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). 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,

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

where:

$P_{\text{stagnation}}$ is the stagnation pressure

$\rho \;$ is the fluid density

$v$ is the velocity of fluid

$P_{\text{static}}$ 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.

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

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]

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

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

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

where:

$p_{t}$ is the stagnation pressure

$p$ is the static pressure

$T_{t}$ is the stagnation temperature

$T$ is the static temperature

$\gamma$ 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 $\gamma$ are assumed to be constant and invariant with temperature (a thermally perfect fluid).

See also

• Hydraulic ram

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

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

External links

• Pitot-Statics and the Standard Atmosphere
• F. L. Thompson (1937) The Measurement of Air Speed in Airplanes, NACA Technical note #616, from SpaceAge Control.