Game Development Reference
In-Depth Information
In Equation (5.12) the quantity
A
is a characteristic body area. Its definition will differ
depending on the body geometry. For most objects, the characteristic area is taken to be the
frontal area. For a sphere, the frontal area would be the cross-sectional area,
pr
2
. Density is a
measure of how heavy the fluid is and is defined as the mass of a fluid per unit volume. Water,
for example, will have a greater density than air. In SI units, density will have units of
kg/m
3
,
velocity will have units of
m/s
, and area will have units of
m
2
. In the English system of units,
density will have units of
slug/ft
3
, velocity will be in
ft/s
, and area will have units of
ft
2
.
Drag Coefficient
The drag coefficient,
C
D
, is a nondimensional number that is used to evaluate drag force. Being
nondimensional means that drag coefficient has no units, it's just a number. Generally speaking,
the drag coefficient for an object will not be a constant, but will be a function of the density of
the fluid, the velocity at which the object is traveling, and the size of the object. Based on a lot
of experimental and theoretical research, it was found that the drag coefficient of an object
could be expressed in terms of a quantity known as
Reynolds number
,
Re
.
(
)
CC
=
Re
(5.13)
D
D
Reynolds number is another nondimensional quantity that is used to characterize the
nature of a fluid flow. It is defined as the ratio of the fluid density,
r
, object velocity,
v
, and char-
acteristic length of the object,
L
, divided by the viscosity of the fluid,
m
.
rvL
m
Re
=
----------
(5.14)
As was the case with the characteristic area in Equation (5.12), the definition of character-
istic length can vary depending on the situation, but it is commonly taken to be the body length
parallel to the direction of the fluid flow. For a sphere, the characteristic length would be the
diameter of the sphere. The viscosity of a fluid is a measure of how “thick” a fluid is. Maple syrup,
for example, would have a higher viscosity than would air. The value of Reynolds number can
vary widely from numbers less than one to numbers in the tens of millions. As an example, a 0.1
m
cannonball traveling at 100
km/hr
at sea level would have a Reynolds number of about 194,000.
As an example of how drag coefficient can vary with Reynolds number, Figure 5-6 shows a
plot of the drag coefficient of a sphere. At low Reynolds numbers (low densities, low velocities,
and/or small objects), the drag coefficient decreases with increasing Reynolds number. Over a
range of Re = 1000 to Re = 250,000, the drag coefficient is more or less constant. A sharp drop in
drag coefficient at around Re = 250,000 occurs due to the transition from laminar to turbulent
flow, a topic we will discuss in the next section.
