Abstract

The objective of this work is to model computationally Bingham and Herschel-Bulkley viscoplastic fluids using stabilized mixed velocity/pressure finite elements. Numerical solutions for these viscoplastic flows are presented and assessed. The regularized viscoplastic models due to Papanastasiou is used. In the discrete model, the Orthogonal Subgrid scale (OSS) method is used.

In this part II , numerical solutions for two problems of Bingham and Herschel-Bulkley confined flows are presented. The solutions obtained validate the methodology proposed in part I of this work.

Abstract

This work presents a methodology for the solution of the Navier-Stokes equations for Bingham and Herschel-Bulkley viscoplastic fluids using stabilized mixed velocity/pressure finite elements. The theoretical formulation is developed and implemented in a computer code.

Viscoplastic fluids are characterized by a 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 are presented. A review of the viscoplastic rheological models is included, with a detailed description of these models. The regularized viscoplastic models due to Papanastasiou are described. Double viscosity regularized models are proposed as an alternative to the models commonly used.

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.

The methodology proposed in this work provides a computational tool to study confined viscoplastic flows, common in industry.

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

In this paper we analyze a stabilized finite element method to approximate the convection diffusion equation on moving domains using an ALE framework. As basic numerical strategy, we discretize the equation in time using first and second order backward differencing (BDF) schemes, whereas space is discretized using a stabilized finite element method (the orthogonal subgrid scale formulation) to deal with convection dominated flows. The semi-discrete problem (continuous in space) is first analyzed. In this situation it is easy to identify the error introduced by the ALE approach. After that, the fully discrete method is considered. We obtain optimal error estimates in both space and time in a mesh dependent norm. The analysis reveals that the ALE approach introduces an upper bound for the time step size for the results to hold. The results obtained for the fully discretized second order scheme (in time) are associated to a weaker norm than the one used for the first order method. Nevertheless, optimal convergence results have been proved. For fixed domains, we recover stability and convergence results with the strong norm for the second order scheme, stressing the aspects that make the analysis of this method much more involved.

Abstract

In this paper we analyze a stabilized finite element approximation for the incompressible Navier–Stokes equations based on the subgrid-scale concept. The essential point is that we explore the properties of the discrete formulation that results allowing the subgrid-scales to depend on time. This apparently “natural” idea avoids several inconsistencies of previous formulations and also opens the door to generalizations.

Abstract

In this article, we analyze some residual-based stabilization techniques for the transient Stokes problem when considering anisotropic time–space discretizations. We define an anisotropic time–space discretization as a family of time–space partitions that does not satisfy the condition h2⩽Cδt with C uniform with respect to h and δt. Standard residual-based stabilization techniques are motivated by a multiscale approach, approximating the effect of the subscales onto the large scales. One of the approximations is to consider the subscales quasi-static (neglecting their time derivative). It is well known that these techniques are unstable for anisotropic time–space discretizations. We show that the use of dynamic subscales (where the subscales time derivatives are not neglected) solves the problem, and prove optimal convergence and stability results that are valid for anisotropic time–space discretizations. Also the improvements related to the use of orthogonal subscales are addressed.

Abstract

In this paper, we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows. We show that, only if the space of subscales is taken orthogonal to the finite element space, this definition is physically reasonable as the coarse and fine scales are properly separated. Then we compare the diffusion introduced by the numerical discretization of the problem with the diffusion introduced by a large eddy simulation model. Results for the flow around a surface-mounted obstacle problem show that numerical dissipation is of the same order as the subgrid dissipation introduced by the Smagorinsky model. Finally, when transient subscales are considered, the model is able to predict backscatter, something that is only possible when dynamic LES closures are used. Numerical evidence supporting this point is also presented.

Abstract

This paper presents a monolithic formulation framework combined with an anisotropic mesh adaptation for fluid–structure interaction (FSI) applications with complex geometry. The fluid–solid interfaces are captured using a level-set method. A new a posteriori error estimate, based on the length distribution tensor approach and the associated edge based error analysis, is then used to ensure an accurate capturing of the discontinuities at the fluid–solid interface. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. The presence of the structure will be taken into account by means of an extra stress tensor in the Navier–Stokes equations. The system is solved using a stabilized three-field, stress, velocity and pressure finite element (FE) formulation. It consists in the decomposition for both the velocity and the pressure fields into coarse/resolved scales and fine/unresolved scales and also in the efficient enrichment of the extra constraint. We assess the accuracy of the proposed formulation by simulating 2D and 3D time-dependent numerical examples such as: falling disk in a channel, turbulent flows behind an airfoil profile and flow behind an immersed vehicle.

Abstract