Conservation of Mass

The finite element formulation of the Lagrangian form of the mass conservation equation is given by:(1)

\frac{d ρ}{d t} | x = - (ρ / V) \frac{d V}{d t} | x

When transformed into the ALE formulation it gives:(2)

Applying a Galerkin variation form for the solution of 式 2:(3)

Where, $ψ$ is the Weighting function.

Using a finite volume formulation

Where, $ψ$ =1

$ρ$ = constant density over control volume $V$ .

Therefore:(4)

\int_{V} \frac{\partial ρ}{\partial t} d V + \int_{V} ρ \frac{\partial ν_{K}}{\partial x_{K}} d V = 0

Using the divergence theorem leads to:(5)

\int_{V} \frac{\partial ρ}{\partial t} d V + \int_{S} ρ (ν {}_{j}\cdot n_{j}) d S = 0

Further expansion gives:(6)

\frac{d}{d t} \int_{v} ρ d V = \int_{s} \underset{m a s s f l u x}{\underset{︸}{ρ (w_{j} - v_{j}) n_{j} d S}}

This formula is still valid if density

ρ

is not assumed uniform over volume

V

The density,

ρ_{i}

, is given computed:(7)

ρ_{i} = \frac{1}{2} ρ_{I} {1 + η s i g n (ϕ_{i})} + \frac{1}{2} ρ_{J} {1 - η s i g n (ϕ_{i})}

Where, $0 \leq η \leq 1$ is the upwind coefficient given on the input card.

If $η$ =0, there is no upwind.

Therefore, $ρ_{i} = \frac{ρ_{I} + ρ_{J}}{2}$ .

If $η$ =1, there is full upwind.

The smaller the upwind factor, the faster the solution; however, the solution is more stable with a large upwind factor. This upwind coefficient can be tuned with parameter from /UPWIND keyword (not recommended, this keyword has been obsolete as of version 2018).

For a free surface: $ρ_{J}$ = $ρ_{I}$