# 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 speed is null and all kinetic energy has been transferred into pressure energy. 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 velocity of fluid
${\displaystyle P_{\text{static}}}$ is the static pressure at any point.

At a stagnation point, the velocity 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 fluid 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 fluid is assumed to be calorically perfect. For such fluids, specific heats and ${\displaystyle \gamma }$ are assumed to be constant and invariant with temperature (a thermally perfect fluid).