On conservation laws of Navier–Stokes Galerkin discretizations

Charnyi, Sergey; Heister, Timo; Olshanskii, Maxim A.; Rebholz, Leo G.

doi:10.1016/j.jcp.2017.02.039

Cited by 114 publications

(125 citation statements)

References 35 publications

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“…For this reason, we call the '2D(u)u + (div u)u' formulation of the nonlinearity the EMAC formulation (energy, momentum, angular momentum conserving). In most common Galerkin methods for incompressible flow problems, such as mixed finite element methods, the divergence constraint is only enforced weakly [9], and in [4] we show how this is the main cause of conservation law violation if standard formulations of the nonlinearity are used. However, with the EMAC formulation, all of these conservation laws are obeyed in Galerkin discretizations.…”

Section: Introductionmentioning

confidence: 86%

“…In the recent work [4], the authors showed that if the NSE system was discretized by a Galerkin method using the reformulation u · ∇u + ∇p = 2D(u)u + (div u)u + ∇P, with P = p − 1 2 |u| 2 and D denoting the rate of deformation tensor, then each of energy, momentum, angular-momentum, 2D enstrophy, helicity, and total vorticity would all be correctly balanced, even if the method does not enforce the divergence constraint strongly. For this reason, we call the '2D(u)u + (div u)u' formulation of the nonlinearity the EMAC formulation (energy, momentum, angular momentum conserving).…”

Section: Introductionmentioning

confidence: 99%

“…However, with the EMAC formulation, all of these conservation laws are obeyed in Galerkin discretizations. The overall efficiency of the formulation and it clear superior performance over traditionally used finite element formulations for the NSE in velocity-pressure variables (for flow examples where conservation properties are of importance) have been recently reported in [4,14].…”

Section: Introductionmentioning

confidence: 99%

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Efficient discretizations for the EMAC formulation of the incompressible Navier–Stokes equations

Charnyi¹,

Heister²,

Olshanskii³

et al. 2019

Applied Numerical Mathematics

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We study discretizations of the incompressible Navier-Stokes equations, written in the newly developed energy-momentum-angular momentum conserving (EMAC) formulation. We consider linearizations of the problem, which at each time step will reduce the computational cost, but can alter the conservation properties. We show that a skew-symmetrized linearization delivers the correct balance of (only) energy and that the Newton linearization conserves momentum and angular momentum, but conserves energy only up to the nonlinear residual. Numerical tests show that linearizing with 2 Newton steps at each time step is very effective at preserving all conservation laws at once, and giving accurate answers on long time intervals. The tests also show that the skew-symmetrized linearization is significantly less accurate. The tests also show that the Newton linearization of EMAC finite element formulation compares favorably to other traditionally used finite element formulation of the incompressible Navier-Stokes equations in primitive variables.

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient discretizations for the EMAC formulation of the incompressible Navier–Stokes equations

Charnyi¹,

Heister²,

Olshanskii³

et al. 2019

Applied Numerical Mathematics

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“…The nonlinear term has been written in its convective form, which is most commonly encountered in computational practice. In the following, we consider the energy, momentum, and angular momentum conserving form for this term described in detail in

N L_{e m a c} false(boldu false) = 2 boldu \cdot bold-italicε false(boldu false) + ((), \nabla \cdot boldu) boldu - \frac{1}{2} \nabla false| boldu {false|}^{2} .

…”

Section: Numerical Treatmentmentioning

confidence: 99%

“…The last term of Equation was absorbed in the pressure in the work of Charnyi et al by redefining the pressure as

p^{*} = p - \frac{1}{2} false| boldu {false|}^{2}

, which has no physical meaning. Here, it is explicitly included in the formulation, avoiding the implementation of nonphysical Neumann conditions at outflow boundaries.…”

Section: Numerical Treatmentmentioning

confidence: 99%

Wall‐modeled large‐eddy simulation in a finite element framework

Owen

Chrysokentis

Ávila

et al. 2019

Numerical Methods in Fluids

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Summary This work studies the implementation of wall modeling for large‐eddy simulation in a finite element context. It provides a detailed description of how the approach used by the finite volume and finite differences communities is adapted to the finite element context. The new implementation is as simple and easy to implement as the classical finite element one, but it provides vastly superior results. In the typical approach used in finite elements, the mesh does not extend all the way to the wall, and the wall stress is evaluated at the first grid point, based on the velocity at the same point. Instead, we adopt the approach commonly used in finite differences, where the mesh covers the whole domain and the wall stress is obtained at the wall grid point, with the velocity evaluated at the first grid point off the wall. The method is tested in a turbulent channel flow at Reτ=2003, a neutral atmospheric boundary layer flow, and a flow over a wall‐mounted hump, with significant improvement in the results compared to the standard finite element approach. Additionally, we examine the effect of evaluating the input velocity further away from the wall, as well as applying temporal filtering on the wall‐model input.

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Evaluating the roles of detailed endocardial structures on right ventricular haemodynamics by means of CFD simulations

Sacco

Paun

Lehmkuhl

et al. 2018

Numer Methods Biomed Eng

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Computational modelling plays an important role in right ventricular (RV) haemodynamic analysis. However, current approaches use smoothed ventricular anatomies. The aim of this study is to characterise RV haemodynamics including detailed endocardial structures like trabeculae, moderator band, and papillary muscles. Four paired detailed and smoothed RV endocardium models (2 male and 2 female) were reconstructed from ex vivo human hearts high-resolution magnetic resonance images. Detailed models include structures with ≥1 mm cross-sectional area. Haemodynamic characterisation was done by computational fluid dynamics simulations with steady and transient inflows, using high-performance computing. The differences between the flows in smoothed and detailed models were assessed using Q-criterion for vorticity quantification, the pressure drop between inlet and outlet, and the wall shear stress. Results demonstrated that detailed endocardial structures increase the degree of intra-ventricular pressure drop, decrease the wall shear stress, and disrupt the dominant vortex creating secondary small vortices. Increasingly turbulent blood flow was observed in the detailed RVs. Female RVs were less trabeculated and presented lower pressure drops than the males. In conclusion, neglecting endocardial structures in RV haemodynamic models may lead to inaccurate conclusions about the pressures, stresses, and blood flow behaviour in the cavity.

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On conservation laws of Navier–Stokes Galerkin discretizations

Cited by 114 publications

References 35 publications

Efficient discretizations for the EMAC formulation of the incompressible Navier–Stokes equations

Efficient discretizations for the EMAC formulation of the incompressible Navier–Stokes equations

Wall‐modeled large‐eddy simulation in a finite element framework

Evaluating the roles of detailed endocardial structures on right ventricular haemodynamics by means of CFD simulations

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