Formula & Calculator
Isentropic Pressure Ratio
Relates static pressure to stagnation pressure for isentropic compressible flow as a function of Mach number.
Interpretation
Isentropic pressure ratio: p₀/p = (1 + ((γ−1)/2)·M²)^(γ/(γ−1)). It relates stagnation pressure to static pressure. Example: M=2, γ=1.4 → p₀/p = (1+0.2×4)^(3.5) = 1.8^3.5 ≈ 7.82.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| p0/p | Stagnation-to-static pressure ratio | |
| γ | Ratio of specific heats | |
| M | Mach number |
What it means
The isentropic pressure ratio is used to find the stagnation pressure from static pressure and Mach number, or vice versa. It is derived from the isentropic relations and is essential for compressible flow calculations in nozzles, diffusers, and wind tunnels. Stagnation pressure represents the pressure the fluid would reach if decelerated isentropically to rest. This ratio is also used in the normal shock relations. Understanding this relation is crucial for designing supersonic inlets and for predicting pressure losses in high‑speed flows.
Worked example
Isentropic Pressure Ratio – Two Examples
Real‑World| Parameter | Value |
|---|---|
| γ | 1.4 |
| M | 0.5 |
| Parameter | Value |
|---|---|
| M | 2.0 |
Common mistakes
- Isentropic pressure ratio: p₀/p = (1 + ((γ−1)/2)·M²)^(γ/(γ−1)).
- γ: Specific heat ratio.
- Stagnation pressure p₀: Total pressure – higher than static.
- Valid for isentropic flow (no shocks).
- Exponent check: γ/(γ−1) ≈ 3.5 for air.
Applications
The isentropic pressure ratio, p₀/p = (1 + ((γ−1)/2)·M²)^(γ/(γ−1)), gives the ratio of stagnation pressure to static pressure. It is essential for computing pressure losses in inlets, compressors, and nozzles, and for designing engine intakes. Engineers use it to determine the pressure recovery of an intake, to size the compressor, and to assess the effect of Mach number on engine performance. The ratio also appears in shock wave calculations. By understanding isentropic pressure ratios, aerospace engineers can design efficient air‑breathing propulsion systems that maximise pressure recovery and minimise losses.
- Inlet and intake design for subsonic/supersonic aircraft
- Compressor pressure ratio and performance analysis
- Nozzle expansion ratio and thrust optimisation
- Wind tunnel calibration and stagnation pressure measurement
- Engine cycle performance modelling
Frequently Asked Questions
It relates the static pressure to the stagnation pressure for isentropic compressible flow. It is essential for calculating pressures in nozzles, diffusers, and wind tunnels.
p0 = stagnation pressure (Pa)
p = static pressure (Pa)
γ = specific heat ratio
M = Mach number
It determines the pressure drop in a nozzle, the pressure recovery in a diffuser, and the forces on aerodynamic surfaces.
As M increases, p0/p increases rapidly. For M = 1, p0/p = 1.893; for M = 2, it is 7.824; for M = 5, it is 529. (for γ=1.4).
- Applying the isentropic relation across a shock wave, where the flow is no longer isentropic.
- Using the wrong value of γ for the gas.
- Confusing static and stagnation pressures.
Air at M = 2, static pressure p = 50 kPa, γ = 1.4. p0 = 50 × (1 + 0.2×4)^(1.4/0.4) = 50 × (1.8)^(3.5) = 50 × 7.824 = 391.2 kPa.
With a pitot tube aligned with the flow. The measured pressure is p0 (if the flow is brought to rest isentropically).
For isentropic flow, p0/p = (T0/T)^(γ/(γ−1)). This follows from the isentropic relations.
The pressure ratio across the nozzle determines the exhaust velocity and hence the thrust. A higher pressure ratio generally increases thrust.
Across a shock, stagnation pressure decreases (because the flow is non‑isentropic). The total pressure loss is related to the entropy increase.