Abstract

The objective of this work is to develop and evaluate a methodology for the solution
of the Navier-Stokes equations for Bingham Herschel-Bulkley viscoplastic fluids using stabilized
mixed velocity/pressure finite elements. The theoretical formulation is developed and
implemented in a computer code. Numerical solutions for these viscoplastic flows are presented
and assessed.
Viscoplastic fluids are characterized by minimum shear stress called yield stress.
Above this yield stress, the fluid is able to flow. Below this yield stress, the fluid behaves as a
quasi-rigid body, with zero strain-rate.
First, the Navier-Stokes equations for incompressible fluid and two immiscible fluids
considering free surface are presented. A review of the Newtonian and non-Newtonian rheological
models is included, with a detailed description of the viscoplastic models. The regularized
viscoplastic models due to Papanastasiou are described. Double viscosity regularized
models are proposed.
The analytical solutions for parallel flows are deduced for Newtonian, Bingham, and
Herschel-Bulkley, pseudoplastic and dilatant fluids.
The discrete model is developed, and the Algebraic SubGrid Scale (ASGS) stabilization
method, the Orthogonal Subgrid scale (OSS) method and the split orthogonal subscales
method are introduced. For the cases of flows with a free surface, the simplified Eulerian
method is employed, with the level set method to solve the motion of the free.
A convergence study is performed to compare the ASGS and OSS stabilization
methods in parallel flows with Bingham and Herschel-Bulkley fluids. The double viscosity
regularized models show lower convergence error convergence than the regularized models
used commonly.
Numerical solutions developed in this work are applied to a broad set of benchmark
problems. They can be divided into three groups: Bingham flows, Herschel-Bulkley flows
and free surface flows.
The solutions obtained validate the methodology proposed in this research and compare
well with the analytical and numerical solutions, experimental and field data.
The methodology proposed in this work provides a computational tool to study confined
viscoplastic flows, common in industry, and debris viscoplastic flows with free surface.
Keywords: stabilized finite elements, incompressibility, level set method, viscoplastic
fluid, Bingham model, Herschel-Bulkley model, debris flow, dam break.

Abstract

Debris flows overriding steep valleys can cause a significant decrease in bed friction resistance due to undrained excess pore water pressure, leading to an exponential increase in both destructiveness and volume. This study develops a two-phase numerical model based on Smoothed Particle Hydrodynamics to simulate the progressive entrainment behavior of debris flow accurately. The fluid and bed-sediment materials are modeled using the non-Newtonian Bingham-type Herschel-Bulkley-Papanastasiou (HBP) constitutive model. The mass erosion behavior of debris flow is achieved and augmented by incorporating the Drucker-Prager (DP) softening model, which accounts for variations in the pore water pressure ratio across different saturation states. A straightforward phase-change approach is implemented according to the mutation of effective viscosity to prevent any minute displacements of viscoplastic materials when subjected to steep inclinations. The multi-phase model has been compared with the large scale flume experiments conducted by the United States Geological Survey. The 3-D numerical results obtained from the rigid bed, dry and wet erodible bed exhibit a good agreement with the experimental data, encompassing flow momentum feedback and erosion patterns. This paper initially attempts to simulate the entrainment of multiple phases in a steep valley by incorporating viscoplastic flow.

Abstract