Airfoil of Infinite Span in Hypersonic Flow

By taking into account the sirmlarity rules of Sec. 4-2-3, specific profile theories have been developed for flow about wing profiles (slender bodies) that depend on

the values of the incident flow Mach number. For Mam < 1 the subsonic flow is described in Sec. 4-3-2, for > 1 the supersonic flow in Sec. 4-3-3, and for Маж – 1 the transonic flow in Sec. 4-3-4. For very high Mach numbers of incident flow, that is, Мйоп > 1, the theory of supersonic flow does not lead to satisfactory results. For this case of incident flow with hypersonic velocity {Max > 4), a few statements on a profile theory of hypersonic flow will be made. First, the following considerations will be based on a slender profile, pointed in front.

Theory of small deflections in hypersonic flow Through a concave deflection by the angle d > 0, a compression flow is produced that can be computed according to the theory of the oblique shock. Conversely, an expansion flow is formed behind a convex deflection by the angle d < 0 that can be treated as a Prandtl-Meyer corner flow. The fluid mechanical quantities before and behind the deflection will be marked by the indices 1 and 2, respectively. The deflection angle is assumed to be small!#!<!, which means that the velocities before and behind the deflection differ only by a small perturbation velocity. The range of Mach numbers of the hypersonic flow considered here is Ma{ > 1 and Ma2 > 1. The pressure coefficients cp = Apjqx of the pressure change Ap – p7 —pb relative to the dynamic pressure before the deflection qx = (Qi/2)Ui, are obtained as [53]

d – <0 (d<0) (4-58Z?)

In either case, the pressure coefficient at small deflections of a hypersonic flow is given as

Cp = $2f(Ma1 її) (4-59)

where Max її is the similarity parameter of hypersonic flow. The parameter will be discussed later in more detail in connection with the hypersonic similarity rule.

For large values of Max її > 1, the expressions

cp = (7 + 1)тЯ (Max ~* °°) (4-6Qa)

= – /2- (-Магїї>—^) (4-60b)

у(Махїї)2 7-І)

are valid. The latter formula indicates that after deflection, vacuum (p2 = 0) is obtained for values of —Max її> 2/(y — 1). In Fig. 4-40, the pressure coefficient in relation to the square of the deflection angle ср/її2 is plotted as a function of the hypersonic similarity parameter Max її by curves 1 and 2. For comparison, the supersonic approximation of Eq. (443a) for high Mach numbers is

shown as curve 3. This approximation agrees better with the expansion flow than with the compression flow. The deviations are too large, however, to adopt this approximation as the pressure equation for hypersonic flow with small deflections.

Inclined flat plate in hypersonic flow By setting її = ±a in Eqs. (4-58a) and (4-58h), a being the angle of attack, the pressure distributions on the lower and upper surfaces of an inclined flat plate in hypersonic flow can be easily computed. They are constant over the chord. The lift is then obtained from the resultant pressure distribution of the lower and upper surfaces. The lift coefficient is obtained as

cL = a2F(Maaa a) (4-62a)

Figure 440 Pressure coefficients at hypersonic flow (7 = 1.4). (1) Expansion: lower sign, from Eq. (4-58b). (2) Compression: upper sign, from Eq. (4-58c). (3) Supersonic approximation from Eq. (4-61).

cl — (7 + l)**2 (Mdoo °°) (4-62b)

In Fig. 441, this result is presented for various Mach numbers of the incident flow Max = Mcioo according to Linnel [53]. It can be seen that the lift coefficient for a fixed angle of attack decreases sharply with increasing Mach number and that the hypersonic theory deviates from the supersonic theory. The curves for Маж = 0 (incompressible flow) and Max = 00 mark the limiting cases.

Hypersonic similarity rule Specific similarity rules were established in Sec. 4-2-3 for subsonic, transonic, and supersonic flows. With their help, flows about geometrically similar bodies can be related to each other. Such a similarity rule also exists for hyper­sonic flow. It was first presented by Tsien [98] and proved to be completely general

by Hayes [98]. The relation between pressure coefficient and deflection angle and Mach number is expressed in Eq. (4-59). For symmetric incident flow, the deflection angle is proportional to the thickness ratio t/c. In this case the Mach number Mat becomes the incident flow Mach number Max. Hence, in analogy to Eqs. (4-35) and (4-36), the following expressions are obtained for the pressure and drag coefficients:

cp = 62/ ^5Mzo=,^ (4-63)

cD = 53F(5 Mcioo) (4-64)

Hypersonic flow over a blunt profile The flow pattern in the vicinity of the nose of a body in hypersonic incident flow is sketched in Fig. 4-42. Keeping in mind the

Figure 441 Lift coefficient of the flat plate vs. angle of attack a for various Mach numbers (y = 1.4). Hypersonic theory for small angles of

attack according to Linnell. (——– ) Hypersonic

theory, Eq. (4-62a), Mam -*■ с/, = (7 + l)a2.

