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Aerospace
Density compression across a normal shock
Ratio of density behind a normal shock to that ahead of it.
- Ref
- AER-40
- Rev
- v1
- Field
- Aerospace
- Result
- ρ2/ρ1
Mass conservation across the shock gives ρ1V1 = ρ2V2, so the density ratio equals the velocity ratio V1/V2 = (γ+1)M1²/[2 + (γ−1)M1²]. It rises with M1 but approaches a finite limit of (γ+1)/(γ−1), which is 6 for air with γ = 1.4. Even a very strong shock cannot compress air by more than about six times in density, because most of the energy goes into temperature.
Ratio of density behind a normal shock to that ahead of it. Mass conservation makes the velocity ratio V1/V2 equal to this value. For fixed γ the compression approaches a finite strong-shock limit of (γ + 1)/(γ − 1), which is 6 for air.
Calculator
- Velocity ratio V₂/V₁0.375
- Strong-shock limit, (γ + 1)/(γ − 1)6
- Share of the limit reached0.4444
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The gas is compressed to 2.6667 times its upstream density and slowed to 37.5 % of its upstream speed.
- This is 44.44 % of the strong-shock limit of 6.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| M1 Upstream Mach number | 1.01 to 6 | 1.017 | 5.268 | 4.252 | +10% | +10.7 % |
| γ Heat-capacity ratio | 1.1 to 1.67 | 2.282 | 3.5 | 1.218 | +10% | −8.41 % |
Sweep of Upstream Mach number
| Upstream Mach number | Density ratio |
|---|---|
| 1.01 | 1.017 |
| 1.634 | 2.088 |
| 2.258 | 3.029 |
| 2.881 | 3.745 |
| 3.505 | 4.264 |
| 4.129 | 4.639 |
| 4.753 | 4.913 |
| 5.376 | 5.115 |
| 6 | 5.268 |
Sweep one variable
Variables and how each one contributes
Density ratio
What the equation gives you. Everything else feeds it.
Heat-capacity ratio
The heat-capacity ratio cp/cv of the gas, about 1.4 for air. A lower γ (hotter, more complex gas) changes the compressible-flow exponents and the sizes of the shock and expansion effects. Raising it by 10 % (to 1.54) moves the result by −8.41 %, from 2.667 to 2.442.
Upstream Mach number
The upstream Mach number, above 1. Raises the density ratio, rapidly at first and then toward a limit. Raising it by 10 % (to 2.2) moves the result by +10.7 %, from 2.667 to 2.951.
Worked example with the values above
- Numerator: (γ + 1)M1² = 2.4 × 4 = 9.6.
- Denominator: 2 + (γ − 1)M1² = 3.6.
- ρ2/ρ1 = 2.66667.
Assumptions and limits
- A steady normal shock in a calorically perfect gas, with the same γ on both sides and no heat addition.
- Upstream Mach number above 1; the downstream flow is subsonic.
Common mistakes
- Using the upstream Mach number on the downstream side.
- Using total pressure where static pressure is needed, or the reverse.
- Applying normal-shock formulas to the full Mach number of an oblique shock.
Statistics and uncertainty
Normal-shock relations are exact for a perfect gas. At high Mach numbers, dissociation and real-gas effects change the pressure and temperature ratios by several per cent, and weak shocks are close to isentropic, so small pressure losses are sensitive to inputs.
Where it is used
- Estimating shock losses in inlets, nozzles and supersonic tunnels.
- Interpreting pitot readings behind a bow shock.
Maths you need
- Rankine–Hugoniot relations: conservation of mass, momentum and energy across a shock
- Stagnation pressure loss: a measure of the irreversibility of the shock
- Entropy: a measure of irreversibility