# The Effect of Blade Angle Deviation on Mixed Inflow Turbine Performances
## 2 Materials and Methods
### Flow Simulations in a Mixed Inflow Turbine  
The three-dimensional Navier-Stokes equations consist of conservation equations of mass (8), momentum (9), and energy (10), supplemented by the ideal gas laws (11). The flows in the turbomachines are mainly gaseous and generally consist of air, which allows us to neglect the volume forces in the system of equations.  
\[\frac{\partial}{\partial t}\iiint\rho.dv+\iint\rho V\;dA=0 \tag{8}\]  
\[\iint(\rho V.dA)V+\iiint\frac{\partial(\rho V)}{\partial t}.dv=\iiint\rho f.dv- \iint p.dA \tag{9}\]  
\[\iiint q\rho.dv-\iint pV.dA+\iiint\rho(f.V)dv=\iiint\frac{\partial}{\partial t }\left[\rho\left(e+\frac{V^{2}}{2}\right)\right]dv+\iint\rho(e+\frac{V^{2}}{2} )V.dA \tag{10}\]  
\[p=\rho RT \tag{11}\]  
Generally, all the flows in the turbomachines are turbulent. The RANS method is used, and the statistical treatment of turbulence in this method assumes that each instantaneous turbulent quantity can be decomposed into a mean amount and a turbulent fluctuation. This decomposition in the middle and in the fluctuating parts, applied to the Navier-Stokes equations, brings up new terms related to turbulence. The loss of information by the average operation leads to a system with more unknowns than equations and which requires closure models to complete it. The standard \(k\)-\(\epsilon\) turbulence model is used because it gives numerical results closer to experimental results in turbomachinery flows; specifically, that the \(y+\) value for the near-wall node has to be in the range of 20 to 100. This model is based on the eddy viscosity concept, which assumes that the Reynolds stresses \(-\rho\overline{u_{i}u_{j}}\) can be expressed in terms of the mean velocity gradients and the eddy or the turbulent viscosity \(\mu_{t}\), in a manner analogous to the viscous stresses \(\tau_{ij}\) for laminar Newtonian flows:  
\[\tau_{ij}=u\left(\frac{\partial U_{j}}{\partial X_{i}}+\frac{\partial u_{i}}{ \partial X_{j}}\right)-\frac{2}{3}\mu\delta_{ij}\frac{\partial U_{k}}{ \partial X_{k}}, \tag{12}\]  
\[-\rho\overline{u_{i}u_{j}}=\mu_{t}\left(\frac{\partial U_{j}}{\partial X_{i}} +\frac{\partial u_{i}}{\partial X_{j}}\right)-\frac{2}{3}\mu_{t}\delta_{ij} \frac{\partial U_{k}}{\partial X_{k}}-\frac{2}{3}\delta_{ij}\rho_{k} \tag{13}\]  
Figure 4: The mesh refinement near the wall [42].  
This model assumes that the eddy viscosity \(\mu_{t}\) is related to the turbulent kinetic energy \(k\) and its dissipation rate \(\varepsilon\) presented by the relation (14):  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon} \tag{14}\]  
where \(C_{\mu}=0.09\) and \(k\), \(\varepsilon\) are defined by the Equations (15) and (16):  
\[\frac{\partial(\rho k)}{\partial t}+\
abla.\left(\rho\dot{k}\dot{U}\right)= \
abla.\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\
abla k\right]+P_{k} -\rho_{\varepsilon} \tag{15}\]  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\
abla.\left(\rho\dot{k}\dot{U} \right)=\
abla.\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\
abla \varepsilon\right]+\frac{\varepsilon}{k}(C_{1\varepsilon}P_{k}-C_{2 \varepsilon}\rho_{\varepsilon}). \tag{16}\]  
In this model \(P_{k}\) is the turbulence production and \(\sigma_{k}=1.00\), \(\sigma_{\varepsilon}=1.30\), \(C_{1\varepsilon}=1.44\), \(C_{2\varepsilon}=1.92\) are constants determined experimentally.