Game Development Reference
In-Depth Information
For a ship, the viscous boundary layer can be quite thick, up to several feet near the end
of the ship depending on its length, and the mass of water that gets accelerated is sig‐
nificant. Therefore, when doing any sort of analysis that involves the acceleration of the
ship, you need to consider added mass, too. The calculation of added mass is beyond
the scope of this topic. We should also point out that, unlike mass, added mass is a tensor
—that is, it depends on the direction of acceleration. Further, added mass applies to
both linear and angular motion.
Added mass is typically expressed in terms of an added mass coefficient, which equals
the added mass divided by the mass of the ship. Some methods actually integrate over
the actual hull surface, while others approximate the hull as an ellipsoid with proportions
matching the ship's. Using this approximation, the ellipsoid's length corresponds to the
ship's length while its width corresponds to the ship's breadth. For longitudinal motion
(that is, linear motion along an axis parallel to the ship's length), the added mass coef‐
ficient varies nearly linearly from 0.0 at a breadth-to-length ratio of 0 (the ship is in‐
finitely thin) up to 0.5 at a breadth-to-length ratio of 1 (a sphere).
When the added mass coefficient is expressed as a percentage of the ship's mass, virtual
mass can be calculated as mv = m (1 + xa ), where m is mass, and xa is the added mass
coefficient—for example, 0.2 for 20%. For typical displacement ship proportions, the
longitudinal added mass ranges from about 4% to 15% of the mass of the ship. Con‐
servative estimates generally use 20%.
Guidance speeds
To provide some guidance, Table 16-1 provides common ship types and appropriate
speed ranges. This will help guide you in properly simulating the resistance of your
Table 16-1. Some vessels and their speeds
Vessel type
Speed (knots)
Horsepower (hp)
Aircraft carrier
Oil tanker
200-foot yacht
35-foot recreational boat
35-foot speedboat
40-foot sailboat
Note that at a certain speed, for non-planing hulls, there is a theoretical limit to how
fast a boat can go. This speed is called the hull speed . At the hull speed the bow and stern
waves reinforce each other, and there is a rise in wave-making resistance. This can be a
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