Free Surface and Moving Boundary Problems

Descriptions given in Section 4 of this chapter about the imposition of boundary conditions are mainly in the context of finite element models that use elements. In models that use Hermite elements derivatives of field variable should also be included in the set of required boundai conditions. In these problems it is necessary to ensure tluit appropriate normality and tangen-tiality conditions along the boundaries of the domain are satisfied (Petera and Pittman, 1994). 

Modelling of steady-state free surface flow corresponds to the solution of a boundary value problem while moving boundary tracking is, in general, viewed as an initial value problem. Therefore, classification of existing methods on the basis of their suitability for boundary value or initial value problems has also been advocated. 

The general class of free boundary flow problems can, however, be modelled using the volume of fluid (VOF) approach (Nichols et ai, 1980). The main concept in this technique is to solve, simultaneously with the governing flow equations, an additional equation that represents the unknown boundary. Three different versions of this method are described in the following sections. 

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The meshless methods have been successfully used in free-surface flow and moving boundaries problems. The methods have also proven to be very powerful in dealing with interfacial flow problems enabling simulation of multiphase and multi-fluid flows. The effect of surface tension has also been investigated. 

In our pursuit of a theoretical description of such problems, we are led to the following ubiquitous situation In order to solve for the bulk flow field (governed by the Navier-Stokes equations, for instance), the position of the free surface and boundary conditions there, need to be known the flow field, and other interfacial forces, act to move the interface to a new position which in turn affects the flow field. In general, this coupling is nonlinear and in many problems of practical interest no steady state exists. In fact, severe flow regimes can arise as in the breakup of Newtonian liquid jets where a topological singularity is encountered in finite time. 

K.N. Christodoulou and L.E. Scriven, Discretization of free surface flows and other moving boundary problems, J. Comp. Phys. 99,39-55, (1992). 

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