The forces acting on the laser cladding molten pool are briefly described, including surface tension, viscous shear force, gravity, and shielding gas pressure, and the formation mechanism of the cladding layer is briefly analyzed from the perspective of tissue growth and molten pool flow. At the same time, the energy distribution laws and equations of different heat source models used in laser cladding simulation are summarized, including surface Gaussian heat source, surface annular heat source, Gaussian body heat source, ellipsoidal heat source, and combined body heat source. On this basis, the research progress in the numerical simulation of the temperature field and flow field of the laser cladding molten pool at home and abroad in recent years is classified and reviewed, and the advantages and disadvantages of various heat source models are analyzed. The applicable environment of different heat sources and the distribution laws of the obtained temperature field and flow field are summarized. In addition, the research methods of the free liquid surface of the molten pool are summarized, and the verification methods of the numerical simulation models of the temperature field and flow field are summarized. At the same time, in view of the problems existing in the numerical simulation research of the laser cladding molten pool, they are summarized from the aspects of numerical models and boundary conditions, and finally the future development direction is prospected.

Laser cladding is a new process for surface modification and repair of materials. It adds cladding material to the substrate surface by spreading or feeding powder, and uses a high-energy laser beam to quickly melt the cladding layer material to form a cladding layer with good metallurgical bonding on the substrate surface, thereby changing the composition of the substrate surface and improving the surface properties of the material [1]. During the laser cladding process, the molten pool is non-steady and will be disturbed by Marangoni flow, powder injection, powder delivery, turbulence in the shielding gas, and variable processing parameters [2]. A large number of studies have shown that the Marangoni force driven by surface tension has an important influence on the flow of the molten pool [3-4], which is also a key factor in determining the morphology and dilution of the cladding layer. The flow behavior in the molten pool has a direct impact on the evolution of the material structure. The fluid dynamics and geometric evolution of the liquid molten pool are directly related to the mechanical properties of the additive manufacturing material. Since the molten pool is formed in an extremely short time and the molten pool size is small, it is almost impossible to accurately monitor the instantaneous evolution of the molten pool in real time during the experiment. Therefore, with the development of computer technology, numerical simulation of the dynamic flow behavior inside the molten pool through finite element simulation has become the mainstream. In the numerical simulation of laser cladding molten pool, a reasonable heat source model is the key to obtaining accurate numerical simulation results. Generally, the corresponding heat source model is established according to the laser heat source required by the actual working conditions. The extinction length of the material mainly depends on the absorption coefficient of the material to a specific laser wavelength. According to the extinction length of the material to the laser, the heat source model in the laser cladding process can be divided into surface heat source and body heat source. If the laser acts on the surface of the material, the laser energy decays to 0 after a short distance of transmission. It can be assumed that the energy is completely absorbed on the surface of the material, and the absorption of the laser energy by the matrix material can be called surface absorption; if the transmission depth is deep, even exceeding the material thickness, it can be called body absorption [5].

This paper first briefly describes the formation mechanism of laser cladding molten pool, and then classifies and reviews the research progress of laser cladding molten pool numerical simulation under several widely used heat source models, summarizes the research status of molten pool temperature field and flow field numerical simulation under different heat sources, and finally looks forward to the molten pool simulation problem of laser cladding.

1 Formation mechanism of laser cladding molten pool

During the laser cladding process, the energy density of the laser input is high. Heat conduction and convection control the physical evolution of the molten pool and directly determine the temperature field and flow field distribution in the molten pool. The laser cladding molten pool can reach equilibrium in a very short time, in which there is a large temperature gradient and rapid cyclic convection. The focused laser beam is irradiated onto the metal substrate, the substrate temperature rises, and it melts to form a molten pool. The nozzle sprays the metal powder into the molten pool synchronously. The liquid metal in the molten pool convects under the action of Marangoni tension. The temperature inside the molten pool gradually becomes uniform. The melt flows to the edge of the molten pool, reaches the surface of the molten pool, and solidifies to form a cladding layer. Song et al. [6] analyzed the formation of the molten pool, the internal convection pattern, and the solidification behavior of the cladding layer.

