Profile Theory of Transonic Flow

Both approximation theories for subsonic and supersonic flows discussed in Secs.

4- 3-2 and 4-3-3 fail when the incident flow velocity approaches the speed of sound. In this case the flow becomes of the mixed type; that is, both subsonic and supersonic velocities exist in the flow field. At certain points the flow therefore passes the speed of sound. In transonic flow fields of this kind, shock waves are formed in most cases, and theoretical treatment is made much more difficult.

Drag-critical Mach number First, the limiting Mach number should be established up to which the theory of subsonic flow of Sec. 4-3-2 is still valid. In the case of a wing profile at subsonic incident flow velocity {Маж < 1), Fig. 4-13a demonstrated that the lift slope can no longer be described by the linear theory at higher subsonic Mach numbers. The results on the neutral-point position of Fig. 4-136, and in particular those on the drag coefficients of Figs. 4-14 and 4-15, confirm this fact, which is caused by flow separation on the profile. Depending on the profile shape (thickness ratio, camber ratio, nose ratio) and the angle of attack, a critical Mach number Ma„cr can be established up to which no significant flow separation occurs. This will be designated as the drag-critical Mach number. It can be defined, for
instance, as the Mach number at which the drag coefficient cD is higher by ЛcD = 0.02 than at Маж = 0.6.

The physical reason for flow separation at higher subsonic Mach numbers is that shock waves are formed when sonic velocity is reached locally on the profile and exceeded over a certain range. The critical Mach number Ma„CT is understood, therefore, to be the Mach number of the incident flow at which sonic velocity is reached locally on the profile. The critical pressure coefficient at the critical Mach number Мажст is cpCT. The critical Mach number Mz«cr is obtained by setting for cpcr the highest underpressure сртїп that occurs at the body. For slender bodies, cpmin is small and МаЖСТ is close to unity. In this case, based on streamline theory of compressible flow, neglecting higher-order terms, cpCT becomes

2 1 – Mai

7 + 1 Mai, cr

Cp min

From Eq. (4-38), cDmin is a function of Mach number. Introducing Eq. (4-38) into Eqs. (4-53a) and (4-53b) yields

In Fig. 4-28, cpcr from Eq. (4-53a) is shown versus as curve 1. For a given

wing profile, Maeccr is determined by the intersection of curve 1 according to Eq. (4-53&) with curve 3 according to Eq. (4-38); see also Fig. 4-16. More simply, MzooCr can be obtained by starting from Eq. (4-54). This relationship is given as curve 2. .

The value of Cpmin depends strongly on the profile shape and the angle of attack. It is obtained from the velocity distribution of potential flow with Cpmin =—2wmax/£/«,. The maximum pressures for various profiles in incompressible flow are plotted in Fig. 2-34 against the thickness ratio. The critical Mach numbers for chord-parallel flow are shown in Fig. 4-29 for several profiles as functions of

Figure 4-28 Illustration of determination of drag-critical Mach number MaXCI of a wing profile; 7= 1.4. Curve 1 from Eq. (4-53), curve 2 from Eq. (4-54), curve 3 from Eq. (4-38).

profile thickness 5 = t/c and relative thickness position Xt = xt/c. As would be expected, the critical Mach number decreases sharply with increasing thickness ratio for all profiles.

Physical behavior of transonic profile flow When a wing profile is exposed to an incident flow velocity high enough to form areas of local supersonic velocity in its vicinity, shock waves are formed in the ranges where the velocity is reverted from supersonic to subsonic. In these shock waves, pressure, density, and temperature change very strongly. The strong pressure rise in the shock wave frequently leads to flow separation and consequently to a complete change of the flow pattern. This effect causes a strong increase in the drag (pressure drag).

