A Buckingham-Pi dimensionless analysis for melt pool stability and defect prediction in additive manufacturing

Melt pool instabilities limit the reliability of additive manufacturing. Here, we demonstrate that a minimal Buckingham– π framework, supplemented by a normalized enthalpy (NE) metric, consolidates process outcomes across heat source settings (power, speed, spot) and material properties. IN738LC was processed on an EOS M290; single-track and bulk responses, melt pool geometric features, part relative density ( ρ ∗ ), and areal roughness parameters S a and S z , were quantified and subsequently mapped onto compact NE–dimensionless number spaces after the normalized-enthalpy metric had been calibrated using an effective absorptivity inferred from the measured melt pool depth. The recoil number cleanly delineates modes: Recoil ≲ 2 (conduction → stable keyhole) maintains ρ ∗ ≳ 99 % with low S a , whereas Recoil ≳ 4 –5 marks an unstable keyhole with spatter and porosity. Within this map, favorable transport balances are Re ≲ 100 , We < 1 , small Ca and Oh , and Fo > 0 . 1 ; external convection remains negligible ( Nu ≪ 1 ). Rather than VED, we advocate working directly in Π –space ( NE , Recoil , Re , We , Ca , Oh , Fo , Nu ) —to define, compare, and transfer qualifiable process windows across machines and alloys. • Depth–calibrated absorptivity ( η ) closes the energy balance—no sensors. • A compact NE+ Π recoil map collapses depth, density, and Sa. • Portable thresholds: Π recoil ≈ 2 (conduction → stable keyhole), 4–5 (unstable). • NE embeds spot size, speed, and thermophysics; replaces ad hoc energy–density heuristics. • Actionable recipe: set NE ≈ Φ ; keep Π recoil ∼ 1 – 2 ; aim R e ≲ 100 , We < 1, Fo > 0.1.