AIRFOIL OF INFINITE SPAN IN COMPRESSIBLE FLOW (PROFILE THEORY)

4- 3-1 Survey

Now that a basic understanding of the compressible flow over wings (slender bodies) has been established in Sec. 4-2, the airfoil of infmite span will be discussed. On the basis of the similarity rules of Sec. 4-2-3, it turns out to be expedient to study pure subsonic and supersonic flows (linear theory) first, that is, flows with subsonic and supersonic approach velocities (Ma„ ^ 1), Secs. 4-3-2 and 4-3-3. The validity range of linear theory for Маж < 1 is limited by the critical Mach number Maxcr for the drag of Sec. 4-3-4. Later, transonic flow (nonlinear theory) will be discussed, at which the incident flow of the wing profile has sonic velocity {Маю & 1). Lastly, in Sec. 4-3-5, a brief account of hypersonic flow will be given, characterized by incident flow velocities much higher than the speed of sound {Маж >1).

4-3-2 Profile Theory of Subsonic Flow

Linear theory (Prandtl, Glauert) The exact theory of inviscid compressible flow leads to a nonlinear differential equation for the velocity potential for which it is
quite difficult to establish numerical solutions in the case of arbitrary body shapes. For slender bodies, however, particularly for wing profiles, this equation can be linearized in good approximation, Eq. (4-8). For such body shapes, explicit solutions are therefore feasible. In these cases, the physical condition has to be satisfied that the perturbation velocities caused by the body are small compared with the incident flow velocity. This condition is satisfied for wing profiles at small and moderate angles of attack. Linear theory of compressible flow at subsonic velocities leads to the Prandtl-Glauert rule. It allows the determination of compressible flows through computation of a subsonic reference flow. As discussed in Sec. 4-2-3, this subsonic similarity rule (version II) consists essentially of the following.

For equal body shapes and equal incident flow conditions, the pressure differences in the compressible flow are greater by the ratio l/Vl —Mai, than those in the incompressible reference flow. Here, Mam = ихаж is the Mach number, with Uoc the incident flow velocity and the speed of sound. Hence, the pressure distribution over the body contour from Eq. (4-23) becomes

P(x) ~ Poo = ■ …… -1 = [PincOO – P°° 3 (4’37)

Vl – Ma-^

Here the quantities of compressible flow are left without index, those of the incompressible reference flow have the index “inc.”

For the dimensionless pressure coefficient, the formula of the translation from incompressible to subsonic flow is obtained as

Cp = : cpinc (version II) (4-38)

?oo у l — Ma2^

Here it has been assumed that profile contours and angles of attack of compressible flow and of the incompressible reference flow are equal; that is,

ZincW = Z{X) (4-3 9 д)

«inc = a (4-396)

where X = xjc and Z = zjc are the dimensionless profile coordinates according to Eq. (2-2).

An experimental check of Eq. (4-38) is given in Fig. 4-11 for the simple case of a symmetric profile of 12% thickness in chord-parallel flow. Agreement between theory and experiment is very good in the lower Mach number range. At higher Mach numbers some differences are found. In Fig. 4-11, the values of the local sonic speed (Ma =1) are included, showing that sonic speed is first reached locally at. Мдет = 0.73.

The lift, obtained by integration of the pressure distribution over the profile chord, increases with the transition from incompressible to compressible flow as l/Vl — Mai, because of Eq. (4-38).

The expression for the lift coefficient is given in Table 4-1, which also contains the transformation formulas for the other lift-related aerodynamic coefficients. For

Figure 4-11 Pressure distributions of the profile NACA 0012 at chord-parallel incident flow for several subsonic Mach numbers Mam. Theory according to the subsonic similarity rule, Eq. (4-38); measurements from Amic [88]; Ma= 1 (wc=a) signifies points where the speed of sound is reached locally.

incompressible flow, the determination of neutral-point position, zero-lift angle, zero-moment coefficient, and angle of attack and angle of smooth leading-edge flow has been discussed in Sec. 24-2. For lift slope and neutral-point position of the skeleton profile, the values found for the inclined flat plate are valid, namely, {dcLldanc — 2/Г and (xNlc)inc = -, respectively.

In Fig. 4-12, the theoretical lift slopes are plotted against the incident flow Mach number.

