. Munk’s Theorem of Stagger

Munk’s theorem of stagger states that “the total drag of a multiplane system does not change when the elements are translated parallel to the direction of the wind, provided that the circulations are left unchanged.” Thus the total induced drag depends only on the frontal aspect. To illustrate this, let us consider a lifting element of length ds1 placed at O, and another lifting element of length ds2 placed at A in a plane parallel to x = 0, as shown in Figure 8.19.

Let the circulations of the elements ds1 and ds2 are Г1 and Г2, respectively. The normal AH drawn in the plane parallel to x = 0 makes an angle ф2 with OB and:

e = Фі – I

If dwn is the induced velocity at A along AH, using Equation (8.64), we can express dwn as:

dwn = dv sin e + dw cos e

sin в cos (Фі — ф2)

2r2

If there is a flow of velocity V along xO, the drag induced by ds1 on ds2 is:

To get the drag induced on ds1 by ds2 let us replace в with —в, the angle of stagger. Then:

Thus the total drag mutually induced on the pair of lifting elements becomes:

p Г1Г2ds1ds2 cos (Ф1 + Ф2) 2n n2

which is independent of the angle of stagger. This yields Munk’s theorem of stagger, that is:

“the total drag of a multiplane system does not change when the elements are translated parallel to the direction of the wind, provided that the circulations are left unchanged."

When the system is unstaggered (that is, when в = 0):

d2D12 = d2D21

and thus if the lifting systems are in the same plane normal to the wind, the drag induced in the first by the second is equal to the drag induced in the second by the first. This result constitutes Munk’s reciprocal theorem.

The total mutual induced drag is:

p Г1Г2 cos (Ф1 + Ф2) 2n n2

where Ф1 is the angle between the plane containing the normals to the element ds1 and the projection of the line joining the elements on the plane normal to the wind and ф2 is the angle between the plane containing the normals to the element ds2 and the projection of the line joining the elements on the plane normal to the wind.