^{1}

^{1}

Nowadays, as the development of Computational Fluid Dynamics (CFD) and the numerical wave tank (NWT) has advanced, numerical analysis has become increasingly useful and powerful for the ship designing and ship hydrodynamics. In this study, a momentum source wave-maker and an analytical relaxation wave absorber were embedded into 2D RANS equation model with RSM turbulence closure scheme to establish the NWT for ship designing and hydrodynamics. The VOF (volume-of-fluid) method was applied to accurately capture the water free surface. The body force-weighted scheme is chosen for pressure interpolation and the second order upwind scheme for discretization of the momentum equation. In order to calculate convection and diffusion fluxes through the control volume faces, PISO algorithm is adopted for pressure-velocity coupling. The momentum source function for wave generation and the analytical relaxation function for wave absorption were deduced for constructing the NWT (numerical wave tank). The proposed NWT was then validated by the laboratory measurements of Umeyama and the analytical solution, indicating that the constructed NWT is effective and accurate.

Recently, the computer technology and numerical methods used in marine science and technology have evolved a lot including these aspects: ocean wave power generation, marine resources exploitation, and marine transportation. All these are related to the numerical simulation of ocean waves. The ocean waves can be categorized into linear waves, high order stokes waves, solitary waves, irregular waves, and so on. The physical methods are very limited due to the aspects of high cost (for establishing the experimental wave tank and for flow measuring), inconvenience for maintenance and management, and hardness for monitoring of some specific flow data leading to the ignorance of some important details [

With the application of static boundary wave-maker, waves are generated at the inlet boundary given specified velocities and the water surface elevation derived from any wave theory formulation [

With the application of internal wave-maker, waves are generated by applying a source function or a source line within a designated region inside the computational zone not interacting with reflected waves and being able to be combined with various wave absorber methods. For the source line method, waves are generated at a single point in the wave propagation direction, by adding, at each time step, an incremental water surface elevation computed by the resolution of the model equations. Larsen and Dancy [

In this study, we deduced the momentum source functions of the RANS equations for the internal wave-maker and the wave absorber based on analytical relaxation method mentioned above. FLUENT software was chosen as the base solver due to its popularity and applicability to water wave problems. All the numerical simulations were run in parallel using series of Intel XEON E5-2640V3 processors (2.60 GHz). To evaluate the application of this method to generate a target wave train in a vertical 2-dimensional channel of constant depth, the numerical results were compared to the laboratory measurements by Umeyama [

The incompressible fluid motion due to the water wave propagation can be described by the mass conservation equation, momentum conservation equations, and RANS equations:

in which

The VOF method is used to track the free water surface. The main idea of VOF method is to define a function

As shown above, the water free surface displacement is not a variable of the RANS equations; it is impossible to get the exact relation expression between the water free surface displacement and its momentum source function of the RANS equations for generating the target water waves. However, the water free surface displacement is an unknown variable of the depth-integrated equations and Wei et al. (1999) derived the mass source function based on the Boussinesq approximation for generating water waves. Their idea can be embedded into the RANS equation, and the momentum source function can be calculated via the appropriate numerical technique. In other words, the mass source function of the continuity equation, in which the water free surface is a key variable, will be transformed into the momentum source function of the RANS equations for generating the target waves in this study.

Wei et al. used the following linear Boussinesq equations to derive the mass source function to generate target water waves over constant water depth:

in which

The velocity vector can be expressed as

For internal wave generation the mass source function

in which,

In order to transform the mass source function derived above into a momentum source function for internal wave generation, the linear Boussinesq equations (

By demanding (

in which

By substituting the

By integrating the above equation, the momentum source function can be expressed as follows.

Using the relation above and the source mass function derived in Wei et al., the x-directional component of the momentum source function can be derived as follows.

Similarly, we can get the y-directional component of the momentum source function.

According to the Fourier transform, the momentum source function for a linear monochromatic wave can be simplified as follows.

