Saturday, August 18, 2007

Aerodynamics for high speed R/C cars

This great article taken from Dave Beeby:

A car of this type needs to take advantage of the air it is ploughing through, without getting dragged back by it too much. I have designed an undertray system to create downforce using the air which flows between the car's chassis floor and the road. This is an extremely efficient way of ensuring the car stays on the road, rather than getting pushed off it and flipped into the air. Wings, as used on RC cars normally, are hardly wings at all, usually just a bent piece of plastic with no aerofoil section. These are fine for low speed applications, up to say 30MPH. When you are gunning for some crazy speeds, this type of inefficiency is not acceptable - there are far better ways to apply downforce. I want to use 'wings' (if at all) only to trim the aerodynamic balance of the car, NOT as a primary source of downforce. The information below shows how this primary source works.

The aerodynamic undertray

This is the part of the car which is designed to 'suck' the chassis to the ground, to counteract the lift generated by the body. It is a channel, a few millimetres deep, with a constriction about half way along its length. The constriction has a width of one half of the initial channel width. Air travelling beneath the chassis is forced to speed up as the channel width decreases. It does this because it takes less energy to speed up than it does to compress - at these sorts of speeds air is almost incompressible. Fast flowing air is at a lower pressure than slower flowing air, so the pressure of the atmosphere pushes the car down trying to equalise the low pressure under the chassis. Of course, some air will be sucked into the channel from the sides (leakage), meaning that the channel pressure may not be as low as theory predicts. Once the air passes through the constriction, it must be returned to 'ambient' airspeed as smoothly as possible. This is the function of the diffuser at the rear of the undertray.

Firstly, I designed the shape of the undertray with the above principles in mind. I then modelled the shape of the undertray channel using 3 intersecting curves, each one a true analytical function. At the leading edge, the channel sides are linear. The next stage is a large-radius circular arc, and the diffuser section is again linear. Because these sections are exact, continuous mathematical functions, I was able to know the channel width accurately as a fuction of distance from the leading edge, without having to make thousands of measurements. I then applied some simple fluid-dynamics theory (namely, the incompressiblity of air, leading on to the Bernoulli equation) to compute the air pressure as a fuction of distance from the leading edge. This could then be integrated, numerically, over the channel area to get a value for the total downforce, which is a function of the speed of the car.

Using vorticity-diffusion theory I also tried to put a value on the thickness of the fluid boundary layer present in the channel. This exists because the air immediately in contact with the chassis surface can not move, due to friction (viscosity), relative to the chassis. Of course, the viscosity of air is very low, but the thickness of the channel is also small. If the boundary layer is close to the size of the channel depth, then the calculated airspeed in the channel will grossly overestimate the true speed, and we will predict a value of downforce which is too big. Luckily, the boundary layer works out to be less than 1/10th of the thickness of the channel (at 60MPH), and this is at the back of the channel where it will have grown to its largest size. With increasing speed, the boundary layer thickness decreases - which should bring the computed downforce nearer to the true value. However at high speeds there will likely be more 'leakage', so downforce could be reduced by this effect.

Calculated results

The graph below shows how the channel pressure varies as you go from the front to the back of the channel. The different curves are for different car speeds. Notice how the pressure is lowest right where the channel is at its narrowest, and therefore where the airflow is most rapid.

Now a graph of the total downforce on the car, as the speed increases. The downforce goes up as the square of the car's speed - similar to the drag on the car (another issue altogether). A good point to note is that this large amount of downforce has been achieved for a tiny drag penalty, if you compare with the drag you would get from a wing mounted on the body which could generate this level of force. This is how I am hoping to beat the record speed set by other cars, which could have had access to a greater driving power than I do. Not that I'm lacking in power you understand - but neither am I driving through treacle.

With the equation of the line plotted, you can easily work out the downforce for any speed. At the record speed of 111MPH there will be a downforce equivalent to 7.7kg (17lb)! Assuming no leakage!

The Proto2 chassis, ready to run in 12 cell mode. This is the state of the car in November 2002. Note the large diffuser section at the rear, and the new suspesion up front.

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