Boundary layer approximation for melt film dynamics in laser fusion cutting

During laser fusion cutting of sheet metal parts, a focused laser beam is traversed relatively to the processed part. The material is molten due to the heat input of the laser beam and driven out of the cutting kerf by a gas jet aligned coaxially with the laser. The edges of the remaining cutting kerf show a striation pattern which affects the quality of the cut. The amplitude, wavelength and shape of these striations depend on the dynamics of the motion of the thin melt film inside the interaction zone. Instabilities in the thickness of the melt film that emerge at the cutting front propagate to the sides and solidify to an irregular surface. Understanding effects that lead to the emergence and stirring up of instabilities is crucial to derive measures for high quality cuts with a drastically reduced striation depth. To simulate the behavior of the thin melt film, the underlying incompressible Navier Stokes equations have to be solved with high accuracy using a well chosen set of boundary conditions. The surface of the melt film moves with a velocity of up to several meters per second from top to bottom side while the thickness of the melt film varies from a few to around one hundred micro meters perpendicular to the laser beam. These scales prevent the use of classical finite element or finite volume approaches to solve the mathematical problem numerically accurate without proper reduction techniques. To develop a simulation that is able to depict the behavior of the melt film, the Navier Stokes equations are transformed to conformal coordinates and subjected to scaling analysis. A perturbation series expansion is performed and the equations are integrated in radial direction using a quadratic ansatz for the mass flux in azimuthal and axial direction. The resulting system of partial differential equations can be solved numerically and describes significant properties of the dynamics of the melt film with high temporal and spatial resolution. The physical mechanisms that lead to striation formation can be investigated by analysis of the model structure as individual physical phenomena can be selectively altered in the simulation. The presented method leads to a simulation that provides support in the evaluation of measures to reduce the striation depth like modulation of laser power or beam shaping optics. The presented combination of model reduction techniques is adaptable to any boundary layer problem of similar type.