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Aerospace
Stagnation-to-static density ratio
Ratio of the stagnation density to the local density in isentropic flow of a calorically perfect gas with Mach number M and heat-capacity ratio γ > 1.
- Ref
- AER-33
- Rev
- v1
- Field
- Aerospace
- Result
- ρt/ρ
In isentropic flow brought to rest, the stagnation temperature ratio is Tt/T = 1 + (γ−1)M²/2, and the density ratio is that raised to the power 1/(γ−1): ρt/ρ = [1 + (γ−1)M²/2]^(1/(γ−1)). At Mach 1 in air the ratio is 1.577, so the density at the sonic point is 63 % of the stagnation value. At low M the change is small, which justifies incompressible flow below Mach 0.3.
Ratio of the stagnation density to the local density in isentropic flow of a calorically perfect gas with Mach number M and heat-capacity ratio γ > 1. Stagnation density is the value reached if the flow is slowed to rest isentropically.
Calculator
- Stagnation-to-static temperature ratio, 1 + (γ − 1)M²/21.2
- Stagnation-to-static pressure ratio1.893
- Density change from stagnation, 1 − ρ/ρt0.3661
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.
- At Mach 1 the density is 63.39 % of the stagnation value, a drop of 36.61 %.
- The temperature falls to 83.33 % of its stagnation value, and the pressure to 52.83 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| M Mach number | 0 to 5 | 1 | 88.18 | 87.18 | +10% | +8.98 % |
| γ Heat-capacity ratio | 1.1 to 1.67 | 1.539 | 1.629 | 0.08973 | +10% | −1.31 % |
Sweep of Mach number
| Mach number | Density ratio |
|---|---|
| 0 | 1 |
| 0.625 | 1.207 |
| 1.25 | 1.974 |
| 1.875 | 3.785 |
| 2.5 | 7.594 |
| 3.125 | 14.99 |
| 3.75 | 28.38 |
| 4.375 | 51.22 |
| 5 | 88.18 |
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 −1.31 %, from 1.577 to 1.557.
Mach number
The local Mach number. Raises the density ratio, slowly at first and steeply at high Mach number. Raising it by 10 % (to 1.1) moves the result by +8.98 %, from 1.577 to 1.719.
Worked example with the values above
- Temperature ratio: 1 + (γ − 1)M²/2 = 1 + 0.2 × 1 = 1.2.
- Exponent: 1/(γ − 1) = 2.5.
- ρt/ρ = 1.2^2.5 = 1.57744.
Assumptions and limits
- Steady, adiabatic, reversible flow of a calorically perfect gas, with no shocks or friction.
- A constant heat-capacity ratio, and quasi-one-dimensional flow where an area enters.
Common mistakes
- Using the subsonic branch of the area–Mach relation for a supersonic nozzle, or the reverse.
- Entering static temperature where the formula needs stagnation temperature.
- Applying the area-velocity relation at M = 1, where it is singular.
Statistics and uncertainty
Isentropic relations are exact for a perfect gas. Real nozzle flow departs from them: boundary layers and non-equilibrium chemistry make mass flow a few per cent lower, and for hot combustion gas γ is a function of temperature, not a constant.
Where it is used
- Sizing nozzles, inlets and diffusers.
- Referring engine data to standard conditions for comparison.
Maths you need
- Stagnation conditions: the state reached if the flow is slowed to rest isentropically
- Choking: the flow is sonic at the minimum area and the mass flow is maximal
- Mach number: M = V/a