# Heave plate test

This test checks the vertical hydrodynamic loads applied on an element.

## 1 Test description

The model used for this test is a cylinder with a heave plate at the bottom, with similar dimensions to the columns of the OC7 model.
The test uses the Loads only setting.

## 2 Model

The model is illustrated in the figure below.

$$R_C = 6\text{ m}$$
and the heave plate has a radius
$$R = 12\text{ m}$$
and a vertical length
$$L=6\text{ m}$$
, extending from
$$z_B = -20\text{ m}$$
to
$$z_T = -16\text{ m}$$
, where
$$z=0$$
corresponds to the mea sea level
The drag and inertia coefficients for perpendicular loading on the heave plate are
$$C_D = 0.6$$
and
$$C_I=1.63$$
The drag and inertia coefficients for vertical loading on the heave plate are
$$C_{D,HP}=4.8$$
and
$$C_{I,HP}=1.67$$
The water density is
$$\rho = 1026.9\text{ k}\cdot\text{m}^{-3}$$
The wave dirction is the y-axis.

## 3 Analytical solution

The wave kinematics are evaluated following Regular waves theory. The loads are computed using the Morison equation
The drag and inertia loads on the heave plate use the wave kinematics of the middle of the element representing the heave plate, located at
$$z=-17\text{ m}$$
.
The drag and inertia loads in the perpendicular direction (i.e. y-axis) are given by

$$F_D=\rho C_D\cdot R\cdot v_y|v_y|\cdot L$$
and

$$F_I=\rho C_I\cdot\pi R^2\cdot a_y\cdot L$$

where
$$v_y$$
and
$$a_y$$
are the wave particle velocity and acceleration in the y-direction, respectively.

The drag and inertia loads in the verticat direction (i.e. z-axis) are given by
$$F_{D,HP}=\frac{1}{2}\rho C_{D,HP}\cdot\pi R^2\cdot v_z|v_z|\cdot L$$
and
$$F_{I,HP}=\rho C_{I,HP}\cdot\frac{4}{3}\pi R^3\cdot a_z\cdot L$$

where
$$v_z$$
and
$$a_z$$
are the wave particle velocity and acceleration in the z-direction, respectively.

Note: for this test, we check the drag and inertia loads at the node at the bottom of the heave plate. In Ashes, the loads on an element are divided in two and applied at both ends of the element. Therefore, to obtain the values from the test, the loads obtained from the above equations must be divided by 2.

The Froude-Krylov forces on the top and bottom ends of the heave plate are computed using the dynamic pressures at the centre of the bottom and top of the heave plate
$$p_B$$
, and
$$p_T$$
,respectively, computed at
$$z_B$$
and
$$z_T$$
. Their values are equal to
$$F_{FK,b}=\pi R^2\cdot p_B$$
and
$$F_{FK,t}=\left(\pi R^2-\pi R_C^2\right)\cdot p_T$$

## 4 Results

A simulation of 30 seconds is run for each load case. The results are considered passed if the results from Ashes are within 2% of the analytical solution.

The report for this test can be found on the following link: