2011
DOI: 10.1002/fld.2270
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An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

Abstract: SUMMARYWe develop in this paper a discretization for the convection term in variable density unstationary Navier-Stokes equations, which applies to low-order non-conforming finite element approximations (the so-called Crouzeix-Raviart or Rannacher-Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L 2 stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiment… Show more

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Cited by 40 publications
(79 citation statements)
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“…The first one is the time-shift of the densities in the prediction step: thanks to this time-shift, the densities satisfy (12b) of the preceding correction step and therefore the convection operator vanishes for constant velocities (i.e.ũ n+1 =cste), which is shown to be a necessary condition to ensure the conservation of the kinetic energy [8,1]. Second, the pressure correction step couples the mixture and dispersed phase mass balance equations (12b) and (12c); this coupling preserves the affine relation between ρ n+1 and z n+1 through the equation of state (12d), with coefficients only depending on the pressure (taken at the same time level).…”
Section: Time Semi-discretizationmentioning
confidence: 99%
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“…The first one is the time-shift of the densities in the prediction step: thanks to this time-shift, the densities satisfy (12b) of the preceding correction step and therefore the convection operator vanishes for constant velocities (i.e.ũ n+1 =cste), which is shown to be a necessary condition to ensure the conservation of the kinetic energy [8,1]. Second, the pressure correction step couples the mixture and dispersed phase mass balance equations (12b) and (12c); this coupling preserves the affine relation between ρ n+1 and z n+1 through the equation of state (12d), with coefficients only depending on the pressure (taken at the same time level).…”
Section: Time Semi-discretizationmentioning
confidence: 99%
“…the quantities (|σ| u n+1 K,σ ρ n+1 σ ) σ∈E(K) appearing in the discrete mass balance (first relation of (15)). We refer to [1,9] for a detailed construction of this approximation. Equation (12a) is discretized similarly to the momentum balance (17), i.e.…”
Section: Discrete Equationsmentioning
confidence: 99%
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“…For this result to be valid, the necessary condition is that the convection operator vanishes for a constant velocity, i.e. that the following discrete mass balance over the diamond cells is satisfied [1,16]:…”
Section: Compressible Barotropic Navier-stokes Equationsmentioning
confidence: 99%
“…the quantities (F K,σ ) σ∈E(K) appearing in the discrete mass balance (7). We do not give here this set of coefficients, and refer to [1,18,31] for a detailed construction of this approximation.…”
Section: Compressible Barotropic Navier-stokes Equationsmentioning
confidence: 99%