# Aerodynamic Analysis of a Bluff Body Under Yaw-induced Motion in Turbulent Flow Regime

Ahmad Fawaz, Alban Leroyer, Emmanuel Guilmineau

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Ahmad Fawaz, Alban Leroyer, Emmanuel Guilmineau. Aerodynamic Analysis of a Bluff Body Under Yaw-induced Motion in Turbulent Flow Regime. 60th 3AF International Conference on Applied Aerodynamics., Feb 2026, Paris, France. hal-05527137

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# Aerodynamic Analysis of a Bluff Body Under Yaw-induced Motion in Turbulent Flow Regime

Ahmad Fawaz<sup>(1)</sup>, Alban Leroyer<sup>(1)</sup>, Emmanuel Guilmineau<sup>(1)</sup>

<sup>(1)</sup> Nantes Université, École Centrale de Nantes, CNRS, LHEEA, UMR 6598, F-44000 Nantes, France

Correspondence: emmanuel.guilmineau@ec-nantes.fr

## ABSTRACT

The present study provides a numerical aerodynamics analysis of cubical bluff bodies, representing a simplified geometry of cable cars, in the turbulent flow regime. Two cube geometries are employed: one with vertical sharp edges and the other featuring rounded vertical edges. The aerodynamics analysis is decomposed into two interlinked stages: static and dynamic analysis. In the static analysis, the cubes are positioned at different yaw incidences and the aerodynamic characteristics are investigated for various yaw angles. The yaw angle at which the global pressure coefficient ($C_p$) reaches its minimum on the cube lateral face (windward side) is denoted as the critical yaw angle ($\beta_{crit}$). Thereafter, a dynamic analysis is carried by yawing the cubes around the predefined $\beta_{crit}$ following a sinusoidal motion law. Static and dynamic CFD (computational fluid dynamics) simulations were conducted using the ISIS-CFD code with the assistance of the $k-\omega$ SST turbulent closure model of URANS (Unsteady Reynolds-Averaged Navier-Stokes) equations. In order to ensure the fidelity of the CFD simulations, the numerical results are compared to the experimental findings obtained using the S620 wind tunnel located at ENSMA, Poitiers, France. The results show that the numerical and experimental results exhibit identical behaviors in the static and dynamic analysis with acceptable discrepancies. This highlights the ability of the CFD simulations to reproduce the main aerodynamic features of bluff-body flows observed experimentally.

## 1. INTRODUCTION

Bluff bodies are widely dispersed in our life and play a crucial role in many engineering applications such as the transport sector (e.g., cable cars [1]). When airflow passes over bluff bodies (e.g., cube, cylinder, etc.), complex aerodynamics phenomena are typically observed in their vicinity, including flow separation-reattachment [2], shear layer instabilities [3] and vortex shedding [4]. Therefore, understanding the fluid-

structure interaction (FSI) between these bluff bodies and surrounding turbulent flows is essential to mitigate several issues such as safety and reliability concerns. With the fast development in computational techniques, the investigation of FSI has become increasingly accessible thanks to CFD (computational fluid dynamics) simulations, enabling detailed analysis of complex aerodynamic phenomena. The CFD simulations complement experiments by offering a more flexible and cost-effective FSI investigation across diverse conditions [5].

The geometry of a bluff body has a pronounced effect on its aerodynamic response [6]. For cylindrical bluff bodies, the flow behavior strongly depends on the Reynolds number (Re), as the flow separation points vary with the flow regime [7]. However, for cubical bluff bodies, the aerodynamics exhibit strong sensitivity to the angle of incidence while generally insensitive to the Re [8]. Due to their sharp edges, cubical bluff bodies experience flow separation primarily at the upstream edges, unlike cylindrical bodies [9]. In the literature, several studies have investigated the aerodynamics over bluff bodies. For instance, Obasaju [10] experimentally studied the aerodynamics of a square section cylinder at different incidence in the turbulent flow regime. Bearman and Trueman [11] reported from experiments that the drag coefficient increases up to nearly 3.0 when the depth exceeds half of the width of rectangular cylinders, with the flow significantly affected by the rear-edge corners. Based on experiments, Rockwell [12] found that the fluctuation amplitude of the pressure in proximity of the downstream edges is minimal when the angle of incidence attains 15°. Moreover, Chen and Liu [13] examined experimentally the surface pressures and frequencies of vortex shedding at non-zero incidence angles of a square cylinder. Huang et al. [14] experimentally studied the influence wide range of incidence angles between 0° and 45° on the vortex shedding of a square cylinder, revealing that at 15° the subcritical and supercritical regimes are separated.
The objective of the present work is to propose a high fidelity numerical approach using the ISIS-CFD with AGR (adaptive grid refinement) techniques for the purpose investigating the unsteady flow separation-reattachment phenomena on the lateral face (windward side) of the cubical bodies with vertical sharp and round edges subjected to yawing motion. For validation purpose, the numerical results are compared to experimental findings obtained using the S620 wind tunnel located at ENSMA, Poitiers, France [15].

