## A 3D Euler-Lagrange model for simulating the multiphase wave bottom boundary layer

In the last several decades a combination of theory, experiments and numerical simulation have contributed significantly to our understanding of the physical processes that drive sediment transport in the wave bottom boundary layer (WBBL). This has led to increased capability and skill of practical models which can be used to predict large scale and long term coastal morphology. However, such models can require a degree of calibration, and can still fail under unusual or extreme events.

As available computational resources continue to grow, new types of models can be applied to study the physical processes in the WBBL with fewer degrees of closure, empiricism, and guesswork. As part of the SINBAD project, we have developed a three dimensional, multiphase Euler Lagrange model, capable of simulating sediment transport over a wide range of conditions in the WBBL. The model is unique in that it fully couples sediment transport and morphological change to the Lagrangian motion of individual sediment particles. Particle motion is computed by evaluating the sum of hydrodynamic and inter-particle forces acting on each particle, and advancing particle positions velocities and angular velocities velocities over time as:

The particle size is considered to be on the order of or smaller than the Eulerian grid spacing. Thus, expressions for these various forces acting on the particles employ closures developed from theory, experiments and fully resolved simulation (DNS). Inter-particle collisions are handled using a soft-sphere discrete element model (DEM) that treats the binary collision between two particles using a spring-dashpot-frictional slider analogy (Cundall & Strack, 1979). The sediment motion is coupled to the Eulerian fluid motion by solving the volume filtered Navier-Stokes equations (Anderson & Jackson, 1967; Capecelatro & Desjardins, 2013):

The above form of the conservation equations accounts for the effects of both momentum exchange and volume displacement of the flow by the particles. A new, Cartesian grid implementation of the algorithm of Shams et al. (2011) is used to solve these governing multiphase equations in a 3D, finite volume framework.

The model allows us to investigate answer a number of important questions about the multiphase WBBL, particularly the nature of the interactions between mobile sediment and the turbulent, oscillatory flow. Below we show one example of our investigations into the dynamics of vortex sand ripples. An initial flat bed of random close-packed particles is generated by sedimentation. The regular oscillatory free stream shown in the figure on the right is generated by adding a time dependent body force to both the flow and particles. Rolling grain ripples begin to develop after just a few oscillations. They subsequently grow and merge until three well developed vortex ripples exist in the domain.

In the figure above we examine the behaviour of the Lagrangian sediment for a single period near the end of the simulation, when there are three well developed ripples in the domain. Each subfigure, a-h, shows a single side view of the rippled bed, with each grain coloured by it’s instantaneous x velocity. The timing of each figure corresponds to the letters in the figure on the left. There is a significant amount of sediment in suspension, and a large sediment cloud travels back and forth over the ripple crests. In the ripple troughs, a large vortex is generated and subsequently ejected during each half cycle. The sediment velocities can reach up to 1.75 time the max free stream velocity during the maximum velocity phases (see subplots a,f) as the flow accelerates over the ripples and, due in part to the vortex, there is a reversal of the flow in the troughs prior to the free stream (see subplots d and h). Such a mechanism is fundamental to the vortex ripple migrating and evolving over time towards its equilibrium. A time animation of this is shown in the figure below.