Quite often we are required to ramp boundary condition values so as not to shock the system, most commonly in SimScale for compressible flows using a ramped velocity inlet. Here is a good resource for a suitable function:
the excel formula:
= < Final Value>(1/2-1/2COS(PI()*(< current time>/< end time>)))
This makes a smooth ramp from zero to the desired value, from zero time to the ramped end time. When ramping from a non zero value, simply add the desired initial value as a constant term and use the end value minus the initial value as the ‘final value’ in the above function.
Hope this can be of help to somebody someday.