Abstract :
[en] Two-dimensional flows around cylinders and spheres (axisymmetric) rising at constant velocity and crossing
a free surface, are analyzed numerically using the Particle Finite Element Method (PFEM). The overall
transient regime is investigated, ranging from the free-surface deformation and the wake dynamics when
the cylinder is below the initial free-surface level, to the interface crossing and the film drainage when it is
above.
The PFEM is well suited to describe these different flow features. First, it is a Lagrangian method and
it therefore naturally captures the surface deformation thanks to the nodal displacements. Moreover, it
can handle topological changes through the combination of the Delaunay triangulation and the alpha -shape
technique, respectively used to re-triangulate the cloud of fluid particles and to identify the fluid boundaries.
In particular, the interface crossing and the subsequent drainage of the thin film above the cylinder/sphere
can be robustly simulated because of these particular features of the method. Nonetheless, the method
has limitations stemming from the alpha-shape technique used to identify the domain boundary. First, the
traditional implementation of this alpha-shape technique in the PFEM is limited to uniform meshes, such that
the use of the PFEM can be very expensive in terms of computing time when a fine mesh is required.
Moreover, it introduces mass conservation errors due to the deletion of existing, or addition of new, fluid
elements.
To overcome the limitations of the PFEM, a novel mesh adaptation algorithm is proposed. A local
target mesh size is prescribed according to geometric and/or physics-based criteria and particles are added
or removed to approximately enforce this target mesh size. Additionally, a new boundary recognition
algorithm relies on the tagging of boundary nodes and a local alpha-shape criterion that depends on the target
mesh size. The method allows thereby reducing mass conservation errors at free surfaces and improving the
local accuracy through local mesh refinement, and simultaneously offers a new boundary tracking algorithm.
The possible extension of the proposed algorithm to three-dimensional tetrahedral meshes is then considered
theoretically. In particular, the problematic case of slivers, i.e., very flat tetrahedra that are not removed
by the 3D Delaunay triangulation and that can strongly deteriorate the local accuracy of the solution, is
discussed.
The novel mesh adaptation algorithm is tested on six two-dimensional validation cases. The first three
cases, i.e., the flow around a (static, rotating, or oscillating) cylinder at Reynolds numbers below or equal
to 200, the lid-driven cavity flow at Reynolds numbers of 100 and 400, and the flow around an impulsively
started cylinder at a Reynolds numbers of 9500, do not feature a free surface and mainly illustrates the
mesh refinement capability. The last three test cases consist in the sloshing problem in a reservoir subjected
to forced oscillations, the fall of a 2D liquid drop into a tank filled with the same viscous fluid, and the
rise of an impulsively started cylinder toward the free surface at constant velocity. These last three cases
demonstrate the more accurate representation of the free surface and a corresponding reduction of the error
in mass conservation.
Then, the novel algorithm is applied to the main case of interest: a 2D cylinder or a 3D axisymmetric
sphere pulled out of a liquid bath at constant velocity and crossing the free surface. Different aspects of
the physics are investigated, including the free-surface elevation and the total drag, as well as the boundary
layer, wake, and film drainage dynamics. In particular, the dependencies of these flow features on the
different flow parameters, i.e., the Reynolds and the Froude numbers, and geometrical parameters, i.e., the
pool width and the release depth, are investigated in details. Comparisons with the literature, as well as
with in-house experiments of a rising cylinder in oil, are performed. In particular, the latter highlights the
limitation of the present two-dimensional approach to represent real three-dimensional cases, despite rather
good agreement for sufficiently high cylinder aspect ratios.
Finally, a mathematical model is developed for the description of the drainage dynamics in the thin film
at the cylinder/sphere apex. The model relies on the observation that the film thickness during interface
crossing is almost uniform around the apex. Combining the radially integrated Navier-Stokes equations and
an assumed velocity profile, the film model enables to describe the transition from the inertia-to-gravity to
the viscous-to-gravity dominated regime. The model involves only one calibrated parameter. It is validated
using different PFEM simulations of a cylinder or a sphere crossing the free surface at constant velocity.
In particular, it is found to predict qualitatively very well, despite small quantitative discrepancies, the
variation of the film thickness for a given range of Froude and Reynolds numbers.
This work concludes with several perspectives for future work.
