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.
Turbulence; Large eddy simulation; Stabilization; Variational multiscale
In this work, we propose Runge–Kutta time integration schemes for the incompressible Navier–Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. The segregated Runge–Kutta methods are motivated as an implicit–explicit Runge–Kutta time integration of the projected Navier–Stokes system onto the discrete divergence-free space, and its re-statement in a velocity–pressure setting using a discrete pressure Poisson equation. We have analysed the preservation of the discrete divergence constraint for segregated Runge–Kutta methods and their relation (in their fully explicit version) with existing half-explicit methods. We have performed a detailed numerical experimentation for a wide set of schemes (from first to third order), including implicit and IMEX integration of viscous and convective terms, for incompressible laminar and turbulent flows. Further, segregated Runge–Kutta schemes with adaptive time stepping are proposed.
Time integration;Runge–Kutta;incompressible Navier–Stokes;pressure-segregation;high-order;adaptive time stepping
The variational multiscale method thought as an implicit large eddy simulation model for turbulent flows has been shown to be an alternative to the widely used physical-based models. This method is traditionally combined with equal-order velocity-pressure pairs, since it provides pressure stabilization. In this work, we consider a different approach, based on inf-sup stable elements and convection-only stabilization. In order to do so, we consider a symmetric projection stabilization of the convective term using a orthogonal subscale decomposition. The accuracy and efficiency of this method compared with residual-based algebraic subgrid scales and orthogonal subscales methods for equal-order interpolation is assessed in this paper. Moreover, when inf-sup stable elements are used, the grad-div stabilization term has been shown to be essential to guarantee accurate solutions. Hence, a study of the influence of such term in the large eddy simulation of turbulent incompressible flows is also performed. Furthermore, a recursive block preconditioning strategy has been considered for the resolution of the problem with an implicit treatment of the projection terms. Two different benchmark tests have been solved: the Taylor-Green Vortex flow with Re = 1600, and the Turbulent Channel Flow at Re = 395 and Re = 590.
Large eddy simulation, turbulence, variational multiscale, block recursive preconditioning, grad-div stabilization
In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system is carried out using mixed finite element (FE) schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint. These additional terms introduced at the discrete level have been proved to act as implicit LES models. In order to perform meaningful wall-unresolved simulations, we consider a weak imposition of the boundary conditions using a Nitsche’s-type scheme, where the tangential component penalty term is designed to act as a wall law. Next, segregated Runge–Kutta (SRK) schemes (recently proposed by the authors for laminar flow problems) are applied to the LES simulation of turbulent flows. By the introduction of a penalty term on the trace of the acceleration, these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms. SRK schemes are excellent for large-scale simulations, since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level, leading to two uncoupled systems. The pressure system is a Darcy-type problem that can easily be preconditioned using a traditional block-preconditioning scheme that only requires a Poisson solver. At the end, only coercive systems have to be solved, which can be effectively preconditioned by multilevel domain decomposition schemes, which are both optimal and scalable. The framework is applied to the Taylor–Green and turbulent channel flow benchmarks in order to prove the accuracy of the convection-stabilized mixed FEs as LES models and SRK time integrators. The scalability of the preconditioning techniques (in space only) has also been proven for one step of the SRK scheme for the Taylor–Green flow using uniform meshes. Moreover, a turbulent flow around a NACA profile is solved to show the applicability of the proposed algorithms for a realistic problem.
Large eddy simulation, turbulence, variational multiscale, wall models, Runge-Kutta
In this article, we develop a dynamic version of the variational multiscale (D‐VMS) stabilization for nearly/fully incompressible solid dynamics simulations of viscoelastic materials. The constitutive models considered here are based on Prony series expansions, which are rather common in the practice of finite element simulations, especially in industrial/commercial applications. Our method is based on a mixed formulation, in which the momentum equation is complemented by a pressure equation in rate form. The unknown pressure, displacement, and velocity are approximated with piecewise linear, continuous finite element functions. To prevent spurious oscillations, the pressure equation is augmented with a stabilization operator specifically designed for viscoelastic problems, in that it depends on the viscoelastic dissipation. We demonstrate the robustness, stability, and accuracy properties of the proposed method with extensive numerical tests in the case of linear and finite deformations.
piece‐wise linear interpolation, stabilized methods, transient dynamics, tetrahedral finite element, viscoelasticity
The discontinuous Galerkin (DG) method has found widespread application in elliptic problems with rough coefficients, of which the Darcy flow equations are a prototypical example. One of the long-standing issues of DG approximations is the overall computational cost, and many different strategies have been proposed, such as the variational multiscale DG method, the hybridizable DG method, the multiscale DG method, the embedded DG method, and the Enriched Galerkin method. In this work, we propose a mixed dual-scale Galerkin method, in which the degrees-of-freedom of a less computationally expensive coarse-scale approximation are linked to the degrees-of-freedom of a base DG approximation. We show that the proposed approach has always similar or improved accuracy with respect to the base DG method, with a considerable reduction in computational cost. For the specific definition of the coarse-scale space, we consider Raviart–Thomas finite elements for the mass flux and piecewise-linear continuous finite elements for the pressure. We provide a complete analysis of stability and convergence of the proposed method, in addition to a study on its conservation and consistency properties. We also present a battery of numerical tests to verify the results of the analysis, and evaluate a number of possible variations, such as using piecewise-linear continuous finite elements for the coarse-scale mass fluxes.
