A Diamond DA40 Instructor Pilot reported experiencing a rough running engine; accompanied by vibrations; resulting in an immediate return to the departure airport.

Narrative

Separate preflight inspections performed by both Student Pilot (SP) and Flight Instructor (IP). No abnormal conditions noted; nor water/debris observed in fuel samples. Oil levels checked; IP noticed engine oil near max line but within acceptable limit. Engine cowling and exhaust openings looked clean; no leaks or abnormal residue observed by either crew member.During climb out; SP and IP both verbalized the recognition of an abnormal vibration and roughness from the engine; which could also be felt strongly in both sets of rudder pedals. IP took controls and notified ATC of possible engine roughness.ATC instructed to enter right downwind for runway XXR(shortened); cleared to land. Vibration continued during flight in the pattern; IP landed aircraft safely on XXR and taxied back to the ramp; no additional abnormal conditions or annunciations observed by crew while in flight or during taxi. No further assistance provided or necessary from ATC.After engine shutdown; IP performed an additional inspection of the engine compartment; and observed visible wet oil in the exhaust opening on lower surface of the engine cowling and on the nose wheel strut.