Sérgio Frey

Stabilized finite element methods. II: The incompressible Navier-Stokes equations

Abstract Stabilized methods are proposed and analyzed for a linearized form of the incompressible Navier-Stokes equations. The methods are extended and tested for the nonlinear model. The methods combine the good features of stabilized methods already proposed for the Stokes flow and for advective-diffusive flows. These methods also generalize previous works restricted to low-order interpolations, thus allowing any combination of velocity and continuous pressure interpolations. A careful design of the stability parameters is suggested which considerably simplifies these generalizations.

Stabilized finite element methods. I: Application to the advective-diffusive model

Some stabilized finite element methods for the Stokes problem are reviewed. The Douglas-Wang approach confirms better stability features for high order interpolations. Next, the advective-diffusive model is approximated in the light of various stabilized methods, a global convergence analysis is presented and numerical experiments are performed. Biquadratic elements produce better numerical results under all stabilized methods examined. The design of the stability parameter is confirmed to be a crucial ingredient for simulating the advective-diffusive model, and some improved possibilities are suggested. Combinations of these methodologies are given in the conclusions and will be examined in detail in the sequel to this paper applied to the incompressible Navier-Stokes equations.

Galerkin Least-Squares Finite Element Approximations for Isochoric Flows of Viscoplastic Liquids

The flow of viscoplastic liquids is studied via a finite element stabilized method. Fluids, such as some food products, blood, mud, and polymer solutions, exhibit viscoplastic behavior. In order to approximate this class of liquids, a mechanical model, based on the principles of power expended and mass conservation, is exploited with the Papanastasiou approximation for Casson equation employed to model viscoplasticity. The approximation for the nonlinear set of partial differential equations is performed, using a stabilized finite element methodology. A Galerkin least-squares strategy is employed to avoid the well-known difficulties of the classical Galerkin method in isochoric flows. It circumvents the Babuska-Brezzi condition and handles the asymmetry of the advective operator in high advective flows. Some two-dimensional (2D) viscoplastic flows through a 4:1 planar expansion, for a range of Casson (0⩽Ca⩽10) and Reynolds (0⩽Re⩽50) numbers, have been investigated, paying special attention to the characterization of vortex length and unyielded regions. The numerical results show the arising of regions of unyielded material throughout the flow, strongly affecting the vortex structure, which is reduced with the increase of the Casson number even in flows with considerable inertia.

Galerkin Least-Squares Multifield Approximations for Flows of Inelastic Non-Newtonian Fluids

The aim of this work is to investigate a Galerkin least-squares (GLS) multifield formulation for inelastic non-Newtonian fluid flows. We present the mechanical modeling of isochoric flows combining mass and momentum balance laws in continuum mechanics with an inelastic constitutive equation for the stress tensor. For the latter, we use the generalized Newtonian liquid model, which may predict either shear-thinning or shear-thickening. We employ a finite element formulation stabilized via a GLS scheme in three primal variables: extra stress, velocity, and pressure. This formulation keeps the inertial terms and has the capability of predicting viscosity dependency on the strain rate. The GLS method circumvents the compatibility conditions that arise in mixed formulations between the approximation functions of pressure and velocity and, in the multifield case, of extra stress and velocity. The GLS terms are added elementwise, as functions of the grid Reynolds number, so as to add artificial diffusivity selectively to diffusion and advection dominant flow regions—an important feature in the case of variable viscosity fluids. We present numerical results for the lid-driven cavity flow of shear-thinning and shear-thickening fluids, using the power-law viscosity function for Reynolds numbers between 50 and 500 and power-law exponents from 0.25 to 1.5. We also present results concerning flows of shear-thinning Carreau fluids through abrupt planar and axisymmetric contractions. We study ranges of Carreau numbers from 1 to 100, Reynolds numbers from 1 to 100, and power-law exponents equal to 0.1 and 0.5. Besides accounting for inertia effects in the flow, the GLS method captures some interesting features of shear-thinning flows, such as the reduction of the fluid stresses, the flattening of the velocity profile in the contraction plane, and the separation of the boundary layer downstream the contraction.

Flow of elasto-viscoplastic liquids through a planar expansion-contraction

The steady flow of incompressible elasto-viscoplastic liquids through a planar expansion–contraction is investigated. A novel constitutive model is employed to describe the mechanical behavior of the flowing liquids. Numerical solutions of the constitutive and conservation equations were obtained via a finite element method to investigate the role of elasticity, yield stress, and inertia. The fields of velocity, stress, elastic strain, and rate of strain were obtained for different combinations of the governing parameters. It was observed that these fields, as well as the shape and position of the yield surface, are all strong functions of elasticity, yield stress, and inertia. The trends observed agree well with previous numerical and visualization results available in the literature. The present work offers a detailed study on the effects of elasticity, presenting, in particular, the fields of elastic strain.

