Validation Case: Free Heave Motion of a Floating Cylinder

The free heave motion of a floating cylinder validation case belongs to fluid dynamics. This test case aims to validate the following parameter:

Six degrees of freedom (6DoF) solver

In this project, a floating cylinder is positioned one inch off the equilibrium level. After releasing the cylinder, the transient heave decay is evaluated. The simulation results of SimScale were compared to experimental\(^1\) and theoretical\(^2\) results.

The geometry is a straight cylindrical body in a pseudo-2D water tank. The centroid of the cylinder is located in the center of the tank, at coordinates (0, 0, 0).

Details of the tank dimensions are provided in Table 1:

Dimension

Measurement \([m]\)

\(L\)

20

\(H\)

2.4384 (8 feet)

\(t\)

0.2

Diameter of the cylinder

0.1524 (6 inches)

Length of the cylinder (z-direction)

0.2

Table 1: Dimensions of the water tank and cylinder

Note

SimScale requires a domain with volume to perform simulations. Therefore, we are going to use a pseudo-2D approach for this validation case.

The meshes used in this project contain a single cell along the z-direction. An empty 2D boundary condition will be applied to both sides of the domain, so the z-direction won’t be resolved.

Mesh and Element Types: The mesh used in this validation case is a hexahedral mesh created locally and imported into SimScale. In Table 2, an outline of the cases is presented:

Case

Mesh Type

Cells

Element Type

Translation

Rotation

1-degree-of-freedom motion

blockMesh

84200

3D hexahedral

Linear (x-direction)

Fixed orientation (no rotation)

3-degree-of-freedom motion

blockMesh

84200

3D hexahedral

Planar (x-y plane, normal to the z-axis)

About the z-axis

Table 2: Overview of the mesh and the motion constraints of the cylinder

Find below the 84200 cells hexahedral mesh. It contains a single cell in the z-direction.

Simulation Setup

Material:

Air

Viscosity model: Newtonian

\((\nu)\) Kinematic viscosity: 1.55e-5 \(m^2/s\)

\((\rho)\) Density: 1.192 \(kg/m^3\)

Water

Viscosity model: Newtonian

\((\nu)\) Kinematic viscosity: 1.06e-6 \(m^2/s\)

\((\rho)\) Density: 999 \(kg/m^3\)

The surface tension between air and water is set to 0.07 \(N/m\)

Boundary Conditions:

Figure 3 will be used as a reference for the definition of the boundary conditions:

The following boundary conditions are used:

Boundary

Boundary Type

Velocity \([m/s]\)

Pressure \([Pa]\)

Phase Fraction

Top

Custom

Pressure inlet-outlet velocity

Total pressure (0)

Inlet-outlet

Sides and Bottom

Wall

Slip

Zero gradient

Zero gradient

Front and Back

Empty 2D

Empty 2D

Empty 2D

Empty 2D

Cylinder wall

Wall

No-slip

Zero gradient

Zero gradient

Table 3: Summary of the boundary conditions used for all cases

Solid Body Motion:

Center of mass: coordinates (0, 0, 0)

Mass: 1.824 \(kg\)

\((\rho)\) Density: 1000 \(kg/m^3\)

Gravity: 9.81 \(m/s^2\) in the negative x-direction

Motion constraints as defined in Table 2

The mass of the cylinder body is such that, at equilibrium, exactly half of the cylinder is immersed.

Initial Conditions

In the experimental setup\(^1\), the cylinder is initially displaced by 1 inch below the equilibrium level. After releasing the cylinder, it’s free to move only in the heave direction.

Figure 4 shows the initial condition in SimScale, to replicate this effect:

Result Comparison

SimScale multiphase simulation results are compared to experimental\(^1\) and theoretical\(^2\) data. The phenomenon of interest is the transient heave decay response. In both references, the constraints ensure a pure heave motion (1DoF).

A mesh independence study was conducted with three meshes, to assess the mesh effects on the results. Moreover, besides the 1DoF simulation, we also present simulation results modeling 3DoF (free translation in the x and y-directions and free rotation about the z-axis).

In all of the plots shown below, the displacements have been normalized in the following way:

Where \(x_0\) is the initial shift from equilibrium (one inch) and \(x\) is the current shift from equilibrium for any given instant. When the normalized displacement stabilizes at 0, the system reaches equilibrium.

Furthermore, time was non-dimensionalized in the following way:

$$Nondimensionalized\ time = t\sqrt {\frac{g}{a}}\tag{2}$$

Where \(t\) is time, in seconds, \(g\) is the gravitational acceleration in \(m/s^2\), and \(a\) is the radius of the cylinder, in meters.

Firstly, a comparison of the heave response from different meshes is given in Figure 5, using a 1DoF approach:

The figure shows that, using a coarse mesh, the baseline heave displacement deviates a lot from the zero line. The moderate mesh also has minor deviations, while the fine mesh attains the best oscillation behavior about the baseline, due to the high resolution of the air-water interface.

The comparison of numerical simulation, using a fine mesh, with theoretical and experimental data is presented in Figure 6. Great agreement was observed:

Finally, comparing the simulation results from SimScale for 1DoF and 3DoF, we get:

The animation below shows SimScale results for the 1DoF simulation. Red represents the water phase.

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