A Novel Explicit Filter for the Approximate Deconvolution in Large-Eddy Simulation on General Unstructured Grids: A posteriori tests on highly stretched grids

This paper introduces a novel explicit filter for Large-Eddy Simulations on general unstructured grids that combines face-averaging and recursive filtering with multi-objective optimization to overcome the spectral deficiencies of conventional filters on highly stretched meshes, thereby significantly improving prediction accuracy and stability in turbulent flow simulations.

The Gist

Imagine you are trying to listen to a symphony orchestra, but you are sitting in a room with a very strange, uneven floor. Some parts of the floor are smooth and flat, while others are stretched out like long, thin strips of rubber.

Large-Eddy Simulation (LES) is like trying to record that symphony. In fluid dynamics (the study of how air and water move), the "music" is the swirling motion of turbulence. To make the recording manageable, scientists use a filter to separate the big, loud notes (the large swirls we can see) from the tiny, high-pitched squeaks (the tiny swirls we can't resolve).

The problem is, when the "floor" (the computer grid) is stretched out like those rubber strips, the old filters used by scientists start to act crazy. They get confused, distort the music, and sometimes even cause the whole recording to crash and stop working.

Here is a simple breakdown of what this paper does to fix that problem:

1. The Problem: The "Stretched Floor" Effect

Think of the computer grid as a net used to catch fish.

• Uniform Grid: A net with perfect, square holes. Easy to use.
• Stretched Grid: A net where some holes are tiny squares, but others are long, skinny rectangles (like a stretched-out piece of taffy). This is necessary to simulate things like air flowing over a car wing or a plane, where you need tiny details near the surface and big, empty spaces far away.

The paper shows that the old filters (the "Laplace" and "Simple" filters) were designed for square nets. When you force them onto a stretched net, they get confused.

• The Laplace Filter: It's like a filter that tries to smooth things out by looking at neighbors. On a stretched net, it gets so confused by the long, skinny cells that it starts creating wild, fake vibrations (oscillations). It's like trying to smooth a wrinkle in a long scarf by pulling too hard on one end—it just rips the fabric. This caused the computer simulations to crash.
• The Simple Filter: It's a bit more stable, but it treats the long direction and the short direction differently. It's like a camera that focuses perfectly on the left side of the room but makes the right side blurry. This leads to inaccurate predictions of how the wind or water actually behaves.

2. The Solution: A "Smart, Recursive" Filter

The authors built a new filter specifically designed to handle these stretched, uneven nets. They call it a "Novel Explicit Filter."

Here is how