*2.3. Turbulence Modelling*

To model the turbulent flow behaviour in the present work, the scale-adaptive simulation shear stress transport (SAS–SST) turbulence model was used. The model was found to perform better than the conventional RANS formulations in similar cases during the underlying project work. It introduces the Von Karman length scale into the turbulence scale equation to adapt to different turbulence structure sizes, while using base RANS equations in areas where the flow behaves more similar to steady state [16]. The SAS–SST are as follows:

$$\frac{\partial k}{\partial t} + \frac{\partial u\_j k}{\partial x\_j} = P\_k - \beta^\* \omega k + \frac{\partial}{\partial x\_j} \left[ (\nu + \frac{\nu\_t}{\sigma\_k}) \frac{\partial k}{\partial x\_j} \right] \tag{4}$$

$$\frac{\partial\omega}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} (\overline{\mathbf{u}}\_j \omega) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ (\nu + \frac{\nu\_t}{\sigma\_\omega}) \frac{\partial \omega}{\partial \mathbf{x}\_j} \right] - \beta \omega^2 + \mathbb{C}\_\omega + aS^2 (1 + P\_{SAS}) \tag{5}$$

$$
\hbar\nu\_{\rm l} \propto \frac{k}{\omega'} \quad P\_{\rm SAS} = \overline{\xi}\_2 \kappa \frac{L}{L\_{\rm vK,3D}} \quad L\_{\rm vK,3D} = \kappa \frac{S}{L l^{\prime\prime}}.\tag{6}
$$

In these equations, *S* and *U* are generic first- and second-velocity derivatives, respectively. The SAS–SST model builds on the SST k-omega model by implementing an extra production term in the *ω*-equation, *PSAS*. This term is attuned to transient fluctuations in the flow. In regions with a fine mesh where the flow is becoming unsteady, *LvK*,3*<sup>D</sup>* is reduced, increasing the production term. This will result in a large *ω* and, therefore, reduced *k* and *ν<sup>t</sup>* values. In this way, the unsteadiness is not dampened, but is instead included as a part of the turbulence that is being resolved, leading to greater accuracy. A reduction of the turbulent viscosity dissipation occurs, which makes the momentum equations interpret the flow as transient rather than steady [17].
