Run with code_saturne v9.1.0 - EDF's open-source CFD code

At a glance

The Windsor squareback at 2.5° yaw - Case 1 of the AutoCFD workshop - was simulated with code_saturne v9.1, the open-source finite-volume CFD solver developed by EDF, on two mesh levels (6.3 M and 37.4 M cells) using steady-state k-ω SST RANS. The results are compared with the AutoCFD4 workshop results, which gather 43 contributions from commercial and open-source solvers on this same reference automotive case.

• Aerodynamic coefficients ($C_D$, $C_{D,\text{base}}$, $C_L$, $C_S$, $C_{M_y}$): both code_saturne simulations sit within the scatter of AutoCFD4 contributions, alongside Star-CCM+, Fluent, ConvergeCFD, Cadence Fidelity and OpenFOAM results.
• Wall pressure and mean wake structure: the $C_p$ distributions, the yaw-induced asymmetry and the overall organisation of the recirculation zone are correctly reproduced.
• Limitations of steady RANS: the discrepancies observed on base pressure, velocity recovery in the wake and turbulence level are the same as those found in the steady RANS contributions of the AutoCFD4 workshop, across all solvers.
• Effect of mesh refinement: moving from 6.3 to 37.4 million cells barely changes the mean results; residual discrepancies are primarily driven by turbulence modelling rather than spatial resolution.
• SST-DDES outlook: the hybrid RANS-LES approach is the natural next step to capture the anisotropic turbulent structures driving wake dynamics.

Why the Windsor squareback benchmark?

The Windsor body is a classical reference geometry for automotive aerodynamics validation: a generic squareback, deliberately simplified to isolate the dominant physics of base flows, without rotating wheels, engine bay or underbody complexity. The goal is not to reproduce a production vehicle, but to focus the analysis on a massively separated wake structured by the shear layers shed from the base edges and by a large low-pressure recirculation zone behind the rear face. When the body is in yaw, the wake symmetry plane is broken: the shear layers and the recirculation zone develop differently on each side of the base, generating the side forces and aerodynamic moments that make this case particularly discriminating for turbulence models.

The reference experimental data come from Varney et al. at Loughborough University and cover all the metrics required for a comprehensive aerodynamic validation: integrated force coefficients, wall pressure distributions on the body and base, and 2D PIV fields in several wake planes. The AutoCFD workshop reuses these data and organises a community-wide blind comparison: each participant uses the same geometry, the same reference conditions, the same meshes and the same post-processing protocol before submitting results to a centralised dashboard. The AutoCFD4 dashboard currently gathers 43 contributions from OpenFOAM, Star-CCM+, Fluent, ConvergeCFD, Cadence Fidelity and several academic or institutional codes, in steady-state RANS, hybrid RANS-LES and LES configurations. The benefit of the benchmark is therefore not only the comparison with experiment: it also makes it possible to position a solver and a modelling strategy relative to the current CFD state of the art on a massively separated automotive wake case.

Case setup

Geometry and flow: The Windsor body follows the generic squareback proportions defined by the workshop: length $L = 1.044$ m, width $W = 0.389$ m, height $H = 0.289$ m and a ground clearance of 0.05 m. The model is mounted on four cylindrical stilts reproducing the experimental rig configuration and oriented at $\psi = 2.5°$ yaw. The upstream conditions correspond to a uniform flow at $U_\text{ref} = 40$ m/s with density $\rho = 1.225$ kg/m³, giving a Reynolds number $Re_L \approx 2.9 \times 10^6$ based on the body length. The reference area is $S_\text{ref} = 0.112$ m² and the moment reduction point is placed at the origin, in accordance with the AutoCFD benchmark definition.

Meshing strategy: Two mesh levels are considered in order to separate discretisation effects from limitations intrinsic to the turbulence model. The coarse mesh contains 6.3 million cells, of which about 75 k on the body, while the medium mesh reaches 37.4 million cells with about 285 k surface cells. Both configurations target a $y^+ \approx 30$, corresponding to the classical range for wall-function use in industrial automotive aerodynamics. The objective of this refinement study is not to chase a strict asymptotic convergence, but to identify which quantities are still mesh-sensitive and which are already limited by the steady RANS model.

Numerical setup: The simulations are performed in incompressible steady-state RANS with the k-ω SST turbulence model and automatic wall functions. The pressure-velocity coupling relies on SIMPLEC with a local pseudo-time step in order to accelerate steady-state convergence. The convective terms are discretised with a second-order centred scheme. On both meshes, around 3000 iterations are sufficient to stabilise the residuals as well as the time history of the aerodynamic coefficients. The computations are carried out with code_saturne v9.1 deployed via the official Docker image on HPC infrastructure.

