Generation and Evolution of Cnoidal Waves in a Two-Dimensional Numerical Viscous Wave Flume
The generation and propagation of water waves in a numerical wave flume with Ursell numbers (<i>Ur</i>) ranging from 0.67 to 43.81 were investigated using the wave generation theory of Goring and Raichlen and a two-dimensional numerical viscous wave flume model. The unsteady Navier–Stoke...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
|
| Series: | Journal of Marine Science and Engineering |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2077-1312/13/6/1102 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849705784001167360 |
|---|---|
| author | Chih-Ming Dong Ching-Jer Huang Hui-Ching Huang |
| author_facet | Chih-Ming Dong Ching-Jer Huang Hui-Ching Huang |
| author_sort | Chih-Ming Dong |
| collection | DOAJ |
| description | The generation and propagation of water waves in a numerical wave flume with Ursell numbers (<i>Ur</i>) ranging from 0.67 to 43.81 were investigated using the wave generation theory of Goring and Raichlen and a two-dimensional numerical viscous wave flume model. The unsteady Navier–Stokes equations, along with nonlinear free surface boundary conditions and upstream boundary conditions at the wavemaker, were solved to build the numerical wave flume. The generated waves included small-amplitude, finite-amplitude, cnoidal, and solitary waves. For computational efficiency, the Jacobi elliptic function representing the surface elevation of a cnoidal wave was expressed as a Fourier series expansion. The accuracy of the generated waveforms and associated flow fields was validated through comparison with theoretical solutions. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mi>r</mi><mo><</mo><mn>26.32</mn></mrow></semantics></math></inline-formula>, small-amplitude waves generated using Goring and Raichlen’s wave generation theory matched those obtained from linear wave theory, while finite-amplitude waves matched those obtained using Madsen’s wave generation theory. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mi>r</mi><mo>></mo><mn>26.32</mn></mrow></semantics></math></inline-formula>, nonlinear wave generated using Goring and Raichlen’s theory remained permanent, whereas that generated using Madsen’s theory did not. The evolution of a cnoidal wave train with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mi>r</mi><mo>=</mo><mn>43.81</mn></mrow></semantics></math></inline-formula> was examined, and it was found that, after an extended propagation period, the leading waves in the wave train evolved into a series of solitary waves, with the tallest wave positioned at the front. |
| format | Article |
| id | doaj-art-dbed82a8698e457882c5728a717f1ac7 |
| institution | DOAJ |
| issn | 2077-1312 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Journal of Marine Science and Engineering |
| spelling | doaj-art-dbed82a8698e457882c5728a717f1ac72025-08-20T03:16:22ZengMDPI AGJournal of Marine Science and Engineering2077-13122025-05-01136110210.3390/jmse13061102Generation and Evolution of Cnoidal Waves in a Two-Dimensional Numerical Viscous Wave FlumeChih-Ming Dong0Ching-Jer Huang1Hui-Ching Huang2Department of Environmental Engineering and Science, Chia Nan University, Tainan City 71710, TaiwanDepartment of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan City 70101, TaiwanDepartment of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan City 70101, TaiwanThe generation and propagation of water waves in a numerical wave flume with Ursell numbers (<i>Ur</i>) ranging from 0.67 to 43.81 were investigated using the wave generation theory of Goring and Raichlen and a two-dimensional numerical viscous wave flume model. The unsteady Navier–Stokes equations, along with nonlinear free surface boundary conditions and upstream boundary conditions at the wavemaker, were solved to build the numerical wave flume. The generated waves included small-amplitude, finite-amplitude, cnoidal, and solitary waves. For computational efficiency, the Jacobi elliptic function representing the surface elevation of a cnoidal wave was expressed as a Fourier series expansion. The accuracy of the generated waveforms and associated flow fields was validated through comparison with theoretical solutions. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mi>r</mi><mo><</mo><mn>26.32</mn></mrow></semantics></math></inline-formula>, small-amplitude waves generated using Goring and Raichlen’s wave generation theory matched those obtained from linear wave theory, while finite-amplitude waves matched those obtained using Madsen’s wave generation theory. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mi>r</mi><mo>></mo><mn>26.32</mn></mrow></semantics></math></inline-formula>, nonlinear wave generated using Goring and Raichlen’s theory remained permanent, whereas that generated using Madsen’s theory did not. The evolution of a cnoidal wave train with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mi>r</mi><mo>=</mo><mn>43.81</mn></mrow></semantics></math></inline-formula> was examined, and it was found that, after an extended propagation period, the leading waves in the wave train evolved into a series of solitary waves, with the tallest wave positioned at the front.https://www.mdpi.com/2077-1312/13/6/1102numerical viscous wave flumecnoidal wavesJacobi elliptic functionsolitary wavesboundary layer flowwave propagation |
| spellingShingle | Chih-Ming Dong Ching-Jer Huang Hui-Ching Huang Generation and Evolution of Cnoidal Waves in a Two-Dimensional Numerical Viscous Wave Flume Journal of Marine Science and Engineering numerical viscous wave flume cnoidal waves Jacobi elliptic function solitary waves boundary layer flow wave propagation |
| title | Generation and Evolution of Cnoidal Waves in a Two-Dimensional Numerical Viscous Wave Flume |
| title_full | Generation and Evolution of Cnoidal Waves in a Two-Dimensional Numerical Viscous Wave Flume |
| title_fullStr | Generation and Evolution of Cnoidal Waves in a Two-Dimensional Numerical Viscous Wave Flume |
| title_full_unstemmed | Generation and Evolution of Cnoidal Waves in a Two-Dimensional Numerical Viscous Wave Flume |
| title_short | Generation and Evolution of Cnoidal Waves in a Two-Dimensional Numerical Viscous Wave Flume |
| title_sort | generation and evolution of cnoidal waves in a two dimensional numerical viscous wave flume |
| topic | numerical viscous wave flume cnoidal waves Jacobi elliptic function solitary waves boundary layer flow wave propagation |
| url | https://www.mdpi.com/2077-1312/13/6/1102 |
| work_keys_str_mv | AT chihmingdong generationandevolutionofcnoidalwavesinatwodimensionalnumericalviscouswaveflume AT chingjerhuang generationandevolutionofcnoidalwavesinatwodimensionalnumericalviscouswaveflume AT huichinghuang generationandevolutionofcnoidalwavesinatwodimensionalnumericalviscouswaveflume |