Constant-Density Flow; the Thickness Problem

Having shown how steady, constant-density flow results are useful at all subcritical M, we now elaborate them for the finite wing pictured in Fig. 7-1. As discussed in Section 5-2 and elsewhere above, it is convenient to identify and separate portions of the field which are symmetrical and
antisymmetrical in z, later adding the disturbance velocities and pressures in accordance with the superposition principle. The separation process involves rewriting the boundary conditions (5-30) as

«* = Tfx + etx~a at г = 0+ ,

do. „dh –

<pz — — Гу + в——– a at z = 0—

dx dx

where h(x, y) is proportional to the ordinate of the mean camber surface, while 2i? (ж, у) is proportional to the thickness distribution. The differential equation is, of course, the three-dimensional Laplace equation

V2<p = 0. (7-9)

As a starting point for the construction of the desired solutions, we adapt (2-28) to express the perturbation velocity potential at an arbitrary field point (ж, у, z),

Here n is the normal directed into the field, and the integrals must be carried out over the upper and lower surfaces of S. Dummy variables (xi, 2/i, zi) will be employed for the integration process, so the scalar distance is properly written

r = V(x — xi)2 + (y — yi)2 + (z — zi)2. (7-11)

In wing problems Zj = 0 generally.

Considering the thickness alone, we have

ip{x, y, z) = <p(X, y, —z)

for all z and the boundary condition

Moreover, no discontinuities of <p or its derivatives are expected anywhere else on or off the ж, y-plane. In (7-10), dS = dzj dyb the values of <p(xi, //і, 0+) and <p(xi, yi,0—) appearing in the integrals over the upper and lower surfaces are equal, while the values of д/дп(1/4ят) are equal and opposite. Hence the contributions from the first term in brackets cancel,

where (7-13) and (7-11) have been – employed. Physically, (7-14) states that the flow due to thickness can be represented by a source sheet over the planform projection, with the source strength per unit area being pro­portional to twice the thickness slope dg/dx. [Compare the two-dimensional counterpart, (5-50).]

Examination of (7-14) leads to the conclusion that the thickness prob­lem is a relatively easy one. In the most common situation when the shape of the wing is known and the flow field constitutes the desired information, one is faced with a fairly straightforward double integration. For certain elementary functions g(x, y) this can be done in closed form; otherwise it is a matter of numerical quadrature, with careful attention to the pole singularity at aq = x, у = y, when one is analyzing points on the wing z = 0. The pressure can be found from (5-31) and (7-14) as

There is no net loading, since Cp has equal values above and below the wing. Also the thickness drag works out to be zero, in accordance with d’Alem­bert’s paradox (Section 2-5). Finally, it should be mentioned that, for any wing with closed leading and trailing edges,

avydaq = Рте — !7le — 0. (7-16)

J chord OX

This means that the total strength of the source sheet in (7-14) is zero. As a consequence, the disturbance at long distances from the wing ap­proaches that due to a doublet with its axis oriented in the flight direction, rather than that due to a point source.

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