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• A finite element model for free surface flows on fixed meshes

  H. Coppola-Owen, R. Codina

  Int. J. Numer. Meth. Fluids (2007). Vol. 54 (10), pp. 1151-1171

  
  

  Abstract

  In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element models for such flows, is that we take advantage of the particularities of free surface flow, instead of considering it a particular case of two‐phase flow. The fact that a given free surface implies a known boundary condition on the interface, allows us to solve the Navier–Stokes equations on the fluid domain uncoupled from the solution on the rest of the finite element mesh. This, together with the use of enhanced integration allows us to model low Froude number flows accurately, something that is not possible with typical two‐phase flow models applied to free surface flow.

  Abstract
  In this paper, we present a finite element model for free surface flows on fixed meshes. The main novelty of the approach, compared with typical fixed mesh finite element [...]

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• Dynamic subscales in the finite element approximation of thermally coupled incompressible flows

  R. Codina, J. Principe

  Int. J. Numer. Meth. Fluids (2007). Vol. 54 (6-8), pp. 707-730

  
  

  Abstract

  In this paper, we propose a variational multiscale finite‐element approximation for the incompressible Navier–Stokes equations using the Boussinesq approximation to model thermal coupling. The main feature of the formulation in contrast to other stabilized methods is that we consider the subscales as transient. They are solution of a differential equation in time that needs to be integrated. Likewise, we keep the effect of the subscales both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation. Apart from presenting the main properties of the formulation, we also discuss some computational aspects such as the linearization strategy or the way to integrate in time the equation for the subscales.

  Abstract
  In this paper, we propose a variational multiscale finite‐element approximation for the incompressible Navier–Stokes equations using the Boussinesq approximation to [...]

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• Time dependent subscales in the stabilized finite element approximation of incompressible flow problems

  R. Codina, O. Guasch, S. Badia, J. Principe

  Comput. Methods Appl. Mech. Engrg., (2007). Vol. 196 (21-24), pp. 2413-2430

  
  

  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 paper we analyze a stabilized finite element approximation for the incompressible Navier–Stokes equations based on the subgrid-scale concept. The essential point [...]

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• A finite element method for the solution of rotary pumps

  G. Houzeaux, R. Codina

  Computers and Fluids (2007). Vol. 36 (4), pp. 667-679

  
  

  Abstract

  We present in this paper a numerical strategy for the simulation of rotary positive displacement pumps, taking as an example a gear pump. While the two gears of the pump are rotating, the intersection between them changes in time. Therefore, the computational domain should be recomputed in some way at each time step. The strategy used here consists in dividing a cycle into a certain number of time steps and obtaining different computational meshes for each of these time steps. The coupling between two consecutive time steps is achieved by interpolating the flow unknowns in a proper way. This geometrical decomposition enables one to have a plain control over the mesh, particularly in the zones of interest, which are the gap between the gears and the casing, and the engagement and disengagement zones of the gears.

  Abstract
  We present in this paper a numerical strategy for the simulation of rotary positive displacement pumps, taking as an example a gear pump. While the two gears of the pump are [...]

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• Analysis of a stabilized finite element approximation of the transient convection-diffusion equation using an ALE framework

  S. Badia, R. Codina

  SIAM J. Numer. Anal. (2006). Vol. 44 (5), pp. 2159-2197

  
  

  Abstract

  In this paper we analyze a stabilized finite element method to approximate the convection‐diffusion equation on moving domains using an arbitrary Lagrangian Eulerian (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 semidiscrete 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 method to approximate the convection‐diffusion equation on moving domains using an arbitrary Lagrangian Eulerian (ALE) [...]

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• Numerical approximation of the heat transfer between domains separated by thin walls

  R. Codina, G. Houzeaux

  Int. J. Numer. Meth. Fluids (2006). Vol. 52 (9), pp. 963-986

  
  

  Abstract

  In this paper, we analyse the numerical approximation of the heat transfer problem between two subdomains that we will consider filled with a fluid and separated by a thin solid wall. First of all, we state the problem in the whole domain with discontinuous physical properties. As an alternative and under certain assumptions on the separating walls, a classical Robin boundary condition between the fluid domains is obtained, thus eliminating the solid wall, and according to which the heat flux is proportional to the temperature difference between the two subdomains. Apart from discussing the relation between both approaches, we consider their numerical approximation, considering different alternatives for the first case, that is, the case in which temperatures are also computed in the solid wall.

  Abstract
  In this paper, we analyse the numerical approximation of the heat transfer problem between two subdomains that we will consider filled with a fluid and separated by a thin [...]

