## Novel immersed boundary method for direct numerical simulations of solid-fluid flows

##### Abstract

Solid-fluid two-phase flows, where the solid volume fraction is large either by geometry or
by population (as in slurry flows), are ubiquitous in nature and industry. The interaction
between the fluid and the suspended solids, in such flows, are too strongly coupled rendering
the assumption of a single-way interaction (flow influences particle motion alone but not
vice-versa) invalid and inaccurate. Most commercial flow solvers do not account for twoway
interactions between fluid and immersed solids. The current state-of-art is restricted to
two-way coupling between spherical particles (of very small diameters, such that the particlediameter
to the characteristic flow domain length scale ratio is less than 0.01) and flow. These
solvers are not suitable for solving several industrial slurry flow problems such as those of
hydrates which is crucial to the oil-gas industry and rheology of slurries, flows in highly constrained
geometries like microchannels or sessile drops that are laden with micro-PIV beads
at concentrations significant for two-way interactions to become prominent. It is therefore
necessary to develop direct numerical simulation flow solvers employing rigorous two-way
coupling in order to accurately characterise the flow profiles between large immersed solids
and fluid. It is necessary that such a solution takes into account the full 3D governing equations
of flow (Navier-Stokes and continuity equations), solid translation (Newton’s second
law) and solid rotation (equation of angular momentum) while simultaneously enabling interaction
at every time step between the forces in the fluid and solid domains.
This thesis concerns with development and rigorous validation of a 3D solid-fluid solver
based on a novel variant of immersed-boundary method (IBM). The solver takes into account
full two-way fluid-solid interaction with 6 degrees-of-freedom (6DOF). The solid motion
solver is seamlessly integrated into the Gerris flow solver hence called Gerris Immersed
Solid Solver (GISS). The IBM developed treats both fluid and solid in the manner of “fluid
fraction” such that any number of immersed solids of arbitrary geometry can be realised. Our
IBM method also allows transient local mesh adaption in the fluid domain around the moving
solid boundary, thereby avoiding problems caused by the mesh skewness (as seen in common
mesh-adaption algorithms) and significantly improves the simulation efficiency. The solver is rigorously validated at levels of increasing complexity against theory and experiment
at low to moderate flow Reynolds number. At low Reynolds numbers (Re 1) these
include: the drag force and terminal settling velocities of spherical bodies (validating translational
degrees of freedom), Jeffrey’s orbits tracked by elliptical solids under shear flow
(validating rotational and translational degrees of freedom) and hydrodynamic interaction
between a solid and wall. Studies are also carried out to understand hydrodynamic interaction
between multiple solid bodies under shear flow. It is found that initial distance between
bodies is crucial towards the nature of hydrodynamic interaction between them: at a distance
smaller than a critical value the solid bodies cluster together (hydrodynamic attraction) and
at a distance greater than this value the solid bodies travel away from each other (hydrodynamic
repulsion). At moderately high flow rates (Re O(100)), the solver is validated against
migratory motion of an eccentrically placed solid sphere in Poisuelle flow. Under inviscid
conditions (at very high Reynolds number) the solver is validated against chaotic motion of
an asymmetric solid body.
These validations not only give us confidence but also demonstrate the versatility of the GISS
towards tackling complex solid-fluid flows. This work demonstrates the first important step
towards ultra-high resolution direct numerical simulations of solid-fluid flows. The GISS will
be available as opensource code from February 2015.