Momentum Transport Force

This scheme is only used with the ALE formulation (Arbitrary Lagrangian Eulerian) and in the CFD version of Radioss. The force is calculated using the relation:(1)

{}^{t r m}F_{i}^{I} = (1 + η_{I}) ρ Φ_{I} (w_{j} - v_{j}) \frac{\partial v_{i}}{\partial X_{j}} V

Where,

$w$: Grid velocity
$ν$: Material velocity
$V$: Element volume
$η$: Upwind coefficient (user-defined, default = 1 for full upwind)

When a Lagrangian formulation is used, the values of $w_{j}$ and $ν_{j}$ are equal. Thus, 式 1 is equal to zero.

Upwinding Technique

An upwinding technique is introduced to add numerical diffusion to the scheme; otherwise it is generally under diffusive and thus unstable. The upwind coefficient used in 式 1 is calculated by:(2)

η_{I} = η s i g n (\frac{\partial Φ_{I}}{\partial X_{j}} (v_{j} - w_{j}))

Development of a less diffusive flux calculation is currently under investigation.(3)

F_{i}^{I} = σ_{i j} \int_{V} \frac{\partial Φ_{I}}{\partial X_{j}} d V

This option is activated with the flag INTEG (only in the CFD version).