ISWFoam: a numerical model for internal solitary wave simulation in continuously stratified fluids

A numerical model, ISWFoam, for simulating internal solitary waves (ISWs) in continuously stratified, incompressible, viscous fluids is developed based on a fully three-dimensional (3D) Navier–Stokes equation using the open-source code OpenFOAM®. This model combines the density transport equation with the Reynolds-averaged Navier–Stokes equation with the Coriolis force, and the model discrete equation adopts the finite-volume method. The k–ω SST turbulence model has also been modified according to the variable density field. ISWFoam provides two initial wave generation methods to generate an ISW in continuously stratified fluids, including solving the weakly nonlinear models of the extended Korteweg–de Vries (eKdV) equation and the fully nonlinear models of the Dubreil–Jacotin–Long (DJL) equation. Grid independence tests for ISWFoam are performed, and considering the accuracy and computing efficiency, the appropriate grid size of the ISW simulation is recommended to be 1/150th of the characteristic length and 1/25th of the ISW amplitude. Model verifications are conducted through comparisons between the simulated and experimental data for ISW propagation examples over a flat bottom section, including laboratory scale and actual ocean scale, a submerged triangular ridge, a Gaussian ridge, and slope. The laboratory test results, including the ISW profile, wave breaking location, ISW arrival time, and the spatial and temporal changes in the mixture region, are well reproduced by ISWFoam. The ISWFoam model with unstructured grids and local mesh refinement can effectively simulate the evolution of ISWs, the ISW breaking phenomenon, waveform inversion of ISWs, and the interaction between ISWs and complex topography.

Numerical Investigation of Internal Solitary Wave Forces on Submarines in Continuously Stratified Fluids

A numerical model of internal solitary waves in continuously stratified fluids is developed by introducing a density transport equation to the three-dimensional Navier–Stokes equation and adopting the fully nonlinear models of the Dubreil-Jacotin-Long equation to obtain the initial field of the ISW. The corresponding turbulence model has also been modified to ensure that it considers the variable density field. Comparisons between numerical simulation results and experimental results show that the total resistance, the nondimensional pressure coefficient, and the nondimensional friction coefficient for the standard submarine model proposed by the Defense Advanced Research Projects Agency under different flow field conditions are highly consistent with the experimental results. The model established is used to numerically analyse the forces and moments of the standard submarine model encountering ISWs at different submergence depths. The influence of the rotation centre position on the moment is discussed, and the position range of the optimal rotation centre is proposed.

Optimal Control Theory for a System of Partial Differential Equations Associated with Stratified Fluids

In this paper, we investigate the existence of an optimal solution of a functional restricted to non-linear partial differential equations, which ruled the dynamics of viscous and incompressible stratified fluids in R3. Additionally, we use the first derivative of the considered functional to establish the necessary condition of the optimality for the optimal solution.

Dynamic soliton–mean flow interaction with non-convex flux

The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to non-convex flux is studied in the framework of the modified Korteweg–de Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamically evolving mean flows on the propagation of localised waves – essentially ‘soliton steering’ by the mean flow – is considered. A recent theoretical and experimental study of this new type of dynamic soliton–mean flow interaction for convex flux has revealed two scenarios where the soliton either transmits through the varying mean flow or remains trapped inside it. In this paper, it is demonstrated that the presence of a non-convex cubic hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations is used to formulate a general mathematical framework for soliton–mean flow interaction with non-convex flux. Solitary wave trapping is stated in terms of crossing modulation characteristics. Non-convexity and positive dispersion – common for stratified fluids – imply the existence of localised, sharp transition fronts (kinks). Kinks play dual roles as a mean flow and a wave, imparting polarity reversal to solitons and dispersive mean flows, respectively. Numerical simulations of the mKdV equation agree with modulation theory predictions. The mathematical framework developed is general, not restricted to completely integrable equations like mKdV, enabling application beyond the mKdV setting to other fluid dynamic contexts subject to non-convex flux such as strongly nonlinear internal wave propagation that is prevalent in the ocean.

ISWFoam: A numerical model for internal solitary wave simulation in continuously stratified fluids

A numerical model, ISWFoam, for simulating internal solitary waves (ISWs) in continuously stratified, incompressible, viscous fluids is developed based on a fully three-dimensional (3D) Navier-Stokes equation using the open source code OpenFOAM. This model combines the density transport equation with the Reynolds-averaged Navier-Stokes equation with the Coriolis force, and the model discrete equation adopts the finite volume method. The k-ω SST turbulence model has also been modified accordingly to the variable density field. ISWFoam provides two initial wave generation methods to generate an ISW in continuously stratified fluids, including solving the weakly nonlinear models of the extended Korteweg–de Vries (eKdV) equation and the fully nonlinear models of the Dubreil-Jacotin-Long (DJL) equation. Grid independence tests for ISWFoam are performed, considering the accuracy and computing efficiency, the appropriate grid size of the ISW simulation is recommended to be one-one hundred and fiftieth of the characteristic length and one-twenty fifth of the ISW amplitude. Model verifications are conducted through comparisons between the simulated and experimental data for ISW propagation examples over a flat bottom section, including laboratory scale and actual ocean scale, a submerged triangular ridge, a Gaussian ridge and slope. The laboratory test results, including the ISW profile, wave breaking location, ISW arrival time, and the spatial and temporal changes in the mixture region, are well reproduced by ISWFoam. The ISWFoam model with unstructured grids and local mesh refinement can accurately simulate the generation and evolution of ISWs, the ISW breaking phenomenon and the interaction between ISWs and complex structures and topography.

Layering, Instabilities, and Mixing in Turbulent Stratified Flows

Understanding how turbulence leads to the enhanced irreversible transport of heat and other scalars such as salt and pollutants in density-stratified fluids is a fundamental and central problem in geophysical and environmental fluid dynamics. This review discusses recent research activity directed at improving community understanding, modeling, and parameterization of the subtle interplay between energy conversion pathways, instabilities, turbulence, external forcing, and irreversible mixing in density-stratified fluids. The conceptual significance of various length scales is highlighted, and in particular, the importance is stressed of overturning or scouring in the formation and maintenance of layered stratifications, i.e., robust density distributions with relatively deep and well-mixed regions separated by relatively thin interfaces of substantially enhanced density gradient.