Post-processing of aerodynamic forces: The aerodynamic coefficients are not extracted directly from the standard code_saturne outputs. A dedicated user routine (cs_user_extra_operations.cpp) computes at each iteration the pressure and friction forces on the different zones of the model.

Where do the force and moment coefficients sit?

The AutoCFD4 dashboard positions each contribution relative to the experimental reference, coefficient by coefficient. Adding our two code_saturne simulations - coarse mesh (g1) and medium mesh (g2) - therefore makes it possible to position the solver directly relative to the other CFD approaches of the benchmark, under homogeneous comparison conditions.

Drag: Both code_saturne simulations fall within the AutoCFD4 scatter of the drag coefficient $C_D$. Drag remains slightly under-predicted with respect to experiment, consistently with all the RANS contributions of the workshop. Mesh refinement does not reduce this gap: moving from 6.3 to 37.4 million cells actually leads to a slight further decrease of $C_D$. At this stage, the integrated drag therefore appears essentially mesh-converged. The residual discrepancy stems more from the limitations of RANS modelling of the base flow than from spatial resolution itself.

Base drag: The base drag coefficient $C_{D,\text{base}}$ shows the same trend as the total drag coefficient $C_D$, with a moderate under-prediction relative to the experimental reference. This behaviour is physically consistent: on a squareback, the dominant drag contribution comes directly from the pressure deficit in the rear recirculation zone. Both code_saturne simulations sit within the central scatter of the AutoCFD4 RANS contributions, while hybrid RANS-LES and LES approaches tend overall to move closer to experiment. This hierarchy is consistent with the physics of the case: the unsteady dynamics of the base recirculation and of the turbulent wake structures plays a direct role in the mean base pressure level, and is only modelled in a steady RANS approach.

Lift, side force and pitching moment: The lift $C_L$, side force $C_S$ and pitching moment $C_{M_y}$ coefficients all fall within the scatter of the AutoCFD4 contributions. The side force $C_S$, directly driven by the wake asymmetry induced by the $2.5°$ yaw, essentially matches the experimental reference on both meshes. The $C_L$ and $C_{M_y}$ coefficients show the same overall level of agreement with the other workshop contributions. Like drag, these quantities remain strongly conditioned by the wake structure - in particular by the position of the recirculation zone and the balance of the separated shear layers behind the base.

Effect of mesh refinement: Across all five aerodynamic coefficients, the gap between coarse and medium meshes remains small, and clearly smaller than the scatter observed between the various AutoCFD4 contributions. For the integrated forces and moments, the coarse mesh therefore already appears representative of a resolution level compatible with industrial use of steady RANS on this type of configuration. A sixfold refinement of the total cell count modifies neither the global trends nor the relative positioning of the results with respect to experiment. The residual discrepancies observed must therefore be sought less in spatial resolution than in the intrinsic limitations of steady RANS modelling of the separated wake - a point analysed in the next section through the local pressure, velocity and turbulent kinetic energy fields.

Base flow and wake

We now move away from integrated coefficients to examine the local fields, where the quality of the aerodynamic prediction is actually decided: pressure distributions on the body and wake structure.

Wall $C_p$ along the body: The pressure coefficient distributions along the symmetry plane ($y = 0$) and along the horizontal cut at $z = 0.2595$ m correctly reproduce the pressure levels and gradients measured experimentally over the whole body. On the median plane, both meshes capture the strong suction peak associated with flow acceleration around the front edge, followed by a progressive recompression on the upper part of the body. The second pressure minimum observed near $x \approx -0.2$ m on the upper branch is also correctly reproduced; it is associated with the geometric slope break between the two upper surfaces of the Windsor model. Downstream, the numerical distributions remain in good agreement with measurements up to the region influenced by the base pressure deficit. The horizontal cut highlights the yaw-induced asymmetry, with different pressure levels between the two sides of the body in agreement with the lateral displacement of the wake. The curves from the coarse and medium meshes remain essentially superimposed on both cuts, indicating that the mean wall pressure distribution is already largely converged in this resolution range.

$C_p$ map on the base: Both code_saturne RANS simulations correctly reproduce the overall lateral asymmetry induced by yaw, with a stronger pressure deficit on one side of the base in line with the mean wake orientation observed experimentally. The discrepancy with Varney's measurements lies mainly in the topology of this low-pressure region. The simulations organise the $C_p$ field around a compact, strongly concentrated low-pressure core at the centre of the rear face, whereas the experimental distribution appears more spread out, more laterally stratified and shifted transversally under the effect of wake dynamics. The RANS solutions also under-predict the overall suction level on the base. The medium mesh actually amplifies this trend by widening the central low-pressure region rather than bringing the solution closer to experiment. This signature is not specific to code_saturne: the same trend is found in the majority of the steady RANS contributions of the AutoCFD4 workshop, including with commercial solvers.

