Normal Shock: Downstream State and Pressure Loss

Method & assumptions

1. Stationary, one-dimensional normal shock; calorically perfect ideal gas, constant R and γ. Upstream Mach must exceed 1 for a shock; exactly 1 is the zero-strength limit. Inputs use the shock-fixed frame, absolute pressure and Kelvin.
2. With stations 1 upstream and 2 downstream: p₂/p₁ = 1 + 2γ(M₁² − 1)/(γ + 1); ρ₂/ρ₁ = (γ + 1)M₁²/[(γ − 1)M₁² + 2]; T₂/T₁ = (p₂/p₁)/(ρ₂/ρ₁).
3. M₂² = [(γ − 1)M₁² + 2]/[2γM₁² − (γ − 1)]. Mass conservation gives V₂/V₁ = ρ₁/ρ₂. Total temperature is conserved; total pressure falls and entropy rises.
4. p₀₂/p₀₁ = [(ρ₂/ρ₁)^γ/(p₂/p₁)]^[1/(γ − 1)]; Δs = −R ln(p₀₂/p₀₁). Extremely weak shocks may display zero loss at floating-point precision.
5. No oblique shock, boundary-layer interaction, nozzle location, moving-shock frame conversion, heat transfer, real gas or variable heat capacity. This tool does not determine whether a shock forms. Check the constant-property assumption against the calculated temperatures.

Sources

• NASA Glenn: Normal Shock Wave Equations ↗