## 3 Results and Discussion
### Mean Velocity Profiles  
Figure 4 compares the dimensionless x-velocity (\(U/U_{M}\)) profiles predicted by six low-Reynolds \(k\)-\(\varepsilon\) number models with PIV measurements at x/L = 0.07, 0.27 and 0.47 for Re = 1000, 1750, and 2500, respectively. \begin{table}
\begin{tabular}{c c c} \hline
**Model** & \(k_{bc}\) & \(\varepsilon_{bc}\) \\ \hline AB & \(k_{w}=0\) & \(\varepsilon_{w}=v\big{(}\partial^{2}k/\partial y^{2}\big{)}\) \\ LB & \(k_{w}=0\) & \(\varepsilon_{w}=v\big{(}\partial^{2}k/\partial y^{2}\big{)}\) \\ LS & \(\left(\partial k/\partial y\right)_{w}=0\) & \(\varepsilon_{w}=0\) \\ YS & \(k_{w}=0\) & \(\varepsilon_{w}=2v\big{(}\partial\sqrt{k}/\partial y\big{)}^{2}\) \\ AKN & \(k_{w}=0\) & \(\varepsilon_{w}=2v\big{(}k_{P}/y_{P}^{2}\big{)}\) \\ CHC & \(k_{w}=0\) & \(\varepsilon_{w}=v\big{(}\partial^{2}k/\partial y^{2}\big{)}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Turbulent kinetic energy and dissipation rate boundary conditions of six low-Reynolds number \(k\)-\(\varepsilon\) models. Figure 3: Grid independence test predicted by AB model. At x/L = 0.07 (Figure 4a,d,g), the YS model predicts erroneous results, especially under the jet region (y/L \(<\) 0.94), and the LB model underpredicts the velocity value. The other four turbulence models predict better results, but there are still some problems: the thickness of the wall jet is too thick. In Figure 4d,g, the YS, AKN, and CHC models predict negative dimensionless velocities near the wall, which means that the jets predicted by these three models are attached to the wall too late. At x/L = 0.27 (Figure 4b,e,h), the LB and YS models still predict totally wrong results of velocity profiles. The AKN model provides a good agreement with the measurement data at Re = 1000 while performing poor at Re = 1750 and 2500. At x/L = 0.07 and 0.27, the thickness of boundary layer predicted by six models is thinner than that of PIV measurements, where the jet has not yet separated from the wall. This may be due to a seriously too-low level of \(\varepsilon\) near the wall in separated flows or flows approaching separation [37]. When the flow does not separate from the wall (at x/L = 0.07 and 0.27), the results obtained by AB and LS models are more accurate. At x/L = 0.47 (Figure 4c,f,i), all six models predict negative dimensionless velocities near the wall, indicating that the boundary layer predicted by the six models starts to separate from the wall. These models underpredict the position of separation of the wall jet. In this situation, only the LS model shows the best agreement of the boundary layer with the PIV measurements. Most of the low-Reynolds number \(k\)-\(\varepsilon\) models (except the LB and YS models) predict almost the same velocity profiles in the recirculation region (y/L < 0.9) but cannot predict well in the boundary layer (0.97 < y/L) and shear layer (0.9 < y/L < 0.94) between the jet region and recirculation region. There is a large velocity gradient in the boundary layer and shear layer, which leads to a strong anisotropic effect. The isotropic Boussinesq hypothesis has some errors in those regions. The LB model provides the worst agreement with experiment data. The velocity predicted by the LB model is always too large in the recirculation region. In general, the LS model performs best in predicting velocity profiles. Because the extra \(D\) and \(E\) terms of the \(k\) and \(\varepsilon\) equations in the LS model are expressed as \(2\
u t\left(\partial\sqrt{k}/\partial y\right)^{2}\) and \(2\
u t_{t}\left(\partial^{2}\overline{U}/\partial y^{2}\right)^{2}\), respectively, rather than most of the turbulence models used in this paper, the extra terms \(D\) and \(E\) are equal to 1. The \(D\) term in the LS model is added to consider that the dissipation of kinetic energy is not isotropic in the viscous sublayer, and the \(E\) term is aimed to make the simulation result of kinetic energy fits better with some experiments. The two extra terms can cover the shortage of isotropic Boussinesq hypothesis in some ways [38].