The aerodynamic theory for incompressible flow over thin airfoils at small angles of attack was presented in Chapter 4. For aircraft of the period 1903-1940, such theory
was adequate for predicting airfoil properties. However, with the rapid evolution of high-power reciprocating engines spurred by World war II, the velocities of military fighter planes began to push close to 450 mi/h. Then, with the advent of the first operational jet-propelled airplanes in 1944 (the German Me 262), flight velocities took a sudden spurt into the 550 mi/h range and faster. As a result, the incompressible flow theory of Chapter 4 was no longer applicable to such aircraft; rather, high-speed airfoil theory had to deal with compressible flow. Because a vast bulk of data and experience had been collected over the years in low-speed aerodynamics, and because there was no desire to totally discard such data, the natural approach to high-speed subsonic aerodynamics was to search for methods that would allow relatively simple corrections to existing incompressible flow results which would approximately take into account the effects of compressibility. Such methods are called compressibility corrections. The first, and most widely known of these corrections is the Prandtl – Glauert compressibility correction, to be derived in this section. The Prandtl-Glauert method is based on the linearized perturbation velocity potential equation given by Equation (11.18). Therefore, it is limited to thin airfoils at small angles of attack. Moreover, it is purely a subsonic theory and begins to give inappropriate results at values of Moo = 0.7 and above.
Consider the subsonic, compressible, inviscid flow over the airfoil sketched in Figure 11.3. The shape of the airfoil is given by у = f(x). Assume that the airfoil is thin and that the angle of attack is small; in such a case, the flow is reasonably approximated by Equation (11.18). Define
P2 – і – мі
so that Equation (11.18) can be written as
Let us transform the independent variables x and у into a new space, £ and ij, such that
П = РУ [11.36b]
Moreover, in this transformed space, consider a new velocity potential ф such that
</>(£, rj) = Рф(х, у) [11.36c]
To recast Equation (11.35) in terms of the transformed variables, recall the chain rule of partial differentiation; that is,
Зф 3ф 3§ 3ф dt]
дх 3£ dx dr] dx
dip d0 3§ 30 dr]
dy Э§ 3у dr] dy
From Equations (11.36a and b), we have
Differentiating Equation (11.41) with respect to x (again using the chain rule), we obtain
З20 1 d2ф
3×2 “ W
Differentiating Equation (11.42) with respect to y, we hnd that the result is
Substitute Equations (11.43) and (11.44) into (11.35):
Examine Equation (11.45)—it should look familiar. Indeed, Equation (11.45) is Laplace’s equation. Recall from Chapter 3 that Laplace’s equation is the governing relation for incompressible flow. Hence, starting with a subsonic compressible flow in physical (x, у) space where the flow is represented by ф(х, у) obtained from Equation (11.35), we have related this flow to an incompressible flow in transformed (£, і)) space, where the flow is represented by ф(%, і) ‘) obtained from Equation (11.45). The relation between ф and ф is given by Equation (11.36c).
Consider again the shape of the airfoil given in physical space by у = f(x). The shape of the airfoil in the transformed space is expressed as r) — q(^). Let us compare the two shapes. First, apply the approximate boundary condition, Equation
(11.34) , in physical space, noting that df/dx = tan в. We obtain
df _дф _ 1 дф _ дф 00 dx ду p ду 3 г)
Similarly, apply the flow-tangency condition in transformed space, which from Equation (11.34) is
dq_ _ дф
°°d$ ~ dr)
Examine Equations (11.46) and (11.47) closely. Note that the right-hand sides of these two equations are identical. Thus, from the left-hand sides, we obtain
Equation (11.48) implies that the shape of the airfoil in the transformed space is the same as in the physical space. Hence, the above transformation relates the compressible flow over an airfoil in (x, y) space to the incompressible flow in (£, >)) space over the same airfoil.
The above theory leads to an immensely practical result, as follows. Recall Equation (11.32) for the linearized pressure coefficient. Inserting the above transformation into Equation (11.32), we obtain
2Й 2 3ф 2 1 3ф 2 1 3ф ^ ^ 49]
Question: What is the significance of 3$/3£ in Equation (11.49)? Recall that ф is the perturbation velocity potential for an incompressible flow in transformed space. Hence, from the definition of velocity potential, дф/ді; = й, where й is a perturbation
velocity for the incompressible flow. Hence, Equation (11.49) can be written as
From Equation (11.32), the expression f—2u/V00) is simply the linearized pressure coefficient for the incompressible flow. Denote this incompressible pressure coefficient by Орд. Hence, Equation (11.50) gives
or recalling that f = ,/F— M^, we have
Equation (11.51) is called the Prandtl-Glauert rule; it states that, if we know the incompressible pressure distribution over an airfoil, then the compressible pressure distribution over the same airfoil can be obtained from Equation (11.51). Therefore, Equation (11.51) is truly a compressibility correction to incompressible data.
Consider the lift and moment coefficients for the airfoil. For an inviscid flow, the aerodynamic lift and moment on a body are simply integrals of the pressure distribution over the body, as described in Section 1.5. (If this is somewhat foggy in your mind, review Section 1.5 before progressing further.) In turn, the lift and moment coefficients are obtained from the integral of the pressure coefficient via Equations (1.15) to (1.19). Since Equation (11.51) relates the compressible and incompressible pressure coefficients, the same relation must therefore hold for lift and moment coefficients:
The Prandtl-Glauert rule, embodied in Equations (11.51) to (11.53), was historically the first compressibility correction to be obtained. As early as 1922, Prandtl was using this result in his lectures at Gottingen, although without written proof. The derivation of Equations (11.51) to (11.53) was first formally published by the British aerodynamicist, Hermann Glauert, in 1928. Hence, the rule is named after both men. The Prandtl-Glauert rule was used exclusively until 1939, when an improved compressibility correction was developed. Because of their simplicity, Equations (11.51) to (11.53) are still used today for initial estimates of compressibility effects.
Recall that the results of Chapters 3 and 4 proved that inviscid, incompressible flow over a closed, two-dimensional body theoretically produces zero drag—the well – known d’Alembert’s paradox. Does the same paradox hold for inviscid, subsonic,
compressible flow? The answer can be obtained by again noting that the only source of drag is the integral of the pressure distribution. If this integral is zero for an incompressible flow, and since the compressible pressure coefficient differs from the incompressible pressure coefficient by only a constant scale factor, p, then the integral must also be zero for a compressible flow. Hence, d’Alembert’s paradox also prevails for inviscid, subsonic, compressible flow. However, as soon as the freestream Mach number is high enough to produce locally supersonic flow on the body surface with attendant shock waves, as shown in Figure 137b, then a positive wave drag is produced, and d’Alembert’s paradox no longer prevails.
Example 11.1 | At a given point on the surface of an airfoil, the pressure coefficient is —0.3 at very low speeds. If the freestream Mach number is 0.6, calculate Cp at this point.
From Equation (11.51),
^ _ Cp, o _ ~~0-3
p ~~ Vl – M2 ~ y/ – (0.6)2
Example 1 1.2 | From Chapter 4, the theoretical lift coefficient for a thin, symmetric airfoil in an incompressible
flow is с/ = 2ла. Calculate the lift coefficient for Mx = 0.7.
From Equation (11.52),
C‘ ~ У1 – Ml ~ Vl – (0.7)2
Note: The effect of compressibility at Mach 0.7 is to increase the lift slope by the ratio 8.8/27Г = 1.4, or by 40 percent.