## 3 CFD Setup  
### Governing Equations  
The incompressible flow is modeled by the URANS equations that are expressed in Cartesian coordinates as follows:  
\[\frac{\partial\overline{u}_{j}}{\partial x_{j}}= 0 \tag{2}\] \[\frac{\partial\rho\overline{u}_{i}}{\partial t}+\frac{\partial \rho\overline{u}_{j}\overline{u}_{i}}{\partial x_{j}}= -\frac{\partial\overline{p}}{\partial x_{i}}+\frac{\partial \sigma_{ij}}{\partial x_{j}}+\frac{\partial\tau_{ij}}{\partial x_{j}} \tag{3}\]  
with:  
* \(\rho\) the fluid density. * \(\overline{p}\) the mean pressure. * \(\overline{u}_{i}\) the mean velocity vector. * \(\sigma_{ij}\) the viscous stress tensor computed using a Newtonian linear constitutive equation. * \(\tau_{ij}=-\rho\overline{u_{i}u_{j}^{\prime}}\) the Reynolds stress tensor (an instantaneous variable \(a\) is decomposed following the Reynolds decomposition in a mean part \(\overline{a}\) and a fluctuating part \(a^{{}^{\prime}}\)). Two types of closure models are considered to compute the Reynolds stress tensor: models based on [PERSON]'s assumption and models based on an explicit algebraic formulation. [PERSON]'s assumption considers a closure formulation of the Reynolds stress tensor (see Equation (4)) by analogy with the Newtonian linear constitutive equation. This assumption reduces the six unknown components of the Reynolds stress tensor to one scalar \(\mu_{t}\), classically called the eddy viscosity. In order to compute the eddy viscosity, various approaches based on an algebraic formulation or one- or two-equation models have been developed [41]. \[\tau_{ij}=\mu_{t}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_ {j}}{\partial x_{i}}\right)-\frac{2}{3}\rho k\delta_{ij} \tag{4}\]  
Figure 1: Evidence of harmful structural loading of the turbine runner blades during the normal start-up and shut-down procedures. Signals recorded with the onboard instrumentation [37]. The explicit algebraic models require formulating the Reynolds stresses as a function of the anisotropy tensor \(a_{ij}\) (see Equation (5)). In the general 3D case, the anisotropy tensor can be expressed as the combination of ten tensors and five invariants [42]. \[\overline{u_{i}u_{j}}=k\left(a_{ij}+\frac{2}{3}\delta_{ij}\right) \tag{5}\]  
In the present paper, three models based on [PERSON]'s assumption have been considered. Each of them requires solving two additional equations in order to compute the eddy viscosity. The first one is the SST \(k-\omega\) model proposed by [PERSON] [43]. This model solved a transport equation for the turbulent kinetic energy \(k\) (Equation (6)) and a transport equation for the turbulent eddy frequency \(\omega\) (Equation (7)), which is in fact a blending between the \(\omega\)- and \(\epsilon\)-equations (in order to limit the paper length, the values of the coefficients used in the turbulence models are not given in the paper, except the case when the value differs from the standard one. The readers are kindly requested to refer to the cited papers.). \[\frac{\partial\rho k}{\partial t}+\frac{\partial\rho ku_{j}}{ \partial x_{j}}= P_{k}-\beta^{*}\rho k\omega+\frac{\partial}{\partial x_{j}} \left[\left(\mu+\sigma_{k}\mu_{t}\right)\frac{\partial k}{\partial x_{j}}\right] \tag{6}\] \[\frac{\partial\rho\omega}{\partial t}+\frac{\partial\rho\omega u_{j}}{ \partial x_{j}}= \frac{\alpha}{v_{t}}P_{k}-\beta\rho\omega^{2}+\frac{\partial}{ \partial x_{j}}\left[\left(\mu+\sigma_{\omega}\mu_{t}\right)\frac{\partial \omega}{\partial x_{j}}\right]+2\left(1-F_{1}\right)\frac{\rho\sigma_{\omega 2}}{ \omega}\frac{\partial k}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}} \tag{7}\]  
The eddy viscosity is then computed as:  
\[\mu_{t}=\frac{\rho a_{1}k}{\max\left(a_{1}\omega,SF_{2}\right)} \tag{8}\]  
The formulation of the SST model can include:  
* A production limiter for the turbulence production term \(P_{k}\) (see Equation (9)), which avoids the build-up of turbulent kinetic energy in stagnation regions [44].