To demonstrate these processes, the pressure distribution on a wing profile is given in Fig. 4-30д for various Mach numbers from measurements in reference [89]. The pressure distribution is steady for Mach numbers at which the maximum velocity on the profile contour is everywhere smaller than the local sound speed, wc<a. In the present case, this holds up to Маж ~ 0.6. Up to Mam «0.6 the pressure rise at the rear end of the profile is as steady as the pressure drop is in front. For higher Mach numbers, Маж > 0.7, at which the sonic velocity is exceeded locally, wc > a, the pressure rise behind the pressure minimum occurs unsteadily in a shock wave. The height of the pressure jump increases with Mach number. This abrupt pressure rise is very undesirable with respect to the boundary layer, which tends to separate even at a steady pressure rise. In most cases, the shock wave causes separation of the flow from the wall and thus a strong drag rise, as is obvious from the curve of the drag coefficient versus Mach number of Fig. 4-30b see also Figs. 4-14 and 4-15.

In Fig. 4-31, a Schlieren picture and an interferometer photograph from Holder [33] are shown of a wing of angle of attack a — 8° in a flow field of Mam = 0.9. The formation of the shock wave and a strong separation immediately behind the shock are clearly noticeable.

The flow pattern in the transonic velocity range, which is, in general, quite complicated, is displayed schematically in Fig. 4-32 for a biconvex profile in symmetric incident flow. Pressure distributions and streamline patterns are given over a range of increasing Mach number. Figure 4-32a represents the incompressible case, Fig. 4-32b the subsonic case in which the “sonic limit” has not yet been exceeded anywhere. Figure 4-32c-e demonstrates the formation of the shock wave after the “sonic limit” of the pressure distribution (critical pressure) has been passed. Figure 4-32f and g represents the typical pressure distribution of supersonic flow that was previously shown in Fig. 4-24.

The formation of shock waves in the transonic range also has a strong effect on the lift. This is demonstrated schematically in Fig. 4-33, in which the solid curve represents a typical measurement of the relation between lift coefficient and Mach number, whereas the dashed line corresponds to the linear theory according to Fig.

4- 20a. For a better understanding of the measured lift curve, the positions of the shock wave and the velocity distributions on the profile for the points А, В, C, D, and E are shown in Fig. 4-34. At Mach number Ma^ — 0.75 (point A), a shock wave does not yet form because the velocity of sound has not been exceeded

Figure 4-30 Measurements on a wing profile at subsonic incident flow from [89], angle of attack a = 0°. (a) Pressure distribution at various Mach numbers, (b) Drag coefficient vs. Mach number.

Figure 4-33 Lift coefficient of wing vs. Mach number. Solid curve: typi­cal trend of measurements. Dashed curve: theory according to Fig,

4-20c.

Figure 4-34 Transonic flow over a wing profile at various Mach numbers; angle of attack a= 2°, from Holder. The points А, В, C, D, and E correspond to the lift coefficients of Fig. 4-33. (a) Position of shock wave, (b) Velocity distribution on profile.

significantly on either side of the profile. Up to this Mach number, the flow is subsonic and the lift follows the linear subsonic theory (Prandtl, Glauert). At Ma„ = 0.81 (point B), the velocity of sound has been exceeded significantly on the front portion of the profile upper surface. A shock wave at the 70% chord is the result. The lower surface is still covered everywhere by subsonic flow. Up to point В, the lift increases with Mach number. At Mach number 0.89 (point C), the velocity of sound is also exceeded over a large portion of the lower surface. A shock wave therefore forms on the lower surface near the trailing edge. This changes the velocity distribution over the profile considerably, resulting in a marked lift reduction. At Mach number Ma^ = 0.98 (point D), the two shock waves on the upper and lower surfaces are considerably weaker than at Mz» = 0.89 and are located at the trailing edge. The lift, therefore, is again larger than at point C. Finally, at Маж = 1.4 (point E), pure supersonic flow has been established with a velocity distribution typical for supersonic flow. The magnitude of the lift now corresponds to the linear supersonic theory (Ackeret).

