Adaptive finite element simulations of the surface currents in the North Sea

Abstract

The Navier-Stokes equations are solved for the pressure and flow of the surface currents in the North Sea. The solution algorithm applied is the nodal adaptive mesh and adaptive time method. The Navier-Stokes equations are split in four equations which are solved sequentially. The first equation, which is solved implicitly, is the diffusion equation. The second equation, which is solved explicitly, is the convection equation. The third equation, which is solved implicitly, is the pressure correction equation and the fourth equation, which is solved explicitly, is the velocity correction equation. The two equations, the diffusion and the pressure correction equation which are solved implicitly, are symmetric, linear and positive definite. The implicit equations are therefore solved by a symmetric conjugate gradient algorithm. The symmetric conjugate gradient algorithm is performed node by node without storage of the equation matrix. The nodal solution algorithm therefore permits the solution of larger problems as compared to algorithms which apply an assembled and stored equation matrix. The coefficients in the equation matrix in the nodal algorithm are generated whenever needed in the matrix-vector multiplication in the conjugate algorithm. The initial finite element grid is obtained by adapting the grid to the coastline. A solution is first obtained at a low Reynolds number. The solution is then scaled and used as a start vector for the computation at a higher Reynolds number. At several time steps in the iteration process, the Reynolds number and the Courant number are computed for each element. An element is refined if the element Reynolds number is greater than 1 and an element is recoarsed if the element Reynolds number is much less than 1. The time step is adjusted simultaneously with adapting the grid to the solution. The time step is computed to ensure that the largest element Courant number is less than 0.5. The simulation results demonstrate that vortices may develop at the coasts outside England and outside Germany.