Since, according to Eq. (4-37), the pressure distributions over a body at various Mach numbers are affine to the incompressible pressure distribution, it follows immediately that the position of the resultant aerodynamic force in the subsonic range (as long as no shock waves are formed) is equal to that in incompressible flow. Also, the drag in the subsonic range is determined by the same processes as in incompressible inviscid flow; that is, it is equal to zero.

Comparison with test results In Fig. 4-13, the most important results of the subsonic similarity rule are compared with measurements of Gothert [88]. For 5 symmetric

Figure 4-13 Lift slope {a) and neutral-point position (b) of NACA profiles of various thickness tjc vs. Mach number, for subsonic incident flow, from Multhopp; measurements from Gothert; neutral-point position as distance from the c/4 point.

wing profiles of thickness ratios tjc — 0.06, 0.09, 0.12, 0.15, and 0.18, lift slopes are plotted in Fig. 4-13a and neutral-point positions in Fig. 4-1 Зі?, both against the Mach number of the incident flow. For comparison, the theory with (dcLlda)inc = 5.71 is drawn as a straight line in Fig. 4-1 За.*

In the lower Mach number range, agreement between theory and measurement is very good, with the exception of the profile of 18% thickness. The theoretical curve follows the experimental data up to a certain Mach number, which shifts toward Mao, = 1 with decreasing profile thickness. The differences between theory and experiment beyond this Mach number are caused by strong flow separation. This fact can also be seen in the presentation of the drag coefficients of the same profiles in Fig. 4-14a.

According to the present linear theory for very thin profiles, the neutral-point position should be independent of Mach number. The experimental results of the profiles of Fig. 4-13& show, however, a considerable dependence of the neutral – point position on the Mach number when the profile thickness increases.

For the same symmetric profiles that have just been discussed with regard to lift slope and neutral-point position, the dependence of the drag (= profile drag) on the angle of attack a and on the Mach number of the incident flow Max is demonstrated in Fig. 4-14. The behavior of the curves for the drag coefficient CDp(Ma^), with tjc as the parameter, is characterized by the near independence of cDp from the Mach number in the lower Mach number range, whereas a very steep

”’Presented in double-logarithmic scale is dcj^jda vs. (1 —Malo).

Figure 4-14 Profile drag of NACA profiles of various thickness vs. Mach number, for subsonic incident flow, from measurements of Gothert. (a) Symmetric incident flow, a=0°. (b) Asymmetric incident flow, a = 4°.

drag rise occurs when approaching Ma„ = 1. This drag rise results from flow separation, caused by a shock wave that originates at the profile station at which the speed of sound is locally exceeded. The associated incident flow Mach number is designated as drag-critical Mach number Maxcr. In the case of chord-parallel incident flow (a = 0) the drag rise and, therefore, Mz«,cr, occur closer to Ma„ — 1 for thin profiles than for thick ones (Fig. 4-14a). For a profile with angle of attack (а Ф 0), the profile thickness has a negligible influence on the drag rise, as seen in Fig. 4-14b. As would be expected, the drag rise shifts to smaller Mach numbers with increasing angle of attack of the profile. The effect of the geometric profile parameters of relative thickness ratio, nose radius, and camber on the trend of the curves cDp{Maao) is shown in Fig. 4-15.

Attention should be called to the test results reported by Abbott and von Doenhoff, Chap. 2 [1], and by Riegels, Chap. 2 [50].

In summary, it can be concluded from the comparison of theory and experiment that the subsonic similarity rule (Prandtl-Glauert rule) is always in good agreement with measurements before sound velocity has been reached locally on the profile, that is, when no shock waves and corresponding separation of the flow can occur. Since these two effects are not covered by linear theory, the drag-critical Mach number is at the same time the validity limit of linear profile theory. Determination and significance of the critical Mach number Ma«, cr will be discussed in detail in Sec. 4-3-4.

Higher-order approximations (von Karman-Tsien, Krahn) From the derivation of the linear theory (Prandtl, Glauert), it can be concluded that the deviations of this approximate solution from the exact solution are increasing when the Mach number approaches Ma — l. The same is shown in the pressure-distribution measurements of

Fig. 4-11. Several efforts have been made, therefore, to improve the Prandtl-Glauert approximation. Steps in this direction have been reported by von Karman and Tsien [96], Betz and Krahn [7], van Dyke [99], and Gretler [29]. By the von Karman-Tsien formula, the computation of a compressible flow about a given profile is reduced to the determination of an incompressible flow about the same profile. The result is given here without derivation:

It can be seen immediately that this equation becomes the Prandtl-Glauert formula [Eq. (4-38)] for small values of cp*пс. According to von Karman-Tsien, the underpressures assume larger values and the overpressures smaller values than according to Prandtl-Glauert. In Fig. 4-16, the von Karman-Tsien rule and the Prandtl-Glauert rule are compared with measurements on the profile NACA 4412. Obviously, for the higher Mach numbers the von Karman-Tsien rule is in markedly better agreement with experiment than the Prandtl-Glauert rule.

