Chapman–Jouguet condition

Source:^[11]

The Rayleigh line equation and the Hugoniot curve equation obtained from the Rankine–Hugoniot relations for an ideal gas, with the assumption of constant specific heat and constant molecular weight, respectively are

${\displaystyle {\begin{aligned}{\frac {{\tilde {p}}-1}{{\tilde {v}}-1}}&=-\mu \\{\tilde {p}}&={\frac {[2\alpha +(\gamma +1)/(\gamma -1)]-{\tilde {v}}}{[(\gamma +1)/(\gamma -1)]{\tilde {v}}-1}},\end{aligned}}}$ 

where ${\displaystyle \gamma }$  is the specific heat ratio and

${\displaystyle {\tilde {p}}={\frac {p_{2}}{p_{1}}},\quad {\tilde {v}}={\frac {\rho _{1}}{\rho _{2}}},\quad \alpha ={\frac {q\rho _{1}}{p_{1}}},\quad \mu ={\frac {m^{2}}{p_{1}\rho _{1}}}.}$ 

Here the subscript 1 and 2 identifies flow properties (pressure ${\displaystyle p}$ , density ${\displaystyle \rho }$ ) upstream and downstream of the wave and ${\displaystyle m}$  is the constant mass flux and ${\displaystyle q}$  is the heat released in the wave. The slopes of Rayleigh line and Hugoniot curve are

${\displaystyle {\begin{aligned}{\frac {d{\tilde {p}}}{d{\tilde {v}}}}&={\frac {{\tilde {p}}-1}{{\tilde {v}}-1}},\\[3pt]{\frac {d{\tilde {p}}}{d{\tilde {v}}}}&=-{\frac {[(\gamma +1)/(\gamma -1)]{\tilde {p}}+1}{[(\gamma +1)/(\gamma -1)]{\tilde {v}}-1}}.\end{aligned}}}$ ⋅

At the Chapman-Jouguet point, both slopes are equal, leading the condition that

${\displaystyle {\tilde {p}}={\frac {\tilde {v}}{(\gamma +1){\tilde {v}}-\gamma }}.}$ 

Substituting this back into the Rayleigh equation, we find

${\displaystyle \mu =\gamma {\frac {\tilde {p}}{\tilde {v}}}.}$ 

Using the definition of mass flux ${\displaystyle m\equiv \rho _{1}u_{1}=\rho _{2}u_{2}}$ , where ${\displaystyle u}$  denotes the flow velocity, we find

${\displaystyle M_{2}={\frac {u_{2}}{c_{2}}}=1}$ 

where ${\displaystyle M}$  is the Mach number and ${\displaystyle c}$  is the speed of sound, in other words, downstream flow is sonic with respect to the Chapman-Jouguet wave. Explicit expression for the variables can be derived,

${\displaystyle {\begin{aligned}{\tilde {p}}_{\pm }&=1+\alpha (\gamma -1)\left\{1\pm \left[1+{\frac {2\gamma }{\alpha \left(\gamma ^{2}-1\right)}}\right]^{\frac {1}{2}}\right\},\\{\tilde {v}}_{\pm }&=1+{\frac {\alpha \left(\gamma -1\right)}{\gamma }}\left\{1\mp \left[1+{\frac {2\gamma }{\alpha \left(\gamma ^{2}-1\right)}}\right]^{\frac {1}{2}}\right\},\\\mu _{\pm }&=\gamma +\alpha (\gamma ^{2}-1)\left\{1\pm \left[1+{\frac {2\gamma }{\alpha \left(\gamma ^{2}-1\right)}}\right]^{\frac {1}{2}}\right\}.\end{aligned}}}$ 

The upper sign applies for the Upper Chapman-Jouguet point (detonation) and the lower sign applies for the Lower Chapman-Jouguet point (deflagration). Similarly, the upstream Mach number can be found from

${\displaystyle M_{1\pm }=\left[1+{\frac {\alpha \left(\gamma ^{2}-1\right)}{2\gamma }}\right]^{\frac {1}{2}}\pm \left[{\frac {\alpha \left(\gamma ^{2}-1\right)}{2\gamma }}\right]^{\frac {1}{2}}}$ 

and the temperature ratio ${\displaystyle {\tilde {T}}=T_{2}/T_{1}}$  can be found from the relation ${\displaystyle {\tilde {T}}={\tilde {p}}{\tilde {v}}}$ .

• Taylor–von Neumann–Sedov blast wave
• Zeldovich–Taylor flow

• Bowen, E. J. (1958). "David Leonard Chapman 1869–1958". Biographical Memoirs of Fellows of the Royal Society. 4: 34–44. doi:10.1098/rsbm.1958.0004.
• Jacques Charles Emile Jouguet (1871–1943) (in French), annales.org
• Ginzburg, V. L. (1994). "Yakov Borissovich Zel'dovich. 8 March 1914 – 2 December 1987". Biographical Memoirs of Fellows of the Royal Society. 40: 430–441. doi:10.1098/rsbm.1994.0049.