3.2.1.1. Standard k-ε model

The velocity of the turbulence can be calculated by forming a model for the corresponding kinetic energy using the following Eq. (4)

$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho k u\_i)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial(k)}{\partial \mathbf{x}\_j} + P\_k + P\_b - \rho \varepsilon - Y\_M + S\_k \tag{20}$$

The length scale is represented by ε, which is the rate of dissipation and can be calculated from the Eq. (4)

$$\frac{\partial(\rho\epsilon)}{\partial t} + \frac{\partial(\rho\epsilon u\_i)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_\epsilon} \right) \frac{\partial(\epsilon)}{\partial \mathbf{x}\_j} + \mathbf{C}\_{1\epsilon} \frac{\epsilon}{k} (\mathbf{P}\_k + \mathbf{C}\_{3\epsilon} \mathbf{P}\_b) - \mathbf{C}\_{2\epsilon} \rho \frac{\varepsilon^2}{k} + \mathbf{S}\_\epsilon \tag{21}$$

where Pk represents the turbulence kinetic energy due to the mean velocity gradients, Pb represents the turbulence kinetic energy due to buoyancy, and YM represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate.

### 3.3. Mean: Line analysis

The difficulty in the design of the compressor is because it involves two vital phases of the design. One being the 1D design of the compressor and the other is the deep numerical analysis of the design. The difficulty in the first phase is overcome by using mean line theory. In principle, the mean line theory follows the preliminary design that is carried out by neglecting the air flow variations in radial direction and the location of the mean blade radius is considered for the analysis.