The forces in the molten pool are complex. The main driving force for the flow of fluid in the molten pool is the Marangoni flow generated by the balance between buoyancy and surface tension gradient and viscous shear force [7]. Shi Jianjun [8] analyzed the three-dimensional force of the molten pool. The force analysis of the molten pool is shown in Figure 1, which mainly includes surface tension Fγ, viscous shear force Fμ, gravity G and shielding gas pressure Fp, and θ is the substrate deflection angle. Under the combined action of surface tension, gravity, viscous shear force and shielding gas pressure, the molten metal forms a dynamically balanced metal molten pool. Among them, the surface tension has the greatest impact on the molten pool, and the flow of the fluid in the molten pool is mainly driven by Marangoni convection driven by surface tension.

2 Research progress of heat source model for numerical simulation of laser cladding

2.1 Surface Gaussian heat source

Currently, most numerical simulations of laser cladding use Gaussian heat source model, as shown in Figure 2. The laser energy is normally distributed in space, with more in the center and less at the edge, which is consistent with the actual processing process. However, the energy distribution in the depth direction of the molten pool is ignored, so it is not suitable for working conditions with a deeper molten pool.

The expression of the heat flux density equation is: See formulas (1) and (2) in the figure. Where: q(r) is the surface heat flux at radius r, W/m2; R is the distance from the center of the spot, m; c is the heat flux concentration coefficient, m2; qm is the maximum heat flux at the center of the heat source, W/m2; P is the laser power, W; η is the laser utilization rate.

Gaussian surface heat source is suitable for processing conditions with small molten pool width and depth and cladding layer thickness. For the numerical simulation of molten pool temperature field, Wang Zhijian et al. [10] used Gaussian surface heat source to numerically simulate the solidification process of single-pass single-layer laser molten pool of TC4 titanium alloy. The study found that in single-pass laser cladding, due to the rapid heat transfer at the tail, the heat at the front end of the molten pool is more concentrated than that at the rear end, and the molten depth is greater. With the increase of laser power, the molten pool molten depth and the range of the heat affected zone gradually increase due to the increase in energy input. Pant et al. [11] established a molten pool mixing model based on the finite element method and studied the heat transfer behavior of the molten pool during laser deposition melting. The results show that the molten pool is elliptical at the front and comet-shaped with a stretched tail. The width of the molten pool increases with the increase of laser power (as shown in Figure 3). The cooling rate increases with the increase of scanning speed, and increasing the laser power will increase the temperature gradient in the molten pool, and the cooling rate will increase accordingly.

In addition, some scholars have studied the three-dimensional morphology simulation of the molten pool under Gaussian surface heat source. Fallah et al. [12] proposed a finite element model transient simulation to predict the size and morphology evolution of the molten pool during laser powder deposition. The results showed that the simulated predicted molten pool profile was close to the experiment,
but no specific analysis was made on the temperature field and flow field of the molten pool. Gao et al. [13] established a three-dimensional numerical prediction model for single-pass processing during laser cladding. Using Gaussian distribution heat source and based on the birth and death unit method, the geometric shape of the cladding layer does not need to be preset. The transient temperature field and geometric structure of the cladding layer are calculated simultaneously. The obtained cladding shape is in good agreement with the experimental results, as shown in Figure 4. In addition, they also analyzed the influence of process parameters on the temperature field and geometric shape of the cladding layer.

Some scholars will preset the three-dimensional shape of the cladding layer in advance when using Gaussian surface heat source to simulate the molten pool flow field. Liu Han et al. [14] established a three-dimensional model based on the actual deposition layer size contour in the numerical simulation study of the temperature field and stress field in the laser deposition three-dimensional forming process. On this basis, a finite element model of the silk-powder synchronous laser deposition molten pool was established, and the distribution law of the molten pool flow field was obtained. Two symmetrically distributed circulations are formed on the cross section of the molten pool, and two radial circulations are generated, one strong in front and one weak in the back. The fluid distribution on the upper surface of the molten pool shows a law of diffusion from the center to the edge. Li et al. [15] established a multi-field coupling model of the disk laser cladding process based on COMSOL software, and calculated the thermal physical properties of the material using the CALPHAD method. Using a Gaussian surface heat source, the interaction between the laser beam and the powder and the stress conditions inside the molten pool were comprehensively considered, and the change law of the temperature field and flow field during the disk laser cladding process was obtained. The molten pool is ellipsoidal, and the highest temperature occurs at the rear of the center of the molten pool. In the early stage of cladding, the flow rate of the molten pool is low, and heat conduction plays a major role in the energy transfer of the molten pool; as the cladding process proceeds, the flow rate of the molten metal in the molten pool accelerates, and heat convection plays a major role at this time, as shown in Figures 5 and 6.

