My PhD in 5 Pictures: NF Morrison

NF Morrison, Teaching Fellow

My thesis title: Computations of flows of elastic liquids

What does this mean?

The aim was to design and validate a bespoke computational method for three-dimensional simulation of elastic liquids, and use it to study phenomena which are hard to simulate.  The broader field is CFD (computational fluid dynamics), but with a particular focus on classes of non-Newtonian fluid for which the governing equations are complicated and problematic to solve via the standard approaches deployed in commercial software.  The method involved a Lagrangian finite-element mesh.  Here "finite-element" means that the fluid domain is partitioned into a mesh consisting of a large number of separate “elements” (tetrahedra in this case), and “Lagrangian” means that the vertices of the mesh are considered as material points within the fluid itself, so that the elements deform along with the fluid as it flows (as opposed to an Eulerian mesh with elements remaining fixed in the laboratory frame of reference).  Because a Lagrangian mesh becomes increasingly deformed as the flow evolves, a bespoke adaptive mesh reconnection and improvement algorithm was integrated within the method.  Validation was via comparison to established “benchmark” problems in the 2D or axisymmetric case.  The fully 3D method was then applied to investigate asymmetric phenomena which had not been fully explained previously, for example the viscoelastic drift of sedimenting particles in pipe flow.

1

The main geometry under investigation was that of a rigid spherical particle sedimenting under gravity in an infinite cylindrical vertical pipe.  In Stokes flow, the sphere falls vertically with no sideways drift, due to reversibility.  In Newtonian flow with inertia, there is a sideways migration to a particular location (the Segré-Silberberg effect), whereas in viscoelastic flow there is a sideways drift towards the pipe wall, sometimes referred to as “negative lift.”  The traditional benchmark problem sets the sphere’s radius (\(a\)) to be half that of the pipe (\(2a\)), with the sphere falling along the pipe’s axis, and the asymmetric problem (as shown) has the sphere initially offset by \(2\varepsilon a\).

2

​The mesh reconnection algorithm was based primarily on properties of the Delaunay triangulation, which for any given set of vertices has the property that the circumsphere of each tetrahedron contains no vertex.  In practice, this involves considering local “flips” between the two configurations shown.  On the left there are two tetrahedra sharing a common face, \(ABC\).  The “2-3 flip” removes this face and adds a new edge \(DE\), forming instead three tetrahedra as shown on the right.  The “3-2 flip” is the same but in reverse.  Combining these with additional heuristic measures, the computational method was able to preserve mesh quality (in the sense of reducing the largest dihedral angle).

3

​Delaunay triangulation is optimal in 2D in the sense of maximizing the smallest angle in any triangle, but this property does not hold in 3D.  Consequently the Delaunay mesh in 3D can include some tetrahedral shapes with “bad” aspect-ratios for flow simulation purposes, known as “slivers.”  A sliver is a very flat tetrahedron without any particularly short edges.  In the example shown, the four vertices are nearly coplanar. On the left, a sliver is shown from two perspectives; on the right it is shown within its circumsphere.  The avoidance of slivers is considered a major goal of high-quality 3D mesh generation, and effective remedies involve assigning artificial weights to vertices, or adding/removing vertices in the neighbourhood of each sliver.

4

​To investigate dynamic viscoelastic particle drift, it is necessary to consider the build-up of polymer stretch within the flow (in the cross-sectional images, yellow is highly stretched, dark blue is unstretched, and between each image the sphere has fallen by one diameter).  At early times the polymer becomes highly stretched in the narrow gap first, and the sphere initially drifts inwards towards the axis. As the material in the wider gap becomes highly stretched, there is a transition to an outward drift, and this dominates at later times.  On the far side of the pipe there is a build up of stress due to shearing by the back flow through the wider gap between the sphere’s wake and the far wall.

5

​Using an argument which considers the radial pressure gradient due to normal stresses, in both the narrow gap and the wider gap on either side of the sphere, an analytical approximation for the ratio of outward drift velocity to vertical falling velocity was established.  In the case of a FENE-CR fluid this is approximately \(\frac{2c}{3\pi K_S} \min{(1, \mathrm{We})}\), where \(K_S\) is the Stokes drag coefficient, \(c\) is the polymer concentration, and \(\mathrm{We}\) is the Weissenberg number for the flow.  This approximation is linear in concentration (for the dilute case), and provides an accurate agreement to the numerical results (until \(c\) becomes large), constituting a qualitative and quantitative explanation of viscoelastic particle drift.