Calculating FRAM’s Dead Water
49
Dead Water Resistance
The pressure force on the ship along the motion direction due to the internal wave
wake is given by F 1 = ∫ F −pn 1 dS, where integration is over the wetted body surface
and the pressure obtained by the Bernoulli equation, p = −𝜌 0 (𝜕𝜙 0 ∕𝜕t +
1
2
|∇𝜙 0 | 2 +
gy) + const. The time dependent force is obtained by use of Gauss’ theorem and the
transport theorem [11]:
F 1 = −𝜌 0
d
dt ∫ F
𝜕𝜙 0F
𝜕x 1
𝛽 d𝐱 − 𝜌 0 U ∫ F
𝜕𝜙 0
𝜕x 1
𝜕𝛽
𝜕x 1
d𝐱 +
𝜌 0
2 ∫ F
|∇𝜙 0 |
2 𝜕𝛽
𝜕x 1
d𝐱.
(16)
The internal wave dead water resistance coefficient is defined by
C dw =
F 1
1
2
𝜌 0 SU 2
,
(17)
where S denotes the wetted surface area of the ship.
Upper Layer Depth. Draught of Ship Model
The ratio between the ship draught b 0 and the upper layer depth h 0 is an important
parameter of the dead water resistance where a value of b 0 ∕h 0 close to unity produces
a strong wave wake and force. A large ship volume relative to h
3
0
is another important
nonlinearity parameter in the dead water problem [12].
Returning to Nansen [1], regarding the level and vertical extent of the pycnocline,
the water was described as salt at the level of the bottom cock of the engine room, at
4 m depth. This is shallower than the draught of FRAM’s keel of 5 m. The stratification of the sea was not measured. We note: First, the FRAM was in motion during
the observations, where the pycnocline at the subcritical speed becomes uplifted at
the rear of the ship. Second, the pycnocline in the observations eventually had some
vertical extension.
The averaged depth of the pycnocline at rest might have been at 4 m, shallower,
or at 5 m. In the present two-layer calculations we shall assume that the depth of the
interface is at h 0 = 4 m. We include calculations from [11] using h 0 = 5 m.
The FRAM has a length of l 0 = 30 m and a width of w 0 = 11 m which in
nondimensional terms become: l 0 ∕h 0 = 7.5 and w 0 ∕h 0 = 2.75, respectively (h 0 =
4 m). The layer depth ratio in the calculations is put to h 1 ∕h 0 = 18 and the density
ratio put to 𝜌 0 ∕𝜌 1 = 𝜇 = 1 − 𝜖 with 𝜖 → 0. The shape of the model ship is given by
𝛽(x 1 , x 2 ) = −b 0
(
(1 − (2x 1 ∕l 0 )
2
− (2x 2 ∕w 0 )
2
)
.
(18)