An implicit method using contravariant velocity components and its application to calculations in a harbour-channel area

The key problem in the computation of fluid dynamics using fine boundary-fitted grids is how to improve the numerical stability and decrease the calculating quantity. To solve this problem, implicit schemes should be adopted since explicit schemes may bring about a great increase in computation quantity according to the Courant-FrledrichsLewy condition. Whereas the adoption of implicit schemes is difficult to be realized because of the existence of two partial derivatives of surface elevations with respect to variables of alternative direction coordinates in each momentum equation in non-rectangular coordinates. With an aim to design an implicit scheme in non-reetangular ccordinates in the present paper, new momentum equations with the contravariant components of velocity vector are derived based on the shallow water dynamic equations in generalized curvilinear coordinates. In each equation, the coefficients before the two detivatives of surface elevations have different orders of magnitude, i. e., the derivative with the larger ceefficient rnay play a more important role than that with the smaller one. With this advantage, the ADI scheme can then be easily employed to improve the numerical stability and decrease the calculating quantity. The calculation in a harbour and a channel in Macau nearshore area shows that the implicit model is effective in calculating current fields in small size areas.

Implicit numerical scheme based on SMAC method for unsteady incompressible Navier-Stokes equations

An implicit numerical scheme is developed based on the simplified marker and cell （SMAC） method to solve Reynolds-averaged equations in general curvilinear coordinates for three-dimensional （3-D） unsteady incompressible turbulent flow. The governing equations include the Reynolds-averaged momentum equations, in which contravariant velocities are unknown variables, pressure-correction Poisson equation and k- s turbulent equations. The governing equations are discretized in a 3-D MAC staggered grid system. To improve the numerical stability of the implicit SMAC scheme, the higherorder high-resolution Chakravarthy-Osher total variation diminishing （TVD） scheme is used to discretize the convective terms in momentum equations and k- e equations. The discretized algebraic momentum equations and k- s equations are solved by the time-diversion multiple access （CTDMA） method. The algebraic Poisson equations are solved by the Tschebyscheff SLOR （successive linear over relaxation） method with alternating computational directions. At the end of the paper, the unsteady flow at high Reynolds numbers through a simplified cascade made up of NACA65-410 blade are simulated with the program written according to the implicit numerical scheme. The reliability and accuracy of the implicit numerical scheme are verified through the satisfactory agreement between the numerical results of the surface pressure coefficient and experimental data. The numerical results indicate that Reynolds number and angle of attack are two primary factors affecting the characteristics of unsteady flow.