Stagnation pressure

Magnitude

The magnitude of stagnation pressure can be derived from Bernoulli equation for incompressible flow and no height changes. For any two points 1 and 2:

:P_1 + \tfrac{1}{2} \rho v_1^2 = P_2 + \tfrac{1}{2} \rho v_2^2

The two points of interest are 1) in the freestream flow at relative speed v where the pressure is called the "static" pressure, (for example well away from an airplane moving at speed v); and 2) at a "stagnation" point where the fluid is at rest with respect to the measuring apparatus (for example at the end of a pitot tube in an airplane).

Then :P_\text{static} + \tfrac{1}{2} \rho v^2 = P_\text{stagnation} + \tfrac{1}{2} \rho (0)^2

or

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

where: :P_\text{stagnation} is the stagnation pressure :\rho; is the fluid density :v is the speed of fluid :P_\text{static} is the static pressure

So the stagnation pressure is increased over the static pressure, by the amount \tfrac{1}{2} \rho v^2 which is called the "dynamic" or "ram" pressure because it results from fluid motion. In our airplane example, the stagnation pressure would be atmospheric pressure plus the dynamic pressure.

In compressible flow however, the fluid density is higher at the stagnation point than at the static point. Therefore, \tfrac{1}{2} \rho v^2 can't be used for the dynamic pressure. 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.

:\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 is the ratio of specific heats

The above derivation holds only for the case when the gas is assumed to be calorically perfect (specific heats and the ratio of the specific heats \gamma are assumed to be constant with temperature).

Notes

References

• L. J. Clancy (1975), Aerodynamics, Pitman Publishing Limited, London.
• Cengel, Boles, "Thermodynamics, an engineering approach, McGraw Hill,

References

1. {{rp. § 3.2 it is defined as the sum of the free-stream [[static pressure]] and the free-stream [[dynamic pressure]].[https://scienceworld.wolfram.com/physics/StagnationPressure.html Stagnation Pressure] at Eric Weisstein's World of Physics (Wolfram Research)
2. Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London. {{ISBN. 0-273-01120-0
3. [http://www.engineeringtoolbox.com/bernouilli-equation-d_183.html Equation 4], Bernoulli Equation - The Engineering Toolbox
4. Houghton, E.L and Carpenter P.W. ''Aerodynamics'' (2003), Section 2.3.1
5. Clancy, L.J. ''Aerodynamics'', Section 3.12
6. [https://web.archive.org/web/20210324235239/https://www.grc.nasa.gov/WWW/BGH/Images/naca1135.pdf Equations 35,44], Equations, Tables and Charts for Compressible Flow