2.2 Surface annular heat source

The surface annular heat source is a heat source model unique to the numerical simulation of hollow annular laser cladding. It is based on the new hollow annular laser cladding process of “hollow beam and powder feeding in the beam”, which has unique advantages over the traditional “solid laser” cladding. Its basic principle is to convert the solid beam into a hollow annular beam through the beam conversion system [16-17], so that the energy distribution concentration area changes from the center to the outer edge (as shown in Figure 7), which can eliminate the phenomenon of incomplete melting at the edge of the molten channel caused by Gaussian solid laser cladding and improve the disadvantage of poor metallurgical bonding [18].

The energy distribution in its annular area is also like a Gaussian distribution, and the energy distribution function is: See formula (3) in the figure. Where: R0 is the outer diameter of the laser at the focal position, mm; z is the defocus, mm; φ is the angle between the hollow laser beam and the horizontal direction; ξ is the energy peak position coefficient.

Tian Meiling et al. [18] used ANSYS finite element analysis software to simulate the temperature field of the annular hollow laser molten pool and conducted a theoretical analysis of the three-dimensional flow field distribution. The flow field of the hollow laser cladding molten pool showed a symmetrical four-ring flow distribution, as shown in Figure 8. Shi Gaolian [20] used ANSYS finite element software and based on the surface hollow annular heat source model to simulate the transient temperature field of the molten pool of 45 steel sample cladding Fe313 alloy, and obtained the evolution law of the temperature field of the hollow laser cladding molten pool. Due to the heat accumulation effect during the cladding process, the temperature in the molten pool gradually increases with the increase of scanning time and height. The shape, position and energy density distribution of the molten pool and the quality of the formed part will change significantly with the change of duty cycle. Li Guangqi et al. [21] simulated the loading of hollow ring laser based on ANSYS software using APDL language combined with the birth-death unit method and obtained the distribution law of the temperature field of the cladding layer. The overall distribution of the temperature field during the cladding process was “comet-shaped”. In the initial stage of scanning, the spot showed a complete ring shape with the same theoretical energy distribution. As the scanning process progressed, the high-temperature area moved backward as a whole, gradually evolving from a ring shape to a saddle shape, as shown in Figure 9. This confirms the characteristics of the hollow ring laser energy of “low in the middle and high at the edge”. In addition, the surface layer of the temperature field of the cladding layer showed a “deep valley shape”, with high on both sides and low in the middle, and in the depth direction of the cladding layer, the temperature gradually decreased with increasing depth, as shown in Figure 10.

2.3 Gaussian body heat source

In the actual laser cladding process, the laser beam moves at a certain speed, and the energy distribution is not uniform, especially the energy distribution of the light source perpendicular to the scanning direction is quite different, and the surface heat source model cannot penetrate into the molten pool. Therefore, the body heat source came into being. The laser energy of the body heat source is not only deposited on the surface of the powder layer, but also can penetrate into the inside of the cladding layer, which improves the calculation accuracy of the transient temperature field or flow field of the molten pool [22]. Some scholars have established a rotating Gaussian body heat source based on the Gaussian surface heat source model, as shown in Figure 11. The rotating Gauss surface body is formed by rotating the Gauss curve around its symmetry axis. Assuming that the heat source energy is all distributed inside this surface body, the heat flux density in the cross section is Gauss distribution.

The energy distribution function is: see formulas (4) and (5) in the figure. Where: e is the natural base; R0 is the radius of the heat source opening; H is the heat source height; Q is the heat source power.

Gaussian heat source is the most widely used heat source model for numerical simulation of laser cladding molten pool. Zhang Kerong et al. [24] numerically simulated the transient process of laser deep melting welding of TC4 titanium alloy keyhole based on the rotating Gaussian volume heat source model, and further analyzed the influence of different process parameters on the keyhole morphology in combination with experiments. The study showed that with the increase of laser energy density, the increase of laser power or the decrease of spot diameter, the depth of the keyhole increased and the size became wider. The spot diameter is the process parameter with the greatest influence on the keyhole morphology. Sun et al. [25] simulated the powder particles of laser directional energy deposition based on fluent software using a discrete phase model, and analyzed the morphology of the powder deposition layer and the distribution of temperature and velocity fields in combination with the Gaussian heat source model. The results show that in high-speed laser energy direct deposition, the downward flow velocity in the powder action area is dominant due to the pressure caused by powder feeding, as shown in Figure 12.