At the stagnation point of a profile, both theories give the pressure coefficients too high, whereas the Krahn theory, which will not be discussed here, describes the behavior at this point accurately. Also, for Маж 1, Eqs. (4-38) and (440) lose validity, as would be expected from the assumptions made in their derivation. The relationship for the critical pressure coefficient cpcT (Mam) is shown in Fig. 4-16 as a limiting curve (see Sec. 4-34, Fig. 4-28).

4- 3-3 Profile Theory of Supersonic Flow

When a slender body with a sharp leading edge is placed into a supersonic flow field streaming in the direction of the body’s longitudinal axis (Fig. 4-17), the leading edge of this body assumes the role of a sound source in the sense of Fig. 1-9d. As a consequence, Mach lines originate at the sharp leading edge, upstream of which the incident parallel flow remains undisturbed. Only downstream of these Mach lines is the flow disturbed by the body. As an example of this behavior, the flow pattern about a convex profile in supersonic incident flow is shown in Fig. 4-18. The Mach lines, at which the pressure changes abruptly, have been made visible by the Schlieren method. The incident flow velocity can be determined quite accurately, with Eq. (1-33), from the angle of the Mach lines that originate at the profile leading edge.

Linear theory (Ackeret) In analogy to the case of subsonic incident flow of Sec.

4- 3-2, inviscid compressible flow about slender bodies (wing profiles) can be

Figure 4-17 Supersonic flow over a sharp-edged wedge.

computed by a linear approximation theory in the case of supersonic incident flow as well. The linearized potential equation, Eq. (4-8), is valid both for subsonic and supersonic flows. It was Ackeret [1 ] who laid the foundation for this linear theory of supersonic flow. The essential concept of this linear theory is expressed by the requirement that the perturbation velocity и in the x direction is a function only of the inclination of the profile contour area elements with respect to the incident flow direction, of the velocity Um, and of the Mach number Ma„:

u{x) =—– — – Uoc with w(x) = &{x)Ux (441)

Maio — 1

according to the kinematic flow condition (£ > 0: concave; # < 0: convex).

/ dz(x) 4* dx )

The inclinations of the contour on the upper and lower surfaces against the incident flow direction, &u and fy, respectively, are given for slender profiles of finite thickness and pointed nose (see Fig. 4-19) as

Here the upper sign applies to the upper surface of the profile, the lower sign to the lower surface. Equation (443) confirms the supersonic similarity rule (version II) as

Figure 4-19 Geometry and incident flow vector used in the profile theory at supersonic velocities.

derived in Sec. 4-2-3 [see Eq. (4-25)]. For the further evaluation of Eq. (443), it is expedient to separate the profile contours again, as in the case of the incompressible flow in Chap. 2, into the profile teardrop and the mean camber (skeleton) line [see Eq. (2-1)].

Z = – = Z<s> ±Z<f) and X=— (4-44)

c c

Here, as previously in Eq. (2-2), the coordinates have been made dimensionless with the profile chord c. Again, the upper sign applies to the upper surface of the profile, the lower sign to the lower surface.

For the pressure difference between the lower and upper surfaces of the profile (load distribution), Eq. (4-43Й) yields with Eq. (444):

The aerodynamic coefficients are easily obtained from the pressure distribution through integration. The lift coefficient is, from Eq. (2-54a),

і

cL = f Acp{X)dX = …… 4 – [25] (446)

J ІМа% – 1

О

It is a remarkable result that the lift coefficient depends only on the angle of attack a and not at all on the profile shape; that is, the zero-lift direction coincides with the profile chord (x axis). The moment coefficient, referred to the profile leading edge (nose up = positive), becomes, from Eq. (2-55u)*:

The lift-related aerodynamic coefficients are compiled in Table 4-2. They include the lift slope dcLfda and the neutral-point position xN/c = —dcMldcL, of which the dependence on the incident Mach number Max > 1 is demonstrated in Fig. 4-20tf and b. For comparison, the dependencies for the skeleton profile in subsonic incident flow, Ma^ <1, are also shown (see Table 4-1). These results are identical to those of the inclined flat plate. For Ma„ 1, both linear theories presented here fail, because the assumptions made are no longer valid. This is true particularly for

the lift slope, as can be seen from Fig. 4-33. The location of the neutral point is at xNjc = for subsonic flow and at xNfc = for supersonic flow. This marked shift toward the rear when the flow changes from subsonic to supersonic velocities should be emphasized.

