Similarity Rules for Wing Theory at Compressible Flow

Velocity potential (linearization) For slender body shapes (wings) in an incident flow of velocity Uaz in the direction of the x axis (longitudinal axis), the local velocities differ only a little from (Jm in direction and magnitude. The total flow can then be separated into a basic flow and a superimposed perturbation flow by setting

U = – p и V = v W — v: (44)

where u, v, w are the perturbation velocities caused by the wing, for which it is required that

и < Ux v <: w <4

By retaining only the largest terms (linearization), the potential equation of compressible flow for such a flow problem takes the form

where Ma — Ufa is the local Mach number. This equation is valid for subsonic, transonic, and supersonic flow.[24] It is nonlinear in the velocity potential. The components of the perturbation velocities are obtained from Eq. (4-5) as

(4-6)

In analyzing the linearized Eq. (4-5) further, the first term requires particular attention because it changes sign when passing the speed of sound (Ma = 1) and thus changes the mathematical character of the differential equation. By retaining only the linear terms in ufU^, the local Mach number Ma in Eq. (4-5) may be expressed by the Mach number of the incident flow Mz« = U^ja as follows:

(4-7)

For pure subsonic and pure supersonic flow, the simplified potential equation

is obtained by replacing Ma by Ma*, in first approximation.

This differential equation for Ф is now linear. For pure subsonic flow, it is of

the elliptic type, as is the equation for incompressible flow. For pure supersonic flow, however, it is of the hyperbolic type.

When the undisturbed flow velocity is equal to the speed of sound {Mam = 1), a transonic flow results whose velocity field may include stations of Ma = 1. In this case, from Eqs. (4-5) and (4-7), it follows that

This differential equation for Ф is nonlinear. Therefore, the computation of transonic flow fields is considerably more difficult than computations of pure subsonic and pure supersonic fields. The potential equations derived above [(4-8) and (4-9)] will now be used to derive similarity rules for three-dimensional wing theory at subsonic, supersonic, and transonic flows.

Similarity rules for subsonic and supersonic flow For subsonic flows, similarity rules can be derived from Eq. (4-8) according to Prandtl [73], Glauert [27], and Gothert [28], the application of which greatly simplifies the computation of compressible potential flows. These procedures can be applied similarly to supersonic flows; see Ackeret [1]. Of the various possible derivations of these similarity rules, the so-called streamline analogy will be applied.

The similarity rules for a wing in subsonic or supersonic incident flow (Mг» ^ 1) are obtained through a transformation of the potential equation, Eq. (4-8). This transformation is such that the Mach number of the undisturbed flow no longer appears explicitly in the transformed – potential equation. To this end, a transformed reference flow is established for the given flow in a suitable way. The variables of the reference flow are designated by a prime:

x’ =X y’ = cxy z’ = cxz Ф = с2Ф’ U’ac = (4-10)

By introducing these terms into Eq. (4-8), the factor cx > 0 is determined in such a way that the Mach number is eliminated, resulting in

c, = Іl – Ma% Wa„ < 1) (4-1 la)

c, = ]/Ма%, – 1 (ifa„ > 1) (4-1 lb)

These cases can be combined to

cx = У|1 – МаІ! (4-12)

With Eq. (4-11) for the transformed reference flow, the following differential equations are obtained for the velocity potential:

(4-13)

(4-14)

The transformed equation for subsonic flow is identical to the potential equation for incompressible flow, and the transformed equation for supersonic flow is identical to the linear potential equation, Eq. (4-8), at Mach number Маж =J2. This transformation shows that computations of subsonic flows of any Mach number can be reduced to computations of flow at Mz« = 0, and computations of supersonic flows to those at Ma^ = Jl.

The transformation factor сг in Eq. (4-10) remains undetermined for the time being. Its value will be given later.

