High‐Order Methods for CFD

Abgrall, Rémi; Ricchiuto, Mario

doi:10.1002/9781119176817.ecm2112

Cited by 26 publications

(41 citation statements)

References 140 publications

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“…All the other criteria, including the physical one for F r scr = 0.75, provide a very early onset of breaking, with an under-prediction of the breaking wave height, and a milder slope after. Some of the height underprediction may be also related to the under-shoaling characteristics of the Madsen and Sørensen model (1). Looking at the wave set-up, reported on the right in the same figure, we can clearly see the effects of the use of h in the local criterion.…”

Section: Breaking and Set-up Of Monochromatic Waves On A Shorementioning

confidence: 79%

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Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model

2019

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The aim of the present work is to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sørensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave's crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy, that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities. We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model are defined via physical arguments, or through some auxiliary evolution model (see e.g.[24] and references therein), and, eventually, adjusted by means of numerical experiments. An alternative to the eddy viscosity type modeling are the roller approaches, that account more explicitly for the large scale effects of vorticity on the mean flow. Finally, the hybrid approaches exploit the properties of hyperbolic conservation laws endowed with an entropy, and model large scale effects of wave breaking with the dissipation of the total energy across shocks arising in shallow water simulations or augment/modify Boussinesq-type equations to include, for example, additional terms caused by the presence of the surface rollers (see cf. [41]). Independently of the chosen closure strategy, some breaking detection criteria is very often required to trigger the onset of wave breaking. In practice it is this criteria that leads to the activation of the closure. As well explained in [33], historically the first detection criteria were based on quantities computed using information over one full phase of the wave. The possibility of performing accurate phase resolved simulations, has led to the necessity of formulating new detection criteria based on local flow features [38,25,42,23]. More specifically, these criteria rely on a wave-by-wave analysis more efficient to program in the context of phase resolved simulations, and providing a physically correct detection of breaking onset and termination, provided that a sufficiently accurate description of the wave profiles is available. In this...

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Section: Breaking and Set-up Of Monochromatic Waves On A Shorementioning

confidence: 79%

“…with the definition of D(η, q) as shown in (1). The newly introduced term f break , denotes a so-called flag, that allows to switch from a propagation region (f break = 1) to a breaking one (f break = 0).…”

Section: Wave Breaking Closure: a Hybrid Boussinesq Type Modelmentioning

confidence: 99%

Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model

2019

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“…Periodic conditions are enforced at all boundaries of the spatial computational domain. The number of DOFs is kept constant for all simulations and equal to 8 × 10 6 , where the grids are equally spaced Cartesian grids with 100 3 , 60 3 , 50 3 , and 40 3 number of elements for polynomial order = 2, 3, 4 and 5, respectively.…”

Section: Application To Viscous Flows a Taylor-green Vortex Problemmentioning

confidence: 99%

“…Spatially high-order accurate discretizations demonstrate much smaller numerical errors than low-order methods, in terms of dispersion as well as dissipation [1][2][3][4]. Consequently, they tend to be more accurate per degree of freedom (DOF) [5,6].…”

Section: Introductionmentioning

confidence: 99%

Optimized Explicit Runge-Kutta Schemes for Entropy Stable Discontinuous Collocated Methods Applied to the Euler and Navier–Stokes equations

Jahdali

Boukharfane²,

Dalcín³

et al. 2021

AIAA Scitech 2021 Forum

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In this work, we design a new set of optimized explicit Runge-Kutta schemes for the integration of systems of ordinary differential equations arising from the spatial discretization of wave propagation problems with entropy stable collocated discontinuous Galerkin methods. The optimization of the new time integration schemes is based on the spectrum of the discrete spatial operator for the advection equation. To demonstrate the efficiency and accuracy of the new schemes compared to some widely used classic explicit Runge-Kutta methods, we report the wall-clock time versus the error for the simulation of the two-dimensional advection equation and the propagation of an isentropic vortex with the compressible Euler equations. The efficiency and robustness of the proposed optimized schemes for more complex flow problems are presented for the three-dimensional Taylor-Green vortex at a Reynolds number of Re = 1.6 × 10 3 and Mach number Ma = 0.1, and the flow past two identical spheres in tandem at a Reynolds number of Re = 3.9 × 10 3 and Mach number Ma = 0.1.

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“…As before, the global error can be split into the approximation and the truncation error. Note that the consistency/orthogonality relation still implies And we focus on the consistency error |B d (W, E * )|, as shown in [1,2]. Using the continuity of B d provides an estimate of the consistency error, namely, |B d (W, E * )| ≤ C cont W H * where the norms on the right hand side are the natural norms in H div × H 1 , and we have:…”

Section: Appendixa1 Symmetric Enrichmentmentioning

confidence: 99%

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Nouveau

Ricchiuto

Scovazzi

2019

Journal of Computational Physics

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We propose an extension of the embedded boundary method known as "shifted boundary method" to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded. similar indicator function. The main idea behind the IBM is to solve the model equations on the entire domain, and to impose the boundary conditionss via a forcing term. The method was originally designed on Cartesian grids and its development has been an active field of research for some decades now. Two exhaustive reviews were written by Mittal and Iaccarino in 2005 [34], and Sotiropoulos and Yang in 2014 [47]. The main drawback of the IBM is the accuracy of boundary conditions, which are most often only first-order accurate. Some strategies have been proposed in the past to increase the IBM accuracy, at the expense of more involved implementations [18,17,26,27], or to compensate low-order accuracy by means of mesh adaptation [38,6,19].A similar objective is pursued by Embedded Boundary Methods, which still use an immersed description of bodies and boundaries, but solve the model equations only in the regions of interest. Within a finite element context, the cut-cell and cutFEM approach [39,15,20,10,24,48] are most commonly utilized. These techniques usually combine a weak enforcement of the boundary conditions with a XFEM strategy. For example, in CutFEMs, the boundary is reconstructed as the intersection between the boundary surface and the embedding (underlaying) grid. Approaches of this kind often require complex...

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High‐Order Methods for CFD

Cited by 26 publications

References 140 publications

Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model

Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model

Optimized Explicit Runge-Kutta Schemes for Entropy Stable Discontinuous Collocated Methods Applied to the Euler and Navier–Stokes equations

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

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