Short answer: for a 2-liter water bottle rocket, a rounded nose cone can be worst or best, depending on the shape.
So I built five nose cones in FreeCAD, designed to fit on a standard 2-liter bottle. The maximum diameter of the noses were 110 mm, while the bottle body was 109 mm, assuming slight overlap of the nose on the body. Some models are theoretical, and a couple I have actually built for real. From worst to best (see pictures attached):
- Hemisphere nose. Ball shaped. This had the most drag.
- Straight cone with a sharp transition to the body. As expected, this isn't optimal but better than nothing.
- Cone with a rounded nose and rounded shoulder. I have built this, using the conical taper from a 1.5L Crystal Geyser water bottle taped onto the rounded shoulder of a 2L bottle, and a ping-pong ball at the tip.
- Cone with a pointy nose and rounded shoulder. This was the first nose I ever built, taping a rolled paper cone on the Crystal Geyser bottle cone to make a sharp tip.
- Elliptical nose. Best performer of the rounded noses despite having more surface area. Not really possible to build unless 3D printed, but then it would likely have a weight penalty. For my simulation, the ellipsoid major axis was double the two minor axes.
- Ogive nose. Best performer overall by a tiny bit, nearly the same as the ellipsoid. This was a rather blunt ogive, using a curvature of 2X the diameter of the rocket body.
I was surprised that the rounded noses spanned the entire range of drag: the best, worst, and middle performers were all rounded. I have seen all sorts of hand-waving advice online that surface area matters more in subsonic flow. That's true for laminar flow, but we don't have laminar flow in rocket flight, we have turbulent flow. Even though the sphere has the least surface area, it seems that the surface angle of incidence matters more. The pressure distribution shown in the simulation was pretty clear.
Based on these observations, I surmise that the ellipse performs best of the round-noses because it minimizes the area that is nearly perpendicular to the flow while also rapidly reducing the wall angle of incidence for the rest of the nose. The ogive performs best overall because it as zero area perpendicular to the flow. The sphere is worst because it has the most surface area near-perpendicular to the flow. The conical shapes maintain the same angle of incidence throughout the length of the nose, although the cone with the rounded tip has some surface that is perpendicular to the flow, which adds to drag. Sharp transitions add drag, which explains why the straight cone is nearly as bad as the spherical nose, and why the cones with rounded shoulders performed better.
For those interested in the CFD details:
- I used SimFlow 3.1 with OpenFOAM 5. OpenFOAM is an industrial-strength open-source CFD engine, used and funded by several large companies. SimFlow is an expensive graphical front end for OpenFOAM, but it's free if your mesh has less than 100,000 nodes, which is more than enough for my purpose here. Once I start simulating whole rockets, I'm not sure.
- SimFlow supports up to OpenFOAM 4, but OpenFOAM 4 is no longer available as an installation package for Linux 18.04 (OpenFOAM is up to version 7). I found that it worked fine with OpenFOAM 5, at least for this type of problem. SimFlow support told that their new release in a couple of weeks supports up to OpenFOAM 6.
- I used a wedge mesh for this axisymmetric problem. The mesh extended 1m in front and around the nose, with the tail end of the rocket body sticking out the back side of the mesh, to eliminate the effect of tail drag and just examine nose drag.
- I used a solver for incompressible flow (compressible flow simulations aren't needed below mach 0.3, or about 100 m/s), and included turbulence simulation. The walls of the rocket were non-slip (i.e. the walls have boundary layer effects). The air speed was set to 77 m/s, which is typical maximum velocity for a 2L soda bottle rocket.