# An embedded boundary approach for efficient simulations of viscoplastic fluids in three dimensions

## TL;DR

This paper introduces a novel embedded boundary method for simulating three-dimensional viscoplastic fluid flows efficiently on structured grids, enabling detailed analysis of complex flow phenomena around objects.

## Contribution

It presents the first implementation of an embedded boundary algorithm for 3D viscoplastic fluids, with robust viscosity approximation and validation against standard tests.

## Key findings

- Validated the method with standard test cases.
- First 3D simulations of creeping Bingham plastics around objects.
- Provided new insights into flow fields around translating objects.

## Abstract

We present a methodology for simulating three-dimensional flow of incompressible viscoplastic fluids modelled by generalised Newtonian rheological equations. It is implemented in a highly efficient framework for massively parallelisable computations on block-structured grids. In this context, geometric features are handled by the embedded boundary approach, which requires specialised treatment only in cells intersecting or adjacent to the boundary. This constitutes the first published implementation of an embedded boundary algorithm for simulating flow of viscoplastic fluids on structured grids. The underlying algorithm employs a two-stage Runge-Kutta method for temporal discretisation, in which viscous terms are treated semi-implicitly and projection methods are utilised to enforce the incompressibility constraint. We augment the embedded boundary algorithm to deal with the variable apparent viscosity of the fluids. Since the viscosity depends strongly on the strain rate tensor, special care has been taken to approximate the components of the velocity gradients robustly near boundary cells, both for viscous wall fluxes in cut cells and for updates of apparent viscosity in cells adjacent to them. After performing convergence analysis and validating the code against standard test cases, we present the first ever fully three-dimensional simulations of creeping flow of Bingham plastics around translating objects. Our results shed new light on the flow fields around these objects.