Compressible Aerodynamics Calculator

A simple tool to calculate typical compressible aerodynamic properties for isentropic flow, normal shocks, and oblique shocks.

Disclaimer

This calculator is for educational and curiosity purposes only. No gaurantee on accuracy is inferred by a result.

Isentropic Flow Relations (for a Calorically Perfect Gas)

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Normal Shock Relations (for a Calorically Perfect Gas)

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Oblique Shock Relations (for a Calorically Perfect Gas)

M1 =
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Nomenclature
\(A\) Throat area
\(A^*\) Throat area at sonic conditions
\(M\) Mach number
\(M_1\) Mach number upstream of shock wave
\(M_2\) Mach number downstream of shock wave
\(M_{n,1}\) Normal component of the Mach number upstream of shock wave
\(M_{n,2}\) Normal component of the Mach number downstream of shock wave
\(p\) Static pressure
\(p^*\) Static pressure at sonic conditions
\(p_1\) Static pressure upstream of shock wave
\(p_2\) Static pressure downstream of shock wave
\(p_0\) Total pressure
\(p_{0,1}\) Total pressure upstream of shock wave
\(p_{0,2}\) Total pressure downstream of shock wave
\(T\) Temperature
\(T_0\) Total temperature
\(T^*\) Temperature at sonic conditions
\(T_1\) Temperature upstream of shock wave
\(T_2\) Temperature downstream of shock wave
\(\beta\) Wave angle
\(\theta\) Deflection angle
\(\mu\) Mach angle
\(\nu\) Prandtl-Meyer function

Compressible Aerodynamics Solver Notes

All equations used in the above calculations can be found in Ref. 1. Below are the listed equations.

Isentropic Flow Equations
\[ \mu = \sin^{-1} \frac{1}{M} \]
(1)
\[ \nu = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \tan^{-1} \left( \sqrt{\frac{\gamma - 1}{\gamma + 1} (M^2 - 1)} \right) - \tan^{-1} (\sqrt{M^2 - 1}) \]
(2)
\[ \frac{p_0}{p} = \left( 1 + \frac{\gamma - 1}{2}M^2 \right)^{\gamma/(\gamma-1)} \]
(3)
\[ \frac{T_0}{T} = 1 + \frac{\gamma - 1}{2}M^2 \]
(4)
\[ \frac{\rho_0}{\rho} = \left( 1 + \frac{\gamma - 1}{2}M^2 \right)^{1/(\gamma-1)} \]
(5)
\[ \frac{p}{p^*} = \left[ \frac{2 + M^2(\gamma - 1)}{\gamma + 1} \right]^{-\gamma/(\gamma-1)} \]
(6)
\[ \frac{T}{T^*} = \frac{\gamma + 1}{2 + M^2(\gamma - 1)} \]
(7)
\[ \frac{\rho}{\rho^*} = \left[ \frac{2 + M^2(\gamma - 1)}{\gamma + 1} \right]^{-1/(\gamma-1)} \]
(8)
\[ \left( \frac{A}{A^*} \right)^2 = \frac{1}{M^2} \left[ \frac{2}{\gamma + 1} \left( 1 + \frac{\gamma - 1}{2}M^2 \right) \right]^{(\gamma + 1)/(\gamma - 1)} \]
(9)
Normal Shock Equations
\[ M_2^2 = \frac{1 + \left( \frac{\gamma - 1}{2} \right)M_1^2}{\gamma M_1^2 - \left( \frac{\gamma - 1}{2} \right)} \]
(10)
\[ \frac{\rho_2}{\rho_1} = \frac{(\gamma + 1) M_1^2}{2 + (\gamma - 1) M_1^2} \]
(11)
\[ \frac{T_2}{T_1} = \left[ 1 + \frac{2 \gamma}{\gamma + 1} (M_1^2 - 1) \right] \left[ \frac{2 + (\gamma - 1) M_1^2}{(\gamma + 1) M_1^2} \right] \]
(12)
\[ \frac{p_2}{p_1} = 1 + \frac{2 \gamma}{\gamma + 1} (M_1^2 - 1) \]
(13)
\[ \frac{p_{0,2}}{p_{0,1}} = \left[ \frac{1 + \left( \frac{\gamma - 1}{2} \right) M_2^2}{1 + \left( \frac{\gamma - 1}{2} \right) M_1^2} \right]^{\gamma / (\gamma - 1)} \left[ 1 + \frac{2 \gamma}{\gamma + 1}(M_1^2 - 1) \right] \]
(14)

Note: \( M_2 \) in the above equation can be solved using Eq. 10.

\[ \frac{p_{0,2}}{p_1} = \left[ \frac{(\gamma + 1)^2 M_1^2}{4 \gamma M_1^2 - 2( \gamma - 1)} \right]^{\gamma / (\gamma - 1)} \left[ \frac{1 - \gamma + 2 \gamma M_1^2}{\gamma + 1} \right] \]
(15)
Oblique Shock Equations

The relationship between deflection angle, \( \theta \), wave angle, \( \beta \), and upstream Mach number, \( M_1 \), are related by Eq. 16, shown below. As is apparent by using the calculator, the upstream Mach number and either the deflection angle or wave angle must be defined.

\[ \tan \theta = 2 \cot \beta \frac{M_1^2 \sin^2 \beta - 1}{M_1^2 (\gamma + \cos 2 \beta) + 2} \]
(16)
\[ M_{n,1} = M_1 \sin \beta \]
(17)
\[ M_{n,2} = \sqrt{ \frac{1 + [(\gamma - 1)/2] M_{n,1}^2}{\gamma M_{n,1}^2 - (\gamma - 1)/2} } \]
(18)
\[ M_2 = \frac{M_{n,2}}{\sin(\beta - \theta)} \]
(19)
\[ \frac{p_2}{p_1} = 1 + \left( \frac{2 \gamma}{\gamma + 1} \right) (M_{n,1}^2 - 1) \]
(20)
\[ \frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_{n,1}^2}{2 + (\gamma - 1)M_{n,1}^2} \]
(21)
\[ \frac{T_2}{T_1} = \frac{p_2}{p_1} \frac{\rho_1}{\rho_2} \]
(22)

The calculation for \( p_{0,2}/p_{0,1} \) is performed in the same manner as in the normal shock calculation, except the normal component of the upstream Mach number, \( M_{n,1} \), is used.

Miscellaneous Notes

I am aware of a bug in some mobile browsers that do not render the html symbols I use to create the subscripts within the dropdown selectors for Desired Input. In my experience, the subscripts will render if you tap on the selector as if you were going to change the Desired Input.