Three-Dimensional Wings in Steady, Subsonic Flow

7- 1 Compressibility Corrections for Wings

This chapter deals with the application to finite, almost-plane wings of the linearized, small-perturbation techniques introduced in Chapter 5. By way of introduction, we first review the similarity relations which govern variations in the parameter M, the flight Mach number.

In the light of the asymptotic expansion procedure, the principal un­known, from which all other needed information can be calculated, is the first-order term* Ф? in the outer expansion for the velocity potential. The term Ф)’ is connected to the more familiar perturbation velocity potential <p(x, y, z) by (5-28). The latter is governed by the differential equations and boundary conditions (5-29)-(5-30), which we reproduce here (see also Fig. 5-1):

(1 M )<Pxx “b <Pyy “f" <Pzz ——– 0,

for (x, y) on S. (5-30)

The pressure coefficient at any point in the field, including the upper and lower wing surfaces z = 0±, is found from

Cp — 2 <px.

Extending a procedure devised by Prandtl and Glauert for two-dimen­sional airfoils (see Fig. 7-1), Gothert (1940) introduced a transformation of independent and dependent variables which is equivalent to

* The zeroth-order term is, of course, the free stream Фо = x.

124

Where /3 = /l — M2, as in Chapter 5. Equation (7-1) converts (5-29) into the constant-density perturbation equation

(<Po)x0x0 + (‘Po)y0Vo + (<Po)z0zo = 0. (7-2)

Some care must be observed when interpreting the transformed boundary condition at the wing surface. Thus, for example, the first of (5-30) states that just above the wing’s projection on the x, у-plane the vertical velocity component produced by the sheet of singularities representing the wing’s disturbance must have certain values, say Fu(x, y). After transformation, we obtain

(<Po)z0 = Fu(0x о, у о) s FUo(x0, Vo)

at z0 = 0—f—, for (x0, y0) on »S’o, where <S0 is an area of the x0, уо-ріапе whose lateral dimensions are the same as the original planform projection S, but which is stretched chordwise by a factor l/13. (See Fig. 7-2.)

Equation (7-3) and the equiva­lent form for the lower surface state, however, that the “equivalent ” wing in zero-M, constant-density flow has (at corresponding stations) the same thickness ratio t, fractional camber

в, and angle of attack a as the origi – ——–

nal wing in the compressible stream. _____

Fig. 7-2. Equivalent wing planform in zero-Mach-number flow. If sweep is present, tanAo = (1 //3) tan A. The aspect ratio is А о = /ЗА.

The similarity law might be abbreviated

where the semicolon is used to separate the independent variables from the parameters.

By way of physical explanation,[6] Gothert’s extended Prandtl-Glauert law states that to every subsonic, compressible flow over a thin wing there exists an equivalent flow of constant density liquid (at the same flight speed and free-stream ambient conditions) over a second wing, obtained from the first by a chordwise stretching 1//3 without change of surface slope distribution. It is obvious from (5-31) that pressure coefficients at corresponding points in the two flows are related by

(7-5)

Since they are all calculated from similar dimensionless chordwise and spanwise integrations of the Cp-distribution, quantities like the sectional lift and moment coefficients Ci(y), Cm{y), the total lift and moment co­efficients Cl, Cm, and the lift-curve slope дСь/да are found from their constant-density counterparts by the same factor 1//3 as in (7-5). It is of interest in connection with spanwise load distribution, however, that the total lift forces and running lifts per unit //-distance are equal on the two wings, because of the increased chordwise dimensions at M = 0.

Unfortunately, when one is treating a given three-dimensional configura­tion, the foregoing transformation requires that a different planform be analyzed (or a different low-speed model be tested) for each flight Mach number at which loading data are needed. This is not true for two-dimen­sional airfoils, since then the chordwise distortion at fixed a, etc., is no more than a change of scale on an otherwise identical profile; we have already seen (Section 1-4) that such a change has no effect on the physical flow quantities at fixed M.

Measurements like those of Feldman (1948) correlate with the Gothert – Prandtl-Glauert law rather well up to the vicinity of critical Mach number, where sonic flow first appears at the wing surface. They also verify what we shall see later theoretically, that the coefficient of induced drag should be unaffected by M-changes below Afcrit. There exist, of course, more accurate compressibility corrections based on nonlinear considerations which are successful up to somewhat higher subsonic M.

Inasmuch as (5-29) applies also to small-perturbation supersonic flow, M > 1, one might suspect that the foregoing considerations could be extended directly into that range. This is an oversimplification, however, since the boundary conditions at infinity undergo an essential change— disturbances are not permitted to proceed upstream but may propagate only downstream and laterally in the manner of an outward-going sound wave. (The behavior is connected with a mathematical alteration in the nature of the partial differential equation, from elliptical to hyperbolic or “wavelike.”) What one does discover is the existence of a convenient reference Mach number, M = /2, which plays a role similar to M = 0 in the subsonic case. When M = /2, the quantity В = y/M2 – 1 becomes unity and all flow Mach lines are inclined at 45° to the flight direction. Repetition of the previous reasoning leads to a supersonic similarity law

<p(x, V, 2; M, А, г, 0, a) = <P (jj ‘ V, 2; M’ = л/2, BA, T, 0, a) . (7-6)

Pressure coefficients at corresponding points, lift coefficients, etc., are related by

CP = (Cp)*_* (7-7)

Once more the equivalent planform at M = /2 is obtained from the original by chordwise distortion, but now this involves a stretching if the original M < /2 and a shrinking if M > y/2. The process has been likened to taking hold of all Mach lines and rotating them to 45°, while chordwise dimensions vary in affine proportion.

Clearly, Eqs. (7-5), (7-7), and the associated transformation techniques fail in the transonic range where M ^ 1. It has been speculated, because the equivalent aspect ratio approaches zero as M —> 1 and slender-body – theory results for lift are independent of Mach number (Chapter 6), that linearized results for three-dimensional wings might be extended into this range. This is, unfortunately, an oversimplification. Starting from the proper, nonlinear formulation of transonic small-disturbance theory, Chapter 12 derives the actual circumstances under which linearization is permissible and gives various similarity rules. It is found, for instance, that loading may be estimated on a linearized basis whenever the param­eter At1/3 is small compared to unity.

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