## 2 Materials and Methods  
### Jet Motion Control Equation  
When water is ejected from the nozzle at a high speed, the fluid is mostly in a turbulent state. So, its motion can be described by the continuity equation of fluid dynamics and the equation of motion after Reynolds time averaging [31]:  
\[\frac{\partial\rho u_{i}}{\partial t}+\frac{\partial\rho u_{i}u_{j}}{\partial x _{j}}=-\frac{\partial P}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}(- \rho\overline{u^{\prime}_{j}u^{\prime}_{i}})+\frac{\partial\tau_{ij}}{\partial x _{j}}+\rho g_{i} \tag{1}\]  
\[\frac{\partial\rho u_{i}}{\partial x_{i}}+\frac{\partial\rho}{\partial t}=0 \tag{2}\]  
where \(\rho\) is density of fluid; \(P\) is pressure; \(t\) is time; \(u_{i}\) and \(u_{j}\) are the flow velocity of the \(i\) direction and \(j\) direction; and \(x_{i}\) and \(x_{j}\) are the Cartesian coordinates. \(\tau_{ij}\) is the strain rate tensor, and the specific expression is as follows:  
\[\tau_{ij}=\mu(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{k}}{\partial x_{k}}) \tag{3}\]  
The turbulent motion of the water jet is described by a two-path Realizable \(k-\varepsilon\) model which can predict the jet diffusion flow. The Reynolds stress term in Equation (1) is calculated using the following formula:  
\[-\rho\overline{u^{\prime}_{j}u^{\prime}_{i}}=\mu_{t}(\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}})-\frac{2}{3}\delta_{ij} (\rho k+u_{t}\frac{\partial u_{k}}{\partial x_{k}}) \tag{4}\]  
where \(\mu_{t}=\rho C_{u}\frac{k^{2}}{\varepsilon}\) is the turbulent eddy viscosity; \(k\) is the turbulent kinetic energy; and \(\varepsilon\) is the turbulent dissipation rate; \(C_{u}=\left(A_{0}+A_{s}\frac{k\Omega t^{s}}{\varepsilon}\right)^{-1}\),  
\[\left\{ \begin{array}{l}U^{*}=\sqrt{S_{ij}S_{ij}+\Omega_{ij}\Omega_{ij}}\\ \Omega_{ij}=\frac{1}{2}(\frac{\partial u_{i}}{\partial x_{j}}-\frac{\partial u _{j}}{\partial x_{i}})\\ A_{s}=\sqrt{6}\cos(\phi)\\ \phi=\frac{1}{3}\cos^{-1}(\sqrt{6}\omega)\\ \omega=\frac{S_{ij}S_{jk}S_{jk}}{(\frac{s}{\sqrt{2}})}\\ S=\sqrt{2S_{ij}S_{ij}}\\ S_{ij}=\frac{1}{2}(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}})\end{array} \right. \tag{5}\]  
The Realizable k-epsilon model is widely used as a turbulence model for simulating complex flows [32; 33]. This model, boasting strong reliability, wide application range and high computational efficiency, can predict the flow behavior of fluids better to a certain extent. According to the actual calculation results, the results calculated by choosing the Realizable k-epsilon model are very close to those of the k-omega model. Therefore, the k-epsilon model is finally selected to ensure computational efficiency in this paper. In the Realizable \(k-\varepsilon\) model, turbulence kinetic energy and dissipation rate are, respectively, solved using the transport equations as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{j})}{\partial x_{ j}}=\frac{\partial}{\partial x_{j}}((\mu+\frac{\mu_{t}}{\sigma_{k}})\frac{ \partial k}{\partial x_{j}})+G_{k}+G_{b}-\rho\varepsilon \tag{6}\]\[\frac{\partial\rho e}{\partial t}+\frac{\partial\rho eu_{j}}{\partial x_{j}}=\frac{ \partial}{\partial x_{j}}((\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}})\frac{ \partial e}{\partial x_{j}})+\rho C_{1}Se-\rho C_{2}\frac{\epsilon^{2}}{k+ \sqrt{v\epsilon}}+G_{1\epsilon}\frac{\epsilon}{k}C_{3}G_{b} \tag{7}\]  
where \(G_{k}=-\rho\overline{u^{\prime}_{j}u^{\prime}}_{i}\frac{\partial u_{j}}{ \partial x_{i}}\) and \(G_{b}=\gamma g_{i}\frac{\mu_{t}}{\Pr_{t}}\frac{\partial T}{\partial x_{i}}\), respectively, represent the average velocity gradient and the function of the air; \(\Pr_{t}=0.85\) is the Prandtl number of turbulent energy; \(\gamma=-\frac{1}{\rho}(\frac{\partial\rho}{\partial T})\) represents the characteristic expansion coefficient.