A new numerical method is developed to solve the unsteady incompressible Navier-Stokes Equations with the second-order accuracy in time and space. In contrast to the SIMPLE algorithms, the present formulation directly solves the discrete x-and y-momentum equations simultaneously in a coupled form. It is found that the present formulation retrieves some cross convection terms overlooked by the conventional iterative methods, which contributes to accuracy and fast convergence. The finite volume method(FVM) is applied to solve on the fully staggered grid the vector-form momentum equations. The preconditioned conjugate gradient squared method(PCGS) has proved very efficient in solving the associate linearized large, sparse block matrix system.
Excellency of the present algorithm has been demonstrated in terms of accuracy and convergence rate by a few bench mark problems such as the Taylor problem, the driven square cavity flow, and the cavity flow with an oscillating wall. Comparison with the SIMPLE algorithm has also indicated that the present momentum coupling method is not only accurate but also robust in solving unsteady as well as steady flow problems. As a further check, we have solved successfully flow past a circular cylinder at Reynolds number 1000 and flow by an impulsely starting cylinder flow at Re=9500.
To study flow problem by this algorithm, we first solved the square cavity flow with an oscillating top wall, in the range of parameter that makes the time and the convection derivative terms about equal order of magnitute. In addition, the vortex shedding by a circular cylinder near the critical Reynolds number and the flow around rotationally oscillating circular cylinder is also investigated.