## 2. TEST CASE DESCRIPTION

In the present study, two cubical geometries are employed: one with vertical sharp edges and the other with vertical round edges as illustrated in Figures 1a and 1b, respectively. Both cube geometries have a side length of 0.3 m (D), with the round edges configuration having a fillet radius (r) of 0.02 m (r=D/15).

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Cube geometries with vertical (a) sharp and (b) rounded edges.</figcaption></figure>

The cubes with sharp and round vertical edges are placed in an airflow surrounding domain with an inlet flow velocity of $U_\infty = 7$ m/s, resulting in a Reynolds number (Re) of $1.3 \times 10^5$ based on the cube side length D. Therefore, the FSI phenomena are investigated in the turbulent flow regime. The numerical study is decomposed into two coupled stages: static and dynamic analysis. In the static analysis, the cube is fixed at a given yaw angle $\beta$ from the free-stream, as depicted in Figure 2a, and the aerodynamic characteristics are investigated for various yaw incidences. The yaw angle at which the global pressure coefficient ($C_p$) reaches its minimum on the windward side of the cube is defined as the critical yaw angle ($\beta_{crit}$). For the sharp edges configuration, the mean flow begins to reattach near the trailing edge of that lateral face at $\beta_{crit}$.

In the dynamic analysis, the cube is forced to oscillate following a sinusoidal law defined from the period of oscillation T, the mean angle of motion $\beta_0$ and the amplitude angle of motion $\Delta\beta$, as expressed in Eq. (1)

and illustrated in Figure 2b. The oscillation period T is set to 2.5 s, and the oscillation amplitude is fixed at 8°. The value of $\beta_0$ is assigned equal to $\beta_{crit}$, enabling the investigation of unsteady flow separation-reattachment phenomena along the cube lateral face. Lastly, the values of the study's input parameters are provided in Table 1.

$$ \beta = \beta_0 + \Delta\beta \cos\left(2\pi \frac{t}{T}\right) \quad (1) $$

<figure><img src="image_3.png" /><figcaption>**Figure 2:** (a) Top view of the static configuration : cube at a fixed yaw angle $\beta$, (b) Top view of the dynamic configuration : oscillating yaw angle $\beta$.</figcaption></figure>

<table><caption>**Table 1:** Input parameters of the study</caption><thead><tr><td>Parameter (unit)</td><td>Value</td></tr></thead><tbody><tr><td><i>Re</i></td><td>1.3 × 10<sup>5</sup></td></tr><tr><td><i>D</i> (m)</td><td>0.3</td></tr><tr><td><i>U</i><sub>∞</sub> (m.s<sup>-1</sup>)</td><td>7</td></tr><tr><td><i>T</i> (s)</td><td>2.5</td></tr><tr><td><i>β</i> (°)</td><td>[0° - 20°]</td></tr><tr><td><i>Δβ</i> (°)</td><td>8°</td></tr></tbody></table>

