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

This paper proposes and validates a dynamic, term-by-term variational multiscale stabilized formulation within an incremental pressure-correction framework that enables robust, equal-order interpolation simulations of incompressible turbulent aerodynamics across laminar to turbulent regimes, successfully capturing complex flow features in large-scale external-aerodynamics configurations like the Ahmed body and Formula 1 cars.

The Gist

Imagine you are trying to predict how air flows around a car moving at high speed. This isn't just about smooth air; it's about chaotic, swirling turbulence that changes every millisecond. To simulate this on a computer, you have to divide the space around the car into millions of tiny puzzle pieces (a mesh).

The problem is that even with millions of pieces, your computer can't see every tiny swirl of air. It's like trying to watch a hurricane through a window with a grid on it; you see the big storms, but the tiny, chaotic eddies between the grid lines are invisible. If you ignore them, your simulation becomes unstable and crashes, or it gives you the wrong answer.

The Paper's Solution: A "Smart Filter" for Airflow

The authors of this paper developed a new mathematical "smart filter" called a Variational Multiscale (VMS) method. Here is how they explain it using simple concepts:

1. The "Big Picture" vs. The "Hidden Details"

Think of the airflow as having two layers:

• The Resolved Scale: The big, visible swirls that your computer mesh can actually see.
• The Subscale: The tiny, invisible swirls that are too small for the mesh to catch.

Old methods often tried to guess what the tiny swirls were doing using fixed rules (like a rigid recipe). This paper proposes a dynamic approach. Instead of a fixed recipe, the computer calculates what the tiny swirls should be doing right now, based on what the big swirls are doing. It's like having a co-pilot who constantly adjusts the steering based on the road conditions, rather than following a pre-set map.

2. The "Term-by-Term" Strategy

The authors built this method to work with a specific way of solving equations called a "fractional-step" method. Imagine solving a complex puzzle by doing one piece at a time: first the speed, then the pressure.

• The Innovation: They added their "smart filter" directly into each step of the puzzle-solving process without messing up the order.
• The Analogy: Imagine you are baking a cake. Usually, you mix ingredients, then bake. If you need to add a special stabilizer, you might have to restart the whole recipe. This new method lets you sprinkle the stabilizer directly into the batter while you are mixing, ensuring the cake rises perfectly without changing the baking steps. This keeps the process fast and stable.

3. The "Orthogonal" Safety Net

A key feature of their method is "orthogonal projection." Imagine you are trying to separate red marbles from blue marbles in a jar.

• Old way: You might accidentally mix them up or leave some behind.
• This method: It ensures that the "big swirls" (red) and the "tiny swirls" (blue) are kept in completely separate, non-overlapping boxes. This prevents the computer from getting confused and double-counting energy, which keeps the simulation stable even when the air is very turbulent.

4. The Real-World Tests

The authors didn't just do this on paper; they tested it on two very difficult scenarios:

• The Ahmed Body: This is a simple, boxy shape used by scientists as a standard test for car aerodynamics. They tested it at different angles (like tilting the back of a car).

  • Result: The method worked perfectly. It predicted the drag (air resistance) accurately and showed that the computer could handle the chaotic air swirling behind the car without crashing. They found that using a very fine mesh (37 million pieces) gave the most accurate results, but the method stayed stable even on coarser meshes.

• The Formula 1 Car: This is a much harder test. An F1 car is covered in wings, wheels, and curves, creating incredibly complex air patterns.

  • Result: They simulated a real F1 car at racing speeds (200 km/h) without using any "turbulence models" (the usual shortcuts). The method handled the complex, 3D air vortices and the "ground effect" (air sucking the car down) successfully. It produced realistic data about how the air moves and how much force is on the car.

5. Checking the "Music" of the Air

To prove their method was working correctly, they looked at the "spectra" of the air flow.

• The Analogy: Think of the air flow as music. In a real turbulent flow, the energy of the "notes" (eddies) follows a specific pattern as they get smaller (like a specific musical scale).
• The Result: The computer simulation produced a "song" that matched the natural physics of turbulence. The energy dropped off at the right rate, proving that the "smart filter" was dissipating energy correctly, just like real air does.

Summary

In short, this paper presents a new, robust way to simulate turbulent air around vehicles. It uses a dynamic, self-adjusting mathematical filter that separates big air movements from tiny ones. It works on complex, unstructured computer meshes (like the shape of a real car) and remains stable even when the air is extremely chaotic. The authors proved it works on both a standard test block and a highly complex Formula 1 car, showing it can handle real-world engineering challenges without needing to rely on simplified guesses about how turbulence behaves.

Technical Summary

Technical Summary: A Term-by-Term Variational Multiscale Method with Dynamic Subscales for Incompressible Turbulent Aerodynamics

Problem Statement
Accurate turbulence representation is critical in computational aerodynamics for predicting drag, lift, and separation, yet Direct Numerical Simulation (DNS) remains computationally prohibitive at engineering Reynolds numbers. While Reynolds-averaged Navier–Stokes (RANS) models offer efficiency, they often lack predictive capability in complex geometries with separated flows. Large-Eddy Simulation (LES) provides a middle ground but incurs substantial costs for high-Reynolds-number, realistic geometries. Furthermore, industrially relevant computations often rely on fractional-step (pressure-segregation) strategies to solve the incompressible Navier–Stokes equations efficiently on large unstructured meshes. However, there is limited large-scale evidence regarding how Variational Multiscale (VMS) turbulence-resolving formulations behave when embedded within these pressure-correction fractional-step schemes for complex, under-resolved external aerodynamics.