In addition to lift, drag is produced in supersonic frictionless flow. It is called wave drag. The two forces are expressed by

C

D=b {Apfti± A pu$u) dx

о

where Api{x) — Рі(х)—рж and Apu(x)= Pu(x)~P°° are the pressures on the lower and upper surfaces of the profile, respectively, and дг and are the profile inclinations from Eq. (4-42). By using the pressure coefficients from Eq. (4-436) and evaluating the integrals under the, assumption that the profiles are closed in front and in the rear, the lift coefficient cL is obtained as in Eq. (446), and the drag coefficient cD becomes*

*Note that, also in subsonic flow, the wing of finite span has a drag that is proportional to the square of the lift (induced drag, see Sec. 34-2).

Table 4-2 Aerodynamic coefficients of a profile in supersonic incident flow based on the linear theory (Ackeret) Pressure distribution -T-……….. 2-…….. L dZ 1 dXj Lift slope dcL da ! vH l ^ ъ8 >- II Neutral-point position XN C _ 1 ~ 2 Zero-lift angle &Q – 0 I , Zero moment cMa l – 4 Г Z’s’dX І – 1 J 0 ! dcD i del = – f d Wave drag cDo

Figure 4-20 Aerodynamic forces of the inclined flat plate at subsonic and supersonic flows, (a) Lift slope dc^lda. (b) Position of the resultant of the aerodynamic forces хдг. (c) Drag coefficient Cjj.

Replacing a by as in Eq. (446), and by Z^ and as in Eq. (444), results in

It should be noted that the total wave drag is composed of three additive contributions. The first contribution is proportional to c and independent of the profile geometry. It is plotted in. Fig. 4-20c against the incident flow Mach number/ The second and third contributions are independent of the lift coefficient and proportional to the square of the relative camber and the relative thickness, respectively. Consequently, it can be seen directly that the flat plate is the so-called best supersonic profile, because the second and third contributions are equal to zero in this case.

The formulas for the drag rise dcjjfdc2L and for the zero drag cD at <?£, = 0 have been listed once more separately in Table 4-2. A simple explanation of the wave drag will be given for the subsequently discussed case of the inclined flat plate.

Results of linear theory The physical understanding of the last section was applied for the first time by Ackeret [1] to a quite simple computation of the flow over a flat plate in a flow of supersonic velocity Um at a small angle of incidence a. According to Fig, 4-21, the streamline incident on the plate leading edge forms with the plate a corner of angle a that is concave on the lower side of the plate and convex on the upper side. Consequently, an expansion Mach line originates on the upper side and a compression Mach line on the lower side. At the trailing edge, the compression line is above, the expansion line below the plate. Behind the plate the velocity is again equal to £/« and the pressure equal to рж, as it is ahead of the plate. Consequently, there is a constant underpressure pu on the entire upper surface and a constant overpressure pi on the lower surface. The pressure coefficient cp(x) = const follows from Eq. (4-436) with аФ 0 and z(x) = 0. The characteristic difference in the pressure distributions for supersonic and subsonic incident flow is explained in Fig. 4-22. From Fig. 4-22<z, at subsonic velocity the pressure distribution produces a force-resultant N normal to the plate, and in addition, the flow around the sharp leading edge produces a suction force S directed upstream along the plate (see Sec. 3-4-3). The resultant of the normal force N and the suction force S is the lift L, which acts normal to the incident flow direction £/«,. The resultant aerodynamic force has no component parallel to the incident flow direction; in other words, the drag in the frictionless subsonic flow is equal to zero.

For the case of supersonic flow, Fig. 4-226, the force N resulting from the pressure distribution also acts normal to the plate. However, because there is no flow around the leading edge, no suction force parallel to the plate exists here. The normal force N in inviscid flow therefore represents the total force. Separation info components normal and parallel to the incident flow direction establishes the lift L = TV cos a. and the wave drag D = N sin a « La. There is another physical explanation for the existence of drag at supersonic incident flow, namely, that for the production of the pressure waves (Mach lines) originating at the body during its motion, energy is expended continuously.

