# Flow over Bodies

Some closed-form solutions for the potential flow over bodies of revolution are available and are useful as reference results.

We noted that in 2-D the maximum velocity on an ellipse was given by:
umax/U = 1 + t/c

In 3-D the surface velocity over an ellipsoid of revolution is given by:

x is the distance from the midpoint in units so that the length is 2.0 and r is the radius (or in these units, the ratio of diameter to length.)

The maximum velocity is given by:

The figure below (from Schlichting) illustrates the pressure distribution on bodies of revolution. D/L = 0.1

Note that perturbation velocities are much smaller than in 2-D. These velocities may be estimated by superimposing point sources. In this case for an ellipsoid:

Note that the maximum velocity is sensitive to the actual shape, with a paraboloid having about 50% larger perturbations. The results from such a distribution of sources on the axis, slightly underpredicts the velocity perturbations.

The figure below shows the pressure distribution on a typical fuselage shape with D/L = 0.09 computed by a source distribution on the x-axis. Note how the pressure falls near the center of the cylindrical portion of the fuselage.

As indicated below, fuselages in inviscid flow produce a nose-up pitching moment when the angle of attack is increased. This effect is destabilizing and is an important consideration in the sizing of the horizontal tail.

Although the inviscid picture suggests that no lift is produced, the viscous flow actually separates at the back of the fuselage, making the moment somewhat smaller and the lift larger than predicted by inviscid theory. This lift produces induced drag and the fuselage behaves as a low aspect ratio wing.

The figure below shows the effect of angle of attack on fuselage lift, drag, and moment based on experimental data. Also shown is the estimated moment based on inviscid theory.