# Measurement Method of Whole Mechanical Automatic Vertical Drilling Tools
## 2 Methods
### Numerical Simulation  
2.5.1. Numerical Simulation of Torque  
Because of the difficulty in measuring the specific value of the disc valve deflection angle in the experiment, simulation auxiliary experiments were conducted in this section to jointly verify the accuracy of the mathematical model. Firstly, the fluid working condition was simulated by the CFD method, and \(\varphi\) was taken as the independent variable to obtain the curve of M changing with \(\varphi\), which was briefly denoted as M(\(\varphi\)). Then, the working process of the regulating device under the action of fluid was studied. For the double-biased block in the N-MVDT, its motion was simulated by multibody dynamic simulation. It was input as an influencing factor to determine the influence of adding \(M\) on \(\varphi\), and then the ranges of two kinds of vertical drilling deflection angles \(\varphi\) were obtained.  
2.5.2. Control Equation  
A continuous equation was applied in this work based on the finite volume method. The inner flow field of the tool, which is three-dimensional incompressible flow, was described by the Navier-Stokes equation. The turbulent model adopted the \(k\)-\(\varepsilon\) turbulence model, which modifies turbulent viscosity considering the average flow of rotation and rotation flow and can better handle the high strain rate and degree of streamline curvature flow.  
In the standard \(k\)-\(\varepsilon\) turbulence model, \(k\) and \(\varepsilon\) are two basic unknowns, and the corresponding transport equation is  
\[\frac{\partial\rho}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho k u _{i})=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{i}}{\delta_{k }}\right)\frac{\partial k}{\partial x_{j}}\right]+G_{k}+G_{b}-\rho\varepsilon -Y_{m}+S_{k} \tag{14}\]  
\[\frac{\partial\rho}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho k u _{i})=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{i}}{\sigma_{ \varepsilon}}\right)\frac{\partial k}{\partial x_{j}}\right]+C_{1\varepsilon }\frac{\varepsilon}{k}(G_{k}+G_{3\varepsilon}G_{b})-C_{2\varepsilon}\rho\frac {\varepsilon^{2}}{k}+S_{\varepsilon} \tag{15}\]  
where \(G_{k}\) is the generation term of turbulent kinetic energy \(k\) caused by the average velocity gradient, is the generation term of turbulent kinetic energy \(k\) caused by buoyancy, is the contribution of pulsation expansion in compressible turbulent pulsation, \(C_{1\varepsilon}\), \(C_{2\varepsilon}\), and \(C_{1\varepsilon}\) are empirical constants, \(\sigma_{k}\) and \(\sigma_{\varepsilon}\) are the Prandtl numbers corresponding to turbulent kinetic energy k and dissipation rate \(\varepsilon\), respectively, and \(S_{k}\) and \(S_{\varepsilon}\) are user-defined source terms.  
2.5.3. Calculation Model  
To investigate the impact of the newly developed enhanced stability platform, it was imperative to initially simulate the regulating device using computational fluid dynamics (CFD). The three-dimensional model of the regulating device was established, which mainly included the inner cylinder valve, the outer cylinder valve, and the turbine. Then, the fluid domain model of the 3-D model of the regulating device was inversely calculated, and this model was used as the simulation object of the CFD simulation. The process is shown in Figure 12.