As a further example of the pressure distribution on profiles in supersonic flow, a biconvex parabolic profile and an infinitely thin cambered parabolic profile, given by the equations

Z(t) = – X)

Figure 4-2! Inclined plate in supersonic incident flow.

Z<*) = 4^X(1 – X)

are compared in Fig. 4-23. Both profiles are in chord-parallel incident flow, a = 0°. Consequently, from Eq. (446), cL = 0 for either profile. The pressure distributions, as computed from Eq. (443), are given in Fig. 4-23. The zero moment of the teardrop profile is equal to zero, whereas that of the skeleton profile is turning the leading edge down (nose-loaded). The lift-independent share of the wave drag is obtained from Eq. (448b) as

_______ 16 (iY

Cdo~ з Ужитті c/

_______ 64 jhy

~ З У Mai, – 1 N c /

These expressions show that the zero-drag coefficients are proportional to the squares of the thickness ratio t/c and the camber h/c, respectively. In Fig. 4-24, the

Figure 4-23 Pressure distribution at supersonic incident flow for para­bolic profiles at chord-parallel inci­dent flow, (a) Biconvex teardrop profile. (b) Skeleton profile.

Figure 4-24 Pressure distribution on profiles at supersonic incident flow. 1, lower surface; u, upper surface, (a) Inclined flat plate. (b) Parabolic skeleton at angle of attack a = 0°. (c) Biconvex profile at a = 0°. (d) Circular-arc profile, a = 0° ■ (e) Biconvex profile, a¥=0°. (f) Circular-arc profile, а Ф 0°.

pressure distributions of an inclined flat plate (Fig. 4-24я), a parabolic skeleton (Fig.

4- 24Z?), a symmetric biconvex profile, and a circular-arc profile at angle of attack a = 0° (Fig. 4-24c and d), as well as at а Ф 0° (Fig. 4-24e and/), are compared.

Further, a few data should be given about the dependence of wave drag on the relative thickness position for double-wedge profiles and parabolic profiles. The

geometry of parabolic profiles was given by Eq. (2-6). In Table 4-3 the results are compiled, and in Fig. 4-25 the contribution to the wave drag that is independent of cL is plotted against the relative thickness position. For a relative thickness position xt = 0.5, the wave drag of the double-wedge profile is

(4-51)

Thus, the drag of this double-wedge profile is lower by a factor f than that of the parabolic profile (Xt= 0.5). The double-wedge profile (Xt = 0.5) is the profile of lowest wave drag for a given thickness. Data on additional profile shapes are found in Wegener and Kowalke [21].

Information on the remaining aerodynamic coefficients, namely, zero-lift angle and zero moment, is compiled in Fig. 4-26 for skeleton profiles of all possible relative camber positions. The geometric data of the skeleton line were given in Eq. (2-6). For comparison, the coefficients for subsonic velocities are also shown. The zero-lift angle and the zero moment are plotted against the relative camber position in Fig. 4-2бй and b, respectively. In either case the basically different trends at subsonic and supersonic velocities are obvious.

Higher-order approximations (Busemann) The above-stated linear profile theory for supersonic flow, characterized by a local pressure difference (p—pcX) proportional to the local profile inclination & was later extended by Busemann [10] to a higher-order theory by adding terms of $2 and г}3. The pressure coefficient of the extended theory changes Eq. (4-43a) into

(4-52л)

Table 4-3 Wave drag at supersonic incident flow for double-wedge profiles and parabolic profiles (see Fig. 4-25)

Figure 4-25 Wave drag at supersonic flow vs, relative thickness position for double-wedge profile (1) and parabolic profile (2), from [21] (see Table 4-3).

with

The aerodynamic coefficients can be determined from Eq. (4-52), but no details will be given here. For the lift-independent contribution, an additional term is obtained that is proportional to (t/c)3 for symmetric profiles. Theoretical drag values, computed using this theory of second-order approximation, are compared in Fig. 4-27 with measurements by Busemann and Walchner [10]. Good agreement is obtained.

CD Figure 4-27 Drag polais of circular-arc profiles of several thickness ratios tic at Mach number Maoo = 1.47, from measurements of Busemann and Walchner; comparison with second-order approximation theory of Busemann.

With greater accuracy than by the above-illustrated theory of second-order approximation, the supersonic flow about thin profiles can be determined by the method of characteristics. Compare, for instance, the publications of Lighthill [51, 52].