Equation Map - Standard Equations

sonic_velocity

$a = \sqrt{\gamma R T}$

Stagnation Relations and Ratios

stagnation_pressure will return $P$ , $P_{t}$ , or $M$ depending on parameters given .

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

stagnation_temperature will return $T$ , $T_{t}$ , or $M$ depending on parameters given.

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

stagnation_pressure_ratio returns $\frac{P}{P_{t}}$

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

stagnation_temperature_ratio returns $\frac{T}{T_{t}}$

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

stagnation_density_ratio returns $\frac{\rho}{\rho_{t}}$ .

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

Mach Number and Property Relations

mach_from_pressure_ratio solves for $M_{2}$ from the following equation, while pressure_from_mach_ratio will solve for $p_{2}$ .

$\frac{p_{2}}{p_{1}} = \left( \frac{ 1 + \frac{\gamma-1}{2}M_{1}^2}{1 + \frac{\gamma-1}{2}M_{2}^2} \right)^{\frac{\gamma}{\gamma-1}}e^{\frac{-\Delta s}{R}}$

mach_from_temperature_ratio solves for $M_{2}$ from the following equation, while temperature_from_mach_ratio will solve for $T_{2}$ .

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

mach_area_star_ratio returns the ratio of $\frac{A}{A*}$ .

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

mach_area_ratio returns the ratio of $\frac{A_{2}}{A_{1}}$ given two Mach numbers, whi;e mach_from_area_ratio will return the possible mach numbers that satisfy the area ratio.

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

Mass Flux

mass_flux returns the flow rate per unit area while mass_flux_max will return the maximum flow rate per unit area, where $M=1$ .

$\frac{\dot{m}}{A}=M\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{-(\gamma+1)}{2(\gamma-1)}}\sqrt{\left(\frac{\gamma}{R}\right)}\frac{p_{t}}{\sqrt{T_{t}}}$

$\frac{\dot{m}}{A^*} = \sqrt{\frac{\gamma}{R}\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{\gamma-1}}}\frac{p_{t}}{\sqrt{T_{t}}}$