# Stagnation pressure

In fluid dynamics, stagnation pressure (or pitot pressure) is the static pressure at a stagnation point in a fluid flow.[1] At a stagnation point the fluid velocity is zero and all kinetic energy has been converted into pressure energy (isentropically). In an incompressible flow, stagnation pressure is equal to the sum of the free-stream dynamic pressure and free-stream static pressure.[2]

Stagnation pressure is sometimes referred to as pitot pressure because it is measured using a pitot tube.

## Magnitude

The magnitude of stagnation pressure can be derived from a simplified form of Bernoulli Equation.[3][1] For incompressible flow,

${\displaystyle P_{\text{stagnation}}={\tfrac {1}{2}}\rho v^{2}+P_{\text{static}}}$

where:

${\displaystyle P_{\text{stagnation}}}$ is the stagnation pressure
${\displaystyle \rho \;}$ is the fluid density
${\displaystyle v}$ is the speed of fluid
${\displaystyle P_{\text{static}}}$ is the static pressure at any point.

At a stagnation point, the speed of the fluid is zero. If the gravity head of the fluid at a particular point in a fluid flow is zero, then the stagnation pressure at that particular point is equal to total pressure.[1] However, in general, total pressure differs from stagnation pressure in that total pressure equals the sum of stagnation pressure and gravity head.

${\displaystyle P_{\text{total}}=0+P_{\text{stagnation}}\;}$

In compressible flow the stagnation pressure is equal to total pressure only if the fluid entering the stagnation point is brought to rest isentropically.[4] For many purposes in compressible flow, the stagnation enthalpy or stagnation temperature plays a role similar to the stagnation pressure in incompressible flow.

## Compressible flow

Stagnation pressure is the static pressure a gas retains when brought to rest isentropically from Mach number M.[5]

${\displaystyle {\frac {p_{t}}{p}}=\left(1+{\frac {\gamma -1}{2}}M^{2}\right)^{\frac {\gamma }{\gamma -1}}\,}$

or, assuming an isentropic process, the stagnation pressure can be calculated from the ratio of stagnation temperature to static temperature:

${\displaystyle {\frac {p_{t}}{p}}=\left({\frac {T_{t}}{T}}\right)^{\frac {\gamma }{\gamma -1}}\,}$

where:

${\displaystyle p_{t}}$ is the stagnation pressure
${\displaystyle p}$ is the static pressure
${\displaystyle T_{t}}$ is the stagnation temperature
${\displaystyle T}$ is the static temperature
${\displaystyle \gamma }$ ratio of specific heats

The above derivation holds only for the case when the gas is assumed to be calorically perfect. For such gases, specific heats and the ratio of the specific heats ${\displaystyle \gamma }$ are assumed to be constant and invariant with temperature.

## Notes

1. Clancy, L.J., Aerodynamics, Section 3.5
2. Stagnation Pressure at Eric Weisstein's World of Physics (Wolfram Research)
3. Equation 4, Bernoulli Equation - The Engineering Toolbox
4. Clancy, L.J. Aerodynamics, Section 3.12
5. Equations 35,44, Equations, Tables and Charts for Compressible Flow

## References

• L. J. Clancy (1975), Aerodynamics, Pitman Publishing Limited, London. ISBN 0-273-01120-0
• Cengel, Boles, "Thermodynamics, an engineering approach, McGraw Hill, ISBN 0-07-254904-1