Frisbee flight

Flow vis of frisbee boundary layer separation (flow is from the top down) – from Potts, J. 2005

Frisbees are super interesting! They generate lift in a way similar to a wing, but because they are so short in a transverse direction, 3d effects dominate. The easiest way to keep stable flight with this sort of shape (aside from gyroscopic stabilisation i.e. throwing it) is to ensure a turbulent boundary layer – which is why frisbees have ridges! The ridges serve to trip the boundary layer and keep it under control, maintaining a more steady lift distribution. Frisbees without ridges are much more sensitive to angle of attack, and harder to throw properly.

I wanted to model this boundary layer separation – and I still have a lot of learning to do! Below is a Q contour animation of my latest model, using LES with the smallest elements on the order of the Taylor microscale. The frisbee was set up at a 5 degree angle of attack with a 15 m/s headwind, which is identical to the image above. The geometry does not have ridges, but it does have a rotating wall boundary condition (counter-clockwise when viewed from above). The weird upstream artifacts are due to a poor mesh from snappyHexMesh – the sharp transition between cell sizes interferes with the LES algorithm, which uses the cell size as a spatial filter – I’ll see if I can get a structured mesh set up.

The aerodynamic coefficients given by this model are C_l ~ 0.20 and C_d ~ 0.10 – the drag coefficient is close to that measured by Potts but the lift coefficient should be closer to 0.4. This likely has a lot to do with the poor mesh.

Vortex shedding

A quick study looking at the Strouhal number for a medium-length cylinder – above is an animation of Q isosurfaces, coloured by vertical velocity. I ran some 2d preliminary models, followed by a 3d model with a structured mesh that I then refined. The Reynolds number was 51,355, which according to data from Achenbach (1968) is in a transitional range for the coefficient of drag:

My results gave C_d = 0.73, which is higher than for a long cylinder the above chart indicates – likely due to the three-dimensionality of the flow around the ends.

A plot of C_l vs time clearly shows the oscillations of the Karman vortex sheet, at a frequency of about 37 Hz. This gives a Strouhal number of about 0.12, which is not far off experimental values of 0.18-0.50.

Also visible in the plot is that after about 0.4s, the oscillations become unstable and appear to ‘beat’. The RMS value does remain pretty constant at around 0.4, but I’m curious whether this is due to end effects or maybe just the integration schemes used!

Return to BFS flow

I’ve been working to improve my CFD skills, and have set up a workstation running OpenFOAM – what better case to practice on than one I’ve already done before?

I wasn’t entirely happy with my results from the last attempt, thinking most of the error was due to an improper mesh, so I ran the same case using a better mesh, and an assortment of turbulence models in 2d and 3d.

Streamlines in the 2d BFS flow

The reattachment length was found by evaluating the wall shear stress along the bottom wall of the downstream section – where this is zero is the reattachment point. I found the standard k-\epsilon model did not reach mesh convergence at any reasonable point, but both k-\omega and realisable k-\epsilon did, giving x_R/H ~ 4.8 and 5.6 respectively. Both of these are still below the real steady-state value of 7.0, but are much more promising than last time!

I used k-\omega for the 3d case as well – it was developed specifically for internal flows and is supposedly the best RANS model for BFS flow, but it looks like I need to work on my calibration as I got an x_R/H of 4.5! This may also be because I imposed a symmetry condition on the centre of the duct – the flow has been reported as two-dimensional along this plane but there are transient 3d effects to take into account.

Three-dimensionality of the near-wall flow

Perth to Melbourne

That was a long drive! 9 days along the south coast.

It was smooth sailing except for a few hiccups – thankfully I brought some tools and the MX5 workshop manual or it would have been a lot longer than 9 days…


Sheet Metal Guillotine

This was designed for a machine design unit.

Requirements were to cut 3 mm mild steel sheet, up to 2400 mm wide, and be powered by one hydraulic cylinder while still maintaining a straight & level cut.

