## 2 Model Description
### Turbulent Flow Modeling  
Figure 2 illustrates the well-known example of a fluid flowing over a plate with a uniform flow of velocity \(U_{\infty}\). A boundary layer develops close to the plate, and initially this layer is fully laminar. As the flow travels further across the plate, the boundary layer begins to flow more chaotically, i.e., a transition region, until it becomes fully turbulent. The Reynolds number at which the flow becomes turbulent depends on the geometry of the objects impeding the flow [25; 26]. Due to the low velocity near the wall, the Reynolds number is locally low. Subsequently, close to the wall, the flow has a viscous sub-layer in the turbulent region, also known as the laminar sub-layer. As the velocity and in turn the Reynolds number increase, further away from the wall the flow becomes more turbulent. A transition layer forms, also known as the buffer layer. Finally, the flow becomes fully turbulent, forming a turbulent layer.  
As the Reynolds number increases, the temporal and spatial scales of the eddies decrease. As a consequence, it is typically not computationally feasible to resolve them using direct numerical simulations. Instead, the Reynolds-averaged Navier-Stokes (RANS) formulation can be used. This is based on the decomposition of flow variables \(u\) into a time-averaged \(\overline{u}\) and local oscillation \(u^{\prime}\), as illustrated in Figure 3[28; 29; 30]. This operation creates extra unknown variables and the RANS set of equations are not closed, i.e., there are more unknowns than equations. Extra equations are used to close the set; different  
Figure 2: Fluid flow over a flat plate [27].  
sets of closure equations create different turbulence models. Some models introduce extra transport equations as part of the closure set, for example, a so-called two-equation model would use two such equations.  
The velocity of a turbulent flow near a wall is dependent on the distance from the wall. Close to the wall, the velocity profile follows the law of the wall, depicted in Figure 4. The dimensionless velocity \(u^{+}\) is defined as,  
\[u^{+}=\frac{u}{u_{\tau}} \tag{6}\]  
where \(u\) is the local velocity and \(u_{\tau}\) is the friction velocity at the closest wall, calculated by  
\[u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}} \tag{7}\]  
where \(\tau_{w}\) is the wall shear stress and \(\rho\) is the density at the wall. The dimensionless wall distance \(y^{+}\) is defined as,  
\[y^{+}=\frac{u_{\tau}y}{\
u} \tag{8}\]  
where \(y\) is the distance from the nearest wall and \(\
u\) is the kinematic viscosity. Close to the wall, the velocity is low and viscous forces will be dominant. The profile follows \(u^{+}=y^{+}\), this viscous sub-layer is also known as the laminar sub-layer introduced before. As the distance from the wall increases and the flow becomes more turbulent, a transition layer is formed, also known as the buffer sub-layer. Even further out, the flow becomes fully turbulent; the profile in the fully turbulent region can be described by \(u^{+}=\frac{1}{\kappa}\ln y^{+}+C\), where \(\kappa\approx 0.41\), known as the von Karman constant, and \(C\approx 5.0\). This layer is also known as the log-layer.  
Figure 4: Velocity profile near the wall [31].  
Figure 3: Decomposition of the flow variable \(u\) into the mean \(\overline{u}\) and local oscillation \(u^{\prime}\).