This study focused on numerical modeling methods of estimating turbulent characteristics around a scour hole. We applied three different turbulence schemes, $k-\epsilon $, $k-\omega $, and $k-\omega SST$, to determine the most appropriate one for simulating the complex flows around the scour. Our methods from both the physical and numerical models will help engineers to select an appropriate or suitable numerical turbulent modeling scheme around the hole.

#### 2.2.2. Turbulence Modeling

In OpenFOAM library, the RANS turbulence models are based on the concept of the eddy viscosity (

${\nu}_{t}$) introduced by Boussinesq in 1887 [

14]. This method relates the Reynolds stress to the mean properties of the flow by using the eddy viscosity.

The RANS standard

k-

ε model is the simplest and one of the most common turbulence models. It is a two-equation model that includes two extra transport equations to represent the turbulent properties of the flow: the turbulent kinetic energy (

k) and the turbulent energy dissipation (

$\epsilon $). The former and the latter variables determine the energy and the scale of the turbulence, respectively. The transport equations for these two variables were first introduced in [

15] and are presented as

where

G_{k} and

G_{b} are, respectively, the generation of turbulent kinetic energy due to the mean velocity gradient and buoyancy;

S_{k} and

${S}_{\epsilon}$ are the moduli of the mean rate-of-strain tensors of

k and

$\epsilon $, respectively. Finally,

${\sigma}_{k}$ and

${\sigma}_{\epsilon}$ are the diffusion rates for

k and

$\epsilon $, respectively, and are derived from the turbulent mixing length [

16].

Additionally, the dynamic turbulent viscosity in this model is defined as

where

${C}_{1\epsilon}$,

${C}_{2\epsilon}$,

${C}_{3\epsilon}$, and

${C}_{\mu}$ are experimental constants proposed as in

Table 2, which are only applicable to high-Reynolds-number flows [

17].

Because the standard $k-\epsilon $ model is a high-Reynolds-number model, it has a limitation in solving flows close to the wall where the Reynolds number is low because of the small velocity profiles in this region. In the $k-\epsilon $ model, the closest mesh point to the wall is located at the turbulent boundary layer, and modeling of the viscous sublayer and buffer layer is avoided. Therefore, in this study, the wall function was applied to enhance the computing accuracy near the wall. This function is required to model the influence of the grain roughness to the flow. Additionally, to ensure consistency in the research, we enabled this wall function for both the $k-\omega $ and $k-\omega SST$ case studies.

The RANS

$k-\omega $ model is also a two-equation model developed by Wilcox in 1993 [

18] that uses a similar approach and has the same definition of

k in the

$k-\epsilon $ model. However, instead of using turbulent energy dissipation

ε, this model uses the rate of dissipation of energy per unit volume and time, the so-called specific dissipation rate

ω. In contrast to the

$k-\epsilon $ model, the

$k-\omega $ model can be used in the regions close to the wall where turbulence is very low and

k tends to be zero [

12].

In this model, the turbulent viscosity is computed by

The modeled transport equations for

k and

ω in this model are

Here,

${G}_{\omega}$ is the generation of specific dissipation rate due to the mean velocity gradient, and

${S}_{\omega}$ is the modulus of the mean rate-of-strain tensors of

ω. The constants were chosen as

${\sigma}_{k}={\sigma}_{\omega}=0.5$ [

19].

The RANS

$k-\omega SST$ (shear–stress transport) model is a two-equation turbulence model developed by Menter in 1994 [

20] based on the previous two models. Basically, the

$k-\omega SST$ model is similar to the standard

$k-\omega $ model although the former includes several improvements and other constant variables. This method effectively blends the accurate formulation of the

$k-\omega $ model in the near-wall region with the free-stream independence of the

$k-\epsilon $ model in the far-field region, away from the wall. This blending function allows switching the turbulence schemes between near-wall and far-field regions, thus reducing the computational requirement [

9,

10].

The transport equation for turbulent kinetic energy

k in this model is the same as that for the basic

$k-\omega $ model mentioned before, while the equation for

ω can be written as

In this equation, ${D}_{\omega}$ represents the cross-diffusion term, which is an addition to the standard $k-\omega $ model.

#### 2.2.3. Numerical Setup

The domain scale of the numerical modeling was the same as the physical experiments in two dimensions. The physical one was conducted in a 0.6 m wide channel, while the numerical one was considered as a two-dimensional flow, in both horizontal and vertical directions, because the physical model was symmetrical along the centerline of the channel. The bed profiles were imported from the physical experimental results at the equilibrium stage. Because the boundary conditions and computation mesh are important for the numerical modeling, and have a strong impact on the output, they should be selected properly, as presented in

Figure 2. The flow was set up as velocity inlet and pressure outlet at the upstream and downstream boundaries, respectively. The upper boundary, which is the atmosphere in the physical experiment, was set as a symmetry condition. The bottom of the channel was divided into two parts, a smooth wall and a rough wall with respect to the acrylic and sand beds in the physical experiment. The “wall” boundary condition allowed the application of the wall function and a specified roughness as well. The value for roughness was set to zero for the smooth wall, and an equivalent value of

d_{50} = 1.2 mm sand particle for the rough wall.

The mesh was gradually refined toward the bottom to enhance the accuracy near the walls, as shown in

Figure 2. The grid size in the scour region was also meshed smaller than other areas to have better results where the flow is complicated. Errors from the discretization were avoided by selecting an optimal combination of grid and time step. A grid size and time step convergence study was conducted for selecting the appropriate parameters. In this study, the basic

$k-\epsilon $ model was used. Three grid sizes (accounting for the cell closest to the wall) of

$\Delta z=$ 1 mm, 1.5 mm, and 3 mm and three time steps of

$\Delta t=1\times {10}^{-5}$ s,

$1\times {10}^{-4}$ s, and

$1\times {10}^{-3}$ s were compared, as shown in

Figure 3. The horizontal velocities

u_{x} were recorded 4.5 m downstream.

Comparison indicates that a square grid of minimum Δ

z = 1.5 mm (near the wall) and maximum 10 mm (close to the atmosphere) and a time step of Δ

t =

$1\times {10}^{-4}$ s are small enough to obtain results within 0.1% of the higher resolution cases. Simulations ran for 120 s, which proved to be enough for the simulated results to reach the stabilized state. The initial condition values for the model parameters were calculated and set as presented in

Table 3.