# Wing of Finite Thickness at Subsonic Incident Flow

Pressure distribution In this section, the wing of finite span at incident flow of subsonic velocities will be investigated, at zero lift (displacement problem). The pressure distribution of such a wing of finite thickness is of particular interest with regard to the determination of the drag-critical Mach number at high subsonic incident flow. The concept of critical Mach number has already been explained in Sec. 4-3-4. The incident flow velocity of the critical Mach number is the lower limit for the formation of the shock waves, which change the entire flow pattern considerably and, in particular, lead to a strong drag rise (see Figs. 4-14 and 4-15).

 Cp inc Vl — Mai*

The pressure distribution of a three-dimensional wing with symmetric wing profiles at subsonic incident flow is obtained from that of the transformed wing of Eq. (4-69) as

where cp jnc is the pressure distribution of the transformed wing for which the pressure distribution of incompressible flow is to be computed. The computational

Figure 4-51 Lift slope of delta wings of various thicknesses; aspect ratio л = 3, from . Comparison with linear theory.

method was given in Sec. 3-6. The transformation of the wing pianform follows Eqs. (4-66)-(4-68); the thickness ratio 8 = t/c remains unchanged (version II of the subsonic similarity rule of Sec. 4-3-2).

Drag-critical Mach number On three-dimensional wings, contrary to the plane problem, frequently the wing leading or trailing edges are not perpendicular to the incident flow direction. The simplest cast of that kind is the swept-back wing of constant chord and infinite span. This case has been treated previously for incompressible flow in Sec. 3-6-3. The sweepback has a significant influence on the magnitude of the critical Mach number, because only the velocity component normal to the leading edge determines the maximum perturbation velocity on the contour of such wings of finite thickness. From Eq. (4-53д), the critical pressure pCT of the wing in incident flow normal to its leading edge is obtained after multiplication with Oc«i7«cr/2 and with Mr», cr = as

By introducing now, in agreement with the above statement, U=<,CTcosy as the effective velocity instead of I/»cr, and again adopting the dimensionless notation, the critical pressure coefficient of the swept-back wing becomes

Pci~Poc 2 1 —MaloCT cos2 ф

JTi 7+1 Mai

– C/cocr

Here, as in Eq. (4-76), the pressure coefficient of the swept-back wing is referred to the dynamic pressure of the incident flow. The relation cpCT{Ma<xct’) is shown in Fig. 4-52 for у = 0° (see curve 1 of Fig. 4-28) and for <p= 45°. To determine the critical Mach number of the incident flow Ma«,at the curve cpmin is drawn in Fig.

4- 52 up to its intersection with the curve cpcx (see Fig. 4-28).

Swept-back airfoil of infinite span For the determination of the pressure difference (p — Р») of a swept-back wing, it should be observed that (p — pM) is proportional to the dynamic pressure of the effective velocity (goo/2)£/2, cos2 </>. It is also proportional to the thickness ratio or the angle of attack, respectively, determined in the plane of the effective incident flow; that is, it is proportional to (t/c) cos <p. It follows that in incompressible flow,

Pine P°° ~~(Pinc P°°)<p= 0 COS

Referred to the dynamic pressure of the incident flow (q*,I2)U1, the relation between the pressure coefficients becomes (cpmin)inc = cos йпс(сртіп)тс,<р=о – With Eqs. (4-76) and (4-68c) it is

COS, v. –

cpmin “ min)inc, =0 With COS —

By substitution, finally,

The above-explained procedure has been applied to an example in Fig. 4-52. Chosen were two airfoils of infinite span, one unswept and one with a sweepback angle of 45°. For the unswept airfoil, (сртт)іпс, у?=о =“0.2 has been assumed, resulting in a critical Mach number = 0-83. The effect of the sweepback is seen in

a shift of the critical Mach number to a considerably larger value of (Ма„ост)^-45 — 1.13. This shift is caused by three effects. First, the curve cpCT is shifted to the right because of the sweepback; second, by the sweepback, cpmin at Ma<* = 0 is

Figure 4-52 Determination of drag – critical Mach number Ma^ cr for an unswept and a swept-back airfoil of infinite span. (cp minV = o, Mace =o = -0.2.

Figure 4-53 Drag-critical Mach number of the incident flow of swept-back airfoils of infinite span, (a) Effect of pressure coefficient, (b) Effect of thickness ratio (biconvex parabolic shape).

reduced; and third, the rise of cpTnin with Mach number is much weaker for a swept-back wing than for an unswept wing.

An extension of Fig. 4-52 is given in Fig. 4-53a, where the critical Mach numbers of swept-back airfoils of infinite span are presented relative to (£pmm)inc)¥> = 0- For a biconvex parabolic profile,(сртіГі)іпс^=0 = -2(итах/С/о«,)тс = —(8/7г)(//с). Corresponding to the example shown in Fig. 4-53c, the sweepback angle has been evaluated in Fig. 4-53& as a function of the critical Mach number and for several thickness ratios. For 5 = tjc = 0, this function is

 (A 5 -* 0)

Ma„ cr =

Thus, sweepback may raise the drag-critical Mach number of very thin profiles considerably above unity.

