**Rising oil droplets in stratified flows**

Collaborators: Shilpa Khatri, Dustin Kleckner and Francois Blanchette (UC Merced)

During the catastrophic Deepwater Horizon oil spill in 2010, about 5 million barrels of petroleum were discharged from the Macondo Well, 1,500 meters below sea level into the Gulf of Mexico. Since oil is less dense than water, we would naively have expected the oil to quickly rise to the surface and form oil slicks. However, oceanographic studies have estimated that approximately 2 million barrels1 were trapped in the deep sea, primarily in intrusion layers found at depths between 900 and 1,300 meters. These layers were observed to persist over tens of kilometers and for several months.

To this day, the fate of these intrusion layers remains unclear. A recent study4 has identified a 3,200 km2 area on the ocean floor around the well contaminated with hopane (a biomarker of crude oil), suggesting that up to 31% of the trapped oil may have ultimately sunk to the deep ocean. It has been hypothesized that the biomass resulting from the sudden development of oil-degrading bacteria may have acted as a flocculant clumping hydrocarbons into higher density aggregates.

While both the trapping process, the formation of the intrusion layers, and the long-term evolution of this trapped oil remain poorly understood, it has been demonstrated that these trapped intrusion layers are caused by the interaction of the multiphase oil plume – composed of water, dissolved hydrocarbons and small oil droplets – and the inherent oceanic density stratification. To gain a more fundamental understanding of these processes, we study oil droplets rising in density-stratified water, both computationally and experimentally

###### Simulation of two different oil droplets rising through a sharp stratification.

**Numerical modeling of confined active suspensions**

Collaborators: David Saintillan (UC San Diego), Arvind Gopinath (UC Merced)

Publication: **Geometric control of active collective motion, **M Theillard, R Alonso-Matilla, D Saintillan, Soft Matter 2017

Recent experimental studies have shown that confinement can profoundly affect self-organization in semi-dilute active suspensions, leading to striking features such as the formation of steady and spontaneous vortices in circular domains and the emergence of unidirectional pumping motions in periodic racetrack geometries. Motivated by these findings, we analyze the dynamics in confined suspensions of active self-propelled swimmers using a mean-field kinetic theory where conservation equations for the particle configurations are coupled to the forced Navier-Stokes equations for the self-generated fluid flow. In circular domains, a systematic exploration of the parameter space casts light on three distinct states: equilibrium with no flow, stable vortex, and chaotic motion, and the transitions between these are explained and predicted quantitatively using a linearized theory. In periodic racetracks, similar transitions from equilibrium to net pumping to traveling waves to chaos are observed in agreement with experimental observations and are also explained theoretically. Our results underscore the subtle effects of geometry on the morphology and dynamics of emerging patterns in active suspensions and pave the way for the control of active collective motion in microfluidic devices.

###### Active flow in a cylindrical geometry, for increasing concentration of swimmer

**Simulation of incompressible single and two-phase flows**

Collaborator: Frederic Gibou (UC Santa Barbara)

Publications:

**A stable projection method for the incompressible Navier–Stokes equations on arbitrary geometries and adaptive Quad/Octrees**A Guittet, M Theillard, F Gibou, Journal of Computational Physics 2015**Level-set simulations of soluble surfactant driven flows**, C Cleret de Langavant, A Guittet, M Theillard, F Temprano-Coleto, F Gibou, Journal of Computational Physics 2017

We developed a solver for the incompressible Navier-Stokes equations on non-graded Quad-/Oc-tree grids. The main challenge is to ensure the stability of the projection step on adaptive meshes. While this is readily available on uniform grids or in a finite element framework, it is far from being the case on adaptive grids. We constructed a projection method for a Marker and Cell layout, where the differential operators are constructed such that the projection is orthogonal and therefore stable. Besides, the viscous effects are discretized implicitly, making use of a Voronoi partition, producing an unconditionally stable solver. We demonstrated analytically, and confirm numerically, that our solver is unconditionally stable. Our approach, implemented on adaptive Cartesian Octree grids, and using a level-set representation of the irregular geometry, is thus applicable to any prescribed arbitrary and potentially deforming geometry. The unconditional stability allows us to consider relatively large time steps even as the spatial resolution increases,

while the adaptivity of our data structure allows the resolution of computational regions where the quantities of interest are varying rapidly, as e.g., in thin boundary layers. We are therefore able to study highly challenging applications, such as high Reynolds number flows in three spatial dimensions (e.g. a swimming shark), surfactant driven flows or flows in porous media.

Recently, our approach has been extended to the simulation of incompressible two-phase flows. The method is fully sharp, in the sense that, numerically, at the interface, all the fluids parameters are discontinuous while the continuity equations, for the stress and the velocity fields, are enforced exactly at the interface. By using a modified pressure-correction projection method, we were able to alleviate the standard time step restriction incurred by capillary forces, which is known forbeing extremely restrictive, especially for low Reynolds number flows. The solver is validated numerically in two and three spatial dimensions and challenging numerical examples have been simulated.

###### Simulation of 100 air bubble rising in a rectangular tank filled with water: adaptive grid and flow

**Electrostatic of biomolecules – Poisson-Boltzmann equation**

Collaborators: Mohammad Mirzadeh (MIT), Frederic Gibou (UC Santa Barbara)

Publications:

**An adaptive, finite difference solver for the nonlinear Poisson-Boltzmann equation with applications to biomolecular computations**M Mirzadeh, M Theillard, A Helgadöttir, D Boy, F Gibou, CiCP 2013**A second-order discretization of the nonlinear Poisson–Boltzmann equation over irregular geometries using non-graded adaptive Cartesian grids**M Mirzadeh, M Theillard, F Gibou, Journal of Computational Physics 2011

The solvation free energy is a fundamental quantity, characterizing the electrochemical properties and behavior of Biomolecules. It plays a central role in many real-life applications, such as drugs discoveries. Precise estimation of this energy for any given molecules is therefore crucial in both academic and industrial contexts. These estimations are very complex to obtain, as they involve to solve a non-linear equation (Poisson-Boltzmann) with jump conditions prescribed on a complex interface (the solvent excluded surface) and in the presence of multiple length scales.

The computational method we developed address these three points. The surface is constructed from the atomic representation using the level-set method in a simple an efficient way. The Finite Volume Method is used to sharply reproduce the interface conditions. Adaptive non-graded Octree grids are employed to capture the inherent multiscale nature of the problem while maintaining the computational cost low.