Slender Body Theory
A simple theory can be used to estimate the aerodynamic characteristics
of bodies that vary slowly in X-direction: wings of very low aspect ratio
and high sweep or slender fuselages.
In such cases, the rate of change of all quantities in the x direction is
small and the governing equation becomes:
Note that the (1-M2) du/dx term has been dropped because we assume that
everything varies slowly in the X direction. The remaining equation is an
equation for 2 dimensional incompressible flow in the cross-plane, and since
the Mach dependence has dropped out, it is valid for both M<1 and M>1.
The 2-D cross flow can be computed using a conformal mapping method or even
a 2-D panel method. Note that in the cross flow plane, the flow is unsteady
as the span of the swept wing or the diameter of the fuselage changes as
it cuts through the plane. So while the solution to Laplace's equation still
provides the correct velocity distribution, the pressures must be computed
using the unsteady Bernoulli equation.
One particularly simple and useful case is that of a highly-swept, low aspect
ratio wing.
Using the slender body concept we can solve for the lift of a low aspect
ratio wing.
Such low aspect ratio wings tend to produce constant downwash and thus nearly
elliptic loading until the angle of attack gets large.
The rate of change of momentum of the air in the cross-flow plane is equal
to the force per unit length on the wing:
If we consider a certain area in the cross plane
given a velocity w ( 
) then the force becomes:
It can be shown from 2-D unsteady theory (not discussed here) that the effective
area , A is given by a circle around the plate of diameter equal to the
local span, Y.
Thus,
The total force on the wing is then given by:
This force acts normal to the wing plane. So for small angles of attack:
This shows that in the low aspect ratio limit, the lift curve slope differs
from what lifting line theory predicted by a factor of 2. An expression
for lift curve slope derived from second order corrections to lifting line
theory is given by Jones:
where p is the ratio of wing semi-perimeter to wing span.
For a rectangular wing p = (b+c) / b = 1 + 1/AR so CL = 2AR / (AR+3).
This expression is in close agreement with experiment over a wide range
of aspect ratios.
A similar analysis can be done for slender bodies of revolution.
This leads to the result that the lift produced by a body with a cut-off
base is given by:
Note again that these results are independent of Mach number. The Prandtl-Glauert
correction still applies, but the reduction in forces due to effective stretching
in the x-direction just cancels the increase in the velocities (1/1-M2)
that would have been applied in 2-D.