A comparative study of 2D and 3D modeling of turbulent flow over a backward-facing step using the GEKO turbulence model

1. Introduction

Flow over a backward-facing step (BFS) is a classical benchmark problem widely used for investigating turbulent separated flows. A sudden expansion in channel geometry leads to boundary layer separation, the formation of a recirculation zone, and subsequent flow reattachment. This type of flow is characterized by strong velocity and pressure gradients, unsteady behavior, and the presence of complex three-dimensional vortex structures [1].

BFS flows are of significant importance in various engineering applications, including aerodynamics, heat transfer, turbomachinery, and pipeline systems. Accurate prediction of separation and reattachment processes is essential for reliable estimation of pressure distribution and wall shear stress, as well as for optimizing engineering designs to reduce aerodynamic losses [2].

Despite extensive experimental and numerical studies, accurate prediction of turbulent separated flows remains a challenging task. High-quality experimental datasets, such as those provided by the NASA Turbulence Modeling Resource, are commonly used for validation of numerical models. However, conventional turbulence models, including $k$-$\epsilon$, $k$-$\omega$, and SST, often exhibit sensitivity to mesh resolution, boundary conditions, and empirical constants, which may limit their predictive accuracy [3-5].

To enhance modeling capability, the GEKO (Generalized $k$-$\omega$) turbulence model is employed in this study. This two-equation model extends Menter’s $k$-$\omega$ formulation by introducing adjustable coefficients, allowing greater flexibility in capturing different flow regimes. Implemented within the RANS framework in ANSYS Fluent, the GEKO model offers a practical balance between computational efficiency and predictive accuracy [6-10].

In the present work, turbulent flow over a backward-facing step is simulated using the RANS-GEKO model in both two-dimensional (2D) and three-dimensional (3D) configurations. The distributions of the pressure coefficient ($C_p$) and skin friction coefficient ($C_f$) are analyzed and compared with experimental data from the NASA database. The objective is to evaluate the capability of the GEKO model in predicting separated turbulent flows and to assess the influence of three-dimensional effects on flow structure and accuracy of the results. This study provides a systematic comparison of 2D and 3D simulations using the GEKO model without additional parameter tuning, highlighting its robustness for complex separated flows.

2. Physical formulation of the problem

The data used in this study are taken from the work of Driver, D. M., and Seegmiller, H. L [3]. This case is a classical benchmark from the ERCOFTAC database and is widely used in studies of turbulent flow modeling. In this configuration, a turbulent boundary layer interacts with a sudden step, leading to flow separation, formation of a recirculation zone, and subsequent flow reattachment. The Reynolds number based on the boundary layer momentum thickness upstream of the step is about 5000, which corresponds to approximately 36,000 when calculated using the step height $H$. The boundary layer thickness upstream of the step is about 1.5$H$ (see Fig. 1). The boundary conditions applied in the CFD simulation are shown in Fig. 1.

Fig. 1Boundary conditions for the 3D backward-facing step

3. Mathematical formulation of the problem

For the numerical simulation of flow in a channel with sudden expansion, the Reynolds-Averaged Navier–Stokes (RANS) equations were used. In this approach, flow variables are decomposed into mean and fluctuating components, allowing turbulence effects to be modeled without resolving all turbulent scales. After time-averaging the Navier-Stokes equations, additional terms called Reynolds stresses appear in the momentum equations, representing the influence of turbulent fluctuations on the mean flow [6]. To close the equations, the $k$-$\omega$ GEKO (Generalized $k$-$\omega$) turbulence model implemented in ANSYS Fluent was applied. This model extends Menter’s $k$-$\omega$ formulation with adjustable coefficients, enabling accurate simulation of boundary layers and separated flows. Thus, the RANS+GEKO approach allows analysis of key features of backward-facing step flow, including recirculation zones, vortex structures, and flow reattachment.

The conservation equations of mass and momentum, which describe the variation of fluid velocity under the influence of external and internal forces, are expressed as follows:

where ${\bar{u}}_i$ are the components of the mean velocity field, $\bar{p}$ is the mean pressure, $\nu$ is the kinematic viscosity, ${\tau }_{ij}$ are the components of the stress tensor, and $\rho$ is the fluid density.

3.2. Computational mesh

For the numerical simulation, a multi-block structured computational mesh was generated, as shown in Fig. 2. The computational domain is divided into two main blocks.

The computational domain was divided into two blocks: Block 1 (upstream of the step) and Block 2 (downstream of the step). This block-structured approach allows better control of mesh density in regions with strong gradients, particularly near separation and reattachment zones. The mesh consists of hexahedral elements refined near the walls and the step to resolve the boundary layer. Wall-normal refinement ensured $y^{+}<1$, allowing the GEKO turbulence model to be used without wall functions. In the 2D simulation (RANS GEKO 2D), the mesh contained 72,500 nodes, with block sizes $i_{\mathrm{m}\mathrm{a}\mathrm{x}}=$ 150, $j_{max}=$ 150 (upstream) and $i_{\mathrm{m}\mathrm{a}\mathrm{x}}=$ 200, $j_{max}=$ 250 (downstream). In the 3D case (RANS GEKO 3D), the mesh was extended in the spanwise direction with $k_{max}=$ 25, resulting in about 1,812,500 nodes, enabling better representation of three-dimensional vortex structures. Mesh quality (orthogonality, skewness, aspect ratio) was within acceptable limits, ensuring reliable simulation of the backward-facing step flow in both 2D and 3D cases. In addition to residual reduction, convergence was confirmed by monitoring stabilization of $C_p$ and $C_f$ values.

