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Nondimensionalization of Venkatakrishnan

Hello

Transforming my code from non-dimensional to dimensional, I need to make Venkatakrishnan limiter use non-dimensional quantities.

For a component of limiter equation, $\Delta_{2} = \nabla U_{I}\cdot\vec{r_{L}}$ , I wonder if this component is non-dimensionalized based on the $\Delta_{2} = U_{J} - U_{I}$ .

Above equations are referred from

Is my approach correct?

Or, is it better to let the solver as it is and apply some dimensional operations in pre/post processing?