All tests indicate that the processes in the shock wave are markedly affected by the friction layer. This interaction between shock wave and boundary layer is, besides other effects, particularly complicated because the behavior of the boundary layer changes with Reynolds number, but on the other hand, the shock wave depends strongly on the Mach number. Above a certain shock strength, the pressure rise in the shock causes boundary-layer separation which, in addition to the drag rise already discussed, leads to strong vibrations as a result of the nonsteady character of this flow. This phenomenon is also called “buffeting” in aeronautics; see, for example, Wood [109]. Both the Mach numbers of sudden drag rise and of buffeting are influenced by the profile shape and the angle of attack a (see Fig.

4- 35). The so-called buffeting limit restricts the Mach number range for safe airplane operation. By increasing the incident flow Mach number to supersonic velocities, the shock moves to the wing trailing edge and the buffeting effects disappear again. For very thin and slightly inclined profiles, this state can be reached without the shock’s gaming sufficient strength to excite buffeting while it is moving over the profile. The individual phases of the flow in Fig. 4-3 5a are explained by the pressure distributions of Fig. 4-356.

Because of the complicated flow processes above the critical Mach number, a strictly theoretical determination of the buffeting limit is not possible. However, Thomas and Redeker [109] developed a semiempirical method for the determina­tion of the buffeting limit; see Sinnott [84]. A comprehensive experimental investigation of this problem, which is most important for aeronautics, has been reported in detail by Pearcey [69] and Holder [33].

Similarity rule for transonic profile flow So far, analytical determinations of transonic flows with shock waves have succeeded only in a few cases. In some cases, however, a steady transition through the sonic velocity (without shock waves) has also been observed. In this latter case, transonic flows can be treated theoretically by means of an approximation method. They lead to similarity rules for pressure distribution and drag coefficient (Sec. 4-2-3) that are in quite good agreement with

measurements. It can be shown that the transonic similarity rule remains valid even when the flow includes weak shock waves.

Between pressure distribution and drag coefficient of wing profiles of various thickness ratios tjc and at various transonic Mach numbers of the incident flow (Mfloo 1), the following expressions are valid according to reference [103], and extend Eqs. (4-35) and (4-36):

Here, cp is called the reduced pressure coefficient, and cq is the reduced drag coefficient. For the special case Маж = 1 (sonic incident flow), тж — 0 from Eq. (4-57). From this it follows immediately that the pressure coefficient cp is proportional to (t)cf’z in this case and the drag coefficient proportional to {tfcf/3 [see Eqs. (4-35) and (4-36), respectively].

Malavard [103] checked the similarity rules, Eqs. (4-55)-(4-57), in compre­hensive experiments. He clearly verified the transonic similarity rule for pressure distribution and drag coefficient of symmetric biconvex profiles of thickness ratios i/c = 0.06-0.12 at chord-parallel flow of incident Mach numbers of Маж = 0.775-1.00. Plotting of the drag coefficient cD against the Mach number in Fig.

4- 36a shows the well-known strong drag rise near Ma«, = 1 and, moreover, the strong increase of this rise with the thickness ratio tjc.

Theories for the computation of transonic profile flows The transonic profile flow with shock waves can be treated only by nonlinear theory, in contrast to the linear theories of subsonic and supersonic profile flows. There exist numerous trials and methods for the solution of this task. A survey of the more recent status of understanding of theory and experiment for transonic flow is given by Zierep [111]. So far, the hodograph method, the integral equation method, the parabolic method, and the method of characteristics have been applied to computations. Guderley uses mainly the hodograph method, Oswatitsch generally prefers the integral equation method. The many publications quoted in [63, 66, 79, 84-87, 111] show that no generally valid solution has been found for the computation of the pressure distribution of wings on which shock waves form at transonic incident flow. More recent progress has been discussed at the two Symposia Transsonica [67].