# 3. COMPUTATIONAL FLUID DYNAMICS

## 3.1. Numerical simulations setting

Both static and dynamic simulations were conducted using the in-house ISIS-CFD code (part of the FINE™/Marine software) developed by the CNRS and Centrale Nantes. The k-ω SST closure turbulent model of URANS (Unsteady Reynolds-Averaged Navier-Stokes) equations is used to simulate the turbulent air flow around the cube and to evaluate the aerodynamics forces. The AVLSMART scheme [16] is used as a discretization scheme for the transport equations (momentum and turbulence), while the second-order backward difference scheme is used to discretize the time with a time step of 0.00042 s (dimensionless time step = 0.0098) used in the static analysis and a time step of 0.00025 s (dimensionless time step = 0.0058) for the dynamics analysis. At the inlet boundary of the computational domain (presented in Figure 3), a Dirichlet boundary condition with a uniform inlet velocity ($U_\infty$) of 7 m.s<sup>-1</sup> is imposed, while a prescribed pressure is defined at the outlet surface. A mirror (symmetry) condition is imposed on the top horizontal plan of the
computational domains. Wall functions are imposed on the lateral and bottom external faces of the computational domain, whereas no-slip (wall-resolved) conditions are applied on the cube faces. The detailed dimensions of the computational domain are provided in Figure 3. Furthermore, constant air properties were used in the numerical analysis with a density (ρ) of 1.2 kg.m⁻³ and a dynamic viscosity (μ) of 1.8E-05 Pa.s.

<figure><img src="image_4.png" /><figcaption>**Figure 3:** Computational domain used in the CFD analysis.</figcaption></figure>

The Hexpress™ prepossessing software is used to generate unstructured mesh on the computational domain with only hexahedrons. For capturing the high velocity gradient near walls, several inflation layers were built on all cube surfaces with an average first layer height of y⁺=0.3, while y⁺=50 is used on the lateral and bottom external surfaces of the computational domain. For better accuracy, AGR techniques are employed to capture the flow separation zones on the rotating cube. In ISIS-CFD, AGR is carried out through the local division of unstructured hexahedral meshes [17]. The mesh refinement regions are determined based on a metric refinement criterion, which is the flux-component Hessian, defined as the maximum between the velocity and pressure Hessians. The grid is refined iteratively until a predefined threshold is met and the desired mesh density being specified by a minimum cell size of 0.00059 m (D/512).

## 3.2. Mesh dependency study

In order to ensure the reliability of our numerical simulations, a mesh dependency study was performed for the sharp and round edges cases. As illustrated by Figures 4 and 5, five mesh levels were introduced by varying the AGR level between 0.4L and 0.05L. Reducing the AGR level correspondingly lowers the refinement threshold, thereby resulting in the generation of a finer mesh. Initially, a very coarse mesh is employed with the AGR technique disabled during the numerical simulation. The mesh is then refined by decreasing the Hessian alpha multiplier of the AGR, in accordance with the refinement methodology proposed by Celik et al. [18]:

$$r = \left[ \frac{N_{cells,i+1}}{N_{cells,i}} \right]^{(1/3)} \quad (2)$$

where $N_{cells}$ is the number of cells in the mesh, $i+1$ is the finer mesh, $i$ is the coarser mesh and $r$ is the factor. Based on experience [18], the factor $r$ should be higher than 1.3. Thereafter, the results are considered to be mesh independent when the following stopping criterion is satisfied:

$$Er(Cp) = \left| \frac{Cp^i - Cp^{i+1}}{Cp^i} \right| < 1\% \quad (3)$$

where $Er$ is the relative error, $Cp$ is the global pressure coefficient calculated on the lateral surface of the cube and $i$ is the index of the mesh dependency test. For instance, the details of the mesh dependency analysis for the cubes with sharp and round edges at $β = 11°$ and $β = 2°$ are provided in Tables 2 and 3, respectively. It can be observed that the numerical results become mesh independent when the number of cells reach 7,581,731 and 6,198,234 for the sharp and round edges cases, respectively, corresponding to an AGR level of 0.1D.

<table><caption>**Table 2:** Mesh dependency test for the sharp edges case at $β = 11°$.</caption><thead><tr><th>AGR level</th><th>Cells number</th><th>Cp</th><th>Error (Cp) (%)</th></tr></thead><tbody><tr><td>--</td><td>364,225</td><td>-0.5806</td><td>--</td></tr><tr><td>0.4D</td><td>764,499</td><td>-0.6078</td><td>4.68%</td></tr><tr><td>0.2D</td><td>2,106,523</td><td>-0.5961</td><td>1.92%</td></tr><tr><td><b>0.1D</b></td><td><b>7,581,731</b></td><td>-0.5889</td><td>1.20%</td></tr><tr><td>0.05D</td><td>25,878,642</td><td>-0.5852</td><td>0.62%</td></tr></tbody></table>