(——- ) Theory based on Eq. (446), Ma^ -*0:

cl — 27га.

Figure 442 Sketch of a hypersonic flow. Zone A: boundary layer with friction and rotation. Zone B: inviscid layer, but with rotation.

important fact that the leading edge of every body is somewhat—even if very little—rounded, it is obvious that a stagnation point always exists on the nose, and therefore a detached shock wave is formed upstream of the stagnation point in which the approaching hypersonic flow is abruptly reduced to subsonic flow. As a result, extremely high temperatures are produced near the stagnation point, which may lead to dissociation and ionization of the gas and thus to deviations from the properties of ideal gases. The thermic equation of state [Eq. (1-1)] is no longer valid, for instance, and the specific heat capacity cp does not stay constant either.

The dependence of the temperature rise that occurs near the stagnation point after passage of the shock wave on the Mach number is presented in Fig. 4-43 for air. The dashed line is valid for the ideal gas (see Fig. 4-2b) and the solid curves for a

Figure 443 Temperature rise behind normal shock vs. Mach number (temperature before the shock: Г» = 222 К). Curve 1: real gas for several values of the static pressure Curve 2: ideal gas (j — 1,4).

real gas at several values of the static pressure of the incident flow. Because of dissociation, the temperature rise at high Mach numbers is considerably smaller for real gases than for ideal gases.

At larger distances from the stagnation point the shock wave closely approaches the body contour. It is strongly curved, therefore, particularly near the stagnation point (Fig. 4-42). On the body contour itself, a friction (boundary) layer (range A) forms because of the viscosity, the thickness of which is now of the same order of magnitude, however, as the distance between shock wave and the outer edge of the boundary layer (range B). The formation of the boundary layer is governed by the pressure distribution on the body, which, at hypersonic incident flow, is determined mainly by the shape of the shock wave. This, in turn, depends on the body contour and its boundary layer. There prevails, consequently, a very strong interaction between friction layer and shock wave in hypersonic flow.

Another difficulty contributes to the problem. Since the shock wave is curved, the entropy increases in the shock wave are different for each streamline. These increases depend on the shock-wave inclination at the respective stations. Therefore, the flow behind the curved shock is no longer isentropic. This means that the flow behind the shock is no longer irrotational and that the separation into a rotational friction layer and an irrotational outer flow, customary in boundary-layer theory, is no longer possible. On the contrary, the total flow field between shock wave and body contour is now rotational. The friction effects, however, are of significance only in the zone next to the wall, zone A of Fig. 4-42, whereas zone В represents an inviscid, but not irrotational, layer. An important characteristic of hypersonic flow is its small lateral extent. Therefore the flow quantities vary strongly in the lateral direction, whereas they vary only little in direction of the incident flow.[26]

The computations of the flow about a body with a blunt leading edge, and particularly the computation of the shock-wave shape and of the pressure distribution. on the body, are very difficult, even when friction is disregarded, because the flow field contains, side by side, zones of hypersonic, supersonic, and subsonic flow.

In the special case (Mar»-*00, у 1), the incident flow would remain undisturbed up to the body contour and then be deflected in direction of the contour. Thereby a portion of the horizontal momentum would be transmitted to the body wall and thus produce the body drag. This special case is termed Newtonian flow because Newton based his theory for the drag of arbitrary bodies on this concept. It leads to the following expression for the pressure coefficient:

cp = 2 sin2 # (Newtonian approximation) (4-65)

with # being the deflection angled This relationship serves as a rough approxima­tion for the front portion of the body, whereas the above momentum consideration does not give an answer for the rear body portion. In this context the expression aerodynamic shadow is used.

The methods for the exact computation of hypersonic flows are very lengthy and can be handled only with modern electronic computers. Investigations in this field are still in progress, and many aerodynamic problems—particularly those including the deviations from the properties of ideal gases—are not yet completely solved.

Monographs in book form on hypersonic flow are listed in Section II of the Bibliography. Compare also Schneider [82].