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

The numerical simulation of complex flows has been a subject of intense research in the last years with important industrial applications in many fields. In this paper we present a finite element method to solve the two immiscible fluid flow problem using the level set method. When the interface between both fluids cuts an element, the discontinuity in the material properties leads to discontinuities in the gradients of the unknowns which cannot be captured using a standard finite element interpolation. The method presented in this work features a local enrichment for the pressure unknowns which allows one to capture pressure gradient discontinuities in fluids presenting different density values. The method is tested in two problems: the first example consists in a sloshing case that involves the interaction of a Giesekus and a Newtonian fluid. This example shows that the enriched pressure functions permit the exact resolution of the hydrostatic rest state. The second example is the classical jet buckling problem used to validate our method. To permit the use of equal interpolation between the variables, we use a variational multiscale formulation proposed recently by Castillo and Codina in Comput. Methods Appl. Mech. Engrg. 279 (2014) 579–605, that has shown very good stability properties, permitting also the resolution of the jet buckling flow problem in the the range of Weissenberg number 0 < We < 100, using the Oldroyd-B model without any sign of numerical instability. Additional ingredients of the work are the inclusion of a discontinuity capturing technique for the constitutive equation and some comparisons between a monolithic resolution and a fractional step approach to solve the viscoelastic fluid flow problem from the point of view of computational requirements

Abstract

Abstract