<table><caption>**Table 3:** Mesh dependency test for the round edges case at $β = 2°$.</caption><thead><tr><th>AGR level</th><th>Cells number</th><th>Cp</th><th>Error (Cp) (%)</th></tr></thead><tbody><tr><td>--</td><td>362,958</td><td>-0.5746</td><td>--</td></tr><tr><td>0.4D</td><td>761,354</td><td>-0.5963</td><td>3.37%</td></tr><tr><td>0.2D</td><td>1,899,588</td><td>-0.5671</td><td>4.89%</td></tr><tr><td><b>0.1D</b></td><td><b>6,198,234</b></td><td>-0.5575</td><td>1.69%</td></tr><tr><td>0.05D</td><td>24,488,615</td><td>-0.5530</td><td>0.81%</td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>**Figure 4:** Meshes with (a) no AGR and with AGR level of (b) 0.4D, (c) 0.2D, (d) 0.1D and (e) 0.05D for the sharp edges case at $\beta = 11^\circ$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5:** Meshes with (a) no AGR and with AGR level of (b) 0.4D, (c) 0.2D, (d) 0.1D and (e) 0.05D for the round edges case at $\beta = 2^\circ$.</figcaption></figure>

**3.3. Numerical-Experimental comparison**

The numerical results obtained in the static and dynamic analysis are compared with the experimental data reported by [15]. The experiments were performed using

the S620 wind tunnel located in ENSMA, Poitiers, France. For further details about the experimental setup and methodology, the readers are referred to the corresponding study [15]. It is noteworthy to mention that the cross-sectional dimensions of the S620 wind
tunnel are identical to the computational domain presented in Fig.3, as well as the dimensions of the employed cubes in order to maintain the consistency between numerical and experimental approaches. Furthermore, this comparison aims to assess the ability of the current numerical approach to reproduce identical physical phenomena under similar conditions. The percentage of agreement between the numerical and experimental results will reflect the model’s validity and highlight the discrepancies that may arise from modeling errors or experimental uncertainties.

# 4. RESULTS AND DISCUSSIONS

This section presents the results of the static and dynamic analysis by offering a comparison between numerical and experimental findings.

## 4.1. Static analysis

Figures 6a and 6b present the variation of the mean pressure coefficient $C_p$ of the cube windward lateral face evaluated numerically and experimentally function of the cube yaw angle $\beta$ under static conditions for the sharp and round edges cases, respectively. The mean $C_p$ refers to the spatial integration of the local pressure coefficients $C_p$ on the studied side face of the cube. Generally, both numerical and experimental curves exhibit identical trends of initially decreasing to a minimum $C_p$ value, followed by an increase. For the sharp edges case, the critical yaw angle $\beta_{crit}$ obtained numerically is 10°, whereas experiments yielded 8°. Furthermore, for the round edges configuration, the $\beta_{crit}$ is numerically

determined to be 4°, while $\beta_{crit}$ acquired experimentally is 2°. More specifically, for $\beta > \beta_{crit}$, the numerical and experimental curves show good agreement with minimal deviation. By contrast, for $\beta < \beta_{crit}$, a remarkable difference in behaviors is observed between the numerical and experimental results. The reason behind this numerical/experimental deviation can be attributed to the limitations of the k-ω SST in capturing the complex flow behavior around bluff bodies, especially when the flow is separated from the cube lateral face for $\beta < \beta_{crit}$. Additionally, in order to evidently visualize the flow separation-reattachment on the cube lateral face, the mean flow streamlines are plotted in the plane of symmetry at $\beta_{crit}$, $\beta_{crit} - 4$ and $\beta_{crit} + 4$ for both sharp and round edges configurations, as illustrated in Figures 7a and 7b, respectively. For both sharp and round edges cases, the strong influence of yaw incidence on the topology of the wake region is clearly evident. At $\beta_{crit}$, the mean flow streamlines begin to reattach near the trailing edge of the lateral face for the sharp edges case, whereas for the round edges case the flow reattachment cannot be clearly seen. Beyond the $\beta_{crit}$ ($\beta_{crit} + 4$), the flow separation-reattachment phenomena become increasingly pronounced for both configurations, as the reattachment region on the lateral face progressively shifts upstream, moving closer to the initial separation point. This elucidates that for $\beta > \beta_{crit}$, the flow separation-reattachment phenomena become faster as the yaw angle $\beta$ increases. Below the $\beta_{crit}$ ($\beta_{crit} - 4$), the flow remains fully separated from the cube lateral face for both cases.

