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In the early 1970s, Gerald Gregorek wrote a technical note on drag for ESTES (maker of model rockets and parts from what I gather).

The note is still very popular in the model rocket community, and it is referenced in most papers written on model rockets I've encountered. Here is one link to it (please replace with a web archive link if it ever breaks).

One equation in Gregorek's note is especially popular, and that is the equation for drag from the combined nose and body tube assembly. Here it is, with some context:

These curves can be used for all 1" models with 3:1 ogive nose cones, but what about other shapes and model rocket diameters? For other rocket configurations we would have to run more wind tunnel tests or find a mathematical way to determine the drag coefficients. Luckily, aerodynamicists have been working on this problem for many years, so short hand equations are available for our use. The one equation which we can use for the nose cone and body tube is $$C_{D_N} + C_{D_{BT}} = 1.02\ C_f \left[ 1 + \frac{1.5}{(L/d)^{3/2}} \right] > \frac{S_W}{S_{BT}} \tag{8}$$

Here, CDN and CDBT are the drag coefficients due to the nose and body tube of the rocket; Cf is the skin friction coefficient, to be found from plots over a range of Reynolds numbers; L/d is the length-to-diameter ratio; and SW and SBT are the wetted surface area and the body tube reference area, with the transverse cross-sectional area assumed in the equation.

If you've struggled to calculate drag coefficients, you can see the appeal of this equation... But there is one very big problem.

Gregorek doesn't tell you where the equation comes from. There is no rationale for it (understandable, given his audience), and there is no reference for it. All the papers that reference this equation reference only Gregorek and nobody else. Presumably the equation works for some model rockets, but why?

Can anyone say where this equation might have come from? Does anyone in the aerodynamics community recognize it? I'm especially interested in the limits to this equation---when is it valid and when does it stop working? If my rocket is 200 ft high and 12 ft wide but all else is the same, will I still get an adequate ballpark figure for my drag coefficient (assuming I don't care about Mach number or angle of attack dependences, and assuming also that the Reynolds number dependence is adequately captured in the friction coefficient calculation)?

Any pointers to the true source of Gregorek's equation (or even to an alternate equation for bigger rockets) would be appreciated!

Edit to include the bits I found from Organic Marble's hugely helpful comment:

You can arrive at an expression very close to Gregorek's using the expressions given in Ch. 6 of Hoerner's Fluid Dynamic Drag book. See pp. 6-15 thru 6-19 ("Drag of Streamline Bodies"). There, Hoerner gives the total drag of a streamlined shape (like a rocket with an aerodynamic nose) based on wetted area (this matters, see below):

The total drag, based on wetted area is consequently $$C_\mathrm{Dwet} / C_f = 1 + 1.5 (d/l)^{3/2} + 7 (d/l)^3 \tag{28}$$

About the third term, Hoerner says this:

The graph shows that the third term of this equation is practically negligible up to d/l $\approx$ 0.2.

Model rockets are usually slender, and Gregorek's examples all have l/d ratios between 10 and 20, which is to say d/l ratios between 0.1 and 0.05. In this range, and given Gregorek's amateur audience, it would have made sense to drop the third term.

Further, because Hoerner's equation is based on wetted area, and not on the frontal cross-sectional area more commonly used in drag calculations, Gregorek would have had to scale Hoerner's equation by a factor equal to the ratio of the wetted area and the cross-sectional area.

In fact, Hoerner later factors in the ratio of wetted and frontal areas, though the expressions he obtains seem less versatile than Gregorek's, since they involve certain assumptions about the (rocket's) streamline shape, which Gregorek avoids.

The only thing we haven't accounted for in Gregorek's equation at this point is his 1.02 multiplier, which suggests the true drag coefficient is 2% larger than that predicted by the simplified Hoerner equation... Maybe the 2% discrepancy comes from the third term he dropped, and the 1.02 multiplier is a way to correct for it. Maybe someone else has a better idea?

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  • $\begingroup$ Is there a way to cross-post this in the aerodynamics community? $\endgroup$
    – user36480
    Commented Aug 30, 2020 at 19:13
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    $\begingroup$ Looks like it's derived from equations in Chapter VI Paragraph C "Drag of Streamlined Shapes" in Hoerner 1965 Fluid-Dynamic Drag. At least, there is a marked similarity. I found similar equations (at least the same power law) in a book I own, McCormick 1979 Aerodynamics, Aeronautics, and Flight Mechanics, and he references Hoerner, as does Gregorek. $\endgroup$
    – Organic Marble
    Commented Aug 30, 2020 at 22:42
  • $\begingroup$ @Alex explicit cross-posting (posting of nearly identical questions on multiple SE sites) is strongly discouraged in Stack Exchange. One reason is that it can result in answer fragmentation; important information being spread out over multiple locations such that future readers might miss some. Some day SE might have some way to handle this, but right now the only tool I know of is to post a link to this question in chat rooms of other relevant sites. In this case those might be Aviation SE and Physics SE. $\endgroup$
    – uhoh
    Commented Aug 31, 2020 at 0:22
  • $\begingroup$ @Organic Marble You're right!!! I just got the book two nights ago, and it is in fact referenced in Gregorek's technical note, though I thought it was mostly a data collection with bits of theory and nothing like this! It's possible Gregorek got his equation elsewhere, but it does seem you can derive it from the expressions given in Ch. 6 of Hoerner's book. Huge thanks!!!! I bow :) $\endgroup$
    – user36480
    Commented Aug 31, 2020 at 2:10

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I don't see the exact equation that is in the question, but it looks like it's derived from equations in Chapter VI Paragraph C "Drag of Streamlined Shapes" in Hoerner 1965 Fluid-Dynamic Drag. At least, there is a marked similarity. I found similar equations (at least the same power law) in a book I own, McCormick 1979 Aerodynamics, Aeronautics, and Flight Mechanics, and he references Hoerner, as does Gregorek.

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    $\begingroup$ Is it possible to include screen shot(s) or snapshots(s) or at least a text transcription(s) of mentioned equation(s)? I'll be happy to help convert them to MathJax if needed. $\endgroup$
    – uhoh
    Commented Aug 31, 2020 at 4:54
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Model rockets are often not very smooth. The skin friction term comes from the laminar portion of the Cf vs Re No. curve. It is highly unlikely that model rockets have much laminar flow. Fins may not have rounded leading edges or sharp trailing edges. Surface finishes may be very rough, depending upon how much time (and skill) the rocketeer put into the finish. These are probable sources for the conservatism in the equation.

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