The influence of laser characteristics on internal flow behaviour in laser melting of metallic substrates

The absorptivity of a material is a major uncertainty in numerical simulations of laser welding and additive manufacturing, and its value is often calibrated through trial-and-error exercises. This adversely affects the capability of numerical simulations when predicting the process behaviour and can eventually hinder the exploitation of fully digitised manufacturing processes, which is a goal of"industry 4.0". In the present work, an enhanced absorption model that takes into account the effects of laser characteristics, incident angle, surface temperature, and material composition is utilised to predict internal heat and fluid flow in laser melting of stainless steel 316L. Employing such an absorption model is physically more realistic than assuming a constant absorptivity and can reduce the costs associated with calibrating an appropriate value. High-fidelity three-dimensional numerical simulations were performed using both variable and constant absorptivity models and the predictions compared with experimental data. The results of the present work unravel the crucial effect of absorptivity on the physics of internal flow in laser material processing. The difference between melt-pool shapes obtained using fibre and CO$_2$ laser sources is explained, and factors affecting the local energy absorption are discussed.


Introduction
Laser-beam melting of metallic substrates forms the basis of many advanced fusion-based manufacturing processes (such as laser welding, laser cladding, laser metal deposition (LMD), and selective laser melting (SLM)) and has brought new perspectives on advancement of materials processing and manufacturing of high-integrity products. Successful adoption of laser-beam melting in real-world engineering applications requires finding processing windows within which the product quality should meet the intended standards [1]. However, determining the processing window through trial-and-error experiments is challenging and involves significant costs due to the large number of process parameters and the coupling between various physical phenomena. Simulation-based approaches have been recognised as a promising alternative to costly and time-inefficient experiments and can be utilised to reduce the costs of design-space exploration [2,3]. Moreover, numerical simulations can enhance our understanding of the complex transport phenomena in laser material processing that are not easily accessible through experiments [4,5].
Successful adoption of simulation-based approaches for process development and optimisation relies predominantly on adequate modelling of various physical phenomena that occur during laser melting (e.g. laser-matter interaction, heat and fluid flow, and solid-liquid phase transformation) [6].
Assumptions made to develop computational models often affect their reliability, accuracy and performance in predicting and describing the process behaviour. For instance, studies suggest that melt-pool surface deformations affect power-density distribution, leading to changes in the thermal field, Marangoni flow pattern and the melt-pool shape [7][8][9]. Conversely, previous investigations [10][11][12][13] have shown a considerable influence of laser characteristics and power-density distribution on molten metal flow behaviour in laser welding and additive manufacturing. Thus, there seems to be an important bi-directional coupling between laser power-density distribution and melt-pool behaviour. Neglecting such effects in numerical simulations of laser-beam melting can negatively affect the quality of numerical predictions of thermal fields, microstructures and properties of the product [8,14]. Moreover, assumptions made to develop a computational model may necessitate the incorporation of unphysical tuning parameters to obtain agreement between numerical and experimental data [8,15]. This can reduce the model reliability for design-space explorations since a change in process parameters or material properties may require recalibrating the tuning parameters [16,17]. Understanding the influence of such assumptions on numerical predictions is therefore essential and can guide the modelling efforts to enhance the current numerical simulations.
Absorption of laser-energy, energy-density distribution and its variation over time are critical components influencing the modelling of laser-beam melting [18] and depend on a variety of process parameters including the characteristics of the laser system (e.g. laser intensity and wavelength), thermophysical properties of the material, surface roughness and chemistry, and interactions of the melt-pool surface with the laser beam [19][20][21][22]. In the majority of previous studies on laser-beam melting, the absorption of laser energy is assumed to be constant [13], neglecting the unsteady interactions between laser-beam and material surface [23]. Studies have shown that changes in meltpool surface morphology and temperature can affect the local absorptivity of the material [10,[24][25][26].
The Fresnel absorption model [27], which is commonly employed in numerical simulations of laser melting (particularly when the ray-tracing method is used), accounts for the effects of laser-ray incident angle and material refractive index, but neglects the temperature dependence of material absorptivity [20]. In laser melting of metallic substrates, the material often experiences large changes in temperature that can significantly affect the thermophysical properties of the material, including the material absorptivity [28,29]. Moreover, the complex molten metal flow in melt-pools continuously disturbs temperature distribution over the surface [8], affecting the local absorptivity of the material. The Fresnel model cannot reflect the variation of local energy absorption that occurs due to changes in melt-pool surface temperature [12,30,31]. Hence, the Fresnel model cannot describe variations in material absorptivity with sufficient accuracy, particularly in cases where the melt-pool surface deformations are small compared to the melt-pool depth (for instance, laser cladding, conduction-mode laser welding and laser metal deposition).
Realising that in practical applications the laser type is not a control parameter, as commercial machines come with a fixed laser type, the present work focuses on understanding the influence of laser characteristics on complex heat and molten metal flow in laser-beam melting. Such an understanding allows us to explain, for example, the difference between melt-pool shapes obtained using fibre transmissible and CO 2 lasers (i.e. the most widely employed lasers for industrial applications).
High-fidelity three-dimensional numerical simulations are performed using an enhanced laser-beam absorption model that takes into account the effects of laser characteristics, surface temperature, incident angle and base-material composition. The results obtained using the enhanced absorption model for different laser systems and laser powers are compared with those obtained using a constant absorptivity and factors affecting the local energy absorption are discussed. Additionally, experiments are carried out for different laser powers to validate the melt-pool shapes predicted using the present computational model. The results and discussions provided in the present work guide the modelling efforts to improve simulations of fusion-based welding and additive manufacturing.