<figure><img src="image_7.png" /><figcaption>**Figure 6:** Variation of the global $C_p$ on the cube lateral face with respect to the yaw angle in the static analysis.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 7:** Mean flow streamlines at different yaw incidences in the static analysis.</figcaption></figure>

For further comparison between the static results obtained using the sharp and round edges configurations, the pressure coefficient ($C_p$) distribution on the windward lateral face of the cube are presented in Figure 8 at different yaw angles including $\beta_{crit}$. Apparently, the $C_p$ contours exhibit a paramount difference between sharp and round edges cases at $\beta_{crit}$, $\beta_{crit} - 4$ and $\beta_{crit} + 4$. This demonstrates that altering the geometry of the cube,

more precisely its vertical edges, leads to distinct physical phenomena under the same input conditions. In both cases, when the yaw angle exceeds the $\beta_{crit}$, a high pressure zone is developed near the downstream edges (right side of the contours) of the cube's lateral face. This can be attributed to the flow reattachment on that face, as previously shown in Figure 7 for $\beta \ge \beta_{crit}$.

<figure><img src="image_9.png" /><figcaption>**Figure 8:** Pressure coefficient ($C_p$) distribution on the cube lateral face at different yaw incidences in the static analysis.</figcaption></figure>
In order to compare the numerical and experimental results at the local scale, the $C_p$ distributions on the lateral face of the cube (windward side) are examined. Figures 9a and 9b illustrate the $C_p$ contours obtained numerically and experimentally for the sharp and round edges cases, respectively. As depicted, the numerical and experimental $C_p$ contours exhibit generally similar trends, which serves as an additional evidence to demonstrate the ability of the CFD approach in capturing the physical phenomena observed in the experiments. Furthermore, the numerical (CFD) $C_p$ contours are

perfectly symmetric about the central horizontal axis due to the imposition of the symmetry condition on the top horizontal plane, since only half of the computational domain is simulated (see subsection 3.1). Nevertheless, the experimental $C_p$ contours do not present perfect symmetry about the central horizontal axis, as the experimental setup itself is not fully symmetric. The lack of symmetry in the experimental setup is mainly due to the support structure used to hold the cube, which can alter flow field above the cube and thereby disrupting the symmetry of the $C_p$ contours on the cube side face [15].

<figure><img src="image_10.png" /><figcaption>**Figure 9:** Numerical and experimental $C_p$ contours on the cube lateral face (windward side) in the static analysis.</figcaption></figure>

## 4.2. Dynamic analysis

Figures 10a and 10b present the variation of the mean pressure coefficient difference $C_p - C_p(\beta_{crit})$ on the windward side face of the cube function of $\beta - \beta_{crit}$ under dynamic conditions for the sharp and round edges configurations, respectively. Overall, the numerical and experimental curves resemble in their behaviors. In particular, for $\beta - \beta_{crit} \ge 5^\circ$, the numerical and experimental dynamic curves exhibit nearly identical trends with minor deviations. This elucidates the ability of the CFD model in reliably capturing identical physical phenomena observed in the experiments, particularly when flow reattachment occurs on the cube face for $\beta - \beta_{crit} \ge 5^\circ$. Nonetheless, more pronounced discrepancies in both the hysteresis behavior and the magnitude of the numerical and experimental $C_p - C_p(\beta_{crit})$ curves are observed for $\beta - \beta_{crit} < 5^\circ$. These deviations may be attributed to the limitation of the k-ω SST turbulence model in accurately capturing the complex flow topology and the unsteady flow separation phenomena around bluff bodies.

