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• Solution of low Mach number aeroacoustic flows using a Variational Multi-Scale finite element formulation of the compressible Navier–Stokes equations written in primitive variables

  C. Bayona, J. Baiges, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2019). Vol. 344, pp. 1073-1103

  
  

  Abstract

  In this work we solve the compressible Navier–Stokes equations written in primitive variables in order to simulate low Mach number aeroacoustic flows. We develop a Variational Multi-Scale formulation to stabilize the finite element discretization by including the orthogonal, dynamic and non-linear subscales, together with an implicit scheme for advancing in time. Three additional features define the proposed numerical scheme: the splitting of the pressure and temperature variables into a relative and a reference part, the definition of the matrix of stabilization parameters in terms of a modified velocity that accounts for the local compressibility, and the approximation of the dynamic stabilization matrix for the time dependent subscales. We also include a weak imposition of implicit non-reflecting boundary conditions in order to overcome the challenges that arise in the aeroacoustic simulations at low compressibility regimes. The order of accuracy of the method is verified for two- and three-dimensional linear and quadratic elements using steady manufactured solutions. Several benchmark flow problems are studied, including transient examples and aeroacoustic applications.

  Abstract
  In this work we solve the compressible Navier–Stokes equations written in primitive variables in order to simulate low Mach number aeroacoustic [...]

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• Computational aeroacoustics to identify sound sources in the generation of sibilant /s/

  A. Pont, O. Guasch, J. Baiges, R. Codina, A. Hirtum

  Int. J. Numer. Meth. Biomed. Engng (2019). Vol. 35 (1), e3153

  
  

  Abstract

  A sibilant fricative /s/ is generated when the turbulent jet in the narrow channel between the tongue blade and the hard palate is deflected downwards through the space between the upper and lower incisors and then impinges the space between the lower incisors and the lower lip. The flow eddies in that region become a source of direct aerodynamic sound, which is also diffracted by the speech articulators and radiated outwards. The numerical simulation of these phenomena is complex. The spectrum of an /s/ typically peaks between 4 and 10 kHz, which implies that very fine computational meshes are needed to capture the eddies producing such high frequencies. In this work, a large‐scale computation of the aeroacoustics of /s/ has been performed for a realistic vocal tract geometry, resorting to two different acoustic analogies. A stabilized finite element method that acts as a large eddy simulation model has been adopted to solve the flow dynamics. Also, a numerical strategy has been implemented that allows the determination, in a single computational run, of the separate contribution of the sound diffracted by the upper incisors from the overall radiated sound. Results are presented for points located close to the lip opening showing the relative influence of the upper teeth depending on frequency.

  Abstract
  A sibilant fricative /s/ is generated when the turbulent jet in the narrow channel between the tongue blade and the hard palate is deflected downwards through the space between [...]

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• Finite element modeling of nonlinear reaction-diffusion-advection systems of equations

  S. Komala, R. Codina

  Int. J. Numer. Meth. Heat and Fluid Flows (2018). Vol. 28 (11), pp. 2688-2715

  
  

  Abstract

  Purpose This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed. Design/methodology/approach A stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation. Findings The stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model. Originality/value A stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.

  Abstract
  Purpose This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear [...]

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• Stationary and time-dependent numerical approximation of the lid-driven cavity problem for power-law fluid flows at high Reynolds numbers using a stabilized finite element formulation of the VMS type

  A. Aguirre, E. Castillo, M. Cruchaga, R. Codina, J. Baiges

  Journal of Non-Newtonian Fluid Mechanics (2018). Vol. 257, pp. 22-43

  
  

  Abstract

  In this work, a variational multiscale finite element formulation is used to approximate numerically the lid-driven cavity flow problem for high Reynolds numbers. For Newtonian fluids, this benchmark case has been extensively studied by many authors for low and moderate Reynolds numbers (up to Re=10,000), giving place to steady flows, using stationary and time-dependent approaches. For more convective flows, the solution becomes unstable, describing an oscillatory behavior. The critical Reynolds number which gives place to this time-dependent fluid dynamics has been defined over a wide range 7, 300 ≲ Re ≲ 35, 000, using different numerical approaches. In the non-Newtonian case, the cavity problem has not been studied deeply for high Reynolds number (Re > 10, 000), specifically, in the oscillatory time-dependent case. A VMS formulation is presented to be validated using existing results, to determine flow conditions at which the instability appears, and lastly, to establish new benchmark solutions for high-Reynolds numbers fluid flows using the power-law model. Obtained results show a good agreement with those reported in the references, and new data related with the oscillatory behavior of the flow has been found for the non-Newtonian case. In this regard, time-dependent flows show dependence on both Reynolds number and power-law index, and the unsteady starting point has been determined for all studied cases. It is determined that the critical Reynolds number (Rec) that defines the first Hopf bifurcation for Newtonian fluid flow is ranged between 8,100 ≲ Rec ≲ 8, 250, whereas for power-law indexes n=0.5 and n=1.5, it is 7,100 ≲ Rec ≲ 7,200 and 18,250 ≲ Rec ≲ 18,500, respectively.

  Abstract
  In this work, a variational multiscale finite element formulation is used to approximate numerically the lid-driven cavity flow problem for high Reynolds numbers. For Newtonian [...]

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• On hp convergence of stabilized finite element methods for the convection–diffusion equation

  R. Codina

  Sociedad Española de Matemática Aplicada (2018). Vol. 75 (4), pp. 591-606

  
  

  Abstract

  This work analyzes some aspects of the hp convergence of stabilized finite element methods for the convection-diffusion equation when diffusion is small. The methods discussed are classical-residual based stabilization techniques and also projection-based stabilization methods. The theoretical impossibility of obtaining an optimal convergence rate in terms of the polynomial order p for all possible Péclet numbers is explained. The key point turns out to be an inverse estimate that scales as p2. The use of this estimate is not needed in a particular case of (H1-)projection-based methods, and therefore the theoretical lack of convergence described does not exist in this case.