Several iterations led to a design that could be CNC machined from one single 20 x 1800 x 6000 plate, with a few extra small parts.

Double Whitworth Quick Return

This was an experiment of a few different things:

  • A quick return mechanism with a return ratio of 24:1
  • 3d printable herringbone gears
  • The gears can be swapped to change the ratio, without changing any other parts. Ratios of 1:4,6,8,10,12 are possible
  • The whole mechanism was designed for assembly: no more than two parts come together at any stage

I’d been using spiral-faced cams for quick-return purposes and thought maybe there was a better option. The Whitworth quick return (below) is a crank-rocker with one slow stroke and a quick return stroke, constructed by having the distance from crank pivot to rocker pivot “almost” equal to the crank length. The return ratio can be defined as r=\frac{\abs{\theta_{max}-\theta_{min}}}{2\pi}, where \theta_{max} is the crank angle when the rocker reaches it’s maximum, and similar for \theta_{min}.

With a crank length L equal to 90% of the pivot distance y, the WQR has a return ratio of about 6.5. A return ratio of 24:1 requires the ratio L/y to be about 99.2% – a tolerance unachievable with 3d printing and difficult in any other manufacturing method. But the double-crank WQR can be used as an input for the crank-rocker WQR – giving a two-stage (or double) quick return linkage that has a high return ratio while using low-tolerance links. The prototype was a good proof-of-concept but of course, a high return ratio requires very little torque on the output – even slight resistance stops the was a success

Herringbone gears provide the same smooth operation as helical gears, without the additional axial load produced by the helical teeth. They can be modelled in the same way as helical gears, except with a loft extending from the midplane to each face, in opposite directions. The ones for this project range in module from 1.2 mm to 0.8 mm, and came out pretty well, with smooth operation all around.

I wrote a script to size the gears for this project. Given a set of desired ratios, the script performs a brute-force search through m, z space to find a two-stage gearbox that can swap gears to satisfy all the desired ratios, while fitting in the available space. It then outputs a few of the best options for each ratio. For this project I chose 1:4,6,8,10,12 and the selected gears are: (given gear 1 with z1, meshing gear two with z2, at module m denoted as z1|z2_m)

    \begin{align*} 1:4 &- 20|40_{1.2} \times 20|40_{1.2} \\ 1:6 &- 20|40_{1.2} \times 18|54_{1.0} \\ 1:8 &- 20|40_{1.2} \times 18|72_{0.8} \\ 1:10 &- 17|55_{1.0} \times 22|68_{0.8} \\ 1:12 &- 18|54_{1.0} \times 18|72_{0.8} \end{align*}

The 1:10 gearset doesn’t share any gears with any other set, so I didn’t bother actually printing that set. However all four other ratios run smoothly and are easy to interchange.

The mechanism was designed such that only two parts come together at any point. This was a lot easier to assemble than the last big linkage I printed, but required a lot more fasteners. It was easy to change the gears, as the faceplate could be removed without disturbing the rest of the layout. I think that the DFA design made the skeletal aesthetic a lot “noisier” and less aesthetically pleasing than a fully-aligned design as before.

Overall this was a successful experiment and a good way to further improve my printing skills.

Spur Differential

Helical gears use an involute tooth that’s been swept along a helix. At any cross-section perpendicular to the gear axis, the gear profile is exactly the same as a straight spur gear – so a helical gear can be modelled as a loft between the top & bottom faces. The sketch defining the tooth profile gets offset by an additional amount, corresponding to the desired helix angle.

Both of these are spur differentials, printed about 6 months apart.

The white one was designed when I was new to both 3d printing and gears, so the involute is not quite right and the tolerances are sloppy. There is a lot of play and occasional jamming, and it was constructed using screws fixed with loctite.

The orange/blue one is more recent and uses my helical gear model. There is just enough clearance between all rotating parts to allow smooth rotation without any significant backlash. The nuts are all press-fit and sit flush with external surfaces – the shafts are designed such that the screws can be tightened in place without loctite. Overall a much better model!