SUMMARYEver since the expansion of the finite element method (FEM) into unsteady fluid mechanics, the 'consistent mass matrix' has been a relevant issue. Applied to the time-dependent incompressible Navier-Stokes equations, it virtually demands the use of implicit time integration methods in which full 'velocity-pressure coupling' is also inherent. The high cost of such (high-quality) FEM calculations led to the development of simpler but ad hoc methods in which the 'lumped' mass matrix is employed and the velocity and pressure are uncoupled to the maximum extent possible. Resulting computer codes were less expensive to use but suffered a significant loss of accuracy, caused by lumping the mass when the flow was advection-dominated and accurate transport of 'information' was important. In the second part of this paper we re-introduce the consistent mass matrix into some semi-implicit projection methods in such a way that the cost advantage of lumped mass and the accuracy advantage of consistent mass are simultaneously realized. KEY WORDS Incompressible flows Navier-Stokes equations Projection methods Consistent mass DISCRETE PROJECTION METHODS*In this second part of the paper we go from semi-discrete to fully discrete in the form of several finite element methods that we have designed and tested. In addition to implementing some of the projection methods discussed in Part 1, we also introduce a variation on the GFEM that utilizes the consistent mass matrix in a cost-effective way. That is, the object of the second part of this paper, besides implementing some of the techniques derived above, is to put the 'mass' back where it belongsdistributed according to the Galerkin principle-not concentrated at node points as it is in the lumped mass approximation used in our (and many others') most recent FEM schemes for solving the time-dependent incompressible Navier-Stokes equations-see Gresho et al. ' for an explicit time integration method and Gresho and Chan2 for a semi-implicit one.

SUMMARYBeginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier-Stokes equations, the spatial approximation is modified in two ways in the interest of cost-effectiveness: the mass matrix is 'lumped' and all coefficient matrices are generated via l-point quadrature. After appending an hour-glass correction term to the diffusion matrices, the modified semi-discretized equations are integrated in time using the forward (explicit) Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection-dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy. These techniques are described and analysed in some detail, and in Part 2 (Applications), the resulting code is demonstrated on three sample problems: steady flow in a lid-driven cavity at Res10,000, flow past a circular cylinder at Re 5400, and the simulation of a heavy gas release over complex topography.

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SUMMARYThree examples will be presented to demonstrate the performance of the scheme described in Part 1 of this paper.' Two are isothermal (T = 0) and two-dimensional, and one of these is steady and the other time-dependent. The third example involves buoyancy effects, is time-dependent and threedimensional, and is presented in less detail. The paper concludes with a short discussion and some conclusions from both Parts 1 and 2. CONTENTS