Fourier spectral computation of geometrically confined two-dimensional flows

Large-scale flow phenomena in the atmosphere and the oceans are predominantly two-dimensional (2D) due to the large aspect ratio of the typical horizontal and vertical length scales in the flow. The 2D nature of large-scale geophysical flows motivates the use of a conceptual approach known as "2D turbulence". It usually involves the (forced/damped) Navier-Stokes equations on a square domain with periodic boundaries or on a spherical surface. This setup may be useful for numerical studies of atmospheric flow. For the oceans, on the other hand, geometrical confinement due to the continental shelves is of crucial importance. The physically most relevant boundary condition for oceanographic flow is probably the no-slip condition. Previous numerical and experimental studies have shown that confinement by no-slip boundaries dramatically affects the dynamics of (quasi-)2D turbulence due to its role as vorticity source. An important process is the detachment of high-amplitude vorticity filaments from the no-slip sidewalls that subsequently affect the internal flow. The first part of the thesis concerns the development and extensive testing of a Fourier spectral scheme for 2D Navier-Stokes flow in domains bounded by rigid noslip walls. An advantage of Fourier methods is that higher-order accuracy can, in principle, be achieved. Moreover, these methods are fast, relatively easy to implement even for performing parallel computations. The no-slip boundary condition is enforced by using an immersed boundary technique called "volume-penalization". In this method an obstacle with no-slip boundaries is modelled as a porous medium with a small permeability. It has recently been shown that in the limit of infinitely small permeability the solution of the penalized Navier-Stokes equations converges towards the solution of the Navier-Stokes equations with no-slip boundaries. Therefore the penalization error can be controlled with an arbitrary parameter. A possible drawback is that the sharp transition between the fluid and the porous medium can trigger Gibbs oscillations that might deteriorate the stability and accuracy of the scheme. Using a very challenging dipole-wall collision as a benchmark problem, it is, however, shown that higher-order accuracy is retrieved by using a novel 159 post-processing procedure to remove the Gibbs effect. The second topic of the thesis is the dynamics of geometrically confined 2D turbulent flows. The role of the geometry on the flow development has been studied extensively. For this purpose high resolution Fourier spectral simulations have been conducted where different geometries are implemented by using the volumepenalization method. A quantity that is of particular importance on a bounded domain is the angular momentum. On a circular domain production of angular momentum is virtually absent. Therefore the amount of angular momentum carried by the initial flow has important consequences for the evolution of the flow. The results of the simulations are consistent with previous numerical and experimental work on this topic performed in a lower Reynolds number regime. The typical vortex structures of the late time evolution of the flow are explained by means of a minimum enstrophy principle and the presence of weak viscous dissipation. For an elliptic geometry it is shown that strong spin-up events of the flow occur even for small eccentricities. The spin-up phenomenon can be related to the role of the pressure along the boundary of the domain. It is found that the magnitude of the torque exerted on the internal fluid can be scaled with the eccentricity. Furthermore, it is observed that angular momentum production in a non circular geometry is not restricted to moderate Reynolds numbers. Significantly higher Reynolds number flow computations in a square geometry clearly reveal strong and rapid spin-up of the flow. Finally the scale-dependence of the vorticity and velocity statistics in forced 2D turbulence on a bounded domain has been studied. A challenging aspect is that a statistically steady state can be achieved by a balance between the injection of kinetic energy by the external forcing and energy dissipation at the no-slip sidewalls. It is important to note that on a double periodic domain a steady state is usually achieved by introducing volumetric drag forces. Several studies reported that this strongly affects the spatial scaling behaviour of the flow. Therefore it is very interesting to quantify the small-scale statistics in the bulk of statistically steady flow on a domain with no-slip boundaries in the absence of bottom drag. It is observed that the internal flow shows extended self-similar, locally homogeneous and isotropic scaling behaviour at small scales. It is further demonstrated that a direct enstrophy cascade develops in the interior of the flow domain. Some deviations from the classical scaling theory of 2D turbulence developed independently by Kraichnan, Batchelor and Leith may be associated to the presence of coherent structures in the flow. It is, however, anticipated that higher-resolution simulations are required in order to draw more decisive conclusions. The parallel Fourier spectral scheme with volume-penalization is very suitable for pursuing such simulations on high performance machines in the near future. In summary the thesis contributes to both the development of numerical techniques and understanding of wall-bounded two-dimensional flows. The Fourier spectral scheme with volume-penalization is found very suitable for pursuing direct numerical simulations in complex geometries. The high-resolution simulations considered in the thesis clearly reveal that spontaneous production of angular momentum due to interaction with non-circular domain boundaries is present for significantly higher Reynolds numbers than considered previously.