## 3 Modeling Approaches  
### Theoretical Background of LWJ Machining  
According to the first law of thermodynamics, the governing equation of transient heat transfer which describes the process of time-dependent heat conduction in the material domain subject [17], can be written as Equation (1):  
\[\frac{\rho c_{p}\partial T}{\partial t}=\
abla\cdot[k(T)\
abla T]+A+A_{L} \tag{1}\]  
where \(\rho\) is the density of CMCs, \(c_{p}\) is the specific heat capacity of CMCs, \(t\) is the time, \(k\) is the thermal conductivity of CMCs, \(A\) is the laser heat source term, and \(A_{L}\) is melting latent heat. For a three-dimension Cartesian coordinate system, Equation (1) can be rewritten as follows:  
\[\frac{\rho c_{p}\partial T}{\partial t}=\frac{\partial}{\partial x}\bigg{[}k_ {x}(T)\frac{\partial T}{\partial x}\bigg{]}+\frac{\partial}{\partial y}\bigg{[} k_{y}(T)\frac{\partial T}{\partial y}\bigg{]}+\frac{\partial}{\partial z}\bigg{[} k_{z}(T)\frac{\partial T}{\partial z}\bigg{]}+A+A_{L} \tag{2}\]  
The interaction between LWJ and substrate is a complex issue that involves optics, heat transfer theory, and fluid mechanics. To solve the transient heat transfer equation of LWJ machining CMCs, several assumptions are raised to simplify the physical model properly in this paper. 1. According to the total reflection at the air-water interface, the energy distribution of LWJ is uniform on the cross-section of the water jet [18; 19]. A flat distribution of laser intensity at the workpiece is assumed instead of a Gaussian distribution, which is shown in Figure 3.
2. The maximum temperature on the substrate surface during the numerical simulation is the melting point of CMCs. When the temperature of one element reaches the melting point, it is assumed to be ablated from the substrate. 3. Heat conduction and heat convection are considered during LWJ machining. Moreover, heat convection is limited in the stagnation cooling area in this study in view of the selection of the Nusselt number. 4. Considering the significantly forced cooling and scouring effect, the flow field in the molten pool is ignored. Figure 2: (**a**) Schematic and (**b**) actual morphology of CMCs; (**c**) detailed morphology of single SiC fiber; (**d**) actual braided structure of CMCs with EDS test result before processing. According to the assumptions above, the laser energy distribution of LWJ on the substrate can be regarded as a multi-mode fiber laser [20]. Therefore, the heat flux of LWJ can be described as Equation (3):  
\[I(t)=P(t)/(\pi d^{2}/4) \tag{3}\]  
where \(I\) is the time-dependent function of the laser incident energy, \(P\) is the laser output power on the surface of CMCs substrate, and \(d\) is the diameter of the water jet. According to physical assumption (1), Equation (3) can be further simplified to a surface heat flux instead of a volume heat source, as represented as Equation (4):  
\[q_{z=0}=I(t)[n\cdot e_{z}] \tag{4}\]  
where \(n\) is a unit vector normal to the material surface, \(e_{z}\) is a unit vector parallel to the \(z\) axis. On the basis of physical assumption 3, the forced heat convection of the water jet should be considered and calculated. Unlike the loading of the laser heat flux, the forced heat convection of the water jet occurs throughout the LWJ machining on the contact area between the water jet and substrate. The coefficient of convective heat transfer can be expressed as follows:  
\[h_{fw}=Nuk_{w}/l \tag{5}\]  
where \(Nu\) is the Nusselt number, \(k_{w}\) is the thermal conductivity of water, and \(l\) is the feature size. In particular, the Nusselt number indicates a close relationship with the status of the affected zone of LWJ, with a proportion quite to the stagnation cooling area referred to in physical assumption (3). According to [PERSON] and [PERSON] [21], an empirical formula of \(Nu\) is obtained as Equation (6):  
\[Nu=\left\{ \begin{array}{ll}0.715 Re^{1/2}Pr^{0.4}&0.15<Pr<3\\ 0.797 Re^{1/2}Pr^{1/3}&Pr>3\end{array} \right.