Maria Laura Martins-Costa

A local model for the heat transfer process in two distinct flow regions

Abstract In the present work, a model for a local description of the energy transfer phenomenon in two distinct flow regions, one consisting of a Newtonian incompressible fluid and the other represented by a binary (solid-fluid) saturated mixture, is proposed. Compatability conditions at the interface (pure-fluid-mixture) for momentum and energy transfer are also proposed and discussed. A particular case is simulated by using an iterative procedure with a finite-difference approach, in which the inlet temperatures of both the fluid (in the pure-fluid region) and the fluid constituent (in the mixture region) are the only boundary conditions prescribed in the x -direction. Representative results of centerline temperatures and temperature profiles are presented for the fluid in the pure-fluid region (the upper channel) as well as for both constituents in the mixture region (the lower channel), since thermal nonequilibrium is allowed.

On the modelling of momentum and energy transfer in incompressible mixtures

Abstract Although many papers concerning the continuum theory of mixtures have been produced in the last years, the adequate form for the first and second laws of thermodynamics is still a controversial subject. In this paper, the basic principles that govern the evolution of a continuous mixture are postulated and used to develop a systematic procedure to obtain thermodynamically admissible objective constitutive equations. The prescription of five thermodynamic potentials is sufficient to define a complete set of constitutive relations. Two examples concerning the thermal analysis of a binary mixture of rigid solids and fluid flow in a porous channel bounded by two isothermal impermeable plates are presented. They illustrate the possibilities of effective practical use of the proposed theory.

Simulation of momentum and energy transfer in a porous medium nonsaturated by an incompressible fluid

Abstract This work studies the transport phenomena in a nonsaturated flow of a newtonian fluid through a rigid porous matrix. A mixture theory approach is used in the modelling of the fluid flow and the heat, transfer problem. The mixture consists of three overlapping continuous constituents: a solid (porous medium), a liquid and an inert gas, assumed with zero mass density and thermal conductivity. A set of four nonlinear partial differential equations describes the problem which is approximated by means of a Glimms scheme, combined with an operator splitting technique, for the hyperbolic equations and a finite-difference scheme for the elliptic ones.

A local model for a packed-bed heat exchanger with a multiphase matrix

Abstract The transport phenomena in a multiphase solid matrix packed-bed heat exchanger are modelled with the help of the continuum theory of mixtures, a specific tool for multiphase phenomena, which allows a thermodynamically consistent local description. The mixture consists of n + 1 ( n rigid solids and an incompressible fluid) overlapping continuous constituents, whose thermodynamical interaction is ensured by momentum and energy generation terms. An iterative procedure is used to simulate the forced convection in a counter-flow heat exchanger allowing the fluid constituent inlet temperature to be the only boundary condition prescribed in the flow direction. Numerical results considering a two-phase solid matrix are presented for some cases of interest.

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

Forced Convection Flow Through an Unsaturated Wellbore

This work studies the dynamics of the filling up of a rigid cylindrical shell porous matrix by a Newtonian fluid and the heat transfer associated phenomenon. A mixture theory approach is employed to obtain a preliminary local model for nonisothermal flows through a wellbore. The mixture consists of three overlapping continuous constituents: a solid (porous medium), a liquid and an inert gas included to account for the compressibility of the mixture as a whole. Assuming the convection flow on radial direction only, a set of four nonlinear partial differential equations describes the problem. Its hydrodynamic part — a nonlinear hyperbolic system — is approximated by means of a Glimm’s scheme, combined with an operator splitting technique, while an implicit finite difference scheme is used to simulate the thermal part.Copyright

Simulation of a Pollutant Transport in an Atmosphere With Shock Waves

In this work a model for transport phenomena in an environment representing the atmosphere containing a pollutant is presented by considering mass and linear momentum conservation for the air-pollutant mixture as well as the mass balance for the pollutant. The resulting mathematical description consists of a nonlinear system of hyperbolic equations that admits discontinuities in addition to smooth or classical solutions. The Riemann problem associated with a class of problems describing the transport of a pollutant in an ideal gas with constant temperature with a discontinuous mass density distribution as initial condition is discussed. Numerical approximations for this nonlinear system in which the problem is solved subjected to a discontinuous initial condition — a jump, originating, in most cases, shock waves — are obtained by employing Glimm’s method and considered in some numerical simulations.Copyright

An alternative finite element formulation for determination of streamlines in two-dimensional problems

It is well known that the numerical solutions of incompressible viscous flows are of great importance in Fluid Dynamics. The graphics output capabilities of their computational codes have revolutionized the communication of ideas to the non-specialist public. In general those codes include, in their hydrodynamic features, the visualization of flow streamlines - essentially a form of contour plot showing the line patterns of the flow - and the magnitudes and orientations of their velocity vectors. However, the standard finite element formulation to compute streamlines suffers from the disadvantage of requiring the determination of boundary integrals, leading to cumbersome implementations at the construction of the finite element code. In this article, we introduce an efficient way - via an alternative variational formulation - to determine the streamlines for fluid 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 three viscous models: Stokes, Navier-Stokes and Viscoelastic flows.