#### 3.2. Hydrodynamics Analysis

Wind, currents, and waves are the three key factors affecting the stability of the floating structure during wet towing. Hence, the responses of the structures in different environments is a great problem that has been solved in this work. The main goal of this study was to provide proper insight into practical applications such as offshore structure design, installation plans, and transportation formulation. The time histories of caisson motion with six degrees of freedom (6-DOF) were stated in the reference system by adopting the water plane as the origin. Based on 3D potential theory, hydrodynamic moments and forces on the caisson were obtained.

The dynamic equation for a floating body under various complex loadings is stated as:

where

$M$ is floating body mass matrix,

$\Delta M$ is floating body added mass matrix,

${B}_{vis}$ is viscous damping matrix,

${B}_{rad}$ is radiation damping matrix,

${K}_{stillwater}$ is hydrostatic stiffness matrix,

${K}_{mooring}$ is mooring system stiffness matrix,

${F}_{1}$ is first order frequency load matrix,

${F}_{2Low}$ is second order low frequency load matrix,

${F}_{2High}$ is second order high frequency load matrix,

${F}_{wind}$ is wind load matrix,

${F}_{current}$ is current load matrix, and

${F}_{others}$ is the remaining load matrix. Floating body mass matrix is stated as:

where

$\left({x}_{G},{y}_{G},{z}_{G}\right)$ is gravity center position and

${I}_{ij}$ is inertia mass.

Hydrostatic stiffness matrix is stated as:

where

$\left({X}_{B},{Y}_{B},{Z}_{B}\right)$ is buoyancy center position,

$S$ is water plane area, and

${S}_{i}$ and

${S}_{ij}$ are the first and second order moments of the water plane area.

In the current work,

$\Delta M$,

${B}_{rad}$,

${F}_{1}$,

${F}_{2Low}$, and

${F}_{2High}$ are obtained by AQWA-Line software and

${B}_{vis}$ is determined by Morison theory. Wind and current drag were obtained in a similar way. Environment load coefficients are defined as:

where

$F$ is drag force,

$A$ is the area of body incident to flow, and

$v$ is velocity relative to wind or current positions.

Therefore, force is determined as:

In the current work, load coefficients have been determined using Ansys Fluent software.

Figure 4 shows the computational domain obtained based on the findings of Lee et al. [

21] and Liu et al. [

22]. Fore body was 1.5 L away from the velocity inlet and aft body was 2.5 L away from the pressure outlet. In addition, lateral distance between wall and body was 2 L. First-order temporal and second-order convection schemes were applied for temporal and momentum equations, respectively, to be applied to perform spatial discretization. Jin et al. [

23] and He et al. [

24] applied the

$k-\omega $ Shear Stress Transfer (SST) turbulence model to analyze the Reynolds averaged Navier-Stokes (RANS) equation and obtained satisfactory results. In the current work, the

$k-\omega $ SST turbulence model was also applied.

For the verification of the accuracy of the developed method, square cylinder load coefficients applied in the Tang’s [

25] experimental research were obtained, and the obtained results are summarized in

Table 2. The result errors of the two methods were lower than 5%, which meant that the developed numerical method could effectively calculate load coefficients. Environment load coefficients of the investigated caisson are shown in

Figure 5.

The Joint North Sea Wave Project (JONSWAP) spectrum was applied as the wave spectrum. Empirical parameters

$\gamma $ and

$\alpha $, as well as peak frequency, were also applied. Spectral ordinate at any frequency is stated as:

where

${\omega}_{P}$ is peak frequency,

$\gamma $ is peak enhancement factor, and

$\alpha $ is a constant value that depends on wave spectrum peak frequency and wind speed, which can be determined as:

Starting and finishing frequencies are also expressed as:

where

$F(\gamma )$ is a weighting function, and weighting function values against

$\gamma \in \left[1.0,20.0\right]$ are available in the AQWA Theory Manual.