Application of the transformation formulas, Eq. (4-10), to wings of finite span will now be treated. The coordinate system x, y, z of Fig. 4-6 will be used with its X axis parallel to the incident (undisturbed) flow. Equation (4-10) describes the procedure for determining the transformed wing from a given wing of a given Mach number where the flow field about the transformed wing is to be computed, according to the above rules, for subsonic flow at Ma^ = 0 and for supersonic flow at Mdoo =/2. The transformed wing according to Eq. (4-10) is then obtained from the given wing by decreasing or increasing, respectively, the dimensions in the directions normal to the incident flow direction (y and z directions) by the factor d of Eq. (4-12).

•For the wing planform, the following relationships between the transformed

(primed symbols) and	the given data are thus obtained:
Taper:	)! = Я	(4-15a)
Aspect ratio:	Л’ = A tjl – Jfa4!	(4-15b)
Sweepback angle:	cot <p’ — cot 99 УІ і — Main 1	(4-15 c)

Figure 4-6 Wing geometry, (a) Wing plan – form; ctlcri, taper; л~Ь2/А, aspect ratio; A = wing planform area; ip= sweep – back angle. (&) Profile section у = const; Z(ix) = profile contour; h/c = relative cam­ber; f/c = relative thickness; a. = angle of attack.

Figuie 4-7 Application of subsonic and supersonic similarity rules to the example of a tapered, swept-back wing, (a) Given wing, to be computed for Mach numbers Maoo = 0.7, 0.9, 1.1, and 2. (b) Trans­formed wing for these Mach numbers.

From Eqs. (4-15b) and (4-15c), the remarkable relationship

A’ tan <p’ — A tan <p (4-15d)

is obtained, where it is immaterial to which of the pianformcontour lines the sweepback angle is referred, for example, the leading edge or the trading edge.

In Fig. 4-7 the transformation of the wing pianform is explained through the example of a swept-back wing. The crosshatched wing pianform in Fig. 4-7a is the shape of the given wing, the flow field of which is to be determined for the various Mach numbers Max = 0.7, 0.9, 1.1, and 2.0. The corresponding transformed wing planforms are shown in Fig. A-lb, where at Мг« <1 the transformed wings are to be computed for incompressible flow (Маж = 0), and at Маж > 1 for Ma*, = J2.

In Fig. 4-8, the wing pianform transformation as given by Eqs. (4-15a)-(4-l 5c) is explained in more detail. Here, A’/A and cotej’/cotip are plotted versus Маж. Again, the given wing pianform, which is to be computed for the various Mach numbers, has been crosshatched. The open wing planforms represent the trans-

Figure 4-8 Illustration of the applica­tion of subsonic and supersonic similar­ity rules; aspect ratio A and sweepback angle у of the transformed wing vs. Mach number.

formed wings for the corresponding Mach numbers. When the given Mach numbers are Afaoo = 0 and Mz» = y/2, respectively, the transformed and the given wings are identical. Figure 4-8 shows that, in the subsonic range, an increase of Mr» results in a decrease of the aspect ratio whereas the sweepback angle increases. For Mz« 2, the aspect ratio of the transformed wing approaches A! -> 0 and the sweepback angle ip’->90°. In the supersonic range Mr» >y/2, the aspect ratio of the transformed wing increases with Mr» whereas the sweepback angle decreases. In the limit of very large Mr», the aspect ratio of the transformed wing Л’ -*00 and the sweepback angle 0. The remarkable result is found that for large Mach numbers the three-dimensional wing flow field is converted into a two-dimensional field.

The Prandtl-Glauert-Gothert-Ackeret rule is also applicable to asymmetric incident flow (yawed wings); see Truckenbrodt [28]. For the profile cross section and angle of attack of Fig. 4-6b, Eqs. (4-10) and (4-12) lead to the following expressions:

Camber:	^r=-V|l-MdI c c	(4-16rr)
Thickness ratio:	– Vll-Mri! c c	(4-166)
Angle of attack:	ql = a Vll — MrLl	(4-17)

This shows that for Mr» < /2, the transformed wing has less camber, is thinner, and has a smaller angle of attack than the given wing; conversely, for Mr» >y/2 it has more camber, is thicker, and has a larger angle of attack.