Galerkin Least-Squares Solutions for Purely Viscous Flows of Shear- Thinning Fluids and Regularized Yield Stress Fluids

frey@mecanica.ufrgs.br Federal University of Rio Grande do Sul - UFRGS Mechanical Engineering Department 90050-170 Porto Alegre, RS. Brazil momentum, coupled to the GNL constitutive equation for the extra-stress tensor. The finite element methodology concerned herein, the well-known Galerkin Least-Squares (GLS)

A numerical investigation of inertia flows of Bingham-Papanastasiou fluids by an extra stress-pressure-velocity galerkin least-squares method

This article is concerned with finite element approximations for yield stress fluid flows through a sudden planar expansion. The mechanical model is composed by mass and momentum balance equations, coupled with the Bingham viscoplastic model regularized by Papanastasiou (1987) equation. A multi-field Galerkin least-squares method in terms of stress, velocity and pressure is employed to approximate the flows. This method is built to circumvent compatibility conditions involving pressure-velocity and stress-velocity finite element subspaces. In addition, thanks to an appropriate design of its stability parameters, it is able to remain stable and accurate in high Bingham and Reynolds flows. Numerical simulations concerning the flow of a regularized Bingham fluid through a one-to-four sudden planar expansion are performed. For creeping flows, yield stress effects on the fluid dynamics of viscoplastic materials are investigated through the ranging of Bingham number from 0.2 to 100. In the sequence, inertia effects are accounted for ranging the Reynolds number from 0 to 50. The numerical results are able to characterize accurately the morphology of yield surfaces in high Bingham flows subjected to inertia.

An alternative Galerkin formulation for streamline problems

An accurate numerical visualization for streamlines of fluid flows is a fundamental tool in computational fluid dynamics. However, the standard finite element formulation to compute streamlines suffers from the disadvantage of requiring the determination of boundary integrals. This shortcoming requires the implementation of two distinct mappings in the finite element code, one for the interior domain employing two-dimensional elements and another with one-dimensional elements to approximate the boundary domain. We introduce an efficient way to determine the streamlines for the above-mentioned flows, which does not need the computation of contour integrals. In order to illustrate the good performance of the alternative formulation proposed, we capture the streamlines of two classical viscous models: Stokes and Navier-Stokes flows

CLASSICAL GALERKIN SHORTCOMINGS ON THE STEADY HEAT CONDUCTION SIMULATION IN A PLATE WITH THERMAL SOURCE

In most cases, finite element methods are based on Galerkin approximation, which, in the last decades, has been applied to approximate a large range of problems in Engineering [2]. Unlike finite difference methodologies, the Galerkin method does not construct its numerical approximation using differential governing equations. Instead, starting from a variational formulation of the problem studied and a triangulation of its domain employing finite elements, the method builds an approximate solution as a combination of piecewise polynomial functions with compact support - the so-called shape functions - and unknown degrees-of-freedom. The discrete problem generated by Galerkin method consists of a system of linear algebraic equations - represented by a banded matrix, which greatly reduces the cost of its numerical solution. Galerkin method was originally introduced for structural problems in which, assuming some restrictions usually present in engineering practical applications [3,5], gives rise to symmetric elliptic operators and generates rather optimum convergence rates. However, when applied to fluid problems, a slow development of the Galerkin approximation has been verified. Spurious oscillations, locking and other undesirable features appear separately or combined when applying the Galerkin method to thin structures, fluid flows, incompressible media and even in such a heat conduction situation when a temperature-dependent heat source dominates the diffusive operator. The present work studies the Galerkin approximation of the heat transfer process in an opaque three-dimensional plate [7] with a non uniform temperature-dependent source when this heat source dominates the conductive operator. The adopted mechanical model is obtained assuming the existence of a heat transfer from/to the plate following Newtons law of cooling. Besides, an integration of this model on plate thickness direction produces a two-dimensional model in terms of a mean plate temperature. Numerical simulations of the above described problem have attested the instability inherent to Galerkin formulation in the presence of very high source-dominated regimen. Usual strategies in the Engineering practice of dealing with this shortcoming, such as mesh refining or higher order interpolations, proved to be inefficient

Finite elements for thixotropic elasto-viscoplastic flows

This work aims at performing finite element approximations for entry flows of thixo- tropic elasto-viscoplastic fluids, using the structured fluid model introduced in de Souza Mendes (2011). The constitutive equation is based on a modification of the Oldroyd-B viscoelastic equa- tion, with variable viscosity and relaxation time. According to this model, both the relaxation time and the viscosity are functions of a scalar parameter – denoted herein by structure parame- ter – that deals with microscopic changes in the material microstructure. All results are obtained using a four-field GLS-type method, aiming at determining the accurate morphology and position of the unyielded regions in entry flows of elasto-viscoplastic fluids through an one-to-four sudden expansion.