Conical Flow Past a Delta Wing

In general, in a conical flow, all fluid properties are invariant along rays through a common vertex. An example of such flows is a triangular wing in supersonic flow, where disturbances are propagated only in the downstream Mach cone. Notice conical flows are not possible in subsonic flow regime.

Following Shapiro [31] and Ashley and Landhal [1], the governing perturbation potential equation is given by

Conical Flow Past a Delta Wing

Conical Flow Past a Delta Wing

Fig. 7.25 Delta wing in supersonic flow

 

y

 

2 d2ф д2ф д2ф

M0)dX2 + – 0

 

(7.152)

 

(1

 

Conical Flow Past a Delta Wing

where the origin is placed at the vertex, and the free stream is in the x-direction, Fig. 7.25.

Подпись: a Подпись: в x Подпись: y2 + z2, Подпись: в — tan 1 Подпись: (7.153)
Conical Flow Past a Delta Wing

The velocity components (u, v, w) are also governed by similar equations. Following Hayes [27] and Stewart [32], conical coordinates are introduced where

The governing equation for the u-velocity component becomes (see also Ashley and Landhal [1])

Подпись: (7.154)2 д2U 1 2 дU 1 д2U

(1 – a 2)^ + a(1 – 2a V + a^ ^ — 0

In the (n, Z) plane. the Mach cone is represented by a unit circle and the wing by a slit along the n-axis of length 2д — в/tan Л, where Л is the sweep angle.

On the mach cone, u, v, w vanish and on the slit w — —a. Using Chaplygin’s transformation, where

Подпись: (7.155)1 — V1 — a2 d q 1 q

a ’ d a 1 — a 2 a

the above equation is reduced to the standard, two dimensional Laplace equation

Conical Flow Past a Delta Wing

Conical Flow Past a Delta Wing

The velocity components are related through the irrotationality conditions, hence on the slit

= 0, |nl < ml (7.158)

dZ

On the Mach cone

u = 0, on n = 0 (7.159)

Conical Flow Past a Delta Wing Подпись: (7.160)

From the incompressible flow solution for a flow normal to a plate, one expects that u has a square-root singularity at the leading edge. Indeed, for a delta wing with

where C is a real constant (since the boundary conditions are homogeneous).

Obviously, C ^ 1 as m’ ^ 0 to recover the slender body solution for a delta wing with vanishing span.

The determination of C (to satisfy the boundary condition w = – a) is compli­cated. Referring to Ashley and Landhal [1]

(7.161)

Conical Flow Past a Delta Wing Подпись: (7.162)

where E (k) is the complete elliptic integral of second kind with modulus k. The pressure coefficient is given by

Подпись: CL Подпись: n AR C ( 2 Подпись: (7.163)

The lift coefficient is found to be

The above results are consistent with slender body theory for m = 1 (or k = 1) and with supersonic edge theory for m = 1 (or k = 0).

The drag is given by

for supersonic edges. When the leading edge is subsonic, Jones and Cohen [11] accounted for leading edge suction, where

Подпись: (7.165)D = L a – Fs

Подпись: CD Conical Flow Past a Delta Wing Подпись: (7.166)

with their estimate of Fs

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