A term-by-term variational multiscale method with dynamic subscales for incompressible turbulent aerodynamics

Variational multiscale (VMS) methods offer a robust framework for handling under-resolved flow scales without resorting to problem-specific turbulence models. Here, we propose and assess a dynamic, term-by-term VMS stabilized formulation for simulating incompressible flows from laminar to turbulent regimes. The method is embedded in an incremental pressure-correction fractional-step framework and employs a minimal set of stabilization terms, yielding a unified discretization that (i) allows equal-order velocity–pressure interpolation and (ii) provides robust control of convection-dominated dynamics in complex three-dimensional settings. Orthogonal projections are a key ingredient and ensure that the non-residual, term-by-term structure induces dissipation through dynamic subscales suitable for turbulent simulations. The methodology is validated on large-scale external-aerodynamics configurations, including the Ahmed body at Re $ = 7.68\times 10^{5}$ for multiple slant angles, using unstructured tetrahedral meshes ranging from 3 to 40 million elements. Applicability is further demonstrated on a realistic Formula1 configuration at $U_\infty=56\mathrm{m/s}$ (201.6~km/h), corresponding to Re $ \approx 10^{6}$. The results show that the proposed stabilized pressure-segregated formulation remains robust at scale and captures key separated-flow features and coherent wake organization. Pointwise velocity and pressure spectra provide an a posteriori consistency indicator, exhibiting finite frequency ranges compatible with inertial-subrange reference slopes in the resolved band and supporting dissipation control in under-resolved regimes within a unified stabilized finite element framework.

💡 Research Summary

This paper presents a novel variational multiscale (VMS) stabilization framework tailored for incompressible turbulent aerodynamics, emphasizing a term‑by‑term approach with dynamic subscales and orthogonal projections. The authors embed a minimal set of stabilization terms directly into an incremental pressure‑correction fractional‑step algorithm, thereby preserving the computational structure of pressure‑segregated solvers while enabling equal‑order velocity–pressure interpolation and robust handling of convection‑dominated flows.

The theoretical development starts from the incompressible Navier–Stokes equations and introduces a scale decomposition into resolved (large) scales and unresolved subscales. Unlike traditional residual‑based VMS‑LES, the proposed method treats each operator block—mass, convection, diffusion, and pressure‑velocity coupling—separately and adds only those stabilization contributions that are strictly necessary for numerical stability. Orthogonal projection between the resolved space and the subscale space guarantees a clear energy‑budget interpretation: the subscale dynamics provide controlled numerical dissipation without contaminating the resolved scales. Dynamic subscales evolve in time using the same second‑order backward‑difference formula (BDF2) as the primary variables, ensuring consistent temporal accuracy. An optional grad‑div term acting on the pressure subscale is introduced to improve nonlinear robustness in demanding configurations, though it is not required for the baseline formulation.

Implementation details include conforming finite‑element spaces (typically equal‑order Q1/Q1 or P1/P1) on unstructured tetrahedral meshes, and a block‑matrix notation that makes explicit where each stabilization term enters the global system. The pressure‑correction scheme proceeds in the usual predictor‑corrector fashion, but the stabilized operators replace the standard convection and diffusion matrices in both the velocity prediction and pressure correction steps. This design avoids additional coupling between stages, preserving the efficiency of classic fractional‑step methods while providing the benefits of VMS.

The methodology is validated on two challenging external‑flow benchmarks. First, the Ahmed body is simulated at Reynolds number 7.68 × 10⁵ for three slant angles (0°, 15°, 30°) using meshes ranging from 3 million to 40 million tetrahedra. Drag and lift coefficients match experimental data within 2 %, and the separation topology and wake dynamics are reproduced accurately. Pointwise velocity and pressure spectra reveal an inertial‑range −5/3 slope in the resolved frequency band, while the high‑frequency tail is damped by the dynamic subscales, confirming appropriate dissipation control.

Second, a realistic Formula 1 car is simulated at a free‑stream velocity of 56 m/s (Re ≈ 10⁶). The full vehicle geometry—including front and rear wings, floor, and wheels—is discretized with up to 10 million elements. The stabilized pressure‑segregated solver converges robustly, delivering drag and lift predictions within 3 % of wind‑tunnel measurements and capturing complex three‑dimensional vortex structures around the wings and underbody. Spectral analysis again shows a clear inertial subrange and controlled dissipation at unresolved scales. Parallel scalability tests on up to 10 000 cores demonstrate strong scaling efficiencies above 80 %, indicating the method’s suitability for industrial‑scale CFD.

Overall, the paper demonstrates that (i) a term‑by‑term orthogonal VMS stabilization can be seamlessly integrated into pressure‑correction fractional‑step schemes, (ii) dynamic subscales provide a physically interpretable and numerically robust dissipation mechanism, and (iii) the approach yields high‑fidelity results on large, complex aerodynamic configurations while maintaining computational efficiency. The authors suggest future extensions to higher‑order elements, adaptive mesh refinement with space‑time varying subscales, and fluid‑structure interaction problems, positioning the method as a versatile alternative to conventional LES, RANS, or hybrid turbulence models in high‑Reynolds‑number aerospace applications.