What is the optimal streamlined shape?

Question

We all know the typical streamline shape which looks like a tear. But what is the optimum shape (for speeds 0-200 km/h)?

Is the front a half sphere or is it an ovoid? How are the sides and the end formed? Can it be described in a formula or in a bezier curve? What is the drag coefficient compared to standard tear drops? Image with explanations how the shape must be built to minimize the drag are perfect.

And no, I don't accept flat plates and perfectly laminar airflow :). The shape must be built around a pipe with diameter R and must not be longer than 10R.

Answer 1

There's a lot more to it than just minimum drag shape. At 200 kph, you want it to be aerodynamically stable as well. If the pipe is supporting the fairing, and you want to limit the load on the pipe, then the pipe has to be forward of the center of resistance of the fairing.

The shape will depend on temperature, altitude, humidity, the size of the pipe, turbulence and other non-uniform features in the freestream, and manufacturing ability. Getting to 95% of the ideal is pretty easy to achieve for say a 6" pipe. Getting that last 5% can be tricky, requiring very stiff and strong materials and very high tolerances, like +/- .0005 inch overall and no wavyness along the cord.

Here are some common ones. Plots are for Reynolds numbers of 500,000 and 1,000,000. You can adjust as necessary.

Eppler 862

Eppler 863

Eppler 864

Answer 2

The general rule of thumb for laminar flow is that the tail needs to be about 4 times as long as the nose because it is more difficult to slow down a flow than to speed it up. Equally you want to avoid any geometric discontinuities so a sensible way to do this is treat it as circular arcs at a tangent to each other.

The grey cube is 1x1 aligned with the major axes to give a sense of orientation.

Of course there are a few caveats to this:

Firstly, scale matters in fluid flow and you get to a point where the onset of turbulence is inevitable at a given scale and speed whatever you do.

Equally: when you talk about optimisation this, by definition, implies that you are seeking a useful balance between conflicting requirements. Reducing drag is only one requirement. In reality and even in in fluid dynamics the body under consideration has some purpose.

For example: Aircraft need to generate lift, thrust, control and carry some payload, so in practice a lot of aerodynamics is managing drag and turbulence rather than attempting to eliminate it entirely.

Answer 3

The optimal shape of a streamlined wing, boat, wind mill blade, is a balance between several factors:

• The speed, hence the variable wing angle, both in delta wings and angle the wing is attached to the fuselage.
• The density and viscosity of the fluid: e.g., different altitudes of flying require different shapes and configuration.

• The utility of the airfoil or rudder or boat, how much volume it should hold, how wide it should be? If you want a streamlined bus it could be too long for the road!

• The cost of manufacturing: precision and efficiency are expensive.

• The training of the pilot or operator: fighter pilots can handle a razor thin wing tip with minimal dynamic balance and thin envelope of authority, to the advantage of agility and speed of aerobatics; commercial planes have bull-nose wing tip to afford more versatility and better lift at angles near stall.

Basically the teardrop shape is practical in most cases, but as speed increases the shape approaches that of a thin blade with sharp tip to cut its way like a knife.