The second off-center orifice plate type is called the segmental orifice plate, where the hole is not circular but rather just a segment of a concentric circle:

As with the eccentric orifice plate design, the segmental hole should be offset downward in gas flow applications and either upward or downward in liquid flow applications depending on the type of undesired material(s) in the flow stream.
An alternative to offsetting or re-shaping the borehole of an orifice plate is to simply drill a small hole near the edge of the plate, flush with the inside diameter of the pipe, allowing undesired substances to pass through the plate rather than collect on the upstream side. If such a hole is oriented upward to pass vapor bubbles, it is called a vent hole. If the hole is oriented downward to pass liquid droplets or solids, it is called a drain hole. Vent and drain holes are useful when the concentration of these undesirable substances is not significant enough to warrant either an eccentric or segmental orifice:

The addition of a vent or drain hole should have a negligible impact on the performance of an orifice plate due to its small size relative to the main bore. If the quantity of undesirable material in the flow stream (bubbles, droplets, or solids) is excessive, an eccentric or segmental orifice plate might be a better choice.