# Parameter Analysis and Optimization of Annular Jet Pump Based on Kriging Model
## 3 CFD Modeling and Verification
### CFD Model  
The annular jet pump calculation domain is shown in Figure 2. According to the existing research results [4], the initial model parameters are \(w=4\) mm, \(r=21.5\) mm, \(t=2\) mm, \(r_{0}=27.5\) mm, \(r_{t}=38\) mm, \(\alpha=18^{\circ}\), \(L_{t}=179\) mm, \(\beta=5.8^{\circ}\), \(m=2.27\). The sum of \(w\), \(t\) and \(r\) equals \(r_{0}\) in the optimization process. The wall thickness of suction pipe, t, is constant. [PERSON] [23] studied the performance over a large range of design parameters of jet pumps by a 2D simulation model. The result showed that the two-dimensional model still well simulated the flow field characteristics and jet pump performance of the actual model, ignoring the velocity in the z direction. Therefore, the 2D axisymmetric model was adopted in order to reduce the calculation cost. In the analysis and optimization of the jet pump, the suction angle \(\alpha\), the diffusion angle \(\beta\) and the area ratio \(m\) among the structural parameters--as shown in the Figure 2--plus the flow ratio \(q\), are determined as the design variables. The annular jet pump internal flow mechanism is complicated. The assumptions are made in CFD simulation for studying the jet pump:  
1. the mixing process is considered as a steady and incompressible state;
2. the heat transfer of the external environment and the fluid medium is ignored;
3. the solid wall is considered smooth;
4. the buoyancy effect is ignored. The continuity and momentum equations are listed as, because of these assumptions:  
\[\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{5}\]  
\[\frac{\partial\Big{(}\rho u_{j}u_{i}\Big{)}}{\partial x_{j}}=\frac{\partial} {\partial x_{j}}\bigg{[}\mu\frac{\partial u_{i}}{\partial x_{j}}-\rho\overline {u_{i}u_{j}}\bigg{]}-\frac{\partial p}{\partial x_{i}} \tag{6}\]  
where Reynolds stresses are  
\[-\rho\overline{u_{i}u_{j}}=\mu_{t}\bigg{[}\frac{\partial u_{i}}{\partial x_{j }}+\frac{\partial u_{j}}{\partial x_{i}}\bigg{]}-\frac{2}{3}\rho k\delta_{ij} \tag{7}\]  
In these equations, \(u_{i}\) is velocity component, \(x_{i}\) is space coordinate, \(\delta\) is boundary layer thickness, \(\mu\) is dynamic viscosity, \(\mu_{t}\) is turbulent viscosity, and \(k\) is turbulence kinetic energy. Due to the jet pump internal flow complexity, it is necessary for ensuring the simulation accuracy to choose the appropriate turbulence model. The existing research [24] shows that standard wall  
Figure 2: The calculation domain. function and the realizable \(k\)-\(\varepsilon\) model can accurately simulate annular jet pump flow details and calculate the performance index. The realizable \(k\)-\(\varepsilon\) model [25] has advantages in simulating flow field details compared with the standard \(k\)-\(\varepsilon\) mode, such as vortices, and rotation. The modified transport equations for \(k\) and \(\varepsilon\) in the realizable \(k\)-\(\varepsilon\) model are:  
\[ \begin{array}{ll}\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_ {j})}{\partial x_{j}}=&\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{ \partial u_{j}}{\partial k}\right)\frac{\partial(k)}{\partial x_{j}}\right]\\ &+G_{k}+G_{b}-\rho\varepsilon-Y_{M}+S_{k}\end{array}  \tag{8}\]  
\[ \begin{array}{ll}\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial \left(\rho\varepsilon u_{j}\right)}{\partial x_{j}}=&\frac{\partial}{ \partial x_{j}}\left[\left(\mu+\frac{\partial u_{j}}{\partial x_{j}}\right) \frac{\partial\varepsilon}{\partial x_{j}}\right]+\rho C_{1}S\varepsilon\\ &-\rho C_{2}\frac{\varepsilon^{2}}{k+\sqrt{\varepsilon}}+C_{1\varepsilon} \frac{\varepsilon}{k}C_{3\varepsilon}G_{b}+S_{\varepsilon}\end{array}  \tag{9}\]  
where  
\[C_{1}{=\max\left[0.43,\frac{\eta}{\eta+5}\right]},\eta=S\frac{k}{\varepsilon},S=\sqrt{2S_{ij}S_{ij}} \tag{10}\]  
In these equations, \(S\) is strain rate magnitude, \(G_{k}\) represents the generation of turbulence kinetic energy due to the mean velocity gradients. \(G_{b}\) is the generation of turbulence kinetic energy due to buoyancy.