In the numerical simulation of molten pool morphology based on Gaussian heat source. Chai et al. [26] established a numerical model of laser cladding on an inclined substrate based on the cellular automaton method and simulated the influence of different inclination angles on the relative cross-sectional area, width, height and vertex offset of the cladding layer, as shown in Figure 13. The results show that the relative cross-sectional area first increases, then decreases, and then tends to be stable with the increase of the substrate inclination angle; the width of the cladding layer increases with the increase of the inclination angle, and the height first increases and then decreases; with the increase of the substrate inclination angle, the gravity component of the cladding layer becomes larger and larger, and the vertex offset gradually increases.

2.4 Ellipsoidal heat source

The energy distribution in the molten pool in laser cladding is often not a three-dimensional Gaussian rotation body. In order to more accurately simulate the size and shape of the molten pool, an ellipsoidal distribution body heat source is proposed. There are two types of ellipsoidal heat sources: a single ellipsoidal heat source with front-to-back symmetry and a double ellipsoidal heat source with different front-to-back energy distribution. In the early days, some scholars proposed a hemispherical heat source [27], and its energy distribution function is: see formula (6) in the figure. Where: q(x,y,z) is the heat flux density of the point (x,y,z) on the coordinate system; c is the radius of the sphere; Q is the heat input rate.

According to a large number of experimental observations, the actual heat source is not symmetrically distributed front and back. Therefore, the researchers proposed a double ellipsoid heat source (as shown in Figure 14), with the front and rear parts being two 1/4 ellipsoids respectively.

Its front and rear energy distribution functions are: See formula (7) in the figure. Where: qf and qr are the heat flux distribution in the front and rear half ellipsoids respectively; af and ar are the semi-axes of the front and rear half ellipsoids respectively; bh and ch are the other two semi-axes of the front and rear half ellipsoids respectively, and the two short semi-axes of the two ellipsoids are equal; ff and fr are the shares of heat input in the front and rear half ellipsoids respectively, and ff + fr = 1.

Due to the large size of the molten pool generated by the ellipsoid heat source, it is widely used in the numerical simulation of laser processing processes such as laser welding [29-30] and pre-set laser cladding. Hocine et al. [31] analyzed the differences between three heat source models (ellipsoid heat source, double ellipsoid heat source, and cylindrical heat source) in simulating the evolution of the temperature field and the contour of the molten pool in selective laser melting. The results showed that the three heat source models have their own unique advantages in calculating the temperature field and the contour of the molten pool. The cylindrical heat source is suitable for calculating the temperature field of the molten pool, while the ellipsoid heat source has higher accuracy in calculating the contour of the molten pool. Luo Xinlei et al. [32] used ANSYS APDL to simulate the temperature field of single-channel selective laser melting and compared the simulation results under the Gaussian surface heat source and the double ellipsoid heat source. The results show that the double ellipsoid heat source has a better agreement with the experimental results than the Gaussian surface heat source because its energy distribution is closer to the actual laser heat source. In the process of selective laser melting, without changing the laser input energy density, increasing the laser power and scanning speed will significantly increase the depth and width of the molten pool, as shown in Figure 15.

Some scholars have also conducted in-depth research on the temperature field changes of the molten pool under different process parameters. Hao Xiaojie [33] used ABAQUS software to analyze the variation of temperature field during selective laser melting. He used a double ellipsoid heat source, which distributed the laser energy input in a certain volume and applied it to the nodes of the material model in the form of heat flux density. He studied the influence of different process parameters on the temperature field during laser melting. When only the laser power increases, the average heating rate and cooling rate in the molten pool and the size of the molten pool increase accordingly; when only the scanning speed increases, the average heating rate and cooling rate in the molten pool increase steadily, while the size of the molten pool decreases relatively; the scanning spacing affects the remelting effect between the melt channels, while the powder thickness affects the bonding effect between the scanning layers.

