Air Speed Theory Eugene M. Cliff February 15, 1998 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 onedimensional flow (along a streamline) and carry out the required development 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. One Dimensional Flow We consider steady (no change with time), frictionless (no viscous forces) flow along a streamline and so we have dp + d[v /] + gdz=0. (1) Our goal is to integrate this expression. Note that the real work is the integration of the first term; this generally requires that we introduce a P relation..1 Case 1: Fluid at Rest In this case we have V 0 so that (1) takes the form dp = gdz, 1
which we recognize as the hydrostatic equilibrium equation used in the investigation of the model atmosphere and altimeter theory. The required P relation came from the perfect gas law and the assumption of a temperature profile.. Case : Constant Density In this case we have (i.e. a constant) and equ n (1) can be simply integrated to yield P + V /+ gz = C, where C is a constant along the given streamline. For most aircraft applications we neglect the last term ( gz) andwrite P + V /=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 can be interpreted as the pressure at a point where the flow has stopped. Such a point is called a stagnation point and the pressure at such a point is called the stagnation pressure (P o ). Thus, P + V /=P o. () We shall show later that this incompressible flow approximation is pretty accurate for low speed flows but much less so for highspeed flows..3 Case 3: Adiabatic Flow We observe that air is generally a poor conductor of thermal energy so that as we move along the streamline the flow doesn t conduct heat. It turns out that this implies that P and are related by P = K γ,whereγ=7/5 (for air) and K is a constant depending on which streamline we follow. For subsequent analysis we will also need the speed of sound (a) which is computed as: a dp d = γk γ 1 = γkγ = γp.
Neglecting the dzterm in (1) we have which implies d dp = V dv or a d = V dv = M dv V, where M V/a is the Mach number. A term such as d as a relative change in density (i.e. as ). can be interpreted Note that if M <.3then the relative change in density will be less than 10% of the change in speed. In such cases one might neglect the density change and this leads to the incompressible flow approximation. The integral of the dp canbedonebyusingthep=kγ expression to replace the differential dp in terms of. This leads to the result dp = γ P γ 1 so that the integrated form of (1) is γ P γ 1 + V /=C, or from our earlier work a γ 1 + V /=C. As in the earlier (incompressible) case we can evaluate the constant C by imagining the stagnation point (V 0) a γ 1 + V /= a o γ 1, (3) where a o is the speed of sound at the stagnation point. As noted above, our primary objective is to develop the mathematical machinery needed to understand the theory of the airspeed indicator. For this purpose we first re-write (3) as 1+ (γ 1) M = a o a, (4) 3
where, as before M V/a is the local Mach number. Note that along the streamline both V and a will change and so will M. We use our expresssion for the speed of sound to re-write the right-hand side as a o a = P γ 1/γ o. P Combining this with (4) we have the final expression [ P o P = 1+ (γ 1) ] γ/γ 1 M. (5) This is the compressible flow form of the Bernoulli equation. It is valid for steady, frictionless flows without shocks. Note that for air we have γ = 7/5 and the coefficient of M has the value 1/5, while the exponent had the value 7/. 3 Calibrated Airspeed The compressible form of the Bernoulli equation (5) is the basis for calibrating the airspeed indicator. This instrument is driven by the pressure difference (P o P ). It is very important to understand this point. If we have a perfectly calibrated airspeed indicator, then the displayed speed is uniquely related to the pressure difference (P o P ). We do not know the stagnation pressure (P o ) nor the static pressure (P ); we only know the difference. We now work from (5) to show how this is done. We basically use the observation that (P o P )=P ( Po ) 1 to write P [ [ (P o P )=P 1+ γ 1 ( V ] γ/γ 1 ] a ) 1. While we have the appropriate pressure difference on the left, on the right we have the static pressure (P ), the speed of sound (a) and the true airpeed (V ). It s clear that we can not uniquely relate pressure difference to the true air speed; we would have to also know the values of P and a. Our solution is to define an airspeed where we use the standard sea-level values for the unknown quantities P and a. Thatiswehave [ [ (P o P)=P SL 1+ γ 1 ( V ] γ/γ 1 c ) 1], V c <a SL (6) a SL 4
where P SL and a SL are the standard sea-level pressure and speed of sound, respectively. The air-speed indicator inverts this expression and displays the calibrated airspeed V c, when subjected to a given pressure difference (P o P ). Equation (5) is appropriate for flows without shocks. Hence, eq n (6) is used for values of V c a SL. We will not consider the modification needed for V c >a SL. 4 Other Airspeeds At this point we have described the true airpseed (V ) and the calibrated airspeed (V c ). Additionally, we have the static pressure (P ) and the stagnation pressure (P o ). It is clear that the term (V /) (see ()) is also important. We call this the dynamic pressure and use the symbol (q). The incompressible flow result says that the stagnation pressure (also called total pressure) is the sum of static plus dynamic pressures. This is strictly true for incompressible flows and approximately so for low speed flows of compressible fluids, such as air. The equivalent airspeed is defined as the answer to the question: how fast do I have to travel in standard sea level air to have the same dynamic pressure (q) that I currently have? In equation form: SL V e /=V /, where V e is the equivalent airspeed (EAS). It is uniquely related to the dynamic pressure. From this definition it is easy to deduce that V e = σv, where σ is the density ratio (/ SL ). From the perfect gas relations one can also show that q = γpm / and that V e = Ma SL P/P SL. 5