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

TL;DR

This paper introduces a dynamic, term-by-term variational multiscale method with minimal stabilization for simulating incompressible turbulent flows, effectively handling complex three-dimensional aerodynamics from laminar to turbulent regimes.

Contribution

It presents a novel, unified stabilized finite element formulation that allows equal-order interpolation and robust turbulence modeling without problem-specific turbulence models.

Findings

Successfully applied to large-scale external aerodynamics configurations.

Captures key flow features and wake organization in turbulent regimes.

Provides spectral indicators consistent with inertial subrange in turbulence.

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.