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Airspeed Theory

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

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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

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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
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