Prandtl probe and Pitot tube flow calculation - theory

Prandtl probe is used for measuring velocity in a particular point of fluid stream. Actually, with Prandtl probe pressures are measured and velocity is calculated latter. Prandtl probe is designed in a way that both total pressure and static pressure are measured. So what are total pressure and static pressure anyway?
Static pressure is pressure of fluid in flow stream.
Total pressure is pressure of fluid when it is brought to rest, i.e. velocity is reduced to 0.

Incompressible flow

If we look at Bernoulli equation for two points in flow field at the same altitude z₁ = z₂:

where is:
p - pressure
rho - density view table
V - velocity

we can easy get equation for total pressure (when velocity at point 2 is zero, pressure at point 2 is than total):

where is:
p_t - total pressure

Actually, in case of Prandtl probe, there is only one point, not two of flow field. What we do is imagining that flow is in that point stopped without any energy loss and for those two moments we are writing energy equation.

The difference between total and static pressure represents fluid kinetic energy and it is called dynamic pressure (p_d):

So, for velocity measuring with Prandtl probe it is necessary to measure static pressure (measured in a way that flow is not disrupted) and total pressure (measured at point where velocity is 0 - flow is stopped). Difference between these two pressures - dynamic pressure, is used for velocity calculation.

Compressible flow

For gases and larger Mach numbers than 0.1 effects of compressibility are not negligible. For compressible flow calculation, gas is considered as ideal, i.e. ideal gas state equation can be used. Velocity for Mach number M < 1 is calculated using following equiation:

where is:
M - Mach number
kappa - isentropic coefficient. view table

It should be said that for M > 0.7 given equation is not totally accurate.

If Mach number M > 1, than normal shock wave will occur. In that case, Prandtl probe is measuring pressures behind shock wave. Total pressure and static pressure are smaller than in front of the wave. Equation for velocity in front of the wave is given bellow:

where is:
p_ti - total pressure behind normal shock wave (that pressure is measured with Prandtl probe) If you need complete derivation of given equations you can download it here.

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