Modelling of wave generation in a numerical tank by SPH method

Abstract

There have been many mathematical and physical modelling strategies to represent a numerical wave tank that can generate the desired wave spectrums. Presently, one of the recent methodologies that have certain intrinsic capabilities for the investigation of free-surface hydrodynamic problems, namely, Smoothed-Particle-Hydrodynamics (SPH) technique, has been utilized for the modelling of a numerical wave tank. The Navier–Stokes and continuity equations are utilized for governing the fluid motion through Weakly Compressible SPH (WCSPH) approach which couples pressure and density by an equation of state. As one of the major numerical treatments, kernel gradient normalization is included into the present SPH method together with the numerical treatments, namely, well-known density smoothing algorithm, hybrid velocity variance-based free surface (VFS), and artificial particle displacement (APD) algorithms. The generation of regular and irregular waves is performed by a moving boundary at the inlet where natural damping is targeted by utilizing a dissipative beach at the end of numerical wave tank. A wide range of test cases in terms of wave-lengths and steepness ratios have been investigated for the regular wave simulations. Although the wave-maker is forced linearly to oscillate sinusoidally at the inlet of the tank, due to the relatively high wave steepness ratios applied, the non-linear character of the free-surface has been clearly observed with the performed Fast Fourier Transform analyses. Wave energy densities of the SPH results have also been compared with the linear theory expectations per unit wave-length. To scrutinize the conditions for the wave-breaking inception, three additional wave steepness values have been simulated at a single wave-length value. As a further examination of the proposed SPH scheme, JONSWAP irregular wave spectrum has been utilized with both flap and piston type moving boundaries. In the light of performed simulations, the proposed SPH numerical scheme can provide robust and consistent results while generating regular and irregular wave systems in deep water conditions. Furthermore, it is observed that it has the capability of capturing the non-linear characteristics of generated waves with high sensitivity, including the wave-breaking phenomenon.

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Abbreviations

\(c_{0}\) :

Reference speed of sound in water

\(c_{w}\) :

Celerity of the generated waves

d :

Depth of water

f :

Frequency of waves

\(\mathbf{g} \) :

Gravitational acceleration

H :

Wave height

h :

Smoothing length

k :

Wave number

L :

Tank length

p :

Instantaneous fluid pressure

\(\mathbf {r}\) :

Position vector

\(S_\zeta \) :

Energy spectrum

s :

Amplitude of the (flap or piston) wave generator

\(\mathbf {u}\) :

Velocity vector

\(V_{i}\) :

Volume of particle i

\(W_{ij}\) :

Kernel function value between neighbor particles

xz :

2D Cartesian coordinates

\(\beta \) :

Angle of inclination of the tank beach

\(\Delta t\) :

Time step value

\(\delta x\) :

Initial particle distance

\(\zeta \) :

Wave elevation

\(\zeta _a\) :

Wave amplitude

\(\lambda \) :

Wave-length

\(\nu \) :

Kinematic viscosity

\(\rho \) :

Instantaneous fluid density

\(\rho _0\) :

Reference density of fluid at rest

\(\omega \) :

Circular frequency of waves

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Acknowledgements

This work is supported by the Scientific and Technological Research Council of Turkey (TUBITAK) Grant Number 117M091.

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Correspondence to Murat Ozbulut.

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Appendix

Appendix

The time series and FFT analyses of non-breaking regular waves of the remaining test cases given in Table  1 are provided here.

See Figs. 12, 13, 14, 15, 16, 17, and 18.

Fig. 14
figure14

Time series of the wave elevations a and its frequency-domain representation b for \(\lambda = 0.60 [m]\) test case

Fig. 15
figure15

Time series of the wave elevations a and its frequency-domain representation b for \(\lambda = 0.70 [m]\) test case

Fig. 16
figure16

Time series of the wave elevations a and its frequency-domain representation b for \(\lambda = 0.75 [m]\) test case

Fig. 17
figure17

Time series of the wave elevations a and its frequency-domain representation b for \(\lambda = 0.80 [m]\) test case

Fig. 18
figure18

Time series of the wave elevations a and its frequency-domain representation b for \(\lambda = 0.90 [m]\) test case

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Ozbulut, M., Ramezanzadeh, S., Yildiz, M. et al. Modelling of wave generation in a numerical tank by SPH method. J. Ocean Eng. Mar. Energy 6, 121–136 (2020). https://doi.org/10.1007/s40722-020-00163-x

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Keywords

  • SPH method
  • Regular/irregular wave generation
  • Wave breaking
  • Wave energy
  • Free surface hydrodynamics