Light Transport; Low Wing; 2 Turbojet Eng
Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Human-Machine Interface; Situational Awareness
Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT
Flight Crew Became Reoriented; Flight Crew FLC complied w / Automation / Advisory
Human Factors; Environment - Non Weather Related
Descending into LUK on the LNAV 3R we first received a report from tower of a low altitude alert in between the FAF and the intermediate step down fix (ZUGMI at or above 1480). I had the aircraft in LNAV/VS with a commanded vertical speed mode of -700FPM. I glanced at the altimeter and saw 1550. This is above the step down altitude and above the MDA. The copilot and I both quickly agreed we were above MDA. At this point also I looked out and could see the runway. We briefly leveled off and planned to continue visually. At this point we received a too low PULL UP. Aural warning. Immediately we went around. After getting back level and configuring we talked about the approach and decided to do it again with less vertical speed and just stay a bit higher on the glide path. We got down to out MDA without issue and landed.I never saw any altitude any lower than the step down altitude on the approach. LUK airport is sunk in the valleys between some hills and I think I should've maybe thought to have a gentler vertical speed.Not accept a fairly close to minimums LNAV only approach with a lot of turbulence into Lunken when ZZZ is right around the corner. Fly the instrument profile at lowest possible vertical speed and visual only after VDP. Although we never busted any altitudes I could've made it a much gentler descent to MDA at the VDP since the VDP is 2.4 from the runway and there are lots of hills on the approach prior to the runway.IMC until ragged bases around 800 AGL to 1;000 AGL. Some Visual conditions at time of the event. Runway/PAPI were in sight.
Light Transport aircraft flight crew reported getting a terrain warning while descending to LUK airport through uneven terrain.