Methodology
The authors propose a dynamic, term-by-term VMS stabilized formulation tailored for an incremental pressure-correction fractional-step method. The methodology is designed to operate on unstructured tetrahedral meshes using equal-order velocity–pressure interpolation, which requires stabilization to maintain robustness in convection-dominated regimes.

Key methodological components include:

• Orthogonal VMS Framework: The solution is decomposed into resolved scales (finite element space) and unresolved subscales (orthogonal complement). Orthogonal projections ensure a consistent scale separation, allowing the non-residual, term-by-term structure to induce dissipation through dynamic subscales without explicit eddy-viscosity closures.
• Dynamic Subscales: Unlike quasi-static approaches, the subscales are treated as time-dependent variables integrated consistently using the second-order backward differentiation formula (BDF2). This allows the method to recover key features of turbulent energy transfer, including inertial-range dissipation and backscatter effects, while removing restrictive time-step constraints.
• Term-by-Term Embedding: The stabilization is constructed at the fully discrete monolithic level and embedded into the fractional-step stages by modifying existing operator blocks rather than introducing new velocity–pressure cross-couplings.
  • Momentum-side stabilization enters the velocity predictor stage.
  • Pressure-related diagonal contributions (induced by the pressure-gradient subscale) augment the pressure-correction Poisson solve.
  • Subscale-history terms appear only as known right-hand-side contributions, preserving the standard projection structure.
• Minimal Stabilization Layout: The formulation introduces only the terms required for stable equal-order interpolation and convection control. An optional grad–div contribution (acting through the pressure subscale) is included to enhance nonlinear robustness in demanding configurations.
• Implementation: Nonlinearities arising from convection and stabilization parameters are handled via Picard (fixed-point) iteration. Orthogonal projections are realized via a projection–correction strategy involving $L^2$ projections onto the discrete space.

Key Contributions

1. Unified Discretization: The method provides a unified framework that enables stable equal-order velocity–pressure interpolation and robust control of convection-dominated dynamics in complex 3D settings without resorting to problem-specific turbulence models.
2. Integration with Fractional-Step Methods: The work demonstrates how orthogonal, dynamic subscale stabilization can be integrated into incremental pressure-correction schemes without altering their stage-wise structure or efficiency.
3. Scalability and Robustness: The formulation is validated on large-scale external aerodynamics configurations using unstructured meshes ranging from 3 to 40 million elements, demonstrating stability across a wide range of resolutions.

Results
The methodology was validated on two configurations:

1. Ahmed Body ($Re = 7.68 \times 10^5$):

   • Simulations were performed for multiple rear slant angles ($0^\circ, 12.5^\circ, 25^\circ, 35^\circ$).
   • Mesh sensitivity analysis (3.2M to 37.3M elements) showed that while integral loads (drag coefficient) are sensitive to resolution, the formulation remained numerically stable across all levels.
   • The fine-mesh result ($C_D \approx 0.33$) showed competitive agreement with reference LES and experimental data, particularly for the critical $25^\circ$ slant angle.
   • Pointwise velocity and pressure spectra exhibited finite frequency ranges compatible with inertial-subrange reference slopes ($-5/3$ for velocity, $-7/3$ for pressure), serving as an a posteriori consistency indicator for dissipation control.
   • Visualization of coherent structures (Q-criterion) and wake topology confirmed the resolution of separated shear layers and vortex interactions.

2. Formula 1 Configuration ($Re \approx 1.13 \times 10^6$):

   • A realistic, full-scale F1 car geometry was simulated at $U_\infty = 56$ m/s using a single fine mesh of $\approx 3.87 \times 10^7$ elements.
   • No explicit turbulence model was used; robustness relied solely on the stabilized formulation.
   • The simulation successfully captured complex 3D flow interactions, including ground-effect mechanisms, wheel wakes, and multi-element lifting surfaces.
   • Integral coefficients ($C_D$, $C_L$) and spectral diagnostics confirmed the method's ability to handle under-resolved regimes in realistic, multi-component geometries.
   • The inclusion of the optional grad–div term was found critical for nonlinear convergence in this demanding case; without it, iterations tended to stagnate.

Significance and Claims
The paper claims that the proposed stabilized, pressure-segregated formulation remains robust at scale and effectively captures key separated-flow features and coherent wake organization in complex external aerodynamics. By utilizing orthogonal dynamic subscales, the method induces controlled numerical dissipation suitable for turbulent simulations without explicit eddy-viscosity models. The work bridges the gap between controlled benchmark studies and application-driven aerodynamics, demonstrating that VMS-based turbulence-resolving formulations can be successfully applied within efficient fractional-step frameworks on large, unstructured meshes. The spectral diagnostics provided offer a practical tool for assessing dissipation control in large-scale, under-resolved simulations. The authors note that the choice of stabilization parameters (specifically $c_3 = 24$) was empirically determined to provide the most robust nonlinear behavior across the test cases.