# Spinning Paper Sheet Meets CFD

I don’t think anyone would argue with the fact that paper airplanes are simple (and fun!), but what is the simplest paper airplane that can still fly? I give you a single rectangular piece of paper without any folds that will gently spin around its longest horizontal axis if released with a long edge parallel to the ground. Next, what is the simplest Computational Fluid Dynamics (CFD) method that can capture the essence of the spinning paper? I give you the Moving Reference Frame (MRF, also known as the frozen rotor method) option for CFD. Combine the two and you arrive at an interesting simulation of a simple phenomenon.

CFD Simulation of a Rotating Paper SheetVelocity vectors at 90 degrees

Recall that a MRF is an approximation to a spinning motion, so the results from such an approach will not be perfect, but they are nevertheless useful and insightful.

### CFD Model

The CFD model for this simple paper airplane consisted of a zero-thickness, double sided face to represent the paper sheet, embedded within a MRF cylinder, which was embedded in a larger cylinder representing the outer extent of the flow domain. Due to symmetry, and to save resources, I only modeled half of the flow domain.

CFD MRF Symmetry 3D Model of a Rotating Paper Sheet

### Results

To obtain flow results at various angles of the paper sheet I used automation via a Python script to rotate the geometry and run a series of CFD simulations. The lift and drag values are collated in the plots below.

Velocity Vectors for Rotating Paper Sheet at 0 Degrees

Velocity Vectors for Rotating Paper Sheet at 90 Degrees

Pressure Iso-Surfaces for Rotating Paper Sheet at 0 Degrees

Pressure Iso-Surfaces for Rotating Paper Sheet at 90 Degrees

Lift for a Rotating Paper Sheet

Drag for a Rotating Paper Sheet

### Conclusion

The lift and drag show a relatively smooth variation through the range of angles simulated. The lift peaks close to the horizontal position (i.e., 0 and 180 degrees) and approaches zero around a 90 degree inclination to the horizontal - resembling a 'U' shape. Interestingly, the drag resembles a sine wave and has negative values (thrust) between 100 - 180 degrees.

### Notes

The CFD simulation was performed in Caedium Professional using the MRF option for the incompressible, steady-state RANS solver, and the k-omega SST turbulence model.