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• Stabilized finite element approximation of the stationary magneto-hydrodynamics equations

  R. Codina, N. Hernández-Silva

  Computational Mechanics (2006). Vol. 38 (4-5), pp. 344-355

  
  

  Abstract

  In this work we present a stabilized finite element method for the stationary magneto-hydrodynamic equations based on a simple algebraic version of the subgrid scale variational concept. The linearization that yields a well posed linear problem is first identified, and for this linear problem the stabilization method is designed. The key point is the correct behavior of the stabilization parameters on which the formulation depends. It is shown that their expression can be obtained only on the basis of having a correct error estimate. For the stabilization parameters chosen, a stability estimate is proved in detail, as well as the convergence of the numerical solution to the continuous one. The method is then extended to nonlinear problems and its performance checked through numerical experiments.

  Abstract
  In this work we present a stabilized finite element method for the stationary magneto-hydrodynamic equations based on a simple algebraic version of the subgrid scale variational [...]

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• Finite element modeling of the lost foam casting process tackling back‐pressure effects

  G. Houzeaux, R. Codina

  Int. J. Num. Meths. for Heat and Fluid Flow (2006). Vol. 16 (5), pp. 573-589

  
  

  Abstract

  Purpose – To develop a numerical methodology to simulate the lost foam casting (LFC), including the gas back-pressure effects. Design/methodology/approach – Back-pressure effects are due to the interactions of many physical processes. The strategy proposed herein tries to model all these processes within a simple formula. The main characteristic of the model consists of assuming that the back-pressure is a known function of the external parameters (coating, temperature, gravity, etc.) that affects directly the heat transfer coefficient from the metal to the foam. The general framework of the simulation is a finite element model based on an arbitrary Lagrangian Eulerian (ALE) approach and the use of level set function to capture the metal front advance. Findings – After experimental tunings, the model provides a way to include the back-pressure effects in a simple way. Research limitations/implications – The method is not completely predictive in the sense that a priori tuning is necessary to calibrate the model. Practical implications – Provides more realistic results than classical models. Originality/value – The paper proposes a theoretical framework of a finite element method for the simulation of LFC process. The method uses an ALE method on a fixed mesh and a level-set function to capture metal front advance. It proposes an original formula for the heat transfer coefficient that enables one to include back-pressure effects

  Abstract
  Purpose – To develop a numerical methodology to simulate the lost foam casting (LFC), including the gas back-pressure effects. Design/methodology/approach – [...]

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• The Characteristic‐Based Split (CBS) scheme—a unified approach to fluid dynamics

  P. Nithiarasu, R. Codina, O. Zienkiewicz

  Int. J. Numer. Meth. Engng. (2006). Vol. 66 (10), pp. 1514-1546

  
  

  Abstract

  This paper presents a comprehensive overview of the characteristic‐based methods and Characteristic‐Based Split (CBS) scheme. The practical difficulties of employing the original characteristic schemes are discussed. The important features of the CBS scheme are brought out by studying several problems of compressible and incompressible flows. All special consideration necessary for solving these problems are thoroughly discussed. The CBS scheme is presented in such a way that any interested researcher should be able to develop a code using the information provided. Several invicid and viscous flow examples are also provided to demonstrate the unified CBS approach. For sample two‐dimensional codes, input files and instructions, the readers are referred to ‘www.nithiarasu.co.uk’ 

  Abstract
  This paper presents a comprehensive overview of the characteristic‐based methods and Characteristic‐Based Split (CBS) scheme. The practical difficulties of employing the [...]

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• Numerical comparison of CBS and SGS as stabilization techniques for the incompressible Navier–Stokes equations

  R. Codina, H. Coppola-Owen, P. Nithiarasu, C. Liu

  Int. J. Numer. Meth. Engng. (2006). Vol. 66 (10), pp. 1677-1689

  
  

  Abstract

  In this work, we present numerical comparisons of some stabilization methods for the incompressible Navier–Stokes. The first is the characteristic‐based split (CBS). It combines the characteristic Galerkin method to deal with convection‐dominated flows with a classical splitting technique, which in some cases allows us to use equal velocity–pressure interpolations. The other two approaches are particular cases of the subgrid scale (SGS) method. The first, obtained after an algebraic approximation of the subgrid scales, is very similar to the popular Galerkin/least‐squares (GLS) method, whereas in the second, the subscales are assumed to be orthogonal to the finite element space. It is shown that all these formulations display similar stabilization mechanisms, provided the stabilization parameter of the SGS methods is identified with the time step of the CBS approach. This paper provides the numerical experiments for the comparison of formulations made by Codina and Zienkiewicz in a previous article.

  Abstract
  In this work, we present numerical comparisons of some stabilization methods for the incompressible Navier–Stokes. The first is the characteristic‐based split (CBS). [...]

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