# Flow-Induced Strength Analysis of Large Francis Turbine Under Extended Load Range
## 2 Methodology and Mathematical Physics Equations  
### Fluid Dynamics Governing Equations  
The governing equations for turbulent fluid flow in the hydraulic Francis turbine units are based on the fundamental laws of mass, momentum, and energy conservation. Since the water in hydraulic turbines is thought to be incompressible and at a constant temperature, the Reynolds-averaged Navier-Stokes (RANS) equations can be used to explain the fundamental laws of mass and momentum conservation:  
\[\frac{\partial\overline{u_{i}}}{\partial t}+\frac{\partial(\overline{u_{i}u_{j}} )}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_{i}} +\frac{\partial}{\partial x_{j}}[\
u(\frac{\partial\overline{u_{i}}}{\partial x _{j}}+\frac{\partial\overline{u_{j}}}{\partial x_{i}})]-\frac{\partial \overline{u_{i}^{\prime}u_{j}^{\prime}}}{\partial x_{i}}+\overline{f_{i}} \tag{1}\]  
\[\frac{\partial\overline{u_{i}}}{\partial x_{i}}=0 \tag{2}\]  
where \(i\) and \(j\) are indicators which fall in \(1,2,3\), \(x_{i}\) and \(x_{j}\) are the Cartesian coordinate directions, \(\overline{u}_{i}\) and \(\overline{u}_{j}\) are the time-averaged terms of the fluid velocity, and \(\overline{u}_{i}^{\prime}\) and \(\overline{u}_{j}^{\prime}\) are time-fluctuating terms of the fluid velocity. Time is represented by \(t\), fluid density by \(\rho\), time-averaged pressure by \(\overline{p}\), kinematic viscosity by \(\
u\), Reynolds stress tensor by \(\overline{u_{i}^{\prime}u_{j}}\), and time-averaged external forces by \(\overline{f_{i}}\).  
The RANS equations introduce an additional unknown quantity, the Reynolds stress tensor \(\overline{u_{i}^{\prime}u_{j}^{\prime}}\), which accounts for the effects of turbulent fluctuations on the mean flow field [28]. To close the RANS equations, various turbulence models are introduced, which parameterize the Reynolds stress tensor in terms of the mean flow properties and turbulence quantities. The \(k\)-\(\epsilon\) model is based on transport equations for the turbulent kinetic energy \(k\) and its dissipation rate \(\epsilon\), and assumes an isotropic eddy viscosity to relate Reynolds stresses to mean flow gradients [29]. Utilizing transport equations for \(k\) and the turbulent  
Figure 2: The workflow of flow-induced strength analysis of the Francis turbine in this study.  
kinetic energy dissipation rate \(\omega\), the \(k\)-\(\omega\) model improves predictions in near-wall regions and for flows with adverse pressure gradients [30]:  
\[k=\frac{1}{2}\overline{u^{\prime}_{i}u^{\prime}_{j}} \tag{3}\]  
\[\varepsilon=\
u(\overline{\frac{\partial u^{\prime}_{i}}{\partial x_{k}}\frac{ \partial u^{\prime}_{i}}{\partial x_{k}}}) \tag{4}\]  
\[\omega=\frac{\varepsilon}{k} \tag{5}\]  
These turbulent models utilize the Boussinesq hypothesis, introducing the turbulent viscosity \(\upsilon_{T}\) to express the Reynolds stress tensor as  
\[\overline{u^{\prime}_{i}u^{\prime}_{j}}=-\upsilon_{T}(\overline{\frac{ \partial\overline{u}_{i}}{\partial x_{j}}}+\frac{\partial\overline{u}_{j}}{ \partial x_{i}})+\frac{2}{3}k\delta_{ij} \tag{6}\]  
The Shear Stress Transport (SST) model is a hybrid approach that combines the advantages of the \(k\)-\(\omega\) model in near-wall regions with the \(k\)-\(\varepsilon\) model in the free-flow regions, providing enhanced accuracy in separating flows [31]. The SST model is widely used for the turbulent flow simulations for hydraulic turbines and therefore also adopted in this investigation:  
\[\frac{\partial\omega}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline{ u}_{j}\omega)=\frac{\partial}{\partial x_{j}}\left[(\upsilon+\frac{ \upsilon_{T}}{\sigma_{\omega_{2}}})\frac{\partial\omega}{x_{j}}\right  
\[\arg_{2}=\max\left(\frac{2\sqrt{k}}{\beta^{\prime}\omega y^{\prime}},\frac{500v}{y^ {2}\omega}\right) \tag{16}\]  
The other parameters are \(\alpha_{1}=0.31\), \(\alpha_{2}=0.44\), \(\beta^{\prime}=0.09\), \(\beta_{2}=0.0828\), \(\sigma_{k_{2}}=1\), and \(\sigma_{\omega_{2}}=1/0.856\).