A Numerthod for the Simulation of Free Surface Flows of Viscoplastic Fluid in 3D

Abstract

In this paper we study a numerical method for the simulation of free surface flows of viscoplastic (Herschel-Bulkley) fluids. The approax;h is based on the level set method for capturing the free surface evolution and on locally reflned and dynamically adapted octree cartesian staggered grids for the discretization of fluid and level set equations. A regularized model is applied to handle the non-differentiability of the constitutive relations. We consider an extension of the stable approximation of the Newtonian flow equations on staggered grid to approximate the viscoplastic model and level-set equations if the free boundary evolves and the mesh is dynamically reflned or coarsened. The numerical method is flrst validated for a Newtonian case. In this case, the convergence of numerical solutions is observed towards experimental data when the mesh is reflned. Further we compute several 3D viscoplastic Herschel-Bulkley fluid flows over incline planes for the dam-break problem. The qualitative comparison of numerical solutions is done versus experimental investigations. Another numerical example is given by computing the freely oscillating viscoplastic droplet, where the motion of fluid is driven by the surface tension forces. Altogether the considered techniques and algorithms (the level-set method, compact discretizations on dynamically adapted octree cartesian grids, regularization, and the surface tension forces approximation) result in efflcient approach to modeling viscoplastic free-surface flows in possibly complex 3D geometries.Mathematics subject classification: 65M06, 76D27, 76D99.

Efficient numerical schemes for viscoplastic avalanches. Part 1: The 1D case

“…Step 2] Update {V k+1 i } i by solving the linear system defined by the second component of (28). See details at (38)-(42), with δ n i+1/2 and ζ k i+1/2 given by (27)-(a) and (26)-(a).…”

“…Step 4] Set: for all i, Compute H n+1 = H k+1 with the finite volume method determined by the first component of (28), defined in terms of the most recent Lagrange multiplier {µ n+1 i+1/2 } i and taking into account the wet/dry treatment presented before where needed.…”

“…From the mathematical viewpoint, the non-linearity associated to viscoplastic models (such as the Bingham model, as we will see below), leads to feasible but very expensive computational times for the full 3D equations (see e.g. [28]). Consequently, numerous reduced 2D model using the shallow flow approximation have been derived.…”

A Unified Approach for Computing Tsunami, Waves, Floods, and Landslides

“…Discretizations on octree meshes benefit from their regular orthogonal structure on one hand and the embedded hierarchy on the other hand, which make the reconstruction and adaptation process as well as data access fast and easy. Nowadays such grids are widely used in simulations and visualization, see, e.g., [6,8,9,11,15,18]. However, building accurate and stable discretizations on such grids is a challenging task.…”

An octree-based solver for the incompressible Navier–Stokes equations with enhanced stability and low dissipation

“…Fast dynamic remeshing with octree grids makes them a natural choice for the simulation of moving interfaces and free surface flows, see, e.g., [5,6,7,8,9,10,11], octree grids became a standard tool in image processing [12].…”