### 1. Introduction

^{1-3)}. By using the X-ray real-time imaging system, keyhole wall dynamics can be observed

^{4,5)}. Technique to monitor full penetration hole dynamics is similar to the one to observe keyhole dynamics, while several research efforts have been done with vision system. Fabbro et al (2005)

^{6)}analyzed the dynamics of the keyhole and its complete geometry (front wall inclination, top and bottom apertures) by using on axis visualizations through the top and the bottom of the keyhole with a high speed video camera. Blug et al (2011)

^{7-9)}investigated coaxial images of laser full penetration welding, extracted full penetration from coaxial images and developed a close loop control system to improve laser full penetration welding quality. Similarly, Zhang et al (2013)

^{10)}studied the full penetration hole dynamics using an auxiliary illuminant.

^{11)}) and VOF method(Hirt and Nichols, 1981

^{12)}) are widely used to track the interface of keyhole wall. Ki et al. (2002)

^{13)}modeled the keyhole by the level set method and showed an improved total laser absorptivity for both laser drilling and welding. Lee et al. (2002)

^{14)}investigated the the mechanisms of keyhole formation and instability with the volume of fluid (VOF) method. Zhou et al. (2006)

^{15)}studied the defect formation mechanisms in pulsed laser welding of zinc- coated steels with VOF method. Dasgupta et al. (2007)

^{16)}developed an adaptive mesh model to reduce computation load in simulation of laser welding of zinc coated steels. Amara et al. (2008)

^{17)}studied the effect of gas jet on the flow pattern of melt pool. Cho et al. (2009)

^{18)}studied the flow pattern evolution when laser weld change from conduction mode to keyhole welding mode. Zhao et al.

^{19)}investigated the formation of keyhole-induced porosity. Tan et al. (2013)

^{20)}used the ghost fluid method to deal with the jump condition across the liquid vapor phase interface.

^{21-23)}. Geiger et al. (2009)

^{24)}investigated the full penetration welding with Open FOAM

^{®}software package, however, no quantitative analysis was presented in this work.

### 2. Mathematic model

*V*, while laser beam is stationary. A laser beam with a Gaussian profile is used as the heat source, which is focused on the substrate surface with a spot size of 0.4 mm. When laser beam is on, the high power laser beam will heat a small area of the substrate and produces a molten pool. The fluid flow in the molten pool is assumed to be incompressible Newtonian fluid. The fluid flow is considered to be laminar and is driven by the buoyancy force, Manangoni force, and surface tension on the molten pool surface. The thermal-physical properties of the substrate are assumed to be temperature dependent.

### 2.1 Free surface tracking

*et al*.

^{25)}, then it is widely used in surface tracking. Compared with VOF, level set method can calculate the surface position and surface normal in a more precise way. Therefore level set method is used to track the evolution of the molten pool free surface in this work. The general form of the level-set equation is

*V*is the free surface movement velocity caused by evaporation and

_{e}^{26)}, using the coupled level set and volume fluid methods

^{27)}. Wen et al.

^{28)}proposed a simple but effective way when dealing with laser cladding by multiplying level set equation and the mass conservation equation. This equation is achieved by making eq. (1) * ρ equal to eq. (14) * Φ. The same corrected level set equation is used here.

### 2.2 Ray-tracing model

^{7)}

*I*is the incident laser energy at point (

_{i,j}*x*), and the Fresnel absorption coefficient α can be calculated as a function of the incident angle θ in equation

_{i}, y_{i}*I*can be expressed as

_{i,j}*P*is laser power and

*r*is laser beam radius.

### 2.3 Evaporation model

^{29)}and was thoroughly described by Ki

^{13,30)}.

*M*is the Mach number of the flow. The net evaporated mass loss of evaporation can be calculated as,

_{s}is the saturation density at liquid surface temperature

*T*, and β is the modification factor concerning back-scattered flux. The recoil pressure at the keyhole wall can be calculated as

_{l}*p*is the saturation density at liquid surface temperature

_{s}(T_{l})*T*.

_{ι}### 2.4 Heat transfer

*q*represents the energy changes. The energy source is mainly on the liquid/gas interface, and it includes laser energy absorbed

*q*and energy transported due to melting, evaporation and thermal emission as

_{laser}_{e}is the mass loss due to evaporation, and

*L*is the latent heat of evaporation, σ is the Stefan-Boltzmann constant and equal to 5.67×10

^{-8}W/m

^{2}·K, and ε is the material emissivity, δ(Φ) is the derivative of Heaviside function of the level set values called delta function as

### 2.5 Fluid flow

*μ*is the viscosity,

*p*is the pressure. The third term on the right side of Eq. (14) is the damping force when fluid passes through a packed bed. The isotropic permeability

*K*expressed by Kozeny-Carman equation is presented as

*K*is a constant determined by the morphology of the mushy zone. The fourth term on the right side of Eq. (14) is the capillary and Marangoni forces on the L/V interface. Since they are surface force, they can be incorporated into the momentum equation by multiplying with δ(Φ) as defined in Eq. (12).

_{0}### 2.6 Boundary conditions

#### 2.6.1 Continuum method on solid/liquid interface

*ρ*, thermal conductivity

*k*, dynamic viscosity

*μ*, and specific heat C

_{p}in the liquid/vapor transition region are expressed as

*m, l*and s denote mushy, solid and liquid phases and

*f*and

_{l}*f*are the solid and liquid mass fractions of material.

_{s}*f*is defined as

_{l}*T*is solidus temperature and

_{s}*T*is liquidus temperature.

_{l}