Volume Wave Drag

One can compute the wave drag on a body of revolution relatively easily. For a paraboloid of revolution the drag coefficient based on frontal area is:

For a body with minimum drag with a fixed length and maximum diameter, the result is:

Note that even with a fineness ratio (L/D = length / diameter) of 10, the drag coefficient is about 0.1 -- a large number considering that typical total fuselage drag coefficients based on frontal area are around 0.2.

Minimum Drag Bodies

In the 1950's Sears and Haack solved for the shape of a body of revolution with minimum wave drag. These results provide guidance for initial estimates of volume wave drag, even before the detailed grometry is known. Two solutions are shown below.
1. Given maximum diameter and length: 2. Given volume and length:

General Shapes

When the body is does not have the Sears-Haack shape, the volume dependent wave drag may be computed from linear supersonic potential theory. The result is known as the supersonic area rule. It says that the drag of a slender body of revolution may be computed from its distribution of cross-sectional area according to the expression:

where A'' is the second derivative of the cross-sectional area with respect to the longitudinal coordinate, x.

For configurations more complicated than bodies of revolution, the drag may be computed with a panel method or other CFD solution. However, there is a simple means of estimating the volume-dependent wave drag of more general bodies. This involves creating an equivalent body of revolution - at Mach 1.0, this body has the same distribution of area over its length as the actual body.

At higher Mach numbers the distribution of area is evaluated with oblique slices through the geometry. A body of revolution with the same distribution of area as that of the oblique cuts through the actual geometry is created and the drag is computed from linear theory.

The angle of the plane with respect to the freestream is the Mach angle, Sin q = 1/M, so at M=1, the plane is normal to the flow direction, while at M = 1.6 the angle is 38.7° (It is inclined 51.3° with respect to the M = 1 case.)

The actual geometry is rotated about its longitudinal axis from 0 to 2 p and the drag associated with each equivalent body of revolution is averaged.

A comparison of actual and estimated drags using this method is shown below.

At the earliest stages of the design process, even this linear method may not be available. For conceptual design, we may add wave drag of the fuselage and the wave drag of the wing with a term for interference that depends strongly on the details of the intersection. For the first estimate in AA241A we simply add the wave drag of the fuselage based on the Sears-Haack results and volume wave drag of the wing with a 15% mark-up for interference and non-optimal volume distributions.

For first estimates of the volume-dependent wave drag of a wing, one may create an equivalent ellipse and use closed-form expressions derived by J.H.B. Smith for the volume-dependent wave drag of an ellipse. For minimum drag with a given volume:

where t is the maximum thickness, b is the semi-major axis, and a is the semi-minor axis. b is defined by: b2 = M2 - 1. Note that in the limit of high aspect ratio (a -> infinity), the result approaches the 2-D result for minimum drag of given thickness:

CD = 4 (t/c)2 / b

Based on this result, for an ellipse of given area and length the volume drag is:

where s is the semi-span and l is the overall length.

The figure below shows how this works.

Volume-dependent wave drag for slender wings with the same area distribution. Data from Kuchemann.