## 3 CFD Model of the Large Diameter Modular Drill Bit
### Governing Equations and Boundary Conditions for the CFD Simulation
In Equation (1), the expression of \(P_{k}\) is:  
\[P_{k}=\mu_{T}\
abla\mathbf{u}:\Big{(}\
abla\mathbf{u}+(\
abla\mathbf{u})^{T}\Big{)}. \tag{3}\]  
In Equation (2), the coefficient \(C_{1}\) is related to the time-averaged strain rate, \(\mathbf{\tau}\), of the fluid velocity gradient, and its expression is:  
\[C_{1}=max\left[0.43,\frac{\eta}{5+\eta}\right], \tag{4}\]  
where \(\eta\) is the averaged strain rate tensor modulus, whose expression is \(\eta=S\frac{k}{\varepsilon}\). \(S\), and \(\varepsilon\) can be expressed as [33; 34]:  
\[\left\{ \begin{array}{l}S=\sqrt{2\mathbf{\tau}:\mathbf{\tau}}\\ \tau=\frac{1}{2}\Big{(}\
abla\mathbf{u}+(\
abla\mathbf{u})^{T}\Big{)}\end{array} \right.. \tag{5}\]  
In addition, the turbulent dynamic viscosity, \(\mu_{T}\), is:  
\[\mu_{T}=\rho C_{\mu}\frac{k^{2}}{\varepsilon}. \tag{6}\]  
in which \(C_{\mu}\) is a variable that changes with the strain rate and is related to the time-averaged strain rate \(\mathbf{\tau}\). Its expression is:  
\[C_{\mu}=\frac{1}{A_{0}+A_{s}U^{(*)}\frac{k}{\varepsilon}}. \tag{7}\]  
where \(U^{(*)}\) is internal energy, J, which can be expressed as:  
\[U^{(*)}=\sqrt{\mathbf{\tau}:\mathbf{\tau}+\mathbf{\Omega}:\mathbf{\Omega}}. \tag{8}\]  
\[\mathbf{\Omega}=\frac{1}{2}\Big{(}\
abla\mathbf{u}-(\
abla\mathbf{u})^{T}\Big{)}. \tag{9}\]  
As there is no rotational motion when drilling fluid flows in a borehole, for calculating \(U^{(*)}\), the second term in the square root in Equation (8) is zero. In Equation (9), \(\mathbf{\Omega}\) is the time-averaged rotational rate tensor, which is derived from the angular velocity reference system. Other parameters are represented as follows:  
\[\left\{ \begin{array}{l}A_{0}=4.04\\ A_{s}=\sqrt{6}\text{cos}\Big{(}\frac{1}{3}\arccos\Big{(}\sqrt{6}W\Big{)}\Big{)} \\ W=\frac{2\sqrt{2}\textbf{r}(\cdot\textbf{r}\cdot\textbf{r})}{|\textbf{r}|^{3}} \end{array} \right.. \tag{10}\]  
The main parameters for the modular bit model, along with their respective values, are provided in Table 2.  
The velocity of the drilling fluids flowing upward along the borehole and the pressure loss after drilling fluids have flowed through the bit body have enormous implications for the drilling process, as a sufficiently high velocity can ensure effective cooling of the bit and removal of rock cuttings, but high pressure loss may increase the burden of the mud pumps. Thus, the inlet boundary is set as flow rate boundary with the fully developed mass flow in this paper. For wireline coring drilling with an impregnated diamond bit in geological exploration, the velocity of the drilling fluids flowing upward from the borehole should reach at least 0.5 m/s to 1.5 m/s to carry rock cuttings effectively. To analyze flow field characteristics of drilling fluid flowing in the modular drill bit, a bit was designed with an outer diameter of 216 mm. Thus, the inlet flow rate should be within the range of 300 L/min to 600 L/min, corresponding to a velocity of 0.5 m/s to 1.5 m/s. The boundary condition set for the outlet is specified as a pressure boundary, with a value of 0, which means that the pressure at the outlet is consistent with the ambient pressure. The wall boundaries of the modular bit CFD model are set as fixed no-slip boundaries. To calculate the simulation model, a segregated stationary solver was adopted, with the turbulent fluid discretization set to P1 + P1 (first order form function to discretize the velocity and pressure fields) and a relative tolerance set to 0.01. Stabilization and acceleration of pseudo time-stepping were selected. The maximum number of iterations, tolerance factor, and residual factor of the solver were set by default.