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

On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions

Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z ∝ Re0.8 and P ∝ Re2.25 for 5 × 102 ≤ Re ≤ 2 × 104 and Z ∝ Re0.5 and P ∝ Re1.5 for Re ≥ 2 × 104 (with Re based on the velocity and size of the dipole). A critical Reynolds number Rec(here, Rec ≈ 2 × 104) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: Z ∝ Re^3/4, P ∝ Re^9/4, and dP/dt ∝ Re11/4 in agreement with the numerically obtained scaling laws. For Re ≥ Rec the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: Z ∝ Re1/2 and P ∝ Re3/2

Role of turbulent kinetic energy modulation by particle–fluid interaction in sediment pick-up

Reliable prediction of the erosion rate of sediment beds is important for many applications in coastal and river engineering. Theoretical understanding of empirically derived scaling relations is still lacking. This applies in particular for the scaling anomaly between low and high Shields number conditions. In this work, the erosion process is studied from the perspective of the phase-averaged turbulent kinetic energy (TKE) equations. The multi-phase TKE equations are written in a form that allows for a direct comparison with the TKE equation that appears for a stratified single-phase flow under the Boussinesq approximation. This reveals that next to buoyancy destruction, several other TKE modulation mechanisms become important at high Shields numbers and concentrations. Two scaling laws are derived for both moderate and high Shields numbers, and are tested against a wide range of experimental data

The minimum-enstrophy principle for decaying 2D turbulence in circular domains

In several numerical and experimental studies [1, 2] on freely evolving or decaying two-dimensional (2D) turbulence on a square bounded domain it is observed that a flow, initially containing no net angular momentum (L), spontaneously acquires angular momentum by flow-wall interaction. From earlier work, by Li and Montgomery [3], it could be conjectured that on a circular domain with a no-slip boundary angular momentum production is absent. Decaying turbulence experiments in stratified fluids conducted a few years later by Maassen et al. [4] provided additional evidence supporting this conjecture. These observations have recently been confirmed by Schneider and Farge [5] for decaying 2D turbulence with substantially higher initial integral scale Reynolds numbers

Angular momentum generation in bounded 2D turbulent flows

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A Fourier spectral solver for confined Navier-Stokes flow

No-slip boundaries have an important effect on forced and decaying two-dimensional turbulence due to their role as vorticity source. During intensive vortex-wall interactions high-amplitude vorticity filaments are produced. Most of these filaments roll up and form small-scale vortices that are advected into the interior by larger-scale vortices. From a computational point of view, it is a challenge to resolve the multiple temporal and spatial scales. Another challenge is to solve 2D turbulence in different geometries, e.g. square, triangle, circle, or ellipse. In this study we use a fast Fourier spectral technique to simulate the Navier-Stokes equations with no-slip boundary conditions. This is enforced by an immersed boundary technique called "volume penalization." The approach has been justified by analytical proofs of the convergence with respect to the penalization parameter. However, the solution of the penalized Navier-Stokes equations is not smooth on the surface of the penalized volume. Therefore, it is not a priori known whether it is possible to actually perform accurate fast Fourier spectral computations. Convergence checks are reported using a recently revived, and unexpectedly difficult, dipole-wall collision as a test case. It is found that Gibbs oscillations have a negligible effect on the flow evolution, also for 2D flows without the presence of reflection symmetry. Convergence results are reported of the angular momentum production by intensive flow-wall interaction

On the origin of spin-up processes in decaying two-dimensional turbulence

A remarkable feature of two-dimensional turbulence in a square container with no-slip walls is the spontaneous production of angular momentum due to flow-wall interactions, also known as spontaneous spin-up of the flow. In this paper we address the statistics of spin-up and discuss its likely origin. A signature of the spontaneous production of angular momentum is the development of a large-scale circulation cell. It is found that the global turnover time of the flow guides the spin-up process, which can be considered as a relaxation process of the macroscopic flow to an angular momentum containing state. The high turnover rate of the small-scale vortical structures emerging from the no-slip walls apparently has no significant effect on the spin-up rate. The presented data on the spin-up process strongly suggest that spin-up is not the net result of isolated vortex-wall interactions, with its associated pressure fluctuations on the domain boundaries, alone. The rapid spin-up of the flow clearly suggests the attraction to an angular momentum containing state

Spontaneous angular momentum generation of two-dimensional fluid flow in an elliptic geometry

Spontaneous spin-up, i.e. the significant increase of the total angular momentum of a flow that initially has no net angular momentum, is very characteristic for decaying two-dimensional turbulence in square domains bounded by rigid no-slip walls. In contrast, spontaneous spin-up is virtually absent for such flows in a circular domain with a no-slip boundary. In order to acquire understanding of this strikingly different behavior observed on the square and the circle we consider a set of elliptic geometries with a gradual increase of the eccentricity. It is shown that a variation of the eccentricity can be used as a control parameter to tune the relative contribution of the pressure and viscous stresses in the angular momentum balance. Direct numerical simulations demonstrate that the magnitude of the torque can be related to the relative contribution of the pressure. As a consequence, the number of spin-up events in an ensemble of slightly different initial conditions strongly depends on the eccentricity.For small eccentricities strong and rapid spin-up events are observed occasionally, whereas the majority of the runs does not show significant spin-up. Small differences in the initial condition can result in a completely different evolution of the flow and appearance of the end-state of the decay process. For sufficiently large eccentricities all the runs in the ensemble demonstrate strong and rapid spin-up, which is consistent with the flow development on the square. It is verified that the number of spin-up events for a given eccentricity does not depend on the Reynolds number of the flow. This observation is consistent with the conjecture that it is the pressure on the domain boundaries thatdrives the spin-up processes