For 2D simulation of the propagation of the linear waves over constant water depth, the wave direction angle

The analytical relaxation method proposed by Mayer and Madson was used here for wave absorption, and the mechanism of the analytical relaxation method can be described as follows: Within the relaxation domain, the velocity and the pressure will be updated at every time step by the added source. As application to the Navier-Stokes equations, the relaxation algorithm for velocity and pressure can be renewed as

in which the subscript

Then, the source function for the analytical relaxation wave absorbing method can be deduced from the Euler equations by ignoring the water viscosity. The difference forms of momentum equations with and without the additive source can be written as follows.

By subtracting (

More details about the source function can be found in [

The Computational Fluid Dynamics code, FLUENT 13.0 [

Figure

An illustrative sketch of the computational domain.

To validate the proposed model, the numerical results are compared with laboratory measurements of Umeyama. Umeyama performed a series of experiments in a physical wave channel with length of 25m, width of 0.7m, and maximum depth of 1.0m. Waves were generated by placing a piston-type wave-maker at one end of the wave channel and were absorbed by installing a wave absorber at the other end. During all the tests in the channel, the water depth was kept at 0.3m and the wave period was chosen as 1.0s; the parameters for the tests are listed in Table

Input parameters.

W1 | W2 | W3 | |
---|---|---|---|

Wave height ( | 1.03 | 2.34 | 3.61 |

Wave period ( | 1.0 | 1.0 | 1.0 |

Wave steepness | 0.0236 | 0.0536 | 0.0827 |

To check the grid and temporal grid sensitivity of the solutions, the temporal grid sensitivity analysis is carried out first. The medium grid (93.1MB, containing 1000000 quadrilateral cells) is used and three different time steps are considered, which are 0.01 (T/100), 0.005 (T/200), and 0.002 s (T/500). The wave amplitude is set to 0.00515m, the period is 1 s, and the simulation is carried out for a duration of 25 periods. The momentum source region is located at x=0m to x=0.5m. The momentum source distribution for all the numerical tests is shown in Figure

Momentum source distribution map for all cases.

Figure

Verification of the grid and temporal grid sensitivity.

Figure

Comparison of simulated and measured, analytical water surface profile within one wave period.

To demonstrate the effectiveness of the internal wave-maker and the wave absorber, the simulated average wave elevation profiles at time

Comparison between simulated and theoretical free surface elevation.

Figures

Comparison of simulated and theoretical horizontal- and vertical-velocity profiles for case-W1.

Comparison of simulated and theoretical horizontal- and vertical-velocity profiles for case-W2.

Comparison of simulated and theoretical horizontal- and vertical-velocity profiles for case-W3.

In this study, a momentum source internal wave-maker and an analytical wave absorber were embedded into 2D RANS equation model with RSM turbulence closure scheme to simulate the water wave propagation over constant water depth. By comparing the simulated results with the experimental data and the theoretical results, it can be found that the momentum source internal wave-maker can generate well-targeted waves and the analytical relaxation wave absorption method can absorb the wave energy effectively for avoiding the reflected waves generated by the left and right wall boundaries influencing the wave profiles in the working zones. The established NWT can be effectively used in the ship designing and ship hydrodynamic analysis.

The [physical experimental results] data supporting this article are from previously reported studies and datasets, which have been cited. The processed data are available from the corresponding author of the cited article (Motohiko Umeyama, Coupled PIV and PTV Measurements of Particle Velocities and Trajectories for Surface Waves Following a Steady Current, Journal of Waterway, Port, Coastal, and Ocean Engineering, vol. 137, no. 2, pp. 85-94, 2011.) upon request. The [numerical cases] data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This project is supported by the National Natural Science Foundation of China (Nos. 51409032 and 51409031) and the Fundamental Research Funds for the Central Universities (China) (Nos. 3132017006 and 3132016314).