Page 80 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 63

mouth of A pointing directly upstream, the stream being of speed v and of static
pressure p. The air flowing past the holes at C will be moving at a speed very little
different from v and its pressure will, therefore, be equal top, and this pressure will be
communicated to the interior of tube B through the holes C. The pressure in B is,
therefore, the static pressure of the stream.
Air entering the mouth of A will, on the other hand, be brought to rest (in the
ultimate analysis by the fluid in the manometer). Its pressure will therefore be equal
to the total head of the stream. As a result a pressure difference exists between the air
in A and that in B, and this may be measured on the manometer. Denote the pressure
in A by PA, that in B by p~, and the difference between them by Ap. Then
AP=PA-PB (2.17)

But, by Bernoulli's equation (for incompressible flow)

and therefore

(2.18)

1
Ap = -pv2
2
whence
(2.19)
The value of p, which is constant in incompressible flow, may be calculated from the
ambient pressure and the temperature. This, together with the measured value of Ap,
permits calculation of the speed v.*
The quantity $pv2 is the dynamic pressure of the flow. Since PA = total
pressure =PO (i.e. the pressure of the air at rest, also referred to as the stagnation
pressure), and p~ = static pressure = p, then
1
Po-P=p? (2.20)

which may be expressed in words as
stagnation pressure - static pressure = dynamic pressure
It should be noted that this equation applies at all speeds, but the dynamic pressure is
equal to $pv2 only in incompressible flow. Note also that
1
-p? = [ML-3L2T-2] = [ML-'TP2]
2
= units of pressure
as is of course essential.
* Note that, notwithstanding the formal restriction of Bernoulli's equation to inviscid flows, the PitBt-
static tube is commonly used to determine the local velocity in wakes and boundary layers with no app-
arent loss of accuracy.