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A highly efficient numerical approach based on multigrid and preconditioning methods is developed for modeling 3-D 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 the Navier-Stokes equations in a strongly coupled way, and acceleration techniques like multigrid methods 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 time-accurate solutions. To improve the performance of this approach for a small physical time step, a modification to residual smoothing parameters is proposed. After the validation of the numerical algorithm and the turbulence model 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, a three-dimensional time-dependent turbulent flow over a spheroid when it is undergoing a pitch-up maneuver is calculated and compared with experimental data. AbstractA highly efficient numerical approach based on multigrid and preconditioning methods is developed for modeling 3-D steady and time-dependent incompressible flows. The k-u turbulence model is used to estimate the effects of turbulence. The model equations are solved together with the N-S equations in a strongly-coupled way, and acceleration techniques like 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 timeaccurate solution. To improve the performance of this approach for small physical time step, a modification to residual smoothing parameters is proposed.After the validation of the numerical algorithm and the turbulence model 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, a three dimensional timedependent turbulent flow over a spheroid when it is undergoing a pitch-up maneuver is calculated and compared with experimental data.

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In this paper, computational results are presented for three-dimensional high-Reynolds number turbulent flows over a simplified submarine model. The simulation is based on the solution of Reynolds-Averaged Navier-Stokes equations and two-equation turbulence models by using a preconditioned time-stepping approach. A multiblock method, in which the block loop is placed in the inner cycle of a multi-grid algorithm, is used to obtain versatility and efficiency. It was found that the calculated body drag, lift, side force coefficients and moments at various angles of attack or angles of drift are in excellent agreement with experimental data. Fast convergence has been achieved for all the cases with large angles of attack and with modest drift angles.

A highly efficient numerical approach based on multigrid and preconditioning methods is developed for modeling 3-D incompressible turbulent flows. The incompressible Reynolds-averaged Navier-Stokes equations are written in pseudo-compressibility from, then a preconditioning method is used to reduce the wave speed disparity. The k-ω and k-ε turbulence models are used to estimate the effects of turbulence. The model equations are solved together with the N-S equations in a strongly-coupled way, and all the acceleration techniques originally developed for N-S equations are also used for the turbulence model equations. A point-implicit technique is developed to improve the efficiency of the solution of the turbulence model equations.