Abstract
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
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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.
Original language | English |
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Qualification | Doctor of Philosophy |
Awarding Institution |
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Supervisors/Advisors |
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Award date | 3 Jun 2008 |
Place of Publication | Eindhoven |
Publisher | |
Print ISBNs | 978-90-386-1278-2 |
DOIs | |
Publication status | Published - 2008 |