  Abstract
  This work analyzes some aspects of the hp convergence of stabilized finite element methods for the convection-diffusion equation when diffusion is small. The methods [...]

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• Variational multiscale error estimators for the adaptive mesh refinement of compressible flow simulations

  C. Bayona, R. Codina, J. Baiges

  Comput. Methods Appl. Mech. Engrg., (2018). Vol. 337, pp. 501-526

  
  

  Abstract

  This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. The proposed methodology employs the Variational Multi-Scale framework, and specifically, the idea is to use the variational subscales to estimate the error. These subscales are defined to be orthogonal to the finite element space, dynamic and non-linear, and both the subscales in the interior of the element and on the element boundaries are considered. Another particularity of the model is that we define some norms that lead to a dimensionally consistent measure of the compressible flow solution error inside each element; a scaled L2-norm, and the calculation of a physical entropy measure, are both studied in this work. The estimation of the error is used to drive the adaptive mesh refinement of several compressible flow simulations. Numerical results demonstrate good accuracy of the local error estimate and the ability to drive the adaptative mesh refinement to minimize the error through the computational domain.

  Abstract
  This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. [...]

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• Unified solver for fluid dynamics and aeroacoustics in isentropic gas flows

  A. Pont, R. Codina, J. Baiges, O. Guasch

  Journal of Computational Physics (2018). Vol. 363, pp. 11-29

  
  

  Abstract

  The high computational cost of solving numerically the fully compressible Navier–Stokes equations, together with the poor performance of most numerical formulations for compressible flow in the low Mach number regime, has led to the necessity for more affordable numerical models for Computational Aeroacoustics. For low Mach number subsonic flows with neither shocks nor thermal coupling, both flow dynamics and wave propagation can be considered isentropic. Therefore, a joint isentropic formulation for flow and aeroacoustics can be devised which avoids the need for segregating flow and acoustic scales. Under these assumptions density and pressure fluctuations are directly proportional, and a two field velocity-pressure compressible formulation can be derived as an extension of an incompressible solver. Moreover, the linear system of equations which arises from the proposed isentropic formulation is better conditioned than the homologous incompressible one due to the presence of a pressure time derivative. Similarly to other compressible formulations the prescription of boundary conditions will have to deal with the backscattering of acoustic waves. In this sense, a separated imposition of boundary conditions for flow and acoustic scales which allows the evacuation of waves through Dirichlet boundaries without using any tailored damping model will be presented.

  Abstract
  The high computational cost of solving numerically the fully compressible Navier–Stokes equations, together with the poor performance of most numerical formulations [...]

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• Stabilized mixed finite element method for the M1 radiation model

  Q. Schmid, Y. Mesri, E. Hachem, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2018). Vol. 335, pp. 69-90

  
  

  Abstract

  In this work, we present a computational approach for the numerical simulation of thermal radiation. Radiation is modeled by solving a set of two coupled partial differential equations, the so-called M1 model. A Variational Multiscale method is developed for this system, and tested on some illustrative benchmarks. The question of dealing with heterogeneous physical properties is also considered, and treated by means of an immersed method. It combines a level-set approach for representation of interfaces, mixing laws to build effective physical properties and an anisotropic mesh adaptation process, all these ingredients leading to an accurate description of the interface.

  Abstract
  In this work, we present a computational approach for the numerical simulation of thermal radiation. Radiation is modeled by solving a set of two coupled partial [...]

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• Variational multiscale approximation of the one‐dimensional forced Burgers equation: The role of orthogonal subgrid scales in turbulence modeling

  C. Bayona, J. Baiges, R. Codina

  Int. J. Numer. Meth. Fluids (2018). Vol. 86 (5), pp. 313-328

  
  

  Abstract

  A numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. We evaluate the role of the variational subscales in describing the Burgers “turbulence” phenomena. Particularly, we seek to clarify the interaction between the subscales and the resolved scales when the former are defined to be orthogonal to the finite‐dimensional space. Direct numerical simulation is used to evaluate the resulting OSGS‐VMS energy spectra. The comparison against a large eddy simulation model is presented for numerical discretizations in which the grid is not capable of solving the small scales. An accurate approximation to the phenomena of turbulence is obtained with the addition of the purely dissipative numerical terms given by the OSGS‐VMS method without any modification of the continuous problem.

  Abstract
  A numerical approximation for the one‐dimensional Burgers equation is proposed by means of the orthogonal subgrid scales–variational multiscale (OSGS‐VMS) method. [...]

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• Variational Multiscale error estimators for solid mechanics adaptive simulations: An Orthogonal Subgrid Scale approach

  J. Baiges, R. Codina

  Comput. Methods Appl. Mech. Engrg., (2017). Vol. 325, pp. 37-55

  
  

  Abstract

  In this work we present a general error estimator for the finite element solution of solid mechanics problems based on the Variational Multiscale method. The main idea is to consider a rich model for the subgrid scales as an error estimator. The subscales are considered to belong to a space orthogonal to the finite element space (Orthogonal Subgrid Scales) and we take into account their contribution both in the element interiors and on the element boundaries (Subscales on the Element Boundaries). A simple analysis shows that the upper bound for the obtained error estimator is sharper than in other error estimators based on the Variational Multiscale Method. Numerical examples show that the proposed error estimator is an accurate approximation for the energy norm error and can be used both in simple linear constitutive models and in more complex non-linear cases.

  Abstract
  In this work we present a general error estimator for the finite element solution of solid mechanics problems based on the Variational Multiscale method. [...]

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