Supersonic Wing Design
Sweep may be used to produce subsonic characteristics for a wing, even in supersonic flow. At some point, though, sweep is no longer very effective in delaying the effects of compressibility. That is, the difficulties associated with sweep outweigh the advantages as the required sweep angle gets very large. When the Mach number normal to the leading edge becomes greater than 1, the airfoil sections behave according to linear supersonic theory, with the associated wave drag.
For a double wedge: Cd = Cl2 (M2-1)0.5/4 + 4 (t/c)2 / (M2-1)0.5
For a parabolic section: Cd = Cl2 (M2-1)0.5/4 + 16/3 (t/c)2 / (M2-1)0.5
As in 2D, such supersonic wings are more easily analyzed than their subsonic counterparts, though. Consider the point (A) on the wing shown below. Its effect on the flow cannot propagate upstream because disturbances travel at the speed of sound and the freestream is traveling faster than this. This fact is called the law of forbidden signals and implies that disturbances originating at (A) can only affect the darker shaded area. Similarly, points outside the forward-going Mach cone (lightly shaded area) cannot affect the flow at point A.
This means that points on the tips of a supersonic wing can only affect a small part of the wing. The rest of the wing behaves as if it did not know about the wing tips and (except for the effects of sweep and taper) the rest of the wing may be treated as a set of 2-D sections. More detailed analysis shows that in the tip regions behave very much like 2-D sections with their lift curve slope reduced by 50%.
To avoid this loss of lift, the tip sections of supersonic wings are sometimes truncated so that no part of the wing is affected by the tips:
Sections with supersonic leading edges generally have more wave drag than sections with subsonic leading edges which can develop leading edge suction. For wings with sufficient sweep an important part of the design problem is to properly distribute the lift and volume over the length and span. The applet below shows some of the considerations involved in doing this.
Supersonic Wing Design Game
The purpose of this game is to distribute lift over the length and span of a wing to minimize drag. The idea is that there are several approaches tro obtaining a desired lift distribution. Click on the squares to add or remove lift from a particular place. The goal is to achieve an elliptic distribution of lift over the length and span of the wing. The score represents the deviation from the ideal loading. To assist in designing your wing the ideal and actual loadings are shown as row and column totals. Also each cell is labeled with the amount by which adding or removing lift will change the score. Clicking on the design button will automatically select those cells that help most, starting with your current design and proceeding for a number of generations.
Here are some designs with a score of 0.
