Purpose and Scope
Data from both Mercury and Apollo were combined here as a “database” for guessing fairly closely the hypersonic drag coefficient of a blunt capsule shape. A drag curve for a shape like Mercury from subsonic to Mach 24.5 was found in one old textbook, presumably based upon wind tunnel data. Data for Apollo-like shapes was found at selected Mach numbers in another old textbook. These were definitely old wind tunnel data. The intent here is to find a range of drag coefficients and average values suited to the Mach 3 to Mach 25 range of hypersonic flight.
Reference 1 (Miele) had a good graph of drag coefficient versus Mach number for a Mercury capsule shape. That graph was read and input into an Excel spreadsheet for purposes here. There were no specific attributions on that graph to actual wind tunnel data, although that was the implication in context.
Reference 2 (Hoerner) had data at selected Mach numbers for both the Mercury and Apollo shapes, with a range of “tumble-home” angles on the Apollo shape. The reference indicated cited actual wind tunnel data.
Geometric data for both capsules was found after a brief internet search, which pinned down the actual Apollo tumble-home angle, and defined the heat shield radius to diameter (bluntness) ratios for both capsules.
It should be noted that the Mercury shape was fairly “sharp" transitioning from the blunt heat shield to the tapered (tumble-home) afterbody. Apollo had a generous radius at that transition, which does affect the location of the sonic line and subsequent flow field around the afterbody.
The Miele curve for Mercury is given in Figure 1 below, with the Hoerner Mercury and Apollo data spotted upon it. At Mach 4.5, the Miele curve and Hoerner’s Mercury data are the same, probably because they are really from the same test data. The Apollo data from Hoerner are generally very slightly above the Miele curve for Mercury. Over the range of Mach 3 to 25, the average drag coefficient of the Mercury shape is 1.45, while Apollo is 5 to 10% higher at about 1.55.
The two capsule shapes are compared in Figure 2, along with a sketch of the general flow field features. Hypersonically, we would expect that the blunter the object, the higher the drag coefficient. Bluntness is measured by the heat shield radius of curvature to diameter ratio. Apollo has a comparable, but slightly higher, bluntness as shown. The reference area for drag coefficient is, in all cases, the frontal or blockage area of the shape:
A = 0.25*pi*(diameter)^2
The radius of curvature at the transition from heat shield to tapered afterbody would be expected to influence overall drag by its effects on the sonic line location and separated flow zones, and their effects on the strength of the trailing shock wave.
The “tumble-home” angle of the tapered afterbody would also be expected to influence overall drag by its relation to the available Prandtl-Meyer expansion turning angle downstream of the sonic line. This affects both surface pressure coefficient and trailing shock strength. Larger tumble-home angle allows greater angle of attack for lifting during entry, without exposing the tapered side wall to “direct” hypersonic windblast and heating.
For either variable (local radius or “tumble-home”) these effects do not act all in one direction, and they most definitely interact in a very complicated way. For the Apollo shape, Figure 3 clearly shows that increasing the tumble-home angle effectively acts to decrease overall drag.
An educated guess says that the bluntness (radius to diameter ratio) effect is stronger than the 5-10% increase in drag seen here, for Apollo to relative to Mercury. It is apparently offset in part by the more generous transition radius on Apollo, which lets the sonic line occur further downstream, and reduces the Prandtl-Meyer expansion effects, in turn weakening the trailing shock. This would act to decrease drag. Thus the two curves are “close”. The net effect says that Apollo is probably pretty close to Miele’s Mercury curve, just factored up by about 1.07.
Another educated guess says that increasing the tumble-home on the Apollo shape seems to “suck” that sonic line further downstream. This apparently has the effects of lowering the imposed flow turning angle, weakening the Prandtl-Meyer expansion, raising the surface pressure coefficients, and lowering the oncoming Mach number for the trailing shock. This is reflected in lower drag at more tumble-home, and higher drag with less, as shown in Figure 3.
For an Apollo shape, I recommend factor 1.04 drag increase for 5 degrees tumble-home decrease, and factor 1.07 decrease for 5 degrees more tumble home. You could very likely use this same sensitivity on data for a Mercury shape, and not be too far off. But the Mercury data will not directly correlate, due to the effects of the different transition radius. This is shown dramatically in Figure 3, where the actual Miele Mercury data are drag coefficient 1.27 at Mach 24.5.
Use tumble-home drag sensitivities of 0.8% increase per degree below reference, and 1.4% decrease per degree above reference, for figuring changes from the baseline Apollo shape at reference tumble-home 34 degrees. You could probably get away with using this same correction referenced to 20 degrees tumble-home, on the basic Mercury shape, although there is no real data here to actually support that action.
For capsule shapes that have a sharp transition and about 20 degrees tumble-home, use the Miele Mercury curve as it is, which corresponds to an average 1.45 hypersonic drag coefficient for the Mach number range from 3 to 25. Then adjust it for tumble-home effects different from 20 degrees.
For capsules with a radiused transition and about 34 degrees tumble-home, use the Miele curve factored up by about 1.07 as it might represent Apollo. This would average 1.55 from Mach 3 to Mach 25. Then adjust that for tumble-home effects different from 34 degrees.
1. Angelo Miele, “Flight Mechanics”, published by Addison-Wesley, 1962.
2. Sighard F. Hoerner, “Fluid Dynamic Drag”, self-published by the author, 1965.
Figure 1 – Blunt Entry Capsule Hypersonic Drag Data
Figure 2 – Blunt Entry Capsule Shape Comparison
Figure 3 – Capsule “Tumble-Home” Angle Effects