Equation Map - Rayleigh

rayleigh_pressure_ratio

$\frac{p_{2}}{p_{1}} = \frac{ 1 + \gamma M_{1}^2}{ 1 + \gamma M_{2}^2}$

rayleigh_temperature_ratio

$\frac{T_{2}}{T_{1}} = \left( \frac{ 1 + \gamma M_{1}^2}{ 1 + \gamma M_{2}^2} \right) ^2 \frac{M_{2}^2}{M_{1}^2}$

rayleigh_density_ratio

$\frac{\rho_{2}}{\rho_{1}} = \frac{M_{1}^2}{M_{2}^2} \left( \frac{ 1 + \gamma M_{2}^2}{ 1 + \gamma M_{1}^2} \right)$

rayleigh_stagnation_temperature_ratio

$\frac{T_{t2}}{T_{t1}} = \left( \frac{ 1 + \gamma M_{1}^2}{ 1 + \gamma M_{2}^2} \right) ^2 \frac{M_{2}^2}{M_{1}^2} \left(\frac{1 + \left[ (\gamma-1)/2 \right] M_{2}^2 }{ 1 + \left[ (\gamma-1)/2 \right] M_{1}^2 } \right)$

rayleigh_stagnation_pressure_ratio

$\frac{p_{t2}}{p_{t1}} = \frac{ 1 + \gamma M_{1}^2}{ 1 + \gamma M_{2}^2} \left(\frac{1 + \left[ (\gamma-1)/2 \right] M_{2}^2 }{ 1 + \left[ (\gamma-1)/2 \right] M_{1}^2 } \right) ^ {\gamma/(\gamma-1)}$

rayleigh_mach_from_pressure_ratio

$M_{2} = \left[ \left( \frac{p_{2}}{p_{1}} \left( 1+\gamma M_{1}^2 \right) - 1 \right) \frac{1}{\gamma} \right]^{1/2}$

rayleigh_mach_from_temperature_ratio , rayleigh_mach_from_stagnation_temperature_ratio , rayleigh_mach_from_stagnation_pressure_ratio all use equation solvers to get the desired result.

rayleigh_pressure_star_ratio

$\frac{p}{p^*} = \frac{1+\gamma}{1+\gamma M^2}$

rayleigh_temperature_star_ratio

$\frac{T}{T^*} = \frac{M^2 (1+\gamma)^2}{ (1+\gamma M^2)^2 }$

rayleigh_density_star_ratio

$\frac{\rho}{\rho^*} = \frac{1+\gamma M^2}{ (1+\gamma) M^2}$

rayleigh_stagnation_temperature_star_ratio

$\frac{T_{t}}{T_{t}^*} = \frac{2 (1+\gamma)^2 M^2}{ (1+\gamma M^2)^2 } \left( 1+ \frac{\gamma-1}{2}M^2\right)$

rayleigh_stagnation_pressure_star_ratio

$\frac{p_{t}}{p_{t}^*} = \frac{1+\gamma}{1+\gamma M^2} \left( \frac{1+ \left[ (\gamma -1)/2 \right] M^2 }{ (\gamma +1 )/ 2} \right) ^ {\frac{\gamma}{\gamma -1}}$

rayleigh_heat_flux

$q = c_{p} (T_{t2} - T_{t1})$