Finite calculus. A stabilized finite element method for solving the Navier-Stokes equations. Application to ocean turbulence

Abstract

Ocean turbulence is a classic example of turbulent motion and observations of turbulence in the ocean lack in regions of complex topography since in-situ and experimental data are complicated to obtain. As a result, ocean circulation and flow interaction with obstacles such as seamounts and submarine canyons rely heavily on modelling and numerical simulation such that these are the only tools that can provide insights and ideas which are essential to the scientific community.
 
 This research investigated the capability of a stabilized finite element method (SFEM) based on the finite increment calculus (FIC) procedure, which is one of the most promising approaches to numerically simulate turbulent flows with particular interest in its application in simulating oceanic turbulence. The use of the FIC procedure allows solution of a wide range of fluid flow problems without the need of a turbulence model.
 
 The available numerical model was first validated through numerical simulations of various test cases such as Taylor-Couette flow, Ekman spiral, lock-exchange flow and circulation driven by oscillatory forcing over a theoretical submarine canyon model. Excellent qualitative agreement with available numerical and experimental data was obtained through minimal modification of the numerical model using TCL programming codes.
 
 To improve on modeling capabilities, which eventually allowed simulation of oceanic turbulence, a systematic modification of the numerical model was done by introducing additional paramaters on the governing equation, which is the incompressible Navier-Stokes equations. The resulting stabilized finite element equation was subsequently implemented into the numerical model, and the modification produced a coastal ocean model version. It was validated through numerical simulations of flow in coastal areas affected by topography, upwelling flow and submarine canyon. These flows are particularly reliant on numerical simulation, as the extreme nature of their flow makes obtention of accurate and reliable experimental data difficult or nearly impossible.
 
 The FIC approach has shown the potential for providing a reliable, accurate and efficient method for numerical simulation of coastal processes. The various applications demonstrated the good performance of the numerical model, and the computational results agree well with the theoretical and experimental data.