LECTURE III

HELMHOLTZ-KIRCHHOF THEORY OF A DISCONTINUOUS STREAM LINE

THE normal conditions of the aeroplane of a machine flying high up in the air are represented on the Kirchhoff theory by the diagram 4 and 10, representing the state of flow relative to the pilot; the analytical conditions have been discussed in Lecture I. and II., and will be found in § 6 of Report 19.

But there is an advantage in extending the theory to a more general case, and beginning as in Report 19, § 2, with a plane barrier oblique to a stream of finite breadth (Fig. 19), as the extension is not essentially more complicated, and it throws light on the simple case of an infinite stream.

No change is required in the 2 diagram (Fig. 19), except that i, j, j', on aa' are now distinct; but now in the w diagram the outside stream lines come from infinity into view, and the diagram has the advantage of being closed, as in Euclid I. 27, so that the difficulties at infinity are kept under observation.

In the w diagram there are three stream lines in the boundary, which we denote by mi, m2, m3; and = m1 is the outer stream line from I to J ; y = m, the inner skin from J to A, from A along AA' to the branch point B and then to A', and again along A'J'; ma is the other outer stream line from J' back to I.

There is also the curved branch IB of the ma stream line along which the current divides; but as IB lies in the interior of the fluid, and the velocity q and its direction @ both vary along IB, the corresponding part ib in the w and £ diagram must be excluded from the boundary; ib is straight in the w diagram, but curved in the 2 diagram, from b at midway height of a and a', to i at height a.

The branch point B where the velocity is zero requires

when resolved into partial fractions, with the factor appropriate for making w change suddenly by m1 m2, m2 — M3, M3 — M1, as u passes through j, j', i, in accordance with II. (15).

M.F.

E

· The z diagram is now as in Fig. 19, and if c1, c2 denotes the breadth of the stream at J and J', where the velocity has become uniform and equal to the skin velocity Q,

and putting c1 + c2 = c, the breadth of the impinging stream

and the curved part BI of the dividing stream line = M2 separates the stream at I into two parts of breadth c1 and c2.

With the value of 2 as before in II. (27), we must write in the region i>u> a,

=0, q=Q, as required over the skin of IJ and

q

and thence the intrinsic equation is derived of IJ and JA, as given in § 2, Report 19.

The linear scale of the diagram is calculated as explained in § 3 by an integration, expressed in terms of the arbitrary constants a, a', b, j, j', i, in a transcendental form, from which the length AA', AB, BA' is calculated, and the angle a, ß, B', at which the stream from I is received by the plane and leaves it again at J, J'.

The inverse process to determine the constants a, a', b, j, j', i from the dimensions of the diagram would be intractable analytically, and would require the trial and error process.

The thrust T is obtained immediately in this case, and without integration, by the principle of momentum; resolving perpendicular to the plane AA,'

But the determination of L the C.P. is intractable for a stream of finite breadth.

A

B

A'

FIG. 20.

The extreme case of an infinite barrier AA' receiving the impact of a jet of finite width is obtained, as shown in Fig. 20, by making ja, j'a'.

Duplication in AA' with the motion reversed will give the direct impact of two unequal jets with equal velocity, as in Fig. 21.

A simple experimental illustration can be given of these problems with a blade held in a jet of water from a tap, or with impinging gas jets.