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• Two-fluid flow problem for viscoelastic fluids using the level set method and pressure enriched shape functions

  E. Castillo, J. Baiges, R. Codina

  Journal of Non-Newtonian Fluid Mechanics (2015). Vol. 225, pp. 37-53

  
  

  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
  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 [...]

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• On some time marching schemes for the stabilized finite element approximation of the mixed wave equation

  H. Espinoza, R. Codina, S. Badia

  Comput. Methods Appl. Mech. Engrg., (2015). Vol. 296, pp. 295-326

  
  

  Abstract

  In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings. On the other hand, we use Fourier techniques (also known as von Neumann analysis) in order to analyze stability, dispersion and dissipation. Numerical convergence tests are presented for various time integration schemes, polynomial interpolations (for the spatial discretization), stabilization methods, and variational forms. To analyze the behavior of the different schemes considered, a 1D wave propagation problem is solved.

  Abstract
  In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, [...]

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• A three‐field stabilized finite element method for fluid‐structure interaction: elastic solid and rigid body limit

  E. Hachem, S. Feghali, T. Coupez, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2015). Vol. 104 (7), pp. 566-584

  
  

  Abstract

  We propose a full Eulerian framework for solving fluid‐structure interaction (FSI) problems based on a unified formulation in which the FSIs are modelled by introducing an extra stress in the momentum equation. The obtained three‐field velocity, pressure and stress system is solved using a stabilized finite element method. The key feature of this unified formulation is the ability to describe different kind of interactions between the fluid and the structure, which can be either elastic or a perfect rigid body, without the need of treating this last case via penalization. The level‐set method combined with a dynamic anisotropic mesh adaptation is used to track the fluid‐solid interface.

  Abstract
  We propose a full Eulerian framework for solving fluid‐structure interaction (FSI) problems based on a unified formulation in which the FSIs are modelled by introducing [...]

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• Weak imposition of essential boundary conditions in the finite element approximation of elliptic problems with non‐matching meshes

  R. Codina, J. Baiges

  Int. J. Numer. Meth. Engng. (2015). Vol. 104 (7), pp. 624-654

  
  

  Abstract

  In this work, we propose a method to prescribe essential boundary conditions in the finite element approximation of elliptic problems when the boundary of the computational domain does not match with the element boundaries. The problems considered are the Poisson problem, the Stokes problem, and the Darcy problem, the latter both in the primal and in the dual formulation. The formulation proposed is of variational type. The key idea is to start with the variational form that defines the problem and treat the boundary condition as a constraint. The particular feature is that the Lagrange multiplier is not defined on the boundary where the essential condition needs to be prescribed but is taken as a certain trace of a field defined in the computational domain, either in all of it or just in a region surrounding the boundary. When approximated numerically, this may allow one to condense the DOFs of the new field and end up with a problem posed only in terms of the original unknowns. The nature of the field used to weakly impose boundary conditions depends on the problem being treated. For the Poisson problem, it is a flux; for the Stokes problem, a stress; for the Darcy problem in primal form, a velocity field; and for the Darcy problem in dual form, it is a potential. 

  Abstract
  In this work, we propose a method to prescribe essential boundary conditions in the finite element approximation of elliptic problems when the boundary of the computational [...]

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• Numerical modeling of chlorine concentration in water storage tank

  R. Codina, C. Muñoz, J. Baiges, J. Principe

  Int. J. Numer. Meth. Fluids (2015). Vol. 79 (2), pp. 84-107

  
  

  Abstract

  In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution of chlorine concentration. The mathematical model contains several ingredients particularly designed for this problem, namely, a boundary condition to model falling jets on free surfaces, an arbitrary Lagrangian–Eulerian formulation to account for the motion of the free surface because of demand and supply of water, and a coupling of the hydrodynamics with a convection–diffusion–reaction equation modeling the time evolution of chlorine. From the numerical point of view, the equations resulting from the mathematical model are approximated using a finite element formulation, with linear continuous interpolations on tetrahedra for all the unknowns. To make it possible, and also to be able to deal with convection‐dominated flows, a stabilized formulation is used. In order to capture the sharp gradients present in the chlorine concentration, particularly near the injection zone, a discontinuity capturing technique is employed.

  Abstract
  In this paper, we describe a numerical model to simulate the evolution in time of the hydrodynamics of water storage tanks, with particular emphasis on the time evolution [...]

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• Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics

  S. Badia, R. Codina, R. Planas

  Archives of Comp. Meths. Engng. (2015). Vol. 22 (4), pp. 621-636

  
  

  Abstract

  In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing ones is the fact that it always converges to the physical solution, even when it is singular. We have performed a detailed stability and convergence analysis of the formulation in a simplified setting. From the convergence analysis, we infer that a particular type of meshes with a macro-element structure is needed, which can be easily obtained after a straight modification of any original mesh.

  Abstract
  In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this [...]

