The bilge keel damping is usually the largest contribution in the total roll damping, which consists of four components:

Among them, the nonlinear normal force component on bilge keels and the hull pressure component make a major contribution to the bilge keel damping moment [22].

The Keulegan-Carpenter number is used in the estimation of the bilge keel damping, which is defined as:

(7–7)

where and are the period of motion and the amplitude of velocity of periodic motion; and are the distance from the roll center to the tip of the bilge keel and the bilge keel breadth respectively, as shown in Figure 7.2: Ship-Like Cross Section with Bilge, and is the amplitude of roll motion.

ITTC [21] recommends that the drag moment due to the nonlinear normal force on a bilge keel is expressed approximately as:

(7–8)

where and are the water density and the length of the bilge keel respectively, is the correction factor to take into account the increment of flow velocity at the bilge, where is the mid-ship section coefficient of which is the area of the mid-ship cross section.

Note that the drag coefficient depends on the Keulegan-Carpenter number. Substituting Equation 7–7 into Equation 7–8 we have:

(7–9)

In Aqwa time domain analysis, the Keulegan-Carpenter number if further defined as:

(7–10)

where is the roll period at resonance.

By employing the ITTC [21] recommended method, the coefficient of pressure on the front and back hull faces of the bilge keels and the length of the negative pressure region are given by:

(7–11)

The roll drag moment is:

(7–12)

where is the length along the girth of the cross section and is the moment lever.

Combining Equation 7–5, Equation 7–8, and Equation 7–12, the total roll drag coefficient due to bilge vortex shedding and bilge keels is:

(7–13)

if the nonlinear drag moment is expressed as