Mathematical Modelling of Nonlinear Ring Waves of Moderate Amplitude in Fluids
Lead Research Organisation:
Loughborough University
Department Name: School of Science
Abstract
Surface and internal waves in the oceans have a strong effect on offshore structures, underwater cables and submersibles. They also contribute to ocean mixing processes, which are important for climate. Internal waves generated in straits, river-sea interaction areas and by localised topographic features are nearly annular (ring-shaped) in form and often propagate over an underlying current. Our project aims to develop appropriate mathematical models and to use them to study the properties and behaviour of such and similar waves with an emphasis on waves of moderate amplitude.
Recently, it was shown that despite the clashing geometries of the waves and the current, there exists a linear modal decomposition that is different from the known decomposition for plane waves [1]. This decomposition was used to describe analytically the distortion of the wavefronts of weakly-nonlinear surface and internal waves, and to systematically develop and use mathematical models in order to solve some applied problems [2, 3].
The proposed work will be devoted to developing new advanced models for the waves of moderate amplitude, and to using them in order to study the behaviour of both surface and internal waves for different types of stratification and currents, as well as developing our understanding of the two-dimensional aspects of stability of the ring waves. The research will employ a variety of analytical and numerical methods.
References:
[1] K.R. Khusnutdinova, X. Zhang, Long ring waves in a stratified fluid over a shear flow, J. Fluid Mech. 794 (2016) 17-44.
[2] K.R. Khusnutdinova, X. Zhang, Nonlinear ring waves in a two-layer fluid, Physica D: Nonlinear Phenomena 333 (2016) 208-221.
[3] K.R. Khusnutdinova, Long internal ring waves in a two-layer fluid with an upper-layer current, Russ. J. Earth Sci. 20 (2020) ES4006.
Recently, it was shown that despite the clashing geometries of the waves and the current, there exists a linear modal decomposition that is different from the known decomposition for plane waves [1]. This decomposition was used to describe analytically the distortion of the wavefronts of weakly-nonlinear surface and internal waves, and to systematically develop and use mathematical models in order to solve some applied problems [2, 3].
The proposed work will be devoted to developing new advanced models for the waves of moderate amplitude, and to using them in order to study the behaviour of both surface and internal waves for different types of stratification and currents, as well as developing our understanding of the two-dimensional aspects of stability of the ring waves. The research will employ a variety of analytical and numerical methods.
References:
[1] K.R. Khusnutdinova, X. Zhang, Long ring waves in a stratified fluid over a shear flow, J. Fluid Mech. 794 (2016) 17-44.
[2] K.R. Khusnutdinova, X. Zhang, Nonlinear ring waves in a two-layer fluid, Physica D: Nonlinear Phenomena 333 (2016) 208-221.
[3] K.R. Khusnutdinova, Long internal ring waves in a two-layer fluid with an upper-layer current, Russ. J. Earth Sci. 20 (2020) ES4006.
People |
ORCID iD |
| Karima Khusnutdinova (Primary Supervisor) | |
| Nerijus Sidorovas (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/V520068/1 | 01/10/2020 | 31/10/2025 | |||
| 2458723 | Studentship | EP/V520068/1 | 01/10/2020 | 30/09/2024 | Nerijus Sidorovas |
| Description | Current mathematical models used to describe 3D surface waves in fluids are complicated and require expensive computations. The cylindrical Korteweg-de-Vries (cKdV) equation is well known to be a good reduced model for the long axisymmetric ring waves of small amplitude. In this study, our aim is to extend this model to waves of moderate amplitude. Using the method of asymptotic multiple scale expansions, we derived the extended cKdV (ecKdV) equation for the surface ring waves of moderate amplitude from the 2D Boussinesq system, Green-Naghdi equations, and Matsuno's extended system. Currently, the numerics has been done within the scope of the axisymmetric 2D Boussinesq system. The ecKdV model is computationally much more efficient compared to the 2D Boussinesq system, and in a case study we have shown that it better describes moderate amplitude waves than the cKdV model. This study is ongoing and similar work is being done for moderate amplitude internal waves. |
| Exploitation Route | The scientific community can use the efficient reduced models and the numerical approaches developed by us in the studies of small and moderate amplitude waves. |
| Sectors | Aerospace, Defence and Marine,Energy,Manufacturing, including Industrial Biotechology |
| Description | Collaboration with Prof. Wooyoung Choi |
| Organisation | New Jersey Institute of Technology |
| Country | United States |
| Sector | Academic/University |
| PI Contribution | We have developed extended cKdV-type models for small-to-moderate amplitude ring waves within the scope of the 2D Boussinesq system, and some strongly nonlinear systems. We have also performed numerical simulations to compare axisymmetric solutions of the 2D Boussinesq system with those obtained from the reduced model. |
| Collaborator Contribution | Prof. Wooyoung Choi has helped to understand and avoid some numerical instabilities for the 2D Boussinesq system. |
| Impact | A joint publication is currently being prepared. |
| Start Year | 2022 |