Middle (root) section of the swept-back wing The discussions about the effect of wing sweepback presented so far are valid only for the straight airfoil of infinite span (see Fig. 4-52). For folded wings (Fig. 3-74), the favorable sweepback effect (raising of the drag-critical Mach number) is not realized fully in the vicinity of the root section. The middle portion of the wing performs somewhat as if it were unswept. For the computation of the critical Mach number of the middle section of the folded swept-back wing, the following procedure has to be applied: For incompressible flow, the velocity distribution over the root section is given by Eq. (3-187). The maximum velocity over the root section produces the largest underpressure (Cpmin)inc 2(Ищах/^°°)іпс* The Value of (^тпах/^°°)іпс ®f ^ parabolic profile is plotted in Fig. 3-76 against the sweepback angle ipinc. Conversion of (Opmin)inc into Cpmin f°r the various Mach numbers is given by Eq. (4-76), where the sweepback angle also has to be transformed according to Eq. (4-68c). The critical Mach number is then obtained as the intersection of the curves cpmjn and Cpcr °f Fig – 4-52, where for the root section the curve cpcT for ip = 0 has to be taken. The result of this computation is presented in Fig. 4-54, for sweepback angles t/?=0, 45, and —45° and for several relative thickness positions Xt. The dashed curve for <p = ±45° shows the values for the straight swept-back wing. They are valid for sections of the folded wing at large distances from the root. It is clearly seen that the swept-back wing (9?=+45°) has the most favorable critical Mach number of the root section for relative thickness positions of about 30%,

whereas the swept-forward wing (<p = —45°) is most favorable for relative thickness ratios of about 70%. These results show that the critical Mach number of the middle section of folded swept-back wings is, in general, considerably lower than that of the tip section. It follows that the favorable sweepback effect of the straight swept-back wing cannot be fully realized by folded wings.

Investigations of the drag-critical Mach number of folded swept-back wings were made by Neumark . He also studied the influence of finite aspect ratios on the critical Mach number, but no marked differences with the airfoil of infinite span were found; see Fig. 3-71.

Experimental results Raising of the drag-critical Mach number by sweepback has found practical applications of great importance for airplane design. As has previously been shown in Sec. 4-3-2, increasing the critical Mach number produces a shift of the compressibility-caused drag rise to higher Mach numbers (Fig. 4-1 Ad). It must be expected, therefore, that sweepback causes a shift to higher Mach numbers of the strong rise of the profile-drag coefficients with Mach number,

This fact was first realized by Betz in 1939 and has been checked experimentally by Ludwieg . A few of his measurements are plotted in Fig. 4-55, The polars for an unswept and for a swept-back trapezoidal wing (ip = 45°) show the following: The profile drag (cL = 0) of the unswept wing is several times larger at Ma„ = 0.9 than at Ma„ = 0.7. Thus the drag-critical Mach number of this wing lies between Ma* = 0.7 and Mr» = 0.9. For the swept-back wing, however, the profile drag at Ma* = 0.9 is only insignificantly higher than at Mz„ = 0.7. In other words, the critical Mach number of this wing lies above Ma„ = 0.9. Another example of this important swept-back wing effect is demonstrated in Fig. 4-56. Here, from , cDp is shown versus Ma„=> for an unswept and a swept-back wing (<p = 45°). The sweepback effect is manifested by a shift of the onset of the drag rise from about Max — 0.8-0.95. This favorable sweepback effect has been exploited by airplane designers since World War II. The presentation of Fig. 34c, namely, sweepback angle versus flight Mach number, shows very clearly that the sweepback angle of airplanes actually built increases markedly when Mach number Mz» = 1 is approached.

Thick wing at sonic incident flow The subsonic similarity rule of Sec. 4-4-3 leads to useful results in computing the lift for incident sonic flow (Маж = 1). It fads, however, in the computation of the displacement effect of a finitely thick wing at sonic incident flow. The reason is that the pressures on the wing become infinitely high. Compare, for example,  for an account of this difference between the lift problem and the thickness problem in the limiting case Ma„ -+ 1. To obtain useful information on the thickness problem at Mam = 1, nonlinear approximation methods have to be applied. The transonic similarity rule (see Sec. 4-3-4) is particularly well suited for classification and systematic presentation of test results on wings of finite span; see Spreiter . Further information on the theory of transonic flow of wings is found in publications by Keune  and Pearcey  and in reference  on the equivalence theorem of wings of small span in transonic flow of zero incidence.

 Figure 4-55 Polars, lift coefficient C£, and drag coefficient cq at high subsonic incident flow; Mach number Mz« = 0.7 and 0.9, for a straight and a swept-back wing of profile Go 623, from Ludwieg. (a) Straight wing, b = 80 mm, cr = 22.5 mm; Re = ижсг/и = 3.0 * 10s at Ma«, = 0.7, = 3.5- IQ5 at Ma