Transonic Small-Disturbance Flow

11- 1 Introduction

A transonic flow is one in which local particle speeds both greater and less than sonic speed are found mixed together. Thus in the lower transonic range (ambient M slightly less than unity) there are one or more super­sonic regions embedded in the subsonic flow and, similarly, in the upper transonic range the supersonic flow encloses one or more subsonic flow regions. Some typical transonic flow patterns are sketched in Fig. 12-1. Since in a transonic flow the body travels at nearly the same speed as the forward-going disturbances that it generates, one would expect that the flow perturbations are generally greater near M = 1 than in purely subsonic or supersonic flow. That this is indeed so is borne out by experi­mental results like those shown in Figs. 12-2 and 12-3, which show that the drag and lift coefficients are maximum in the transonic range. In the early days of high-speed flight, many doubted that supersonic aeroplanes could ever be built because of the “sonic barrier,” the sharp increase in drag experienced near M equal to unity.

Many of the special physical features, and the associated analytical difficulties, of a transonic flow may be qualitatively understood by con­sidering the simplest case of one-dimensional fluid motion in a stream tube. Combination of the Euler equation

and with the equation of continuity

і (pm = o,

where S(x) is the stream-tube area, yields after some manipulation

dp dS

pU2 S[1 – (C72/a2)]

This relation shows that for U/а — 1 the flow will resist with an infinite force any stream-tube area changes, i. e., it will effectively make the flow incompressible to gross changes in the stream-tube area (but not to curva­ture changes or lateral displacement of a stream-tube pattern). There­fore, the crossflow in planes normal to the free-stream direction will tend to be incompressible, as in the case of the flow near a slender body, so that much of the analysis of Chapter 6 applies in the transonic range to con­figurations that are not necessarily slender. This point will be discussed further below. It is evident that because of the stream-tube area constraint, there will be a tendency for a stronger cross flow within the stream tube and hence the effect of finite span will be maximum near M = 1. From (12-1) it also follows that in order to avoid large perturbation pressures and hence high drag one should avoid large (and sudden) cross-sectional area changes, which in essence is the statement of the transonic area rule discussed in Chapter 6. For the same reason one can see that the boundary layer can have a substantial influence on a transonic pressure distribution, since it provides a region of low-speed flow which is less "stiff” to area changes and hence can act as a “buffer” smoothing out area changes.

From such one-dimensional flow considerations, one practical difficulty also becomes apparent, namely that of wind tunnel testing at transonic speeds. Although a flow of M = 1 can be obtained in the minimum-area section of a nozzle with a moderate pressure ratio, the addition of a model, however small, will change the area distribution so that the flow no longer will correspond to an unbounded one of sonic free-stream speed. This problem was solved in the early 1950’s with the development of slotted – wall wind tunnels in which the wall effects are eliminated or minimized by using partially open walls.

The main difficulty in the theoretical analysis of transonic flow is that the equations for small-disturbance flow are basically nonlinear, in con­trast to those for subsonic and supersonic flow. This again may be sur­mised from equations like (12-1), because even a small velocity change caused by a pressure change will have a large effect on the pressure-area relation. So far, no satisfactory general method exists for solving the transonic small-perturbation equations. In the case of two-dimensional flow it is possible, through the interchange of dependent and independent
variables, to transform the nonlinear equations into linear ones in the hodograph plane. However, solutions by the hodograph method have been obtained only for special simple airfoil shapes and, again, two-dimen­sional flow solutions are of rather limited practical usefulness for transonic speeds. For axisymmetrie and three-dimensional flow, various approxi­mate methods have been suggested, some of which will be discussed below.

11- 2 Small-Perturbation Flow Equations

That the small-perturbation theory for sub – and supersonic flow breaks down at transonic speeds becomes evident from the linearized differential equations (5-29) and (6-21) for the perturbation potential, which in the limit of M —> 1 become

<Pzz = 0, for two-dimensional flow, (12-2)

і (pr + tfrr = 0, for axisymmetrie flow. (12-3)

Thus, both the inner and outer flows will be described by the same differ­ential equation, and it will in general not be possible to satisfy the boundary condition of vanishing perturbation velocities at infinity. For transonic flow it will hence be necessary to consider a different expansion that retains at least one more term in the equation for the first-order outer flow.

In searching for such an expansion we may be guided by experiments. By testing airfoils, or bodies of revolution, of the same shape but different thickness ratios (affine bodies) in a sonic flow one will find that, as the thickness ratio is decreased, not only will the flow disturbances decrease, as would be expected, but also the disturbance pattern will persist to larger distances (see Fig. 12-4).

This would suggest that the significant portion of the outer flow will recede farther and farther away from the body as its thickness tends towards zero. In order to preserve, in the limit of vanishing body thick­ness, those portions of the outer flow field in which the condition of vanishing flow perturbations is to be applied we must therefore “compress ” this (in the mathematical sense). Taking first the case of a two-dimen­sional airfoil with thickness but no lift, we shall therefore consider an expansion of the following form:

where S(x) is the cross-sectional area. The outer flow must therefore be equal to that around the equivalent body of revolution as in the slender- body case, and we have thus demonstrated the validity of the transonic equivalence rule resulting from the form of the first-order term in an asymptotic series expansion as the disturbance level «, and M2 — 1 [, both tend to zero. The approach followed is essentially that taken by Messiter (1957). A similar derivation was given by Guderley (1957).

There is no requirement on aspect ratio except that it should be finite so that 8 A —> 0 as e —> 0, in order for the outer flow to be axisymmetric in the limit. A consequence of this is that slender-body theory should provide a valid first-order approximation to lifting transonic flows for wings of finite (and moderate) aspect ratios. In Fig. 12-3 the slender-body value for the lift coefficient is compared with experimental results for a delta wing of A = 2. It is seen that the agreement is indeed excellent at M = 1.

