# Buoyancy test

This test checks the values for the buoyancy components, see Buoyancy for details on the theory.

## 1 Test description

This test uses a cylinder floating in water. For different wave and initial conditions, the Buoyancy loads and Buoyancy springs from the Total hydro loads sensor are benchmarked against analytical solutions.

## 2 Model

The model is illustrated in the figure below: The model consists of a uniform circular cylinder. The diameter of the cylinder is
$$D = 6\text{ m}$$
and the thickness is
$$t = 0.060\text{ m}$$
. This gives an outer area
$$A = \pi(D/2)^2=28.27\text{ m}^2$$
and a structural area
$$A_S = \pi(D/2)^2 - \pi(D/2-t)^2=1.120\text{ m}^2$$
.
The cylinder is made of steel with a density
$$d = 8500\text{ kg}\cdot\text{m}^{-3}$$
. The length of the cylinder is
$$L = 30\text{ m}$$
, which gives it a mass
$$m_S = A_S\cdot L\cdot d = 285.5\text{ t}$$
. In addition, a point mass of
$$m_P = 195.18\text{ t}$$
is added at the lowest node of the structure, which gives a total mass of
$$m = m_S+m_P=580.69\text{ t}$$
The water density is
$$d_W = 1026.9\text{ kg}\cdot\text{m}^{-3}$$
.
The gravitational acceleration is
$$g=9,80665\text{ m}\cdot\text{s}^{-2}$$

## 3 Analytical solution

### 3.1 Draft 20 m

For this test, the initial draft of the cylinder is
$$l = 20\text{ m}$$
.
The mass of the displaced water is thus
$$M_W=l\cdot A\cdot d_W$$
and the buoyancy force will be
$$F_B = g\cdot M_W=5694\text{ kN}$$

### 3.2 Draft 20 m  with waves

This test is the same as the previous one with the addition of linear regular waves of period
$$T = 10\text{ s}$$
and amplitude
$$A = 2\text{ m}$$
. The time-varying wave elevation at the center of the cylinder will thus be
$$\zeta(t) = A\sin(2\pi t/T)$$
. The buoyancy force will be
$$F_B(t) = g\cdot A\cdot d_W \cdot\left(l+\zeta(t)\right)$$