SUMMARYThe pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no 'equation of state' for an incompressible fluid. It is in one sense a mathematical artefact-a Lagrange multiplier that constrains the velocity field to remain divergence-free; i.e., incompressible-yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time-varying divergencefree velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly welldefined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.

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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.

The stability of a horizontal layer of fluid heated from above or below is examined for the case of a time-dependent buoyancy force which is generated by shaking the fluid layer, thus causing a sinusoidal modulation of the gravitational field. A linearized stability analysis is performed to show that gravity modulation can significantly affect the stability limits of the system. In this analysis, much emphasis is placed on qualitative results obtained by an approximate solution, which permits a rather complete stability analysis. A useful mechanical analogy is developed by considering the effects of gravity modulation on a simple pendulum. Finally, some effects of finite amplitude flows are considered and discussed.

SUMMARYEver since the time of Chorin's classic 1968 paper on projection methods, there have been lingering and poorly understood issues related to the best-r even proper or appropriate-boundary conditions (BCs) that should be (or could be) applied to the 'intermediate' velocity when the viscous terms in the incompressible Navier-Stokes equations are treated with an implicit time integration method and a Poisson equation is solved as part of a 'time step'. These issues also pervade all related methods that uncouple the equations by 'splitting' the pressure computation from that of the velocity-at least in the presence of solid boundaries and (again) when implicit treatment of the viscous terms is employed. This paper is intended to clarify these issues by showing which intermediate BCs are 'best' and why some that are not work well anyway. In particular we show that all intermediate BCs must cause problems related to the regularity of the solution near boundaries, but that a near-miraculous recovery occurs such that accurate results are nevertheless achieved beyond the spurious boundary layer introduced by such methods. The mechanism for this 'miracle' is related to the existence of a higher-order equation that is actually satisfied by the pressure. All that is required then for projection (splitting, fractional step, etc.) methods to work well is that the spurious boundary layer be thin-as has been largely observed in practice.