SUMMARYNew implicit finite difference schemes for solving the time-dependent incompressible Navier-Stokes equations using primitive variables and non-staggered grids are presented in this paper. A priori estimates for the discrete solution of the methods are obtained. Employing the operator approach, some requirements on the difference operators of the scheme are formulated in order to derive a scheme which is essentially consistent with the initial differential equations. The operators of the scheme inherit the fkdamental properties of the corresponding differential operators and this allows a priori estimates for the discrete solution to be obtained. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees boundedness of the solution. To derive the consistent scheme, special approximations for convective terms and div and grad operators are employed. Two variants of time discretization by the operator-splitting technique are considered and compared. It is shown that the derived scheme has a very weak restriction on the time step size. A lid-driven cavity flow has been predicted to examine the stability and accuracy of the schemes for Reynolds number up to 3200 on the sequence of grids with 21 x 21,41 x 41, 81 x 81 and 161 x 161 grid points.KEY WORDS: incompressible viscous flow; numerical methods; non-staggered grids; consistent approximations of operators; operator-splitting technique

Detailed results of numerical calculations of transient, 2D incompressible flow around and in the wake of a square prism at Re = 100, 200 and 500 are presented. An implicit finitedifference operator-splitting method, a version of the known SIMPLEC-like method on a staggered grid, is described. Appropriate theoretical results are presented. The method has second-order accuracy in space, conserving mass, momentum and kinetic energy. A new modification of the multigrid method is employed to solve the elliptic pressure problem. Calculations are performed on a sequence of spatial grids with up to 401 Â 321 grid points, at sequentially halved time steps to ensure grid-independent results. Three types of flow are shown to exist at Re = 500: a steady-state unstable flow and two which are transient, fully periodic and asymmetric about the centre line but mirror symmetric to each other. Discrete frequency spectra of drag and lift coefficients are presented.

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A new numerical method to simulate steady-state isothermal liquid flows with hydraulic cavitation is proposed. The 3D averaged N-S equations with LamBremhorst κ -ε turbulence model are used. The barotropic state equation is developed basing on thermodynamic equilibrium relations. Simulation of such flows faces a lot of numerical difficulties concerned with variations of density, speed of sound and time scale. The method is a hybrid splitting scheme that is a mixture of "density-based" and "pressure-based" approaches. The splitting scheme is the "pressure-based" SIMPLE-type algorithm in the region of incompressible liquid flow without cavitation. The scheme degenerates to the "density-based" algorithm in a compressible region (2-phase state or pure gas). The proposed method differs from both "density-based" preconditioned algorithms and SIMPLE-type methods adapted to the case of cavitating flows. The method has been tested on numerous typical 3D problems with hydraulic cavitation. Results of numerical simulation are in good agreement with experimental data. The algorithm shows high efficiency for the considered problems. The method has been implemented in FloEFD ТМ .

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