Discontinuous Galerkin method, Variational multiscale method, Hybridization, Elliptic problems, Darcy flow
The numerical simulation of physical phenomena and engineering problems can be affected by numerical errors and various types of uncertainties. Characterizing the former in computational frameworks involving system parameter uncertainties becomes a key issue. In this work, we study the behavior of new variational multiscale (VMS) error estimators for the propagation of parametric uncertainties in a Convection–Diffusion–Reaction (CDR) problem. A sensitivity analysis is performed to assess the performance of the error estimator with respect to the mesh discretization and physical parameters (here, the viscosity value and advection velocity). Three different manufactured analytical solutions are considered as benchmarking tests. Next, the performance of the VMS error estimators is evaluated for the CDR problem with uncertain input parameters. For this purpose, two probabilistic models are constructed for the viscosity and advection direction, and the uncertainties are propagated using a polynomial chaos expansion approach. A convergence analysis is specifically carried out for different configurations, corresponding to regimes where the CDR operator is either smooth or non-smooth. An assessment of the proposed error estimator is finally conducted for the three tests, considering both the viscous- and convection-dominated regimes.
Variational multiscale method, A posteriori error estimation, Convection–diffusion–reaction equation, Uncertainty propagation
In this work, we define a family of explicit a posteriori error estimators for Finite Volume methods in computational fluid dynamics. The proposed error estimators are inspired by the Variational Multiscale method, originally defined in a Finite Element context. The proposed error estimators are tested in simulations of the incompressible Navier-Stokes equations, the thermally-coupled Navier-Stokes equations, and the fully-coupled compressible large eddy simulation of the HIFiRE Direct Connect Rig Scramjet combustor.
In this thesis we have developed a path towards large scale Finite Element simulations of turbulent incompressible flows.
We have assessed the performance of residual-based variational multiscale (VMS) methods for the large eddy simulation (LES) of turbulent incompressible flows, showing that VMS thought as an implicit LES model can be an alternative to the widely used physical-based models. This method is traditionally combined with equal-order velocity-pressure pairs, but in this work we also consider an approach based on inf-sup stable elements and symmetric projection stabilization of the convective term using an orthogonal subscale decomposition. Furthermore, we propose a segregated Runge-Kutta time integration scheme in which the velocity and pressure computations are segregated at the time integration level, and that keep the same order of accuracy for both velocities and pressures. Precisely, the symmetric projection stabilization approach is suitable for this time integration scheme. This combination, together with block-preconditioning techniques, lead to problems that can be optimally preconditioned using the balancing domain decomposition by constraints preconditioners. Additionally, we also contemplate the weak imposition of the Dirichlet boundary conditions for wall-bounded turbulent flows.
Four well known problems have been mainly considered for the numerical experiments: the decay of homogeneous isotropic turbulence, the Taylor-Green vortex problem, the turbulent flow in a channel and the turbulent flow around an airfoil.
Finite element method, variational multiscale, large scale solvers, large eddy simulation, turbulence
En esta tesina se estudian distintos modelos de daño para problemas de impacto así como los métodos de resolución de estos mismos. El problema de impacto es uno de los casos que requiere un estudio minucioso para poder evaluar la máxima solicitación en ciertas aplicaciones, sobretodo en el caso de materiales compuestos. Para analizar este tipo de problemas se usan simulaciones numéricas obtenidas mediante el método de los elementos finitos (MEF). Para ello se requiere la definición de un modelo de daño que nos permita describir el comportamiento del material.
Finite element method, Composite materials, Elements finits, Mètode dels, Materials compostos
Flow simulations on porous media, reconstructed from Micro-Computerised Tomography (μCT)-scans, is becoming a common tool to compute the permeability of rocks. In order for the value of this homogenised hydraulic property to be representative of the rock at a continuum scale, the sample considered needs to be at least as large as the Representative Elementary Volume. More-over, the numerical discretisation of the digital rock needs to be fine enough to reach numerical convergence. In the particular case of Finite Elements (FE), studies have shown that simulations should use structured meshes at least two times finer than the original image resolution in order to reach the mesh convergence. These two conditions and the increased resolution of μCT-scans to observe finer details of the microstructure, can lead to extremely computationally expensive numerical simulations. In order to reduce this cost, we couple a FE numerical model for Stokes flow in porous media with an unfitted boundary method, which allows to improve results precision for coarse meshes. Indeed, this method enables to obtain a definition of the pore-grain interface as precise as for a conformal mesh, without a computationally expensive and complex mesh generation for μCT-scans of rocks. From the benchmark of three different rock samples, we observe a clear improvement of the mesh convergence for the permeability value using the unfitted boundary method. An accurate permeability value is obtained for a mesh coarser than the initial image resolution. The method is then applied to a large sample of a high resolution μCT-scan to showcase its advantage.
Finite element method, Permeability upscaling, µCT-Scan, Flow simulations, unfitted boundary method
The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods and was recently introduced for the Poisson, linear advection/diffusion, Stokes, Navier-Stokes, acoustics, and shallow-water equations. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Navier-Stokes flows with moving free-surfaces, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach prevents spurious pressure oscillations in time, which would otherwise be produced if the total active fluid volume were to change abruptly over a time step. In fact, the proposed weighted SBM method induces small mass (i.e., volume) conservation errors, which converge quadratically in the case of piecewise-linear finite element interpolations, as the grid is refined. We present an extensive set of two- and three-dimensional tests to demonstrate the robustness and accuracy of the method.
Immersed boundary, Unfitted finite element method, Approximate boundary, Free-surface flows, Computational fluid dynamics, Shifted Boundary Method.