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...) of (8), the discretization of the Helmholtz-projected Oseen equations (4) would require divergence-free finite elements. As our approach should work with a standard Navier-Stokes solver like NAVIER =-=[6]-=-, where Taylor-Hood elements are available, we have to deal in some way with the Helmholtz projector P . 2. Each step of the Newton-ADI iteration with A0 := A + ωI requires the solution of A T j Zj+1Z...

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...r–Stokes case as well. We will upgrade some ideas and show more efficient realizations in the numerical treatment. Furthermore, we will describe the extension of the finite element flow solver NAVIER =-=[6]-=- regarding a closed-loop simulation. In what follows, the incompressible Navier–Stokes equations are considered, reading in dimensionless form ∂ ∂t v(t, x)− 1 Re Δv(t, x) + (v(t, x) · ∇)v(t, x...

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...) + b(u, u, v) + c(p, v) − 1 ∫ We Γf Hn · v = − Bo We s(ez, v) (3.5) c(q, u) = 0 (3.6) 3.2 Time discretization NAVIER uses the so-called fractional step θ scheme in an operator splitting variant, see =-=[8, 2]-=-. The benefit of using this method is that it decouples two major numerical difficulties of the Navier-Stokes equations, the solenoidal condition and the non-linearity. Therefore in each time step one...

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...a simple velocity update based on the solution of a Poisson equation for the pressure. The splitting approach may also be motivated by decoupling the incompressibility constraint and the nonlinearity =-=[3]-=-. In our computations, we always use inf–sup stable pairs of finite element spaces and hence no stabilization of the incompressibility constraint is considered. If the viscosity ν is small, a stabiliz...

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...erStokes case as well. We will upgrade some ideas and show more efficient realizations in the numerical treatment. Furthermore, we will describe the extension of the finite element flow solver NAVIER =-=[5]-=- regarding a closed-loop simulation. In what follows, we consider the incompressible Navier-Stokes equations (NSE), reading in dimensionless form: ∂ ∂t ~v(t, ~x)− 1 Re ∆~v(t, ~x) + (~v(t, ~x) · ∇)~v(t...

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...for u. In order to determine the pressure uniquely, as usual we impose the condition∫ Ω p = 0. (2) The structure of the solver used in this paper for the computational solution of (1) is described in =-=[4]-=-. It uses algorithms proposed in [3]. The following simplified “QuasiStokes” problem appears as a core subproblem after the time discretization: find (u, p) such that µu− ν∆u +∇p = f in Ω, ∇·u = 0 in ...