# 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)
${}^{trm}F{}_{i}^{I}=\left(1+{\eta }_{I}\right)\rho {\Phi }_{I}\left({w}_{j}-{v}_{j}\right)\text{\hspace{0.17em}}\frac{\partial {v}_{i}}{\partial {X}_{j}}\text{\hspace{0.17em}}V$
Where,
$w$
Grid velocity
$\nu$
Material velocity
$V$
Element volume
$\eta$
Upwind coefficient (user-defined, default = 1 for full upwind)

When a Lagrangian formulation is used, the values of ${w}_{j}$ and ${\nu }_{j}$ are equal. Thus, Equation 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 Equation 1 is calculated by:(2)
${\eta }_{I}=\eta sign\left(\frac{\partial {\Phi }_{I}}{\partial {X}_{j}}\left({v}_{j}-{w}_{j}\right)\right)$
Development of a less diffusive flux calculation is currently under investigation.(3)
${F}_{i}^{I}={\sigma }_{ij}\underset{V}{\int }\frac{\partial {\Phi }_{I}}{\partial {X}_{j}}dV$

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