After the effect of the transformation, Eq. (4-10), on the wing geometry has been discussed, the relationship between the pressure distributions of the given and the transformed wing must be studied.

The dimensionless pressure coefficients cp = (p — p«)/(p«.i7»/2) assume, within the framework of linearization, the approximate form

(4-18 a)

(4-Ш) where the velocities of the incident flow £/«. of the given and transformed flow must be equal.

This leads with Eq. (4-10) directly to

cp = сгс’р (4-19)

The still-unknown transformation factor c2 is determined from the kinematic flow conditions for the two wings (streamline analogy). These are, within the framework of linearized theory,

W = U°°1? w’=V°°l5 (4’20)

where w and w’ are the z components of the perturbation velocity on the profile contour zc and zc, respectively (Fig. 4-6b). Because w= дФ/dz and w’ = ЪФ’jbz, we find with Eq. (4-10):

(4-21)

The meaning of the subsonic and supersonic similarity rules can now be summarized as follows: From the given wing and the incident flow Mach number, the transformed wing is found by multiplying the dimensions of the given wing in the у and z directions and its angle of attack by the factor cx = Vl(l — Mzi)|, whereas the dimensions in the x direction remain unchanged. For subsonic velocities, the flow about the transformed wing is computed from the incom­pressible equations; for supersonic velocity, however, it is computed from the compressible equations for MzTO = /2. If the incident flow velocities are equal for both wings, the pressure coefficients are related by

With regard to practical applications, it is expedient to choose a transformation in which only the dimensions in the у direction (wing planform) are distorted, whereas the dimensions in the z direction (profile and angle of attack) remain unchanged. Such a transformation is obtained from the above version I by removing the distortion in the z direction according to Eqs. (4-16a), (4-16b), and (4-17). Thus, from Eq. (4-22), the pressure coefficient is changed, within the limits of the linearized theory, by the factor Vll — Mai, I, that is, c’p = c’p Vl I —MaL|, and the pressure coefficient becomes

This relationship is shown in Fig. 4-9. Thus, the following version is obtained for the subsonic and the supersonic similarity rule.

From the given wing and Mach number, a transformed wing is formed by multiplying the dimensions of the given wing in the у direction with the factor c — Vl(l —Mala)I, whereas the dimensions in the x and z directions remain unchanged. For the transformed wing thus obtained, the incompressible flow field is computed when the given incident flow Mach number lies in the subsonic range. When the Mach number lies in the supersonic range, however, the flow field about the transformed wing is computed from compressible equations at Mam = V2. For equal incident flow velocities £4 of given and transformed wings, the pressure coefficients are interrelated through Eq. (4-23). From the subsonic and supersonic similarity rules, the following generally valid relationships for the aerodynamic coefficients are obtained: Let the function

Cp = <5’/i h; A", cot f’-j; (4-24)

describe the dependency of the pressure coefficient on the geometric wing data at Maoo = 0 or Max = fl. Then the corresponding dependency of the geometric wing data at an arbitrary Mach number is obtained, because of Eqs. (4-15) and (4-22), in the form:

S = 7 7=- fM; A 1 – Mal ; cot^] 1 — M<&| ; f) (4-25a)

У1- Ma^ I c sl

Here 5 stands for the relative thickness tjc, the relative camber height А/с, or the angle of attack. This equation can be written in a simpler form:

cr= ,_____ І___ (4-256)

]/j 1 — Ma^o) C 4

From this formula for the pressure distribution, the lift coefficient is obtained in corresponding form by integration over the wing surface:

cL = .. 5 —ідІЛ: Л tan93; A ^ 11 — Ma%) (4-26)

Here 5 stands for the angle of attack or for the relative camber height. By going to the limiting case of the airfoil of infinite span (X= 1, ip — О, Л-»°°), the subsonic similarity rule transforms into the well-known Prandtl-Glauert rule of plane flow.

