# Category 12: Fill Equation#

The so-called Fill equation is used by the volume-of-fluid and level-set Eulerian interface tracking in Goma. Basically it is a statement of Lagrangian invariance and is hence a hyperbolic statement of the so-called kinematic equation. Given a velocity field, this equation advances the fill function as a set of material points; hence material surfaces remain ostensibly intact. The boundary conditions in this section are used to specify the level-of-fill at a boundary at which a fluid of a specific phase is flowing into the problem domain.

## F#

```BC = F NS <bc_id> <float1> [float2]
```

### Description / Usage#

(DC/FILL)

This Dirichlet boundary condition specifies a value of the fill or level set unknown field on a node set.

A description of the input parameters is as follows:

 F Name of the boundary condition (). NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value at which the fill or level set unknown will be fixed on this node set. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

### Examples#

An example:

```BC = F NS 100 1.0
```

### Technical Discussion#

This boundary condition finds most of its use in the VOF/FILL interface tracking algorithm where it is used to fix the value of the color function at an inlet or outlet boundary. In the level set formulation, it is used less but is still useful in defining the absolute fixed location of an interface by setting the value assigned to 0 on a node set.

## FILL_INLET#

```BC = FILL_INLET SS <bc_id> <float1>
```

### Description / Usage#

(SPECIAL/FILL)

This boundary condition allows the user to specify a value on a inlet boundary from VOF problems employing discontinuous interpolation of the color function, F.

Description of the input parameters is as follows:

 FILL_INLET Name of the boundary condition. SS Type of boundary condition (), where SS denotes side set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (side set in EXODUS II) in the problem domain. The value of the fill function, F, as it flows across into the domain.

### Examples#

An example:

```BC = FILL_INLET SS 10 1.0
```

### Technical Discussion#

• This boundary condition is useful only in problems involving VOF interface tracking in which the fill function is interpolated discontinuously. In this formulation, communication of the fill function value can only be made by finding the value of the fill function in the element upstream of the current position. While this is a stable formulation for the advective VOF method, it does introduce the complexity of determining which element is actually upstream.

• When there is no element upstream, as in the case of an inlet boundary, this boundary condition must be present to establish the value of the fill function that is flowing across the inlet boundary into the domain. Consequently, this boundary condition should be present on all inlet boundaries of the problem. It sometimes is also useful to have it on outflow boundaries as well, just in case a backflow situation arises.