High performance simulations of yield stress fluids in a structured adaptive mesh refinement framework with embedded boundaries

Viscoplastic fluids are a class of non-Newtonian liquids characterised by their yield stress. Unless an external stress is applied which is larger than this threshold value, the fluid does not flow, but exhibits rigid body behaviour. Above the yield stress, applied forces cause viscous deformation. Such fluids play important roles in a range of fields, notably in wellbore drilling, which is the application that motivated this project. One aspect of this operation requires displacement of drilling fluid by cement in the annulus between casing and geological surroundings, and both of these fluids are viscoplastics. Ensuring that this is done properly is of utmost importance to the overall safety of the drilling operation. Often, numerical simulations are the only viable way of experimenting with the effect of drilling parameters and fluid properties on the flow configuration and resulting behaviour. Unfortunately, the presence of a yield stress leads to a singularity in the apparent viscosity at zero strain. This causes substantial computational expense for the algorithms used to simulate fluid flow numerically, even when regularisation techniques are employed to alleviate the problem. Consequently, most published results in the literature on computational viscoplasticity has been restricted to two-dimensional and steady-state flows. In an attempt to address this, we have applied state-of-the-art techniques from high-performance computational fluid dynamics to the viscoplastic flow problem. Specifically, we utilise spatio-temporal adaptive mesh refinement on structured meshes in this context for the first time. This is achieved through the software framework AMReX, which includes state-of-the-art numerical tools for solving partial differential equations with optimal parallel scaling. The ability to rapidly simulate unsteady viscoplastic flow problems in three dimensions is demonstrated by novel numerical experiments in a lid-driven cavity. In order to investigate flows in more interesting domain geometries and around objects, an embedded boundary algorithm has been developed which works alongside the viscoplastic flow solver. We show how this methodology can be utilised to simulate flow inside non-rectangular objects, and investigate fully three-dimensional viscoplastic flow past several shapes of bodies for the first time.EPSRC Centre for Doctoral Training in Computational Methods for Materials Science grant number EP/L015552/1 BP International Centre for Advanced Materials (BP-ICAM) Extra support towards living expenses through the Aker Scholarshi

Highly parallelisable simulations of time-dependent viscoplastic fluid flow simulations with structured adaptive mesh refinement

We present the extension of an efficient and highly parallelisable framework for incompressible fluid flow simulations to viscoplastic fluids. The system is governed by incompressible conservation of mass, the Cauchy momentum equation and a generalised Newtonian constitutive law. In order to simulate a wide range of viscoplastic fluids, we employ the Herschel-Bulkley model for yield-stress fluids with nonlinear stress-strain dependency above the yield limit. We utilise Papanastasiou regularisation in our algorithm to deal with the singularity in apparent viscosity. The resulting system of partial differential equations is solved using the IAMR code (Incompressible Adaptive Mesh Refinement), which uses second-order Godunov methodology for the advective terms and semi-implicit diffusion in the context of an approximate projection method to solve on adaptively refined meshes. By augmenting the IAMR code with the ability to simulate regularised Herschel-Bulkley fluids, we obtain efficient numerical software for time-dependent viscoplastic flow in three dimensions, which can be used to investigate systems not considered previously due to computational expense. We validate results from simulations using this new capability against previously published data for Bingham plastics and power-law fluids in the two-dimensional lid-driven cavity. In doing so, we expand the range of Bingham and Reynolds numbers which have been considered in the benchmark tests. Moreover, extensions to time-dependent flow of Herschel-Bulkley fluids and three spatial dimensions offer new insights into the flow of viscoplastic fluids in this test case, and we provide missing benchmark results for these extensions.Funding and technical support from BP through the BP International Centre for Advanced Materials (BP-ICAM) which made this research possible

A bi-projection method for Bingham type flows

International audienceWe propose and study a new numerical scheme to compute the isothermal and unsteady flow of an incompressible viscoplastic Bingham medium.The main difficulty, for both theoretical and numerical approaches, is due to the non-differentiability of the plastic part of stress tensor in regionswhere the rate-of-strain tensor vanishes. This is handled by reformulating the definition of the plastic stress tensor in terms ofa projection.A new time scheme, based on the classical incremental projection method for the Newtonian Navier-Stokes equations, is proposed. The plastictensor is treated implicitly in the first sub-step of the projection scheme and is computed by using a fixed point procedure. A pseudo-timerelaxation is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm. This is akey feature of the bi-projection scheme which provides a fast and accurate computation of the plastic tensor.Stability and error analyses of the numerical scheme are provided. The error induced by the pseudo-time relaxation term is controlled bya prescribed numerical parameter so that a first-order estimate of the time error is derived for the velocity field.A second-order cell-centred finite volume scheme on staggered grids is applied for the spatial discretisation.The scheme is assessed against previously published benchmark results for both Newtonian and Bingham flows in a two-dimensional lid-drivencavity for Reynolds number equals 1 000.Moreover, the proposed numerical scheme is able to reproduce the fundamental property of cessation in finite time of a viscoplasticmedium in the absence of any energy source term in the equations.For a fixed value (100) of the Bingham number, various numerical simulations for a range of Reynolds numbers up to 200 000 were performedwith the bi-projection scheme on a grid with 1024x1024 mesh points. The effect of this (physical) parameter on the flow behaviour is discussed