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 |
| Nerijus Sidorovas (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/V520068/1 | 30/09/2020 | 31/10/2025 | |||
| 2458723 | Studentship | EP/V520068/1 | 30/09/2020 | 29/09/2024 | 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. Internal waves within the scope of the Miyata-Choi-Camssa (MCC) model are being studied. Progress has been made to derive extended cKdV model with the account of variable bottom topography. Analytical results have been obtained for plane internal solitons and cnoidal waves of moderate amplitude via the Kodama-Fokas-Liu near-identity transformations. Numerical comparisons are currently underway, and initial results indicate good agreement between the constructed analytical solution and the numerically computed solitons of the MCC model. |
| 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. The constructed analytical solutions provide useful initial conditions and can be used to study the dependence on the parameters of the problem. |
| Sectors | Aerospace Defence and Marine Energy Manufacturing including Industrial Biotechology |
| URL | https://doi.org/10.1016/j.wavemoti.2024.103295 |
| Title | Extended cKdV models for ring waves of moderate amplitude |
| Description | We developed extended cylindrical Korteweg-de Vries (cKdV) models to describe ring waves of moderate amplitude in fluids. Efficient numerical schemes for these models were also developed and provided in the Open Access publication. |
| Type Of Material | Computer model/algorithm |
| Year Produced | 2024 |
| Provided To Others? | Yes |
| Impact | Performance of these models was compared to that of the full parent systems, and was shown to provide a big reduction in computational time for long-time simulations without significant loss of accuracy up to the waves of moderate amplitude. This opens the way for analytical development for waves of moderate amplitude using methods of weakly nonlinear theory (methods for small amplitude waves). |
| URL | https://doi.org/10.1016/j.wavemoti.2024.103295 |
| 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 paper has been published in Wave Motion in 2024. Sidorovas and Khusnutdinova have been invited to present at mini-symposium at the SIAM conference on Nonlinear Waves and Coherent Structures, in Baltimore, USA, in June 2024. |
| Start Year | 2022 |