# Simulation Analysis of Nanosecond Laser Processing of Titanium Alloy Based on Helical Trepanning
## 2 Theory and Two-Phase Flow Heat Transfer Simulation Model  
The interaction between a laser and a titanium alloy during nanosecond laser processing represents a complex coupling process. Owing to the short action time between the laser and material, the solid-liquid phase transition process is ignored in this work, and only the influence of the material's evaporation and removal on processed morphology is considered. Therefore, the two-phase flow heat transfer simulation model analyzed the material removal mechanism considering the two-phase flow and temperature and adopting the following hypotheses:  
* The processing environment is uniform and isotropic in a metal material node. * The metallic phase is transferred to the processing environment by evaporation. * The thermodynamic parameters of the solid-liquid mixing zone in the melt change linearly with temperature. * Heat transfer and radiation in laser movement are considered. * The solid metal is used as a fluid with great dynamic viscosity. The horizontal set heat transfer model of a two-phase flow with the evaporative transfer source term was adopted. The material phase states in the two-phase flow heat transfer simulation model were distinguished based on two parameters, level set function and node temperature. The metal phase and environment were distinguished by the level set coefficient from one to zero. In the metallic phase, those below melting temperature denote the solid phase, those above evaporation temperatures denote the gas phase, and those in the middle represent the liquid phase. The energy conservation, mass conservation,and momentum conservation equations of the two-phase flow heat transfer simulation model are as follows:  
\[\rho C_{p}\bigg{[}\frac{\partial T}{\partial t}+u\cdot\
abla(T)\bigg{]}=\
abla \cdot(k\
abla T)+S-S_{loss}, \tag{1}\]  
\[\
abla\cdot u=\delta(\phi)\dot{m}\bigg{(}\frac{1}{\rho_{g}}-\frac{1}{\rho_{l}} \bigg{)}, \tag{2}\]  
\[\rho\bigg{(}\frac{\partial u}{\partial t}+u(\
abla u)\bigg{)}=\
abla[-PI+u( \
abla u)]+P_{sat}(T)\
abla\phi+\big{(}\rho-\rho_{g}\big{)}, \tag{3}\]  
where \(\rho\) is the mixing density of the metal phase; \(\rho_{g}\) is the gas phase density; \(\rho_{l}\) is the liquid phase density; \(k\) is the thermal conductivity coefficient; \(S\) is the laser heat source; \(S_{loss}\) is the heat loss term; \(\dot{m}\) is the mass loss; \(u\) represents the velocity of fluid; \(P\) is the pressure matrix; and \(P_{sat}\) is the evaporation pressure. The laser heat source \(S\), heat loss \(S_{loss}\), and evaporation pressure \(P_{sat}\) are respectively given by  
\[S=\frac{2p_{max}\eta}{\pi R_{0}^{2}\Big{(}1-\frac{Z}{2}\Big{)}^{2}}e^{(-\frac {2f(y)^{2}}{R_{0}^{2}(1-\frac{Z}{2})^{2}})}\delta(\phi), \tag{4}\]  
\[S_{loss}=-\Big{(}H_{g}\dot{m}+\sigma_{0}\varepsilon_{0}\Big{(}T^{4}-T_{0}^{4} \Big{)}+h(T-T_{0})\Big{)}\delta(\phi)+H_{l}\
abla m, \tag{5}\]  
\[P_{sat}(T)=P_{0}e^{H_{g}\frac{T-T_{sat}}{R_{0}^{2}T_{sat}}}. \tag{6}\]  
In the type, \(p_{max}\) is laser power, \(\eta\) is the laser absorption rate of the material, \(R_{0}\) is the effective radius of the laser beam, \(z\) is defocus quantity, \(f(x)\) is the path function of laser scanning, \(\delta(\phi)\) is the level set function, which indicates that the energy occurs at the interface, \(H_{g}\) is latent heat of evaporation, \(\sigma_{0}\) is the Stefan-Boltzmann constant, \(\varepsilon_{0}\) is radiation coefficient, \(h\) is convective heat transfer coefficient, \(H_{l}\) stands for latent heat of melting, \(R_{g}\) is the universal gas constant, \(T_{sat}\) is the gas-liquid transition temperature, \(H_{g}\dot{m}\) represents heat loss caused by evaporation of metal materials, \(\sigma_{0}\varepsilon_{0}\big{(}T^{4}-T_{0}^{4}\big{)}\) and \(h(T-T_{0})\) respectively represent energy loss caused by thermal radiation and thermal convection, and \(H_{l}\
abla m\) stands for heat loss caused by melting of metal materials.