2.5 Combined heat source

The single volume distribution heat source simplifies the distribution law of the heat source in the depth direction of the molten pool, and does not distinguish the distribution difference of laser energy on the surface and inside of the molten pool [34]. Therefore, combined heat sources are derived, such as segmented body heat source, double ellipsoid cone combined heat source, and combined heat source combining Gaussian surface heat source and body heat source. The combined heat source combines the advantages of the surface heat source and the body heat source, is more in line with the actual working conditions, and has higher simulation accuracy. In the combined heat source, the surface heat source is generally a Gaussian heat flux distribution surface heat source, and the body heat source is generally a linearly attenuated Gaussian cylinder heat source or a rotating body heat source with decreasing heat flux [35].

Cai Haipeng et al. [36] improved the welding heat source on the basis of the moving Gaussian heat source, established a segmented heat source model, used coarse grids and appropriate heat source segmentation to calculate the welding deformation problem, and combined the local refined grid technology to simulate the stress evolution. Wang Qibing et al. [37] used a combined heat source combining the upper part of the double ellipsoid heat source and the lower part of the Gaussian rotating body heat source to simulate the molten pool heat and flow field during the laser-MIG hybrid welding of Invar steel. The results showed that the molten pool temperature field distribution simulated by the combined heat source was basically consistent with the actual experimental results. Xie Yinkai et al. [38] established a combined heat source of a parabolic rotating body heat source (lower half) and a cylindrical heat source (upper half) (as shown in Figure 16) to simulate the specific disturbances of the molten pool size, melt flow and gas-liquid free interface during laser selective melting. In single-pass cladding, scanning speed and powder layer thickness dominate the factors of pore formation. For multi-pass cladding, the factors affecting pore formation are mainly scanning spacing, and the number of pores increases with the increase of scanning spacing.

Wang Yiwen et al. [39] established a three-dimensional symmetrical numerical model for the transient motion and heat and mass transfer of the molten pool based on Fluent software. Using a three-dimensional hemispherical Gaussian heat source, the evolution process and flow behavior of the molten pool liquid/gas interface under different process parameters were analyzed, and the relationship between flow, temperature and molten pool size and surface quality was established, as shown in Figures 17 and 18, respectively. The results show that the morphology of the single-pass cladding layer obtained by the experiment and the simulation is similar. After a stable molten pool is formed, the fluid in the molten pool flows from the high-temperature area to the low-temperature area in a radial shape, and the flow rate gradually increases from the middle to the outside. The camera monitors the flow of the slag in real time and the flow direction of the simulated flow field is consistent.

2.6 Other heat source models

With the further development of computer technology, some scholars have further optimized the existing heat source model according to actual working conditions and established a new heat source model. In addition, numerical simulations under some special processing conditions can also be achieved through specific heat source models, such as broadband laser beam heat source model, hollow ring heat source model, etc.

Lei Dingzhong et al. [40] used TracePro software to simulate and analyze the light path and the distribution of the focused spot light flux W formed by the broadband laser cladding nozzle with powder feeding in the light, and established a three-dimensional mathematical model of the hollow ring broadband laser on the mirror surface. Tseng et al. [41] proposed a laser heat source model based on SYSWELD software, comprehensively analyzed the influence of laser beam characteristics and process parameters on the temperature field and the shape of the cladding layer, and designed a numerical model for laser cladding experimental verification, which can be applied to the numerical simulation of other laser processing processes. Liu et al. [42] established a broadband laser beam heat source model, and its energy distribution function is: see formula (8) in the figure.
Where: I0=αβP/(wd). α is the laser absorption coefficient, α=0.75; β is the power efficiency, β=0.98; P is the laser power; d is the width of the broadband laser spot, d=1.5 mm; w is the length of the broadband laser spot, w=15 mm. Liu et al. [42] studied the temperature field and stress field of a single-pass cladding layer in wide-beam laser cladding, where the temperature field distribution is shown in Figure 19. Combined with the temperature data, the length, width and depth of the molten pool were calculated. At the same time, the effects of process parameters such as laser power and scanning speed on the molten pool size, temperature gradient, cooling rate and solidification rate were discussed. In addition, the thermal stress field distribution of the cladding layer in different directions and on different paths was also studied.