For a wing of high aspect ratio, the first-order theory will provide a poor approximation for thickness ratios of engineering interest. A different expansion is then called for, which does not lead to an axisymmetric outer flow. We therefore introduce in the outer expansion

V = 8y, f = 8z, (12-38)

with 8 chosen as before, (12-11). This then gives the following equation for the first-order outer term:

The matching will prescribe the normal velocity on the wing projection on f = 0, which in view of (12-38) will have all spanwise dimensions reduced by the factor S. Thus, if the limit of e —» 0, and hence 5 —> 0,

is taken with the aspect ratio A kept constant, the projection in the

x, y-plane will have a reduced aspect ratio AS that will shrink to zero in the limit, and the previous case with an axisymmetric outer flow is then recovered. In this case we therefore instead consider the limit of e —> 0 with A —» oo in such a manner that

AS = K2 (12-40)

approaches a constant. The reduced aspect ratio will then be finite and equal to K2. The matching procedure now parallels that for the two­dimensional flow and the choice (12-17) for e gives the boundary condition

Фн(*. V, 0±) = ± U (12-41)

to be satisfied on the wing projection of reduced aspect ratio

K2 = t1I3AM2,3( 7 + 1)1/3 (12-42)

Thus, the solution in this case depends on two transonic parameters K1 and K2. The previous case, for which the transonic equivalence rule holds, may be considered the limiting solution when

K2 -» 0.

As the approach to zero is made the solution defined by (12-39) and (12-41) becomes, in the limit, proportional to K2nK2. The two-dimensional case described by (12-12)-(12-14), or (12-19)-(12-21), is obtained as the limit of

K2 —» oo.

The most general transonic small-perturbation equation is thus

[1 — M2 — M2{ 7 + 1 )<pz]<pxx + <Pyy + <Ргг = 0, (12-43)

with the boundary condition in case of a thin wing

4>z(x, y, 0±) = ±r ~ on wing projection Sw. (12-44) ox

The pressure is given by (12-21) for a thin wing and by

Cp = —2ipx – <p2y – v2 (12-45)

for a slender configuration.

12- 3 Similarity Rules

The first-order terms in the series expansion considered above provide similarity rules to relate the flow around affine bodies.[9] Taking first the two-dimensional case, we see from (12-18) that the reduced pressure coefficient

я [My+ 1)]1/3

Ьр ~ т2/3 Ьр

must be a function of x/c (c = chord) and the parameter

M2 – 1 M2 – 1

fM2(T + 1) ~~ [М2т(У + l)]2/3

only. This conclusion follows because the solution must be independent of scale (see Section 1-4) and К i is the only parameter that enters the boundary value problem defined by (12-12) and (12-15).

The total drag is obtained by integrating the pressure times the airfoil slope, which leads in a similar way to the result that

[M2{ 7 + 1)]1/3„


must be a function of Kx only. The additional factor of r enters because the slope is proportional to r. Of course, (12—48) holds only for the wave drag, so that in order to use it to correlate measurements, the friction drag must be subtracted out. Such an application to biconvex-airfoil drag measurements by Michel, Marchaud, and LeGallo (1953) is illustrated in Fig. 12-5. As may be seen, the drag coefficients for the various airfoil thicknesses, when reduced this way, fall essentially on one single curve, thus confirming the validity of (12—48), and hence the small-perturbation equations.

Transonic similarity rules for two-dimensional flow were first derived by von Kdrmdn (1947b) and Oswatitsch (1947). These rules also included the lifting case.

Rules for a slender body of revolution were formulated by Oswatitsch and Berndt (1950). There is an additional difficulty in this case associated with the logarithmic singularity at the axis. It follows from the formula­tion (12-26)-(12-30) for the outer flow and the matching to the inner flow
as given by (12-28) that, near the body,

$i = ^S'(aO lnp + £i(z).

The transonic parameter

M2 – 1 _ M2 – 1

Kl ~ eM*{7 + 1) M2r2(7 + 1)

thus enters into (ji only. By using (12-33) to calculate the pressure coeffi­cient we find that, on the body,

Cp = —2T2 {^S"(x) In [t2MVt~+7 R] + &'(*) + І(й’)2} * (12-51)

where R(x) = R(x)/t. Thus,

Cp = {— Cp + ^ S"(x) In [т2Мф + l]J (12-52)

is a function of K and x/l only. An application of (12-52) to correlate the measured pressures on two bodies of different thickness ratios, carried out by Drougge (1959), is shown in Fig. 12-6. As may be seen, the correla­tion is almost perfect, except at the rearmost portions of the bodies where boundary layer separation occurs.

From (12-52) one can also construct an expression for the drag, as shown in the original paper by Oswatitsch and Berndt (1950). They found that ^ 1

D = t D– + ~ [S'(l)]2 In [t-WyTT] (12-53)

2P« UQoT

must be a function of Ki only.

Fig. 12-6. Correlation of pressure measurements on two bodies of revolution using the transonic similarity law (12-52). P = Cp + (l/ir)S" In (7 + 1). (Adapted from Drougge, 1959. Courtesy of Aeronautical Research Institute of Sweden.)

It is a fairly straightforward matter to construct corresponding simi­larity rules for configurations of low-to-moderate aspect ratios. For wings of large aspect ratios one obtains results of the same form as for the two­dimensional case, except that now the reduced quantities depend on the second transonic parameter

K2 = t1/3AM2/3( у + 1)1/3

as well as on K. The rules for three-dimensional wings were derived by Berndt (1950) and by Spreiter (1953).