Isentropic flow functions (ideal gas)
\( \displaystyle
\frac{T}{T_0}=\Big(1+\frac{k-1}{2}\mathrm{Ma}^{2}\Big)^{-1}, \quad
\frac{p}{p_0}=\Big(1+\frac{k-1}{2}\mathrm{Ma}^{2}\Big)^{-\frac{k}{k-1}}.
\)
\( \displaystyle
\frac{\rho}{\rho_0}=\Big(1+\frac{k-1}{2}\mathrm{Ma}^{2}\Big)^{-\frac{1}{k-1}}, \quad
\frac{A}{A^{*}}=\frac{1}{\mathrm{Ma}}
\left[\frac{2}{k+1}\left(1+\frac{k-1}{2}\mathrm{Ma}^{2}\right)\right]^{\frac{k+1}{2(k-1)}}.
\)
Symbols
- Ma — Mach number (flow speed / local speed of sound)
- k — specific heat ratio (Cp/Cv)
- T, p, ρ — local static temperature, pressure, density
- T₀, p₀, ρ₀ — stagnation (total) values
- A — area at the location where the Mach number is Ma
- A* — area that would correspond to Ma = 1 (critical/choked area)
Bernoulli equation for incompressible flow is typically applicable when \( \mathrm{Ma} \lesssim 0.28 \) (≈ 2% pressure error).
T/T₀
—
p/p₀
—
ρ/ρ₀
—
A/A*
—
T/T₀
p/p₀
ρ/ρ₀
A/A*
Curves show static-to-stagnation ratios vs Ma for the chosen k. Drag anywhere across the plot to change Ma.