3.1.2. Turbulence Model

A realizable k- turbulence model was selected for its ability to correctly capture the turbulent nature of the flow in Francis turbines [20–22]. This model was selected for its robustness and its improved boundary-layer-solving capacity under strong pressure gradients and flow separation compared to the standard k- model [23]. In addition, the k- turbulence model has a low computational expense when compared to k-*ω* SST. The transport equations for the realizable k- take the following form:

$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho k u\_j)}{\partial x\_j} = \frac{\partial}{\partial x\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial k}{\partial x\_j} \right] + G\_k + G\_b + \rho \epsilon - Y\_M + S\_k \tag{3}$$

$$\frac{\partial(\rho\varepsilon)}{\partial t} + \frac{\partial(\rho\varepsilon u\_j)}{\partial \mathbf{x}\_j} = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_c} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + \rho \mathbb{C}\_1 \mathbb{S}\_\varepsilon - \rho \mathbb{C}\_2 \frac{\varepsilon^2}{k + \sqrt{v\varepsilon}} + \mathbb{C}\_{1c} \frac{\varepsilon}{\tilde{k}} \mathbb{C}\_{\tilde{\mathcal{K}}} \mathbb{G}\_b + \mathbb{S}\_c \tag{4}$$

$$\mathbf{C}\_{1} = \max\left[0.43, \frac{\sqrt{2S\_{i,j}S\_{i,j}\frac{k}{c}}}{\sqrt{2S\_{i,j}S\_{i,j}\frac{k}{c}}+5}\right] \tag{5}$$

where:


The turbulence viscosity *μ<sup>t</sup>* is computed by:

$$
\mu\_t = \rho \mathbb{C}\_{\mu} \frac{k^2}{\mathfrak{c}} \tag{6}
$$

Model variable *Cμ* is defined by:

$$C\_{\mu} = \frac{1}{4.04 + \sqrt{6} \cos \phi \frac{kll^\*}{c}} \tag{7}$$

$$\phi = \frac{1}{3}\cos^{-1}(\sqrt{6}W) \tag{8}$$

$$\mathcal{W} = \frac{S\_{i\bar{j}} S\_{j\bar{k}} S\_{k\bar{i}}}{\bar{S}^3} \tag{9}$$

$$
\tilde{S} = \sqrt{S\_{i\bar{j}} S\_{i\bar{j}}} \tag{10}
$$

$$S\_{i\bar{j}} = \frac{1}{2} \left( \frac{\partial u\_{\bar{j}}}{\partial \boldsymbol{\alpha}\_{\bar{i}}} + \frac{\partial u\_{i}}{\boldsymbol{\alpha}\_{\bar{j}}} \right) \tag{11}$$

$$
\delta L^\* = \sqrt{S\_{ij} S\_{ij} + \Omega\_{ij} \Omega\_{ij}} \tag{12}
$$

$$
\Omega\_{ij} = \Omega\_{ij} - 2\mathfrak{e}\_{ijk}\omega\_k \tag{13}
$$

$$
\Omega\_{i\bar{j}} = \overline{\Omega\_{i\bar{j}}} - \varepsilon\_{i\bar{j}} \omega\_k \tag{14}
$$

where Ω*ij* is the tensor for the mean rate of rotation in a reference frame rotating at an angular velocity *ωk*.