Problem description
As shown in figure 1, a moving laser beam is employed to locally heat and melt the substrate that is made of a stainless steel alloy (AISI 316L) and is initially at an ambient temperature of 300 K.
The gas layer above the plate is included in simulations to track the motion of gas-metal interface.
The influence of laser characteristics on melting of a metallic substrate is studied numerically for fibre and CO 2 lasers, whose wavelengths are different. The laser beam is perpendicular to the substrate surface and has a Gaussian intensity profile.
Three batches of three-dimensional numerical simulations are executed for different laser types and powers using both constant and variable absorptivity models to describe the complex thermal and This assumption is made based on the fact that the density, viscosity and thermal conductivity of argon are very small compared to those of the metal, therefore variations of those gas properties with temperature have negligible influence on the numerical predictions [32].    [37], (b) specific heat capacity [34], (c) thermal conductivity [34], (d) electrical resistivity [38] (e) surface tension [39] and (f) dynamic viscosity [37].

Model formulation
The present computational model is developed on the basis of the finite-volume approach and utilises the volume-of-fluid (VOF) method [40] to locate the gas-metal interface. It is assumed that both the molten metal and argon are Newtonian fluids and that their densities are pressure-independent.
Based on these assumptions, the governing equations for conservation of mass, momentum and energy as well as for the advection of the scalar function φ, which represents the local volume-fraction of the steel phase in a computational cell, are defined as follows: Here, ρ is the density, t the time, u the fluid velocity vector, µ the dynamic viscosity, p the pressure, h the sensible heat, k the thermal conductivity, c p the specific heat capacity, and (ψL f ) the latent heat with ψ being the local liquid volume-fraction and L f the latent heat of fusion. It is assumed that the liquid volume-fraction ψ varies linearly with temperature [41], and its value is calculated as follows: The effective material properties in computational cells were computed as follows: where, ξ corresponds to density ρ, specific heat capacity c p , thermal conductivity k or viscosity µ, and subscripts 'm' and 'g' indicate metal or gas respectively.
F d is a sink term incorporated into equation (2) to damp fluid velocities in the mushy region and to suppress them in solid regions, and is defined based on the enthalpy-porosity technique [42] as where, C is the permeability coefficient (also known as the mushy-zone constant) and is a constant, equal to 10 −3 , employed to avoid division by zero. The value of the permeability coefficient C was set to 10 7 kg m −2 s −2 , based on the criteria proposed by Ebrahimi et al. [43].
A continuum surface force model [44] is used to model forces acting on the gas-metal interface (i.e. capillary force, Marangoni shear force and recoil pressure). Accordingly, the source term F s is incorporated into equation (2) as follows: where, f s is the surface force applied to a unit area, and the term 2ρ/(ρ m + ρ g ) is included to redistribute the surface-forces towards the heavier phase. The surface force f s is determined as follows: where, γ is the surface tension,n the surface unit normal vector (n = ∇φ/ ∇φ ), κ the surface curvature (κ = ∇ ·n), p 0 the ambient pressure, and R the universal gas constant. The third term on the right-hand side of equation (9), f P recoil , is included to model the recoil pressure generated due to vaporisation of the molten metal [45,46].
The source S q and the sink term S l are incorporated into equation (3) to model the laser heat input to the material and heat losses from the material due to convection, radiation and vaporisation respectively, and are defined as follows: Here, a is the absorptivity, P the laser power, r b the radius of the laser beam, R the radial distance from the laser-beam axis in x-y plane, and where, T 0 is the ambient temperature, K b the Stefan-Boltzmann constant, and h c and E are the heat transfer coefficient and the radiation emissivity equal to 25 W m −2 K −1 [47] and 0.45 [48] respectively.
Compared to the total laser energy absorbed by the material, the heat losses from the material due to convection and radiation are quite small; thus, the precise values of h c and E are not critical in the present simulations. The coefficient 0.82 in equation (14) is included based on Anisimov's theory [45] to account for the reduced cooling effect due to metal vapour condensation.
In the VOF method, the energy fluxes applied to the material surface are included as volumetric source terms in the computational cells that encompass the melt-pool surface (i.e. cells with 0 < φ < 1). Hence, melt-pool surface deformations that occur during the process can result in an increase in the total heat input to the material [8]. F q in equation (10) is a dynamic adjustment factor introduced to abate artificial increase in energy absorption due to surface deformations and is defined as where, "∀" indicates the computational domain. The influence of utilising such an adjustment factor on numerical predictions of molten metal flow in laser melting is explained by Ebrahimi et al. [8].