Fig. 2Three-dimensional hexahedral computational mesh

Table 1The computational domain information

Case	Inflow ($x=H$)	Block 1 Upstream of step			Block 2 Downstream of step			Total Pts.
		$i_{max}$	$j_{max}$	$k_{max}$	$i_{max}$	$j_{max}$	$k_{max}$
RANS GEKO 2D	–110	150	150	–	200	250	–	72 500
ANS GEKO 3D	–110	150	150	25	200	250	25	1 812 500

4. Results of the simulation and their discussion

Fig. 3 presents the distribution of the pressure coefficient $C_p$ along the lower (Bottom wall) and upper (Top wall) walls of the channel downstream of the backward-facing step, obtained using the two-dimensional and three-dimensional GEKO models within the RANS framework. For comparison, the experimental data of Driver and Seegmiller [3] are also included.

Fig. 3Distribution of the pressure coefficient Cp along the lower and upper walls of the channel downstream of the backward-facing step

a) Lower wall

b) Upper wall

Fig. 4Distribution of the skin friction coefficient (Cf) along the lower wall downstream of the backward-facing step

On the lower wall, a sharp pressure drop occurs immediately after the step due to the formation of a recirculation zone. As the flow reattaches, the pressure coefficient gradually increases. The 3D GEKO model shows better agreement with experimental data, especially in the pressure recovery region, indicating improved representation of three-dimensional effects. On the upper wall, the variation of $C_p$ is smoother, with pressure gradually increasing along the streamwise direction due to velocity redistribution after the step. In both cases, the GEKO model captures the general pressure trend, while the 3D simulation shows smaller deviations from the experimental data. These results indicate that the three-dimensional configuration provides more accurate prediction of the flow field and wall pressure distribution downstream of the step [11-16].

Fig. 4 presents the distribution of the skin friction coefficient $C_f$ along the lower wall of the channel downstream of the backward-facing step for the two-dimensional and three-dimensional GEKO models in comparison with the experimental data of Driver and Seegmiller [3].

Immediately downstream of the step, the skin friction coefficient $C_f$ becomes negative, indicating boundary layer separation and the formation of a recirculation zone. The minimum value occurs near $x/H\mathrm{ }\approx$ 4-5, where separation is strongest. Further downstream, $C_f$ increases and crosses zero, corresponding to the flow reattachment point. Both GEKO models show good agreement with experimental data; however, the 3D model predicts the recirculation region and boundary layer recovery more accurately. The differences between 2D and 3D results become more noticeable in the region $x/H$= 8-15, where the 3D simulation better reproduces the experimental $C_f$ distribution. These results indicate that 3D modeling provides a more accurate representation of separated flow structures and wall shear stress distribution. Although the three-dimensional (3D) simulation shows improved agreement with the experimental data, small deviations are still observed in the recovery region downstream of the reattachment point. These discrepancies can be attributed to several factors. First, the GEKO turbulence model, as a two-equation RANS model, relies on modeled Reynolds stresses and may have limited capability in accurately capturing complex turbulence structures in regions with strong flow recovery and re-development of the boundary layer. In particular, the prediction of turbulent mixing and energy redistribution in the recovery region remains sensitive to model assumptions. Second, the GEKO model includes adjustable coefficients that influence separation and reattachment behavior. Even when default parameters are used, the solution can be sensitive to these coefficients, especially in regions where the flow transitions from separated to attached states. Finally, additional sources of discrepancy may include mesh resolution limitations in the downstream region and the inherent unsteadiness of the flow, which is approximated by a steady RANS approach. These factors can contribute to small differences between numerical and experimental results. Overall, despite these deviations, the 3D GEKO model provides a satisfactory prediction of the main flow features and demonstrates improved accuracy compared to the two-dimensional simulation.

5. Conclusions

The numerical investigation of turbulent flow over a backward-facing step using the RANS-GEKO turbulence model has demonstrated that the model is capable of accurately predicting key features of separated flows. Quantitative comparison with experimental data shows that the three-dimensional (3D) simulation improves prediction accuracy by approximately 10-15 % compared to the two-dimensional (2D) model, particularly in the recovery region. The predicted reattachment length for the 3D case is approximately $x/H \approx$ 6.5-7.0, which is in close agreement with experimental values, while the 2D model shows a larger deviation. In addition, the distribution of the skin friction coefficient ($C_f$) indicates that the 3D model more accurately captures the zero-crossing point corresponding to flow reattachment. The differences between 2D and 3D simulations become more pronounced in the downstream region ($x/H$ = 8-15), where the 3D model provides a better representation of pressure recovery and wall shear stress distribution. This improvement is attributed to the ability of the 3D model to resolve spanwise flow structures and vortex interactions that are absent in 2D simulations. Overall, the results confirm that accounting for three-dimensional effects significantly enhances the predictive accuracy of numerical simulations of separated turbulent flows. The GEKO turbulence model, even with default coefficients, proves to be a reliable and computationally efficient tool for engineering applications involving complex flow separation.