To further investigate the dynamic response, the $C_p - C_p(\beta_{crit})$ contours obtained experimentally and

numerically at different instants of the yawing motion (1/4, 1/2, 3/4 and 1 period) are presented in Figures 11a and 11b, respectively, for the sharp edges configuration. The experimental $C_p - C_p(\beta_{crit})$ contour at one complete period of the yawing motion is not provided due to its unavailability. The numerical and experimental contours show good agreement in terms of global trends, while exhibiting noticeable discrepancies at the local scale. At the 1/4-period instant of the yawing cycle, the cubes reaches the maximum possible yaw angle ($\beta_{crit} + 8^\circ$) during the dynamics analysis. A high pressure region is observed on the right side of the contours, near the trailing edge of the cube lateral face. This can be attributed to the flow reattachment phenomenon on that face for $\beta > \beta_{crit}$, as previously shown in the static analysis (see Figure 7). At the 3/4-period instant of the yawing motion, when the cubes attain the minimum yaw angle in the dynamic analysis ($\beta_{crit} - 8^\circ$), both numerical and experimental contours exhibit minor pressure variations due to the flow separation from the cube lateral face for $\beta < \beta_{crit}$ (see Figure 7). At 1/2- and 1-period instant of the yawing motion, the cubes have identical yaw angles ($\beta_{crit}$), as defined by the sinusoidal law governing the present yawing motion. Although the
cubes possess the same yaw incidence ($\beta_{crit}$) at the 1/2- and 1- period instants, the contours differ significantly, elucidating that distinct physical phenomena and flow behaviors were obtained at these two instants. This observation highlights a phase lag in the flow separation-

reattachment dynamics near the cube's lateral surface, indicative of unsteady flow behavior with delayed flow separation and flow reattachment on the cube side during the yawing motion.

<figure><img src="image_11.png" /><figcaption>**Figure 10:** Variation of the mean pressure coefficient difference $C_p - C_p(\beta_{crit})$ on the cube lateral face with respect to $\beta - \beta_{crit}$ in the dynamic analysis.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 11:** Numerical and experimental $C_p - C_p(\beta_{crit})$ contours on the cube lateral face (windward side) at different instants in the dynamic analysis.</figcaption></figure>
## 5. CONCLUSIONS

In the present work, a CFD analysis was performed to investigate the aerodynamic behavior of cubical bluff bodies with rounded and sharp vertical edges under static and dynamic conditions. The numerical static and dynamic results were compared with existing experimental data reported in the literature in order to assess the accuracy and reliability of the CFD model. First, a static analysis was conducted to examine aerodynamic characteristic of the cubes at various yaw angles and to identify the critical yaw angle ($\beta_{crit}$) at which the mean pressure coefficient reaches its minimum on the cube windward lateral face. For the sharp edges configuration, the numerical static analysis yielded a $\beta_{crit}$ of 10°, while the $\beta_{crit}$ obtained experimentally is 8°. Furthermore, for the round edges case, the $\beta_{crit}$ was numerically determined to be 4°, whereas $\beta_{crit}$ acquired experimentally is 2°. In general, the CFD and experimental curves exhibit comparable trends in the static analysis, showing a descendant behavior up to the $\beta_{crit}$ value followed by an ascending one. Upon the determination of the $\beta_{crit}$ from the static analysis, a dynamic analysis was performed by yawing the cubes around $\beta_{crit}$ according to a sinusoidal law motion with an amplitude of 8° and a period of oscillations of 2.5 s. The dynamic analysis reveals that the numerical and experimental results exhibit globally similar behaviors with acceptable deviations. This demonstrates the fidelity of the CFD approach in capturing complex aerodynamic phenomena around the employed cubical bluff bodies. Additionally, the static and dynamic analysis indicated that the aerodynamic responses of the cubes with rounded and sharp edge configurations differ significantly, elucidating that even slight modifications in the cube geometry under identical input conditions can lead to different physical phenomena.

Future efforts are still required, particularly through the use of more advanced turbulence models, such as Delayed Detached Eddy Simulation (DDES), to reduce the discrepancies between the CFD and experimental results.

## 6. ACKNOWLEDGMENT

We are indebted to the French Agence Nationale pour la Recherche (ANR) through the ANR project Turb-Cab (ID ANR-22-CE22-0010) for financial support. The computations were performed using HPC resources from GENCI (Grand Equipement National de Calcul Intensif) (Grant-A0182A001299) which is gratefully acknowledged.

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