Velocity profiles and recirculation bubble structure: The transverse profiles of longitudinal velocity $u_x / U_\text{ref}$ are extracted at four wake stations at mid-height of the body ($z = 0.194$ m, from $x - x_{base} = 0.12$ to $0.72$ m). Both simulations correctly reproduce the lateral extent of the velocity deficit as well as the wake offset induced by yaw, with a consistent dissymmetry of the profiles on either side of the median axis. The main discrepancy with measurements concerns velocity recovery in the wake: the PIV profiles climb back towards $u_x / U_\text{ref} = 1$ faster than the RANS simulations, whose deficit remains deeper up to the most downstream stations. The 2D field of $u_x$ in the horizontal plane gives the corresponding spatial reading: the topology of the recirculation bubble is qualitatively correct, but it extends further downstream in the RANS simulations than in PIV. The coarse and medium meshes show similar behaviour both for the profiles and for the 2D velocity field. This behaviour indicates that velocity recovery in the wake is driven by wake modelling rather than by spatial resolution, and the same trend is observed in the RANS contributions of the AutoCFD4 workshop.

Turbulent kinetic energy in the wake: The modelled turbulent kinetic energy field (TKE or $k$) in the horizontal plane at $z = 0.194$ m globally reproduces the structure of the experimental field, with a peak-TKE region offset laterally around $y = -0.18$ m and a wake asymmetry consistent with the orientation imposed by yaw. The peak levels are however slightly under-estimated and the high-TKE region less extended laterally than in the PIV, where comparable values extend up to $y \approx 0.05$ m. This moderate deficit in TKE amplitude and extent is consistent with the gap observed on wake recovery: with less turbulence in the wake, transverse mixing and the refilling of the recirculation zone are slower, which lengthens the recirculation bubble of the RANS simulations relative to experiment. Both meshes show similar behaviour.

Next step: hybrid RANS-LES with DDES

The three discrepancies identified in the previous sections - under-estimation of base suction, over-estimation of recirculation length and TKE deficit in the wake - share a common signature: their amplitude barely changes between the two mesh levels and the same trends appear recurrently in the steady RANS contributions of the AutoCFD4 workshop, regardless of the solver used. This convergence of behaviour points to a limitation linked primarily to wake modelling rather than to numerical discretisation itself.

On this type of base flow, the dominant physics is governed by strongly anisotropic shear layers shed from the base edges, which drive momentum transfer and the organisation of the recirculation zone. In a linear eddy-viscosity RANS approach such as the k-ω SST RANS simulations, these mechanisms are represented from the mean field through a scalar turbulent viscosity. On the Windsor case, this representation consistently leads - across solvers - to a more persistent recirculation zone, a slower velocity recovery and a more localised turbulence than in the experimental measurements.

DDES (Delayed Detached Eddy Simulation) modelling aims precisely at transferring part of the turbulent dynamics from the model to resolved structures in the separated regions of the wake. The model retains a k-ω SST RANS formulation in the attached boundary layers in order to keep the computational cost compatible with industrial automotive Reynolds numbers, then switches to LES behaviour in the wake where large coherent structures can be explicitly resolved. The aim is not to resolve the full turbulent spectrum, as in a complete LES, but to recover the dominant momentum transfer mechanisms and the shear-layer anisotropy that structures the base recirculation.

An SST-DDES simulation is currently running on the medium mesh in order to assess the direct impact of this hybrid approach on the wake structure and on the turbulent transfer mechanisms identified above. The instantaneous Q-criterion visualisation below gives a first glimpse of the turbulent structures resolved in the wake - separated shear layers, longitudinal vortices and three-dimensional dynamics of the recirculation zone - absent in resolved form from the RANS solution.

In closing

On a public automotive benchmark gathering 43 contributions from the main commercial and academic CFD solvers, code_saturne - the open-source solver developed by EDF and distributed under the GPL licence - consistently positions itself within the scatter of steady RANS results of the AutoCFD4 workshop. The aerodynamic coefficients, wall pressure distributions and mean wake structure stand at the same level as the Star-CCM+, Fluent, ConvergeCFD, Cadence Fidelity or OpenFOAM contributions on this case.

Where the simulations depart from the PIV measurements - base pressure, recirculation length or turbulence level in the wake - the discrepancies observed are the same as those found across all the steady RANS approaches of the benchmark, regardless of the solver used. On this case, the identified limitation is therefore that of turbulence modelling applied to a strongly anisotropic and separated wake, and not that of any specific code.

For an industrial team evaluating CFD solutions, the practical reading is direct: on a reference automotive case at industrial Reynolds, code_saturne reproduces the performance level of established commercial solvers while remaining entirely open-source and free to use. The transition to hybrid RANS-LES approaches such as SST-DDES does not require switching tools, only switching modelling levels within the same computational environment.