Airspeed Theory - Air Speed Theory Eugene M Cliff 1...

Air Speed Theory
Eugene M. Cliff
January 22, 2001
1 Introduction
The primary purpose of these notes is to develop the necessary mathematical
machinery to understand pitot-static airspeed indicators and several related
notions of airspeed. In the next section we will begin with a model of
one-dimensional flow (along a streamline) and carry out the required devel-
opment for a compressible-flow form of the Bernoulli equation. From there
we will define a few notions of airspeed and discuss the relations among
them.
2 One Dimensional Flow
We consider steady ( no change with time ), frictionless ( no viscous forces )
flow along a streamline and so we have
d P
ρ
+ d [ V
2
/ 2] + g dz = 0 . (1)
Our goal is to ‘integrate’ this expression. Note that the real work is the
integration of the first term; this requires that we introduce a P − ρ relation.
2.1 Case 1: Fluid at Rest
In this case we have V ≡ 0 so that (1) takes the form
d P = − ρ g dz,
which we recognize as the hydrostatic equilibrium equation used in the inves-
tigation of the model atmosphere and altimeter theory. The required P − ρ
relation came from the perfect gas law and the assumption of a temperature
profile.
1

2.2 Case 2: Constant Density
In this case we have ρ ≡ ˜ ρ ( i.e. a constant) and equ’n (1) can be simply
integrated to yield
P + ˜ ρV
2
/ 2 + ˜ ρgz = C ,
where C is a constant along the given streamline. For most aircraft applica-
tions we neglect the last term (˜ ρgz ) and write
P + ˜ ρV
2
/ 2 = C .
The constant ( C ) has a fixed value along the streamline; along the line P
(the static pressure) and V (the airspeed) change, but in such a way that the
combined term is constant. We can imagine a place along the (extended)
line where the velocity is zero and at such a point ( p ) P ( p ) = C . This means
that the constant of integration C