Chapman–Jouguet detonation waves

Introduction

Chapman–Jouguet (CJ) detonation waves are computed by solving the Rankine–Hugoniot relations across a planar detonation front in a reactive gas mixture. As illustrated in Fig. 13, a detonation wave consists of a leading shock followed by a tightly coupled reaction zone, in which the gas is compressed, heated, and chemically transformed.

Fig. 13 Schematic of a planar detonation wave in the wave-fixed frame assuming the Chapman–Jouguet criterion.

The upstream state is characterized by the velocity \(u_1\), temperature \(T_1\), pressure \(p_1\), and chemical composition \(\boldsymbol{n}_1\). The flow undergoes an abrupt thermodynamic transformation across the wave, driven by shock compression and chemical energy release. The downstream state \((u_2, p_2, T_2, \boldsymbol{n}_2)\) is obtained by solving the conservation equations for mass, momentum, and energy, assuming the products are in chemical equilibrium.

The Chapman–Jouguet condition imposes that the flow becomes sonic relative to the detonation front immediately behind the wave, i.e., \(u_2 = a_2\). This condition selects the unique point where the Rayleigh line is tangent to the equilibrium Hugoniot curve (see Fig. 14), defining the minimum velocity for a self-sustaining detonation in chemical equilibrium.

Fig. 14 Family of admissible solutions defined by the Rankine–Hugoniot and Rayleigh relations for a planar, steady detonation wave.

Governing equations

For a one-dimensional planar shock, the Rankine–Hugoniot relations express the conservation of mass, momentum, and energy across the shock front as:

\begin{equation} p_2 = p_1 + \rho_1 u_1^2 \left( 1-\dfrac{\rho_1}{\rho_2}\right) \quad \text{and} \quad h_2 = h_1 + \dfrac{u_1^2}{2}\left[1- \left(\dfrac{\rho_1}{\rho_2}\right)^2\right], \end{equation}

where \(h\) denotes specific enthalpy. These equations must be closed with the equation of state, which for an ideal gas reads:

\begin{equation} p = \rho R T / W, \end{equation}

where \(R\) is the universal gas constant and \(W\) is the molecular weight of the gas.

Numerical example

We now illustrate how to solve the Rankine–Hugoniot equations in the Combustion Toolbox using the class DetonationSolver() class, part of the +combustiontoolbox.+shockdetonation (CT-SD) subpackage (module). Below is an example for methane–air mixtures at standard temperature and pressure, over a range of equivalence ratios \(\phi \in [0.5, 4.0]\):

% Import packages
import combustiontoolbox.databases.NasaDatabase
import combustiontoolbox.core.*
import combustiontoolbox.shockdetonation.*

% Get Nasa database
DB = NasaDatabase();

% Define chemical system
system = ChemicalSystem(DB);

% Initialize mixture
mix = Mixture(system);

% Define chemical state
set(mix, {'CH4'}, 'fuel', 1);
set(mix, {'N2', 'O2'}, 'oxidizer', [79/21, 1]);

% Define properties
mixArray1 = setProperties(mix, 'temperature', 300, 'pressure', 1, 'equivalenceRatio', 0.5:0.01:4);

% Initialize solver
solver = DetonationSolver('problemType', 'DET');

% Solve problem
[mixArray1, mixArray2] = solver.solveArray(mixArray1);

% Generate report
report(solver, mixArray1, mixArray2);

This script generates two diagnostic figures: one for the species molar fractions in the detonation products as a function of equivalence ratio, and another for the thermodynamic and flow properties across the wave.

Fig. 15 Molar fraction of chemical species downstream of a Chapman–Jouguet detonation in methane–air mixtures as a function of equivalence ratio \(\phi\), at \(T_1 = 300\) K and \(p_1 = 1\) bar.

Fig. 16 Thermodynamic and flow properties downstream of a Chapman–Jouguet detonation in methane–air mixtures as a function of equivalence ratio \(\phi\), at \(T_1 = 300\) K and \(p_1 = 1\) bar.