Update: Karan 2018-12-09

• Brainstormed on a theoretical approach to predict forces and torques on the bottom quad due to the top quad.
• Studied propeller slipstream model

The theoretical approach we follow lies on the assumption that all the disturbances on the quads due to air flow are effectively due to 2 types of forces:

1. Pressure Drag
2. Propeller inefficiency due to incoming flow

The pressure drag is simply as a result of the incoming flow (due to the top quad) exerting forces on the bottom quad which is assumed to be a bluff body. Hence all we need is the velocity field on the bottom quad. This is given by the propeller slipstream model.

Propeller inefficiencies on the other hand are due to the change of the thrust constant used in our code (thrust = constant * rpm^2). The word ‘inefficiency’ is misleading because the propeller performance could even ‘improve’ with oncoming flow (that is the constant could increase). Only experiments will tell.

Propeller slipstream model

Propeller slipstream model is a distance dependent model which characterizes the downwash due to a propeller (a.k.a. propeller slipstream or propwash). This model was chosen because it has distance dependence which makes it superior to many other models including the actuator disk model. The actuator disk model uses momentum theory (only) which predicts contraction of the downstream flow but does not consider diffusive effects due to viscosity of air.

The propwash model takes into account these diffusive effects and hence the airflow of a propeller initially contracts due to dominant pressure forces from the motion of the propeller. After a certain distance in the axial direction, the viscous forces start to dominate and there is a reduction in velocity which results in expansion of the flow (to conserve mass).

Essentially, the downstream flow of a propeller undergoes contraction, reaches a minimum at the ‘efflux’ plane, and then undergoes expansion.

For ease of explanation here are some parameters that I will use for succinctness: Radius/diameter of propeller (Rp/Dp); radial distance from the axis of symmetry or propeller axis (r); axial distance from propeller (z or x, used interchangeably); radius/diameter of the efflux plane flow cross-section (Ro/Do). (Note that Do<Dp)

The propwash is divided into 2 regions: a near-field region and a far-field region. In the near field region, we have dominant pressure forces and the flow contracts. The radial flow profile is also very peculiar – it has a peak at a particular radial distance, rather than at the axis of the propeller. If we give it a thought, it turns out to be logical because the speed of the cross-section of the propeller near the center is low compared to that towards the tip (v=r*$\omega$). A peak is probably obtained because the chord length of the propeller is smaller at the tip (r=Rp) as compared to somewhere a little behind the tip (r<Rp).

The far-field region is divided into 2 zones: a zone of flow establishment (ZFE) and a zone of established flow (ZEF) (I hate the acronyms but these are what are used in the paper and I want to be consistent). In ZFE, the flow still has a peak at r>0, but this peak slowly moves towards the axis (r=0). The axial size of this zone is about 3.25*Do, and at the end of this zone, the peak is finally at the axis. Beyond this zone is the ZEF where we have a single peak. Here the flow is assumed to be gaussian with a variance dependent on ‘z’ and viscosity of air.

The image shown below summarizes all the regions and zones very well.

Since Do<Dp, we know that the ZFE is smaller than 3.25*Dp, where Dp is obviously known to us. The question that remains is to find xo. The paper does provide a formula to obtain xo, but the parameters used in it are empirical (for example, coefficient of thrust) that we have to determine experimentally for our quads.

Roughly, xo lies between 1-3 propeller diameters, so we know that the ZEF assumption would be valid beyond 6-8 propeller diameters.

Propeller inefficiency

These would be characterized by directly doing experiments with a propeller on a thrust stand with oncoming flow (using Jason’s nozzle, if he allows it; if not, then any other fan). The thrust constant would be calculated for various flow velocities which will give us a relation between the two. This then in conjunction with the propwash model could give us accurate estimates of RPM to produce a given thrust, rather than using a constant coefficient.

3. Planned work for the next two weeks

(Planned work is for 2 weeks, because this week me and Trey might not get enough time due to the end-semester exams)

My plan is to use the theoretical knowledge of flow field in the ZEF to predict the forces and torques on the bottom quadrotor. The top quadrotor is assumed to be a single rotor with 4 times the area of its individual propellers. This assumption is reasonable because we are working in the ZEF where the flow is expected to become axisymmetric with a single peak at the axis itself. Once we have the expected forces and torques, we can modify the controller to produce motor thrusts such that they can counter the effect of these forces and moments.