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Review Article

Discrete Energy-Conservation Properties in the Numerical Simulation of the Navier–Stokes Equations

[+] Author and Article Information
Gennaro Coppola

Dipartimento di Ingegneria Industriale,
Università di Napoli “Federico II,”
Napoli 80125, Italy
e-mail: gcoppola@unina.it

Francesco Capuano

Dipartimento di Ingegneria Industriale,
Università di Napoli “Federico II,”
Napoli 80125, Italy
e-mail: francesco.capuano@unina.it

Luigi de Luca

Dipartimento di Ingegneria Industriale,
Università di Napoli “Federico II,”
Napoli 80125, Italy
e-mail: deluca@unina.it

Manuscript received July 2, 2018; final manuscript received February 8, 2019; published online March 6, 2019. Assoc. Editor: Sergei I. Chernyshenko.

Appl. Mech. Rev 71(1), 010803 (Mar 06, 2019) (19 pages) Paper No: AMR-18-1073; doi: 10.1115/1.4042820 History: Received July 02, 2018; Revised February 08, 2019

Nonlinear convective terms pose the most critical issues when a numerical discretization of the Navier–Stokes equations is performed, especially at high Reynolds numbers. They are indeed responsible for a nonlinear instability arising from the amplification of aliasing errors that come from the evaluation of the products of two or more variables on a finite grid. The classical remedy to this difficulty has been the construction of difference schemes able to reproduce at a discrete level some of the fundamental symmetry properties of the Navier–Stokes equations. The invariant character of quadratic quantities such as global kinetic energy in inviscid incompressible flows is a particular symmetry, whose enforcement typically guarantees a sufficient control of aliasing errors that allows the fulfillment of long-time integration. In this paper, a survey of the most successful approaches developed in this field is presented. The incompressible and compressible cases are both covered, and treated separately, and the topics of spatial and temporal energy conservation are discussed. The theory and the ideas are exposed with full details in classical simplified numerical settings, and the extensions to more complex situations are also reviewed. The effectiveness of the illustrated approaches is documented by numerical simulations of canonical flows and by industrial flow computations taken from the literature.

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Figures

Grahic Jump Location
Fig. 1

Energy decay (left) and dissipation rate (right) for the TGV at Re = 1600 for second-order (dotted) and spectral (solid) accuracy on a 1283 grid. The reference solution is a fully de-aliased computation with the same number of effective modes. The reference time is tr=L/U0.

Grahic Jump Location
Fig. 2

Three-dimensional energy spectra for spectral large-eddy simulations of forced homogeneous isotropic turbulence at Reλ = 100 (left) and Reλ = 170 (right), using different formulations for the convective term. A reference direct numerical simulations (DNS) computation is also shown for comparison.

Grahic Jump Location
Fig. 3

Variables layout on a staggered mesh

Grahic Jump Location
Fig. 4

Comparison of energy-conservation properties of different splittings of the convective terms for the compressible inviscid TGV flow. Left: M =0.1. Right: M =0.5. The splittings refer to the Feiereisen form (F), Eq. (60); the Blaisdell form (B), Eq. (61); the KGP form, obtained by combining Eqs. (51)(54) with a weight equal to 1/4; the divergence form (DIV), Eq. (51). The reference time is tr=L/(Mc), where c is the speed of sound.

Grahic Jump Location
Fig. 5

Comparison of energy-conservation properties of different temporal schemes. Left: energy-conservation error for the 2D inviscid double mixing layer. Right: ratio of temporal to physical dissipation rates for a 3D TGV at Re = 1600 (solid lines) and Re = 3000 (dashed lines) for various standard, symplectic and pseudo-symplectic RK methods.

Grahic Jump Location
Fig. 6

Contour plots of the instantaneous streamwise velocity component u/U∞ predicted by LES based on a nondissipative scheme without discrete kinetic energy conservation (left) and with a solver that preserves kinetic energy (right). Shown are 30 levels in the ranges of −0.43 to 2.58). Reproduced from Ref. [104] with permission from AIP Publishing.

Grahic Jump Location
Fig. 7

Results for the flow around ONERA M6 wing. Top: pressure contours on the wing surface for rhoEnergyFoam (left) and rhoCentralFoam (right); shown are 32 levels in the range 0.4≤p/p∞≤1.2. Bottom: spatial distribution of the pressure coefficient along the chord at sections z/b=0.2 (left) and z/b=0.65 (right) for rhoEnergyFoam (solid lines) and rhoCentralFoam (dot-dashed lines), compared to experimental data (circles). Figure courtesy of D. Modesti and S. Pirozzoli.

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