## 2 Numerical Model  
In this paper, the phenomenon of detonation reflection is described by Euler equation [42] without considering dissipation effects such as viscosity, heat conduction and diffusion. An Additive Runge-Kutta (ARK) time step method is used to integrate the governing equations to avoid rigidity of chemical sources. A 9 species, 48 reaction mechanism is adopted. The reacting species are H\({}_{2}\), O\({}_{2}\), H, O, OH, HO\({}_{2}\), H\({}_{2}\)O\({}_{2}\), H\({}_{2}\)O and Ar. A fifth-order Weighted Essentially Non-Oscillatory scheme is used to integrated on convective terms.  
Either with commercial software or self-developed software, the grid subdivision is crucial for discrete area in calculation. At the same time, the grid subdivision is closely related to the calculation format. Thus, the grid subdivision quality has direct influence on the accuracy of the calculation result. The object discussed in this paper is smooth bend with different curvatures and the structure is not regular, which is different from wedge surface and treated as series of micro yuan of wedge surface with continuously changing angle. This is a more complex structure. The wall boundary conditions are difficult to give out in high precision calculation with rectangular grid format and rules. Thus, the body-fitted curve coordinate system is adopted and the curve coordinate will be transformed to the isometric partitioning coordinates in calculation. This can simplify the treatment to geometric boundary conditions. In the curve coordinates, the reaction Euler equation is as follows:  
\[\frac{\partial\widetilde{\mathbf{U}}}{\partial\tau}+\frac{\partial\widetilde {\mathbf{F}}}{\partial\widetilde{\zeta}}+\frac{\partial\widetilde{\mathbf{G}} }{\partial\eta}=\widetilde{\mathbf{S}}\]  
\[\widetilde{\mathbf{U}}=\frac{\mathbf{U}}{\mathbf{J}},\ \widetilde{\mathbf{F}}=\frac{1}{ \mathbf{J}}\big{(}\xi_{x}\mathbf{F}+\xi_{y}\mathbf{G}\big{)},\ \widetilde{\mathbf{G}}=\frac{1}{\mathbf{J}}\big{(}\eta_{x}\mathbf{F}+\eta_{y} \mathbf{G}\big{)},\ \widetilde{\mathbf{S}}=\frac{\mathbf{S}}{\mathbf{J}},\ \mathbf{J}=\bigg{|} \frac{\partial(\xi,\eta)}{\partial(x,y)}\bigg{|},\]  
\[U=\{\rho,\rho u,\rho v,E,\rho Y_{1}\ldots\rho Y_{N}\};\ F(U)=\Big{\{}\rho u, \rho u^{2}+p,\rho uv,(E+p)u,\rho uY_{1}\ldots\rho uY_{N}\Big{\}};\]  
\[G(U)=\Big{\{}\rho v,\rho uv,\rho v^{2}+p,(E+p)v,\rho vY_{1}\ldots\rho vY_{N} \big{\}};\ S=\{0,0,0,0,\omega_{1}\ldots\omega_{N}\}\]  
where J is Jacobi determinant; \(\zeta_{x}\), \(\zeta_{y}\), \(\eta_{x}\), \(\eta_{y}\) are transforming coefficients of curvilinear coordinates and calculating coordinates, respectively; others are similar to traditional Euler equations.  
The whole bent pipe is composed of a turning angle section, a straight pipe section before the turning angle and a straight pipe section after the turning angle. The turning angle among \(\theta=15^{\circ}\),\(20^{\circ}\),\(30^{\circ}\),\(40^{\circ}\),\(45^{\circ}\), \(60^{\circ}\) respectively. The channel width is five times of the cell width and divided into\(1500\times 300\) grids with size of 0.1 mm. The initial conditions are set the same as [43]: the concentration is stoichiometric H\({}_{2}\)/O\({}_{2}\) mixture diluted with 70% argon. The numerical simulation we consider is the evolution of the cellular detonation wave propagating through realistic smooth pipe bends of different angle in two space dimensions at an initial pressure and temperature of 6.67 kPa and 298 K, respectively. In order to simplify the computational initialization process, a cellular detonation is located on the left. The entrance is extrapolation boundary, the exit is Neumann boundary and other boundaries are solid wall. Figure 1 shows a schematic view of the bend.