Absorptivity model
When a laser beam with total energy of P interacts with material, here stainless steel 316L, part of its energy is absorbed by the material for a fraction equal to its absorptivity (a). Numerical simulations developed for laser welding and additive manufacturing commonly assume the absorptivity to be constant [36], which is physically unrealistic, and its value is often regarded as a calibration parameter [13]. The absorptivity should be considered as a material property and not a calibration parameter [13]. In the present work, the amount of laser energy absorbed by the material was modelled according to the absorptivity model proposed by Yang et al. [13] and Mahrle and Beyer [11], which takes into account the effects of laser characteristics, laser-ray incident angle, surface temperature and base-material composition. Accordingly, the absorptivity a for circularly polarised or un-polarised laser radiation was approximated as follows [49]: where, according to the Fresnel's reflection equations, R s and R p are the reflectance for parallel and perpendicularly polarised light [25] defined as Here, θ is the incident angle of the laser ray, and α and β are functions of the refractive index n and the extinction coefficient k of the irradiated material. The values of α, β, n and k were determined as follows [13]: where, e r and e i are the real and imaginary parts of the relative electric permittivityẽ respectively [50], defined as Here, ω p , f and δ are the plasma frequency, laser frequency and damping frequency respectively, defined as follows [11,13]: where, N e is the mean number density of free electrons (the value of which approximately equals to suggest that the incident angle of the laser ray θ negligibly affects the absorptivity up to 40 • but its effect becomes significant for larger incident angles (for instance, in cases where a keyhole is formed).
Additionally, changes in the temperature notably affect absorptivity, making the assumption of constant absorptivity questionable.

Numerical implementation
The present numerical simulations were constructed on the foundation of a proprietary flow solver, ANSYS Fluent [51]. User-defined functions (UDFs) programmed in the C programming language were developed to implement the absorptivity model, source and sink terms in the momentum and energy equations as well as the surface tension model in the simulations. As shown in our previous works [3,8,9,52,53], The central differencing scheme with second-order accuracy and a first-order implicit scheme were employed for spatial discretisation and time marching respectively. A fixed time-step size ∆t was used in the simulations and its value was chosen sufficiently small (10 −8 s < ∆t < 10 −5 s) to achieve a Courant number (Co = u ∆t/∆x) less than 0.2. The PRESTO (pressure staggering option) scheme [54] was used for the pressure interpolation, and the PISO (pressure-implicit with splitting of operators) scheme [55] was used to couple velocity and pressure fields. An explicit compressive VOF method [56] was employed to formulate the advection of the scalar field φ. Each simulation was run in parallel on 16 cores (AMD EPYC 7452) of a high-performance computing cluster with 256 GB memory.