Supercritical profiles For wing profiles operating at high subsonic flight velocities, the drag-critical Mach number Мажсr according to Figs. 4-14a and 4-29 can be shifted to higher values by reducing the profile thickness ratio or by lowering

Figure 4-36 Drag measurements on symmetric profiles in the transonic velocity range at chord-parallel incident flow, from Malavard. (a) Drag coefficient cjj vs. Mach number Max for symmetric profiles of various thickness ratios tjc. (jb) Reduced drag coefficient cq from Eq. (4-56) vs. reduced Mach number rkx from Eq. (4-57) for symmetric profiles of various thickness ratios tjc.

the profile lift coefficient.* Profiles at which the critical pressure coefficient cp cr from Eq. (4-53a) has not yet been exceeded or has just been reached on the suction side (profile upper side) are termed subcritical profiles. On them no shock waves form, and therefore no shock-induced flow separation occurs. Through suitable profile design, local areas of supersonic flow can be created on the profile in which recompression to subsonic flow occurs steadily or in weak shock waves only. On these profiles the pressure rise in the recompression zone is gradual and therefore does not cause flow separation. Transonic profiles designed according to the stated criterion are termed supercritical profiles.

A few more statements should be made about the evolution from subcritical to supercritical wing profiles. In many designs the product of lift-to-drag ratio and Mach number must be optimized. This request may roughly be transferred to the aim to achieve for a given profile thickness ratio at the design Mach number the highest possible lift at fully attached flow conditions. By starting with the pressure distribution la in Fig. 4-37 found on the suction side of the conventional NACA 64A010 profile a gain in lift first may be obtained by further upstream and downstream extension of the minimum suction pressure just along its critical value

*The feasibility of increasing the drag-critical Mach number by sweeping back the wing will be discussed in Sec. 4-44. Figure 4-37 Pressure distributions of various wing profiles, (a) Suction side (upper surface), (b) Pressure side (lower surface). (1) Conventional profile NACA 64A010 at Max = 0.76, a. = 1.2°, measurements of Stivers [65]. (2) Roof-top profile. (3) Supercritical profile of thickness ratio f/c = 0.118 with “rear loading,” from Kacprzynski [65]. Theory: Mz<» = 0.75, cjr = 0.63. Measurements: Маж = 0.77, су, = 0.58.

Figure 4-38 Comparison of the contour of a supercritical profile with a conventional profile (NACA 64, A212), thickness ratio tjc = 0.12.

according to curve 2a. Such profiles are called “roof-top profiles.” In the range of the profile nose, a strong acceleration of the flow is required, which is accomplished by increasing the nose radius. The onset of the recompression needed to match the pressure at the profile trailing edge (pressure at the rear stagnation point in inviscid flow) must be chosen to allow establishment of a pressure gradient over the rear portion of the profile that does not cause flow separation. Chordwise linear recompression according to curve 2a has been found to be good in practical applications. A further marked increase in lift is obtained by admitting a local supersonic flow field on the profile suction side, which means choosing pressure distributions exceeding the critical pressure coefficient. That kind of flow implies a further increase in nose radius, and, in addition, a flattening of the upper surface. In this case, an essentially shock-free or weak shock pressure distribution along the profile chord, allowing recompression without separation, curve 3a, is of decisive importance. The pressure distribution over the rear portion of the pressure side of conventional profiles is little different from that on the suction side (curves la and lb). Thus, the rear portion of such profiles contributes little to the lift. A larger difference in the pressure distribution of upper and lower side, curves la and 3b, is obtained through changing the profile lower contour between the range of maximum thickness and the trailing edge such that a reduced local thickness is obtained. This change means, according to Fig. 4-38, the establishment of a corresponding profile camber. Measures of that kind are known as “rear loading.” At such profile designs, caution is necessary to avoid flow separation in the recompression region, precisely as it was required on the suction side.

A comparison of the geometries of a subcritical and supercritical profile with “rear loading” and thickness ratios tjc = 0.12 is shown in Fig. 4-38. Systematic investigations on profiles with shock-free recompression from subsonic to supersonic flow have been made by Pearcy [69]. The first design intended to produce shock-free supercritical profiles, so-called quasi-elliptic profiles, was conducted by Niewland [65] and confirmed in the wind tunnel (Fig. 4-39). Since then, a number of generally applicable design methods for supercritical profiles have been developed, and profile families have been checked out successfully in the wind tunnel [4, 54, 55].