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• Finite element dynamical subgrid-scale model for low Mach number flows with radiative heat transfer

  M. Ávila, R. Codina, J. Principe

  Int. J. Numer. Meth. Heat and Fluid Flows (2015). (Preprint) Vol. 25 (6), pp. 1361-1384

  
  

  Abstract

  Purpose – The purpose of this paper is to present a finite element approximation of the low Mach number equations coupled with radiative equations to account for radiative heat transfer. For high-temperature flows this coupling can have strong effects on the temperature and velocity fields. Design/methodology/approach – The basic numerical formulation has been proposed in previous works. It is based on the variational multiscale (VMS) concept in which the unknowns of the problem are divided into resolved and subgrid parts which are modeled to consider their effect into the former. The aim of the present paper is to extend this modeling to the case in which the low Mach number equations are coupled with radiation, also introducing the concept of subgrid scales for the radiation equations. Findings – As in the non-radiative case, an important improvement in the accuracy of the numerical scheme is observed when the nonlinear effects of the subgrid scales are taken into account. Besides it is possible to show global conservation of thermal energy. Originality/value – The original contribution of the work is the proposal of keeping the VMS splitting into the nonlinear coupling between the low Mach number and the radiative transport equations, its numerical evaluation and the description of its properties.

  Abstract
  Purpose – The purpose of this paper is to present a finite element approximation of the low Mach number equations coupled with radiative equations to account for radiative [...]

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• First, second and third order fractional step methods for the three-field viscoelastic flow problem

  E. Castillo, R. Codina

  Journal of Computational Physics (2015). Vol. 296, pp. 113-137

  
  

  Abstract

  In this paper, three different fractional step methods are designed for the three-field viscoelastic flow problem, whose variables are velocity, pressure and elastic stress. The starting point of our methods is the same as for classical pressure segregation algorithms used in the Newtonian incompressible Navier–Stokes problem. These methods can be understood as an inexact LU block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level. The final schemes allow one to solve the problem in a fully decoupled form, where each equation (for velocity, pressure and elastic stress) is solved separately. The first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation, whereas the key point for the second order scheme is a first order extrapolation of these variables. The third order fractional step method relies on Yosida's scheme. Referring to the spatial discretization, either the Galerkin method or a stabilized finite element formulation can be used. We describe the fractional step methods first assuming the former, and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work. This discretization in space shows very good stability, permitting in particular the use of equal interpolation for all variables.

  Abstract
  In this paper, three different fractional step methods are designed for the three-field viscoelastic flow problem, whose variables are velocity, pressure and [...]

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• Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows

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

  Comput. Methods Appl. Mech. Engrg., (2015). Vol. 285, pp. 32-63

  
  

  Abstract

  In this work we study the performance of some variational multiscale models (VMS) in the large eddy simulation (LES) of turbulent flows. We consider VMS models obtained by different subgrid scale approximations which include either static or dynamic subscales, linear or nonlinear multiscale splitting, and different choices of the subscale space. After a brief review of these models, we discuss some implementation aspects particularly relevant to the simulation of turbulent flows, namely the use of a skew symmetric form of the convective term and the computation of projections when orthogonal subscales are used. We analyze the energy conservation (and numerical dissipation) of the alternative VMS formulations, which is numerically evaluated. In the numerical study, we have considered three well known problems: the decay of homogeneous isotropic turbulence, the Taylor–Green vortex problem and the turbulent flow in a channel. We compare the results obtained using different VMS models, paying special attention to the effect of using orthogonal subscale spaces. The VMS results are also compared against classical LES scheme based on filtering and the dynamic Smagorinsky closure. Altogether, our results show the tremendous potential of VMS for the numerical simulation of turbulence. Further, we study the sensitivity of VMS to the algorithmic constants and analyze the behavior in the small time step limit. We have also carried out a computational cost comparison of the different formulations. Out of these experiments, we can state that the numerical results obtained with the different VMS formulations (as far as they converge) are quite similar. However, some choices are prone to instabilities and the results obtained in terms of computational cost are certainly different. The dynamic orthogonal subscales model turns out to be best in terms of efficiency and robustness.

  Abstract
  In this work we study the performance of some variational multiscale models (VMS) in the large eddy simulation (LES) of turbulent flows. We consider VMS models obtained by [...]

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• Variational MultiScale based dissipation models for the estimation of Atmospheric Seeing

  J. Baiges, R. Codina

  Computers and Fluids (2015). Vol. 107, pp. 141-154

  
  

  Abstract

  In this work we present a numerical model for the estimation of atmospheric seeing in observation sites. The particularity of the method is that it is based on a Variational Multiscale turbulence model, its main feature being that the numerical mechanisms which are used to deal with stability issues (convection and the inf–sup condition for incompressible flows) do also take care of the modeling of turbulence. Based on this turbulence model, we develop the expressions for the viscous and thermal dissipations, ∊∊num and χnum, which are later used for evaluating the constant of structure of the refraction index Cn2 following the classical model developed by Tatarski. Numerical examples show the behavior of the proposed numerical scheme when applied to turbulent flow practical cases, which include a convective boundary layer, the flow inside a transfer optics room, and a telescope enclosure.

  Abstract
  In this work we present a numerical model for the estimation of atmospheric seeing in observation sites. The particularity of the method is that it is based on a Variational [...]

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