Feng Yiqi [43] established a selective laser melting molten pool fluid mechanics model. Based on the energy attenuation characteristics of the laser inside the powder bed, a laser intensity attenuation body heat source was used in the simulation: see formula (9) in the figure.
The simulation results of the powder spreading model were imported into the molten pool fluid mechanics model to predict the flow behavior of the molten pool, and an in-depth analysis was made on the relationship between the flow behavior, holes and spheroidization effect of the molten pool in multi-pass cladding. The results show that the bottom surface of the additive material has a denser powder distribution than the flat bottom surface. Due to the great uncertainty of the melt pool flow, the spheroidization phenomenon mainly occurs on the bottom surface of the additive material, and the unfused holes are mostly produced at the melt channel necking between multiple cladding layers, as shown in Figure 20.

Song et al. [44] comprehensively considered the attenuation effect of the interaction between the powder jet and the laser and the heat sink effect of the unmelted powder particles entering the molten pool. Based on the COMSOL software, a heat source model was established to simulate the melt flow and the surface tension of the gas-liquid interface. The temperature field and flow field distribution are shown in Figure 21. At the same time, the curvature of the free surface of the molten pool and the size of the cladding layer were predicted. In three different cross-sectional directions, the simulated temperature gradient direction is consistent with the grain growth direction. Experimental verification of the cladding layer width, height and molten pool depth shows that under the process parameters considering the influence of different laser powers, laser scanning speeds and powder feeding rates, the maximum error between the simulation results and the experimental results is 10%.

Xu Jiachao et al. [45] established a three-dimensional mathematical model of a hollow ring laser heat source by combining the idea of ​​a geometric body of revolution, and obtained its mathematical analytical formula as follows: see formula (10) in the figure.

Where: f1 is the energy conversion coefficient, f1≤1; Q is the heat input power, W; μ is the energy peak position, which is usually located at the center of the ring area, that is, μ=(R+r)/2; a is 1/2 of the ring width, that is, (R-r)/2; R and r are the outer diameter and inner diameter of the ring spot, mm; c is the depth of the light source, mm. The relevant parameters of the heat source model were determined experimentally, and the model was loaded based on the COMSOL software to simulate the transient temperature field distribution and thermal cycle curve of the ring laser cladding. The temperature peak and peak valley decrease and increase respectively due to heat accumulation and heat conduction. As the layer height increases, the temperature rise area of ​​the deposited layer becomes flat.

In summary, the applicable environments of several widely used laser heat source models are summarized in Table 1. In the temperature field simulation, the temperature field distribution trends obtained by different models are similar, all in the shape of elliptical comets, and the main difference is the different high-temperature areas; in the flow field simulation, the overall distribution of the molten pool flow field obtained by different heat source models is similar, and the high-speed area is also concentrated in the center of the molten pool. The main difference is that the molten pool size is different, and the heat source model with more dispersed energy distribution obtains smaller melting depth and width. Because the process parameters are complex in the actual cladding process, Table 1 is only for reference, and the heat source model should be reasonably selected according to the actual experimental conditions.

3 Research progress of free liquid surface in numerical simulation of laser cladding

In the laser cladding process, the free liquid surface of the molten pool is in direct contact with the air, which is mainly affected by the surface tension and directly determines the size profile of the cladding layer. At present, the mainstream methods for studying the free surface of the molten pool include the Level Set method based on fixed grids, the Volume of Fluid method, the Coupled Level Set method and the Volume of Fluid method, the Phase Field method, and the Arbitrary Lagrangian-Eulerian method based on moving grids.

3.1 Level Set Method

The Level Set (L-S) method, also known as the isosurface function method[49], uses a distance field function to describe the dynamic interface. The Level Set method was originally proposed to study the interface of multiphase flow, and is now also used in image recognition, interface reconstruction and other fields. Liu et al.[50] used the Level Set method to track the free surface of the molten metal in selective laser melting and found that the unstable disturbance caused by the change of surface tension caused local depressions on the surface of the molten pool, thereby affecting the surface roughness of the cladding layer after forming. However, the numerical dissipation of the L-S method is relatively serious during calculation, which is prone to mass non-conservation problems.

