (q = 10) as the exothermicity increases and with dashed and solid curves employed for M/M cj < 3 and M/M cj > 3, respectively.

Detonation response to isotropic density disturbances as a function of the Mach number for γ = 1.2 and q = (0, 1, 2, 5, 10), with the grey tone used for each curve progressing from darkest (q = 0) to lightest (q = 10) as the exothermicity increases.

Detonation response to isotropic density disturbances as a function of the Mach number for γ = 1.2 and q = (0, 1, 2, 5, 10), with the grey tone used for each curve progressing from darkest (q = 0) to lightest (q = 10) as the exothermicity increases.

The integration domain in the (x, τ) plane including the detonation trajectory x = M d τ, the characteristics lines (dashed lines), and the particle paths (thick dotted lines) with indication of regions of perturbed and unperturbed flow.

The integration domain in the (x, τ) plane including the detonation trajectory x = M d τ, the characteristics lines (dashed lines), and the particle paths (thick dotted lines) with indication of regions of perturbed and unperturbed flow.

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