Formula & Calculator
Stagnation Pressure
Pressure a moving fluid would reach if brought to rest isentropically; measured directly by pitot tubes.
Interpretation
Stagnation pressure: p₀ = p·(1 + ((γ−1)/2)·M²)^(γ/(γ−1)). It is the pressure when the flow is brought to rest isentropically. Example: p=101 kPa, M=2, γ=1.4 → p₀ = 101×1.8^3.5 ≈ 101×7.82 ≈ 790 kPa.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| p0 | Stagnation pressure | Pa |
| p | Static pressure | Pa |
| γ | Ratio of specific heats | |
| M | Mach number |
What it means
Stagnation pressure (total pressure) is the pressure that a fluid would have if brought to rest isentropically. It is a measure of the total energy of the flow. In compressible flows, it decreases across shocks due to irreversibility. This parameter is used in the design of inlets, nozzles, and wind tunnels. It is also measured by pitot tubes (with corrections for compressibility). Understanding p₀ is essential for evaluating pressure recovery in engine intakes and for calculating thrust. The isentropic relation is used to compute p₀ from static pressure and Mach number.
Worked example
Stagnation Pressure – Two Examples
Real‑World| Parameter | Value |
|---|---|
| p | 22,632 Pa |
| γ | 1.4 |
| M | 2.0 |
| Parameter | Value |
|---|---|
| p | 101,325 |
| M | 0.3 |
Common mistakes
- Stagnation pressure: p₀ = p · (1 + ((γ−1)/2)·M²)^(γ/(γ−1)).
- p: Static pressure.
- p₀ > p for M>0.
- Constant during isentropic stagnation.
Applications
Stagnation pressure, p₀ = p·(1 + ((γ−1)/2)·M²)^(γ/(γ−1)), is the total pressure of the flow, representing the pressure the fluid would have if decelerated isentropically. It is used to measure total pressure recovery in inlets, to compute nozzle thrust, and to assess engine performance. Losses in stagnation pressure indicate irreversibilities (shocks, friction). Engineers use this to design efficient intakes and exhaust systems, ensuring maximum thrust and minimal losses. By understanding stagnation pressure, aerospace engineers can optimise the aerodynamic design of the propulsion system.
- Inlet and intake design for maximum pressure recovery
- Nozzle performance analysis and thrust calculation
- Engine cycle analysis (total pressure losses)
- Wind tunnel testing and instrumentation (Pitot tube)
- Supersonic diffuser and shock train design
Frequently Asked Questions
It is the pressure a moving fluid would reach if brought to rest isentropically. It is directly measured by pitot tubes and is used to compute airspeed and flow properties.
p0 = stagnation pressure (Pa)
p = static pressure (Pa)
γ = specific heat ratio
M = Mach number
In a pitot‑static system, the difference between stagnation and static pressure gives dynamic pressure q = p0 − p, from which airspeed can be determined.
- Using the isentropic form across shocks, where stagnation pressure decreases due to entropy generation.
- Confusing static and stagnation pressures.
- Applying the formula for incompressible flow (which is different).
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.
Across a shock, stagnation pressure decreases (p02 < p01) because the process is irreversible. The loss is related to the entropy increase.
Higher Mach numbers give higher stagnation pressures relative to static pressure, but also larger losses across shocks.
With a pitot tube aligned with the flow. The pressure tap at the tube’s tip measures stagnation pressure.
Total pressure includes the kinetic contribution; static pressure is the pressure in the moving fluid frame.
It is used to set the test section Mach number and to compute forces from pressure distributions.