Lift and Induced Drag of a Body of Revolution at Angle of Attack

Подпись: 2 д2ф 1 d M0 ^ + 7 dr Подпись: 1д2ф + Г2 ^2 = 0
Lift and Induced Drag of a Body of Revolution at Angle of Attack

The flow situation is depicted in Fig.7.8. The perturbation potential ф is governed in this case by the three dimensional equation

y

 

Usin a

 

Ucos a

 

Lift and Induced Drag of a Body of Revolution at Angle of Attack

0

Fig. 7.8 Body of revolution at angle of attack: Top, coordinate system; Bottom, bullet at supersonic speed (from http://en. wikipedia. org)

The boundary condition is given by

Подпись:ur dR

ux body dx

In terms of the incoming flow and perturbation velocity, this becomes

U sin a sin в + ^r dR

V U cos a + Ш )body ~ d

Because of linearity, ф can be written as (see Liepmann and Roshko [6])

Lift and Induced Drag of a Body of Revolution at Angle of Attack Lift and Induced Drag of a Body of Revolution at Angle of Attack

ф(х, r, в) = ф0 (x, r) + фа(х, r, в) (7.64)

and фа is solution of the full equation that can be modeled as a doublet distribution along the x-axis. The axis of the doublet is parallel to the z-axis to represent the cross flow, namely

дф0 дф0

фа (x, Г, в ) = 0 = sin в 0 (7.66)

Подпись: dR дфа дфа dx dx d r body Подпись: U sin a sin в Подпись: (7.67)

with the boundary condition on the surface as

This last equation is obtained to second-order O (a2) from the general boundary condition for ф, upon subtraction of the axisymmetric boundary condition for ф0.

Notice that the first term in the left-hand-side is very small for slender configura­tions. In this case

фа = ^X^ sin в (7.68)

r

Подпись: фа Lift and Induced Drag of a Body of Revolution at Angle of Attack Lift and Induced Drag of a Body of Revolution at Angle of Attack Подпись: (7.69)

The strength of the doublet q (x) is proportional to the cross sectional area

Therefore, the cross flow solution at any section is the same as the incompressible flow solution over a cylinder, independent of M0.

The surface pressure coefficient is given by

Подпись:2 1 Г / дф2 /1 дф 21

Подпись: hence Lift and Induced Drag of a Body of Revolution at Angle of Attack Подпись: / dR dx Подпись: (7.71)

U cos ad x (U cos a)2 dr r дв

and

 

Lift and Induced Drag of a Body of Revolution at Angle of Attack

(7.72)

 

Lift and Induced Drag of a Body of Revolution at Angle of Attack

In the latter expression, a2 terms are neglected.

The normal force in the z-direction can be calculated via integration

N = – p0U2 / f {—Cp, a sin в) Rdedx

2 0 0

1 2 I I 2 dR 1 2

= 4a-p0U2 sin2 в R de dx = 2a-p0U2 S(l) (7.73)

2 0 0 dx 2

Hence, the normal force coefficient, using the base area as reference, is given by

Cn = 2a (7.74)

Notice that, if the body is closed at both ends, N = 0.

The calculation of the drag is more delicate, since it is a small quantity (of order a2). For slender bodies, CL — CN = 2a, independent of M0, and the lift slope is dCL/da = 2 for incompressible and compressible flows.

If one considers a cross section of the body at a given x location and let the circulation be defined on the body as

r(x, в) = ивRde, 0 < в < п (7.75)

—в

then

r(x, в) = фa(x, R, в) — фa(x, R, —в)

= 2фa(x, R, в) = 2U sin aR(x) sin в = 2U sin a^JR2(x) — y2 (7.76)

In term of the “span” variable y, this is an elliptic loading with induced drag Coi = at Cl.

It can be shown that at = a/2 and CDi = a2.

Подпись: ArefCot = U 2 + 2 £base U' ^U ' Подпись: dydz Подпись: (7.77)

According to Viviand [7], the induced drag coefficient is given by the same formula as for a finite wing

where the surface integral is performed on £base, the part of the cross flow plane outside the base of the body and approximately perpendicular to the incoming flow, Fig.7.9.

Lift and Induced Drag of a Body of Revolution at Angle of Attack
As seen in Chap. 6, this surface integral can be reduced to a line integral along the body base, Chase, as

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