A formula analogous to Eq. (4-26) for the drag coefficient (wave drag) that is valid, however, only for supersonic flow (see the discussions of Sec. 4-5-5) is given as

cD = — A F2 (A A tail у, Л ІМа% — 1) (4-27)

№4 – і

For wings with zero angle of incidence, 5 is the wing thickness ratio tjc. In this case, the drag coefficient at zero lift cD = c^q is proportional to 52.

Figure 4-9 Illustration of the applica­tion of subsonic and supersonic similar­ity rules (version II): transformation of the pressure coefficients.

The outstanding value of the above formulas lies in their describing the Mach number effect in a simple way. They can, however, also be used to great advantage for the classification of test results.

Transonic similarity rule For flows of velocities near the speed of sound (transonic flows), a similarity rule can be derived after von Karman [103] that is related to those for subsonic and supersonic flows. For wings in a flow field of sonic incident velocity (Mao, = 1), it is obtained from the potential equation, Eq. (4-9).

Contrary to the similarity rules for subsonic and supersonic flows, for which

the dependency of aerodynamic coefficients from the geometric wing parameters

and the Mach number was investigated, only1 the dependency of the aerodynamic coefficients on the geometric parameters must now be studied, because Max = const = 1.

The problem can be posed in the following way: Given is a wing with all geometric data (planform and profile) at an angle of attack zero. What, then, is the geometry of a reference wing, also in an incident flow field of Ma„ = 1, that has an affine pressure distribution equal to that of the given wing? To answer this question, the following transformation is introduced into Eq. (4-9) [see Eq. (4-10)]:

x’ = X y’ = csij v! = c3z Ф = с4Ф’ U’cc = (4-28)

where the quantities without primes refer to the given wing, those with primes to

the reference wing.

Introducing Eq. (4-28) into Eq. (4-9) yields, with

4 = (4-29)

the following nonlinear differential equation for the velocity potential of the transformed flow:

7+ 1 8Ф’ д*Ф’ /Э2Ф’ 32Ф’ _ ffco дх’ дх’* +ду/2 + Ъг’4- J ~

{Маж = Ma’c* = 1)

For an additional relationship between the constants c3 and c4, the kinematic flow conditions, Eq. (4-20), for both wings have to be established.

For chord-parallel incident flow, this relationship is

where 5 = t/c is the thickness ratio of the wing profile, which has been assumed to be symmetric.

Hence, with Eq. (4-29):

The distortion of the geometric data of the wing planform is given by the factor c3 in Eq. (4-28). Hence, the following transformations are valid:

Taper:	;/ = A	(4-33 a)
Aspect ratio:	–	(4-33 b)
Angle of sweepback:	cotу = cot (p	(4-3 3c)

As an example for the transonic similarity rule, the transformation for a swept-back wing is presented in Fig. 4-10.

Transformation of the pressure distribution is obtained in analogy to Eqs. (4-18) and (4-19) merely by replacing c2 by c4s that is, cp=c4cp. With сл according to Eq. (4-23), it follows that

If the pressure distribution is to be related to the geometric parameters, Eq. (4-34), considering Eqs. (4-33a)-(4-33c) leads to

Cp = <32/3 / A tanЛ<51/8. (4-35)

Hence it is shown that the pressure coefficient from the transonic similarity rule is proportional to 52/3, whereas it is proportional to 5 according to the subsonic and supersonic similarity rules of Eq. (4-25).

From Eq. (4-35) the following expression is found for the drag coefficient,

cD = (55/3 F (/, A taпер, .Id1 /3) (4-36)

showing that the drag coefficient is proportional to 5S/3, whereas it is proportional to 52 according to Eq. (4-27).

Figure 4-10 Application of the transonic similarity rule for sonic incident flow to the example of a trapezoidal swept-back wing. (a) Thickness ratio 8 = t/c — 0.05. (b) Thick­ness ratio 5′ = t’/c’ = 0.10.

The formulas for the airfoil of infinite span (X— 1, <p= 0, A -»«>) wiH be given in Sec. 4-3-4 in extended form (Mz„ « 1 instead of Max = 1).