Preconditioned multigrid methods for unsteady incompressible flows.

*(English)*Zbl 0908.76064Summary: A numerical approach based on multigrid and preconditioning methods is developed for modeling three-dimensional steady and time-dependent incompressible flows. The \(k-\omega\) turbulence model is used to estimate the effects of turbulence. The model equations are solved together with Navier-Stokes equations in a strongly coupled way, and acceleration techniques like the multigrid method are also used for the turbulence model equations. For unsteady problems, a dual-time stepping procedure is adopted to satisfy the divergence-free constraint and to obtain a time-accurate solution. To improve the performance of this approach for small physical time steps, a modification to residual smoothing parameters is proposed. The numerical algorithm and the turbulence model are validated first by calculating unsteady inviscid flow around an oscillating cylinder, unsteady laminar flow past a circular cylinder, and steady high-Reynolds number turbulent flow over a \(6:1\) prolate spheroid. Then the three-dimensional time-dependent turbulent flow over a spheroid when it is undergoing a pitch-up maneuver is calculated and compared with experimental data.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76F10 | Shear flows and turbulence |

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

\(k\)-omega turbulence model; Navier-Stokes equations; \(6:1\) prolate spheroid; acceleration technique; dual-time stepping procedure; divergence-free constraint; residual smoothing parameters; oscillating cylinder; circular cylinder; pitch-up maneuver
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\textit{C. Liu} et al., J. Comput. Phys. 139, No. 1, 35--57 (1998; Zbl 0908.76064)

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