Experimental setup and procedure
The experimental setup employed in the present work is shown in figure 5.  without calibration. Figure 6 shows a comparison between the melt-pool shapes obtained from the present computational model with those obtained from experiments using an Yb:YAG laser (λ = 1.030 × 10 −6 m) and different laser powers, which indicates a reasonable agreement (generally less than 5% difference in melt-pool dimensions).
The characteristics of the laser system used in laser melting can affect the absorptivity and hence can change the resulting melt-pool shape. The results of the present computational model are also benchmarked against the experimental data reported by Kell et al. [57] for laser melting of a 1 mmthick steel plate using a CO 2 laser (λ = 1.060 × 10 −5 m) with the energy-density (E = P/(Vd b )) being set to 120 MJ m −2 , and the results are shown in figure 7. To compare the numerically predicted melt-pool shape with experimental measurements, the relative difference between meltpool dimensions (i.e. the melt-pool width and depth) was calculated as follows: where, L indicates the melt-pool depth and width. In this case, the deviation between the numerically predicted and the experimentally measured melt-pool dimensions is less than 2%, demonstrating the reliability of the present computational model in predicting the melt-pool shape. The deviation between the numerical and experimental results can be attributed to uncertainties in modelling temperature-dependent material properties, particularly in the liquid phase, the assumptions made to develop the present computational model as well as uncertainties associated with the experimental measurements.

Melt-pool shape and dimensions
To be able to systematically study the effects of laser characteristics and melt-pool surface deformations on variation of local absorptivity, three batches of simulations are considered for different laser types. For cases in batch 1 and 2, the power density is too low to cause significant vaporisation and surface deformations are small compared to the melt-pool depth. Thus, changes in the absorptivity for a specific laser and material can be attributed primarily to changes in surface temperature.
The laser spot size for the cases in batch 3 is intentionally chosen smaller than that for the cases in batch 1 and 2 to achieve high values of power-density, resulting in significant vaporisation of the material and melt-pool surface deformations compared to its depth. For all three batches, the results obtained using the enhanced absorption model are compared with those obtained using a constant absorptivity. Figure 8 shows the numerically predicted melt-pool dimensions obtained for different laser powers using CO 2 and Nd:YAG lasers (i.e. cases in batch 1 and 2). The melt-pool dimensions obtained using the variable absorptivity model are compared with those obtained using different constant values of the absorptivity. For cases in batch 1 and 2, the power density is too low to cause significant vaporisation and surface deformations are small compared to the melt-pool depth. Thus, changes in the absorptivity for a specific laser and material can be attributed primarily to changes in surface temperature (see figure 3).
For the cases where the CO 2 laser was employed ( figure 8(a-c)), melt-pool dimensions predicted using a constant absorptivity between 0.12 and 0.14 seem to agree with those obtained using the variable absorptivity model. However, the results suggest that employing a constant absorptivity does not necessarily render all the melt-pool dimensions with the same level of accuracy, which means the results are less reliable with respect to those obtained using the variable absorptivity model.
This can be attributed to the fact that changes in local energy absorption due to changes in surface temperature, and changes in total energy absorption over time are both neglected when a constant absorptivity is employed. Surface temperature in the spot region after reaching a quasi-steady-state The results shown in figure 8 suggest that for a certain set of process parameters, a lower laser power is required to obtain a melt-pool with similar dimensions using an Nd:YAG laser with an emission wavelength of λ = 1.064 × 10 −6 m than a CO 2 laser with an emission wavelength of λ = 1.060 × 10 −5 m. This arises because the absorptivity for a CO 2 laser is generally lower than that for an Nd:YAG laser when the incident angle is too small to affect the absorptivity significantly (θ < 40 • , as is suggested by the data shown in figure 3), which is the case in conduction-mode laser melting. For the cases where the Nd:YAG laser was employed ( figure 8(d-f)), using a constant absorptivity of 0.35 can render the melt-pool dimensions with a reasonable resolution. When an Nd:YAG laser with a laser power of P = 700 W is employed, numerically predicted surface temperature in the spot region after reaching a quasi-steady-state condition ranges between 1900 K and 2600 K. For this temperature range, the absorptivity varies between 0.347 and 0.36 according to the variable absorptivity model, and its arithmetic average 0.354 is close to 0.35. Since the melt-pool surface temperature and its distribution are not known a priory and are significantly influenced by the process parameters as well as the complex internal molten metal flow, running trial-and-error tests is indispensable to calibrate the value of constant absorptivity. Running such trial-and-error tests increases the total costs of computational analyses and such ad hoc calibration often lacks generality. Figure 9 shows the numerically predicted melt-pool shapes obtained using both variable and constant absorptivity models for a fibre laser with an emission wavelength of λ = 1.070 × 10 −6 m (i.e. cases in batch 3). The power density for the cases in batch 3 is relatively high, resulting in significant vaporisation of the material and melt-pool surface deformations compared to its depth.
In contrast to the cases in batch 1 and 2, the absorptivity for the cases in batch 3 are affected by both temperature and incident angle of the laser ray (see figure 3).
The results presented in figure 9 show an agreement between the melt-pool dimensions predicted using a constant absorptivity of 0.35 and those obtained using the variable absorptivity model. Consequently, these effects cannot be described adequately when a constant absorptivity model is employed in numerical simulations of laser welding and additive manufacturing. Modelling such phenomena with sufficient accuracy is crucial in numerical simulations of transition from conduction to keyhole mode laser melting as well as those developed to predict solidification microstructure and texture.

