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

TL;DR

This work develops and validates a dynamic, term-by-term variational multiscale (VMS) stabilization for incompressible turbulence within an incremental pressure-correction framework. By employing orthogonal projections and dynamic subscales, the method achieves stable equal-order velocity–pressure discretizations and controlled dissipation without altering the projection structure, enabling robust simulations on large unstructured meshes. Validation on Ahmed body configurations at $Re \\approx 7.68\times 10^{5}$ and a full-scale Formula 1 geometry at $Re\\approx 1.13\times 10^{6}$ shows accurate wake dynamics, surface pressures, and spectral consistency with turbulent trends, while maintaining stability across mesh resolutions. The approach offers a practical, scalable path to turbulence-resolving external aerodynamics on industrial-scale geometries without explicit turbulence models, with clear implications for drag prediction and wake characterization in vehicle-era simulations.

Abstract

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 Formula~1 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.

Figures (20)

• Figure 1: Ahmed body geometry.
• Figure 2: Computational domain dimensions.
• Figure 3: Computational mesh used for the Ahmed simulations, illustrating near-wall refinement and wake resolution.
• Figure 4: Validation of aerodynamic results for the Ahmed body. (Left) Normalized mean velocity profiles at the symmetry plane for a fixed slant angle of $\theta = 25^\circ$, compared against experimental data. (Right) Variation of the mean drag coefficient $C_D$ as a function of the slant angle ($\theta$), illustrating the comparison with experimental and numerical references.
• Figure 5: Surface pressure coefficient ($C_p$) contours on the Ahmed body for different rear slant angles: (a) $\theta= 0^\circ$, (b) $\theta= 12.5^\circ$, (c) $\theta= 25^\circ$, and (d) $\theta= 35^\circ$.

Theorems & Definitions (1)

• Remark 4.1