3.2 Volume of Fluid Method

The Volume of Fluid (VOF) method describes the free interface by defining a volume fraction function, and reconstructs the interface by solving the volume fraction in a single grid. The VOF method has better mass conservation than the L-S method. Ye Chen [51] simulated and predicted the size profile of the cladding layer of laser cladding based on the VOF method, and verified the simulation results by orthogonal experiments. The comparison results of the three groups of data, namely, melting height, melting depth, and dilution rate, showed a deviation within 10%, which proved the accuracy of the numerical model. However, the accuracy of the free interface constructed by the VOF method is not high enough, and the flow in the normal direction of the interface cannot be accurately tracked [52]. Wen Baoxian et al. [53] established a body heat source model of laser energy distribution in the powder bed based on the propagation law of the light beam in the powder medium based on the fluent software, and modified the classic VOF method, and proposed a VOF method that can be used to simulate the collapse phenomenon after powder melting. The calculation results show that the change in the volume of the powder layer will affect the temperature field and velocity field of the molten pool and its surroundings, as well as the final morphology of the workpiece.

3.3 Coupled Level Set Method and Fluid Volume Method

The Coupled Level-set with VOF (CLSVOF) method combines the advantages of the L-S method and the VOF method, and has good interface reconstruction accuracy and mass conservation. Wei et al. [54] combined the L-S method and the VOF method to propose a coupled multiphase flow model to study the heat and mass transfer during laser hot wire deposition and the flow of the free surface. The model can capture the subtle fluctuations of the gas/liquid interface as small as about 0.03 mm. Wang Xiangyu et al. [55] used the CLSVOF method to predict the change of the free liquid surface of the molten pool, analyzed the mass transfer inside the molten pool, and proposed a multiphase flow model for simulating the microflow of laser cladding of heterogeneous materials. The deviations between the experiment and the simulation were within 9%. In addition, in the field of selective laser melting, Thorsten Heeling et al. [56] established a numerical simulation model of the molten pool based on the CLSVOF method. When analyzing the molten pool size obtained by simulation and experiment, it was found that the deviation of the molten pool depth increased with the increase of scanning speed, while the deviation of the cross-sectional size decreased with the increase of scanning speed.

3.4 Phase Field Method

The Phase Field (PF) method is based on the Ginzburg-Landau theory and solves the transient changes of the interface through differential equations [57]. Unlike the VOF method, it does not require reconstruction of the interface. Compared with the L-S method, it does not require tedious initialization of the distance function. The calculation amount is relatively small, and it has unique advantages in dealing with free liquid surface problems with smaller scales or high surface tension sensitivity. Jin et al. [58] established a two-dimensional numerical simulation model of laser powder bed melting based on the phase field method and found that the Marangoni effect will cause bubbles to form in the molten pool. The remelting process and increasing the laser power can help eliminate pores, as shown in Figure 22.

3.5 Arbitrary Lagrangian and Euler methods

The Arbitrary Lagrangian-Eulerian (ALE) method tracks the dynamic interface through the interface movement function. It combines the advantages of the two description methods, Lagrangian and Eulerian, and has obvious advantages in dealing with high-precision free liquid surface and fluid-solid coupling problems. Based on the ALE method, Tian et al. [59] used COMSOL software to establish a heat transfer and fluid flow finite element model containing multiple physical parameters, and explored the influence of different process parameters on the dilution rate and the geometry of the molten pool. The results show that within a certain range, the dilution rate is linearly related to the relative energy-to-mass ratio. In addition, with the increase of the relative energy-to-mass ratio, accompanied by the flow of the fluid in the molten pool, the arc-shaped solid-liquid interface at the bottom of the molten pool gradually changes from shallow to deep, as shown in Figure 23. Gan et al. [60] established a multiphase heat and mass transfer model for laser direct deposition, and used the ALE method based on dynamic mesh technology to track the dynamic changes of the molten pool surface, and calculated the molten pool size profile and composition distribution, indicating that convection is the main mechanism of mass transfer of alloy elements in the molten pool.
In summary, the advantages and disadvantages of the above free liquid surface tracking methods are summarized in Table 2.

4 Verification of laser cladding molten pool simulation model

In the study of laser cladding numerical simulation, it is necessary to establish a reasonable numerical analysis model and verify the model. The current model verification is mainly through the acquisition of molten pool temperature, image and other signals, using computer technology for signal processing, and finally comparing and verifying with temperature field and flow field simulation data.