Thermal and fluid flow fields
Soon after exposing the material to laser radiation, a melt pool forms and grows over time and if the boundary conditions allow, reaches a quasi-steady-state condition. Figure 10 shows  the depressed region is about 6 m s −1 due to the large temperature gradients, forming a multi-cellular flow pattern in the thin molten metal layer due to Marangoni flow instabilities [61]. The maximum molten metal velocity predicted for cases in batch 3 is higher than that for cases in batch 1 and 2.
This is primarily attributed to larger temperature gradients induced over the surface, increasing the magnitude of Marangoni shear force. Moreover, for temperatures above a critical value at which the sign of the temperature gradient of surface tension (dγ/dT ) changes from positive to negative (see figure 2(e)), the absolute value of the temperature gradient of surface tension increases with temperature, increasing the magnitude of Marangoni force applied to the molten material.
Due to the recoil pressure and the outward fluid motion on the surface, molten metal accumulates ahead of the depressed region, which is also observed experimentally by Nakamura et al. [62] and simulated numerically by Khairallah et al. [1,36]. Elements of the accumulated liquid volume can be ejected from the pool and form spatters as shown in figure 11(a and c). Spatters are small  The case belong to batch 3 and the variable absorptivity model is utilised.

Conclusions
The influence of laser characteristics on internal molten metal flow in laser-beam melting of a metallic substrate was investigated numerically using a high-fidelity three-dimensional model. An enhanced absorption model that accounts for laser emission wavelength, surface temperature, laser-ray incident angle and material composition was utilised in the model, and the results compared with experimental measurements as well as numerical data predicted using a constant absorption model. The physics of complex heat and molten metal flow in laser melting is described for various test cases with different laser powers, laser emission wavelengths, and power-density distributions.
For conduction-mode laser melting, where surface deformations are small compared to the melt-pool depth, the absorptivity changes primarily because of changes in surface temperature. However, for cases that surface deformations are significant with respect to the melt-pool depth, changes in the absorptivity are affected by both the surface temperature and the laser-ray incident angle.
Changes in the absorptivity affect energy-density distribution over the surface and hence the thermal field over the melt-pool surface, which in turn can influence the Marangoni-driven molten metal flow as well as the distribution of recoil pressure over the surface. These physical processes are tightly coupled to one another, resulting in highly non-linear responses to changes in process parameters.
For laser melting processes with a relatively low power density using a CO 2 or fibre transmissible laser (with an emission wavelength close to 1 µm), the molten metal velocities and surface deformations are relatively small. Because of the small incident angle, the absorptivity for a CO 2 laser is lower than that for a Nd:YAG laser; thus, a lower laser power is required to obtain a melt-pool with similar dimensions using an Nd:YAG laser as compared to a CO 2 laser. Switching to a relatively high power density laser melting process, molten metal velocities increase compared to the low power density processes. For sufficiently high power densities, melt-pool surface deformations become significant, resulting in strongly enhanced laser absorption which in turn further enhances metal vaporisation.
The results of the present work demonstrate that the coupling between these physical processes cannot be rendered with sufficient resolution employing a constant absorptivity model, reducing the range of predictability of the computational models developed to describe the dynamics of melt-pool behaviour in laser welding and additive manufacturing. Moreover, considering absorptivity as a calibration parameter in computational models necessitates trial-and-error simulations, which increases the total costs of computational analyses.
Although the focus of the present work is primarily on laser melting of bare metallic substrates without powder layers, the fundamental laser-matter interaction mechanisms described here are similar to those in laser melting of powder beds. The enhanced laser-absorptivity model employed in the present work can also be utilised in numerical simulations of melt-pool behaviour in laser melting of powder beds, provided that multiple reflections are included in the model.