Computation Fluid Dynamics Model of a Foster Slug

Recently (well late 2009) at work we acquire the hardware (big/fast multi-core PC with 24Gb of RAM) and software (ANSYS Fluent) to start doing CFD work in house. CFD is Computation Fluid Dynamics. CFD is a manner of taking a fluid volume and breaking it down in to many smaller easier solved regions. The fluid can be a liquid or a gas and can be an internal volume such as flow in a pipe or and external volume such at the volume of air around a passing projectile. Each region if small enough can be relatively easily described mathematically and the fluid interactions computed in fairly straight forward manners. If you then couple each of these small simple regions together with their neighbors and require the solutions on the boundaries between the regions to match you have a mesh. This mesh becomes a huger mathematical array representing these small regions. By doing this we formulated what would normally be a very difficult problem into a form that plays to a computerís strength; lots of simple math done over and over and over.

As my colleague was still coming up to speed on the software, I got him to sneak in a few runs of one of my slug models through the software as a practice model. The model was originally a CAD model of the slug. The CAD model of the slug is then subtracted from a much larger volume to represent a large volume of air around the projectile. The region needs to be large enough that effects of the projectile have minimal interaction with the boundaries of the volume. In this case the volume of 10 diameter of slug infront 20 diameter behind and 15 diameters from the sides.

In the first attempt we did not setup initial and boundary conditions correctly and the results below are for a slug only going ~480fps. The slug itself is a model of a 410 Remington Slugger (did you really think I wouldn't do a 410 slug first). Anyway the first plot is the velocity distribution around the slug and the second is the pressure distribution around the slug at ~480fps.

Velocity Plot ~480fps (click to enlarge)

Pressure Plot ~480fps (click to enlarge)

After working through some more failed attempts and a little help from Fluent's tech-support we got our first good super sonic solution. The following is the Velocity and Pressure plots for the 410 Slugger doing approximately Mach 1.4, which is ~1560fps. The solution is using a relatively coarse grid (to save computation time) and only 1000 iteration through the solver so the solution is good but probably not as accurate as it could be. It does however show the general velocity and pressure distribution around a classic foster slug in flight.

Velocity Plot ~1560fps (click to enlarge)

Some cool things to note in the velocity plot is the well developed bow shock in front of the slug and then the smaller secondary oblique shock waves coming off the very front of the rifle grooves and the skirt at the base of the rifle grooves.

Pressure Plot ~1560fps (click to enlarge)

The big thing to note in the pressure plot is the low pressure along the sides of the slugs caused by bow shock and shock off the leading edge of the rifled grooves. These two shocks waves create a relatively low pressure along the length of the rifled grooves. This supports the idea that the angled groove cannot cause appreciable amount of moment to spin the slug.

Also note the extreme low pressure behind the hollow base. This plays an important role in the stability of the foster slugs and is likely the largest source of the drag stability of slug couple with the forward location of the center of mass.

The next run used a finer mesh and more iterations through the solver and the solution turned out very nicely. The following are streamline plots. Each line in the plot represents the path a "partial" of air might follow as the slug flies past, viewed from the slug's point of view. Both plots are the same but the second plot omits the slug so you can see some of the details of the recirculation that is occurring in the hollow base.

Stream Line Plot (click to enlarge)

Stream Line plot with out slug body (click to enlarge)

I also calculated a torque on the slug due to the angled rifle grooves. Now this torque is only valid for this particular velocity and for a non-spinning slug. The torque will be different at different velocities and spin rates of the slug.

But I used this torque and the mass moment of inertial of a 410 slug to make a crude first order approximation of the spin rate the slug would have at 100 meters down range. This approximation assumes a constant torque and no initial spin as the slug leaves the barrel. The first order approximation arrives at ~2500rpms. Remember the torque due to the angled grooves on the slug diminishes as the slug's spin rate increases and the velocity decreases so this approximation is likely on the high side.

2500rpms also sounds, at first blush, pretty high but in actuality it's really slow for a spin stabilized projectiles. For comparison look at sabot-slug from a modern rifled slug-gun. A 1:28 twist slug-barrel firing a slug going 1900fps results in an initial spin rate of nearly 49,000rpms. About the slowest twist rate you will find is a muzzle-loader set up for shooting round balls, these often have a twist of 1:66. Assuming a muzzle-velocity 1900fps the resulting spin rate is just over 20,000rpm.

Next to come is to run the model at one or more very slight angles of attack (~1-3 degrees). This will let us calculate the center of pressure and restoring moments so we can start to get an idea of the stability forces involved in the drag stability of the slug. I would also like to run the model at a few different velocities so I can establish drag and moment coefficients. This will allow for a more accurate ballistic model of the slug.

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