Game Development Reference

In-Depth Information

Joukouski theroem
to estimate the lift force on rotating objects such as cylinders and

spheres. The Kutta-Joukouski theorem is based on a frictionless idealization of fluid

flow involving the concept of circulation around the object (like a vortex around the

object). You can find the details of this theory in any fluid dynamics text (we give some

references in the
Bibliography
), so we won't go into the details here. However, we will

give you some results.

For a spinning circular cylinder moving through a fluid, you can use this formula to

estimate the Magnus lift force:

F
L
= 2 π ρ L v r
2
ω

where
v
is the speed of travel,
L
is the length of the cylinder,
r
is its radius, and ω is its

angular velocity in radians per second (rad/s). If you have spin,
n
, in revolutions per

second (rps), then ω = 2 π
n
. If you have spin,
n
, in revolutions per minute (rpm), then

ω = (2 π
n
) / 60.

For a spinning sphere moving through a fluid, you can use this formula:

F
L
= (2 π ρ v r
4
ω) / (2 r)

where
r
is the radius of the sphere. Consistent units for these equations would yield lift

force in pounds in the English system or newtons in the SI system. In the SI system, the

appropriate units for these quantities are kg/m
3
, m/s, and m, respectively.

Keep in mind that these formulas only approximate the Magnus force; they'll get you

in the ballpark, but they are not exact and actually could be off by up to 50% depending

on the situation. These formulas assume that 1) there is no slip between the fluid and

the rotating surface of the object, 2) there is no friction, 3) surface roughness is not taken

into account, and 4) there is no boundary layer.

At any rate, these equations will allow you to approximate the Magnus effect for flying

objects in your games, where you'll be able to model the relative differences between

objects of different size that may be traveling at different speeds with different spin rates.

You'll get the look right. If numerical accuracy is what you're looking for, then you'll

have to turn to experimental data for your specific problem.

Similar to the drag data shown in the previous section, experimental lift data is generally

presented in terms of lift coefficient. Using an equation similar to the drag equation,

you can calculate the lift force with the following equation:

F
L
= (0.5 ρ v
2
A) C
L

As usual, it's not as simple as this equation makes it appear. The trick is in determining

the lift coefficient, C
L
, which is a function of surface conditions, the Reynolds number,