Engineering
Supersonic Normal Shock Wave: Downstream Mach & Rankine-Hugoniot Jumps
Calculate supersonic normal shock downstream Mach number M_2 = SQRT((1 + ((g-1)/2)*M1^2) / (g*M1^2 - (g-1)/2)) and static pressure jump P2/P1 = 1 + (2*g/(g+1))*(M1^2 - 1) for compressible gas flows.
Size supersonic jet inlets, Pitot tubes, and rocket nozzle overexpansion shocks!
Across a stationary planar Normal Shock Wave, supersonic flow (upstream Mach M_1 > 1) decelerates discontinuously into subsonic flow (M_2 < 1) across a thin viscous front a few mean-free-paths thick! By the Second Law of Thermodynamics, entropy increases across the shock ONLY when M_1 >= 1—expansion normal shocks (M_1 < 1) are physically impossible!
For an ideal gas of specific heat ratio gamma (1.40 for diatomic air), the Rankine-Hugoniot jump relations give exact downstream Mach number M_2 = SQRT((1 + ((gamma - 1) / 2) * M_1^2) / (gamma * M_1^2 - ((gamma - 1) / 2))) and static pressure ratio P_2 / P_1 = 1 + ((2 * gamma) / (gamma + 1)) * (M_1^2 - 1): at Mach 2.0 in air, flow drops instantaneously to M_2 = 0.5774 while static pressure jumps 4.50x!
| Upstream Mach M_1 (Air gamma = 1.40) | Downstream Subsonic M_2 | Static Pressure Jump P_2 / P_1 | Density Jump rho_2 / rho_1 |
|---|---|---|---|
| Sonic Limit (M_1 = 1.00) | 1.0000 | 1.00x (Acoustic Wave) | 1.000x |
| Supersonic Cruise (M_1 = 2.00) | 0.5774 | 4.50x | 2.667x |
| High Supersonic (M_1 = 3.00) | 0.4752 | 10.33x | 3.857x |
Downstream Subsonic Mach Number M_2 Across a Normal Shock
Enforces supersonic upstream Mach M_1 >= 1 (B2) and gamma > 1 (B3), then evaluates downstream M_2.
=IF(B2 < 1, "Subsonic Flow (Normal Shock Requires M1 >= 1)", IF(B3 <= 1, "Invalid Gamma <= 1", ROUND(SQRT((1 + ((B3 - 1) / 2) * (B2 ^ 2)) / ((B3 * (B2 ^ 2)) - ((B3 - 1) / 2))), 4)))Rankine-Hugoniot Static Pressure Ratio P_2 / P_1
Computes 1 + ((2 * gamma) / (gamma + 1)) * (M_1^2 - 1).
=IF(B2 < 1, "Subsonic Flow (No Normal Shock)", ROUND(1 + ((2 * B3) / (B3 + 1)) * ((B2 ^ 2) - 1), 2))Try the example
Use the original values or change them to test your understanding.