General Solutions to the Navier-Stokes Equations for Incompressible Flow
Waves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed ba...
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The Korean Society of Ocean Engineers
2024-10-01
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| Series: | 한국해양공학회지 |
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| Online Access: | https://doi.org/10.26748/KSOE.2024.051 |
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| author | JangRyong Shin |
| author_facet | JangRyong Shin |
| author_sort | JangRyong Shin |
| collection | DOAJ |
| description | Waves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed based on the assumption of inviscid flow, but these theories are inadequate in explaining the transformation of waves into swells. This study addressed these limitations by analytically deriving general solutions to the Navier–Stokes equations. By expressing the velocity field as the product of a solution to the Helmholtz equation and a time-dependent univariate function, the Navier–Stokes equations are decomposed into an ordinary differential equation and the Euler equations, which are solved using tensor calculus. This paper provides solutions for viscous flow with shear currents when applied to the water wave problem. These solutions were validated through their application to the vorticity equation. The decay modulus of water waves was compared with experimental data, showing a significant degree of concordance. In contrast to other wave theories, this study clarified the process through which waves evolve into swells. |
| format | Article |
| id | doaj-art-3996f00358744f288a514b876da4c661 |
| institution | OA Journals |
| issn | 1225-0767 2287-6715 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | The Korean Society of Ocean Engineers |
| record_format | Article |
| series | 한국해양공학회지 |
| spelling | doaj-art-3996f00358744f288a514b876da4c6612025-08-20T01:47:25ZengThe Korean Society of Ocean Engineers한국해양공학회지1225-07672287-67152024-10-0138531532410.26748/KSOE.2024.051General Solutions to the Navier-Stokes Equations for Incompressible FlowJangRyong Shin0https://orcid.org/0000-0002-0144-2084Hanwha Ocean Co., LTDWaves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed based on the assumption of inviscid flow, but these theories are inadequate in explaining the transformation of waves into swells. This study addressed these limitations by analytically deriving general solutions to the Navier–Stokes equations. By expressing the velocity field as the product of a solution to the Helmholtz equation and a time-dependent univariate function, the Navier–Stokes equations are decomposed into an ordinary differential equation and the Euler equations, which are solved using tensor calculus. This paper provides solutions for viscous flow with shear currents when applied to the water wave problem. These solutions were validated through their application to the vorticity equation. The decay modulus of water waves was compared with experimental data, showing a significant degree of concordance. In contrast to other wave theories, this study clarified the process through which waves evolve into swells.https://doi.org/10.26748/KSOE.2024.051navier-stokes equationseuler equationshelmholtz equationbernoulli’s principlebeltrami flowwater wave |
| spellingShingle | JangRyong Shin General Solutions to the Navier-Stokes Equations for Incompressible Flow 한국해양공학회지 navier-stokes equations euler equations helmholtz equation bernoulli’s principle beltrami flow water wave |
| title | General Solutions to the Navier-Stokes Equations for Incompressible Flow |
| title_full | General Solutions to the Navier-Stokes Equations for Incompressible Flow |
| title_fullStr | General Solutions to the Navier-Stokes Equations for Incompressible Flow |
| title_full_unstemmed | General Solutions to the Navier-Stokes Equations for Incompressible Flow |
| title_short | General Solutions to the Navier-Stokes Equations for Incompressible Flow |
| title_sort | general solutions to the navier stokes equations for incompressible flow |
| topic | navier-stokes equations euler equations helmholtz equation bernoulli’s principle beltrami flow water wave |
| url | https://doi.org/10.26748/KSOE.2024.051 |
| work_keys_str_mv | AT jangryongshin generalsolutionstothenavierstokesequationsforincompressibleflow |