## 5 Numerical Simulation and Results Discussion
### Numerical Results Discussion  
Figure 8 shows that the surface of workpiece is composed of three different arcs. There are several poles and inflection points. Oscillations and fluctuations of energy in the fluid flow have a direct relevancy to the shape of workpiece. According to the structure of multi-segment constrained modules, the profiling constrained flow passage can be divided into five regions: the radius of L1 (X range: 0-0.01259 m) is 0.01 m; L2 (X range: 0.01259-0.02224 m) is a plane; the radius of L3 (X range: 0.02224-0.03649 m) is 0.03 m; L4 (X range: 0.03649-0.04535 m) is a plane; and the radius of L5 (X range: 0.04535-0.06 m) is 0.01 m. For optimizing the constrained flow passage, the velocities of particles and the contact forces between particles and workpiece are analyzed by the computational fluid dynamics (CFD) method. The distance between the observation plane and workpiece is set as 0.005 mm, and principal parameters are listed in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Velocity/m\(\cdot\)s\({}^{-1}\)** & **Dynamic Viscosity/(Pa-s)** & **Particle** & **Volume Fraction/\%** & **Hydraulic Diameter/m** & **Length 1/m** \\ \hline
10 & 2.06 & SiC & 10 & 0.008 & 0.06 \\ \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of abrasive flow. Figure 8: Division of curvature region: 1, inlet: 2, workpiece; 3, constrained flow passage; 4, outlet. Figure 7: Numerical model of profiling constrained flow passage for titanium alloy complex surface: (a) orthogonal view; (b) local enlarged view; and (c) local enlarged view of a region. Velocities of particles in profiling constrained flow passage are shown in Figure 9. In Figure 9a, the X velocities in the L1 and L5 are close to 0. Figure 9b shows that the Y velocities in the L1 are obviously decreased, but the Y velocities in L5 are obviously increased. Because particles vertically enter L1 and leave L5, the X velocities in L1 and L5 are close to 0. Due to the particles moving parallel to the workpiece along with fluid after particles enter L1, the included angle between vectors of particles and vertical direction becomes larger, and the Y velocities in L1 are decreased. The velocity variation in L5 is similar. The X velocities of particles in L2, L3 and L4 are the same, but Y velocities of particles in L2, L3 and L4 are changed slowly. From the flow field profile results, we can conclude the following laws. (a) On the different machined surface, velocities direction of particles participating in cutting are changed. In the plane, the X velocities and Y velocities of particles are not change nearly. (b) On the curved surface, the amplitudes of X velocities and Y velocities are changed with the radius of the curved surface. The X/Y velocities of particles can be obtained by CFD method, but the tangential velocity must be obtained by calculation for further study, as shown in Figure 10. The trend of the tangential velocity in L1 and L5 is similar to the Y velocity, and the maximum in L1 is bigger than in L5. In L2, the tangential velocity is gradually decreased, but, in L4, the tangential velocity is gradually decreased and the changing speed is smaller. In L3, the tangential velocity is gradually increased. On the curved surface, the bigger the radius of the curved surface is, the more quickly the velocity of the particle changes. Figure 9: Velocity vectors of particles: (**a**) X velocities of particles; and (**b**) Y velocities of particles. Based on the numerical simulation results of the tangential velocity, the following regularities can be acquired. Because of the linear loss, the tangential velocity is gradually decreased. On the impact face, the tangential velocity is gradually increased, but, on the non-impact face, the tangential velocity is gradually decreased, and the changing speed is much bigger than the changing speed caused by the linear loss. To obtain the affecting forces of particles on the workpiece, L1 and L5 are evenly divided into 14 planes, respectively. L3 is evenly divided into 12 planes.