Supersonic Leading Edges

Reference 5.24 presents generalized expressions for the lift curve slope of thin, swept, tapered wings operating with supersonic leading and trailing edges. The results are restricted to the case where the Mach line from one tip does not intersect the other half of the wing. The analysis is based on a linearized theory for the surface velocity potential.

For the case (Figure 5.47a) where the Mach line from the apex intersects the tip chord between the leading and trailing edges, the following lengthy equation is obtained for Cj_a.

Figure 5.46 (continued)

Figure 5.47 Wings with supersonic leading edges, (a) Mach line intersects tip chord, (b) Mach line intersects trailing edge.

„ 1 ([4m’k + A'(k-1)]2 Г1 1

“ irflVm’2-1 і 2A'(k2 — 1) Ik m’

kVm’*-1 / _,-l _,4W(A’-1)-A'(* + 3)1

V(fcm’+l)(fcm’-l) V km’ 4km’ + A'(k-l) /1

[4m’к – A'(k – l)]2 / m’+l 4Jtm'(l – A’) + A'(3it + 1)

4A'(k – 1) V k(km’ + 1)COS 4km’- A'(k – 1)

[4m’fc + A'(l + ЗА:)]2 / m’-l 4km'(A’ – 1) + A'(k – 1)1

4A'(k + 1) V k(km’+ 1)COS 4km’+ A'(3k+ 1) J

(5.107)

If the Mach line from the apex intersects the trailing edge inboard of the tip, as shown in Figure 5.47b, CLa is given by

1 ([4m’k + A'(k-1)]2 Г1

L" ВтгЛ/т’2— 11 2A'(k2— 1) Lfc C°S m’

kVm’2-l…-і-l] w[4km’ – A'(k – l)]2 /_ m’+ 1 ] V(Am’-l)(Am’+1)C°S itm’J 4A'(it – 1) ‘ k(km’+ l)i

(5.108)

In applying Equations 5.107 and 5.108, care must be taken to retain the signs of quantities under the radical. For example, if x and у are two arbitrary positive quantities,

V(-Jt)(-y) = V(- l)2xy = – Vxy

In order to use Equations 5.107 and 5.108, the following quantities are defined.

, _ cot Ate cot A

A(l + A)

A(l + A) —4m(l —A) m = cot Л

A = taper ratio (tip chord/root chord) m’ = Вт A’ = BA

Since there is no leading edge suction for a wing with a supersonic leading edge, its leading edges should be sharp to reduce the drag. The drag caused by lift in this case is given simply by

Cd, = Cl tan a (5.109)

To this must be added the wave drag and skin friction drag in order to obtain the total Cd-

A delta wing having the leading edge ahead of the Mach cone from the apex is a relatively simple case to treat. For this configuration, the expression for Cl, is identical to that obtained for a two-dimensional airfoil (Equation 5.76). ‘

For certain extreme cases, the lift of a supersonic wing can be quickly approximated using two-dimensional results. Such a case is pictured in Figure 5.48. Here, a moderately swept, fairly high aspect ratio wing is shown operating at a Mach number of around 2.0. Since pressure disturbances are not propagated outside the Mach cone, the flow over the wing is two dimensional in nature, except for the hatched regions shown within the Mach cones from the apex and tips. Thus, as a first approximation, the lift and wave drag for this wing can be calculated using the corresponding expressions derived for a two-dimensional supersonic airfoil section.

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