4.1 Temperature field verification

Laser cladding molten pool temperature detection is divided into contact detection and non-contact detection [62]. Commonly used contact temperature detection is mainly through thermocouple temperature measurement, and the temperature sensing element is in direct contact with the target to be measured. The advantage is simple operation and high detection accuracy. Li Yanmin et al. [63] used thermocouples to measure the temperature of the substrate, and combined with numerical simulation to analyze the temperature distribution inside the molten pool, and approximately obtained the temperature change of the molten pool. Since the temperature at the center of the laser cladding molten pool is too high, the temperature sensing element cannot measure the temperature at the center of the molten pool, and the long-term high-temperature working environment will greatly reduce the service life of the detection equipment. Therefore, the current mainstream molten pool temperature detection adopts non-contact temperature measurement. The non-contact temperature measurement of the laser cladding molten pool mainly includes monochrome temperature measurement, colorimetric temperature measurement, and image signal acquisition and temperature measurement through CCD [64]. Peng Cheng et al. [65] used ANSYS software to simulate the temperature field distribution during the cladding forming process of titanium alloy thin wall, and designed a hollow ring laser cladding molten pool temperature online detection system using a two-color thermometer, measured the actual temperature, and verified the simulation results. The results show that as the deposition layer accumulates upward, the heat accumulation phenomenon becomes more serious. Forien et al. [66] designed an in-situ detection system for the molten pool in the laser powder bed melting process using high-temperature diode temperature measurement and high-speed imaging technology. They found that the change in the pyrometer signal was related to the pore formation area, and the probability of pore formation increased sharply in the high-temperature signal transition area (5%~95%).

4.2 Flow field verification

The flow field verification of the molten pool mainly includes two types: in-situ detection and non-in-situ detection. In-situ detection mainly uses a CCD camera or a CMOS camera to obtain the surface morphology image of the real-time molten pool during the laser cladding process. After image processing, it is compared with the simulation data for verification. Wirth et al. [67] designed a laser cladding high-speed camera image online acquisition system (as shown in Figure 24) to obtain the flow law of the molten pool surface and the particle movement speed. The analysis found that the local flow direction of the molten pool is affected by the process parameters and has a certain randomness. In most numerical simulations, the assumption that the fluid in the molten pool is laminar flow will have a certain impact on the simulation results. Huang Jiankang et al. [68] used particle tracing method combined with molten pool mirror imaging system to study the flow behavior of TIG welding molten pool surface. By calibrating the mapping relationship between the actual molten pool width and the pixel width of the video data, they calculated that the flow velocity of the molten pool surface was about 12 mm/s (304 stainless steel) and 15 mm/s (Q235 carbon steel). Non-in-situ detection mainly detects the size profile and mechanical properties of the experimental samples, and then compares them with the simulation data for verification. Wu Jiazhu [68] studied the heat flow transfer mechanism of laser direct metal deposition process, measured the sample melting depth and deposition layer height obtained by the experiment, and compared them with the molten pool shape profile data obtained by simulation, verifying that the model has a high prediction accuracy (≥95%).

5 Summary and Outlook

The temperature field and flow field simulation of laser cladding are conducive to revealing the metallurgical dynamic characteristics of the molten pool, but there are still the following problems:

1) In the study of the molten pool flow field simulation, the boundary conditions are not perfect. Generally, only the surface tension, gravity and buoyancy of the molten pool are considered for the forces on the fluid in the molten pool, and the pressure of the protective gas and the impact of the unmelted powder particles on the molten pool surface are considered less.

2) In the process of studying the changes in the temperature field and flow field inside the molten pool, some scholars will preset the shape of the cladding layer in advance or assume that the molten pool is located in the plane when establishing the finite element model, while ignoring the free surface of the molten pool liquid/gas, which limits the accuracy of these models for the analysis of the molten pool movement and liquid/gas interface, as well as the study of the flow mechanism of the molten pool.

3) Most studies are based on horizontal substrates, but the parts that need to be repaired are often complex in shape and on non-horizontal base surfaces. Therefore, laser cladding on non-horizontal base surfaces needs further research.

In view of the above shortcomings, the following improvement measures are proposed.

1) Improve the boundary conditions. The shielding gas pressure is measured experimentally, quantified and added to the surface of the molten pool as a boundary condition.

2) Improve the numerical model. The simulation research of the powder flow field of the laser cladding nozzle is already very mature. We can try to combine the discrete phase model to simultaneously add powder materials to form the cladding layer during the simulation process and establish a suitable multiphase flow heat and mass transfer model.

3) The formation mechanism and evolution process of the cladding layer should be analyzed in combination with the internal force of the molten pool, and a scientific explanation of the flow behavior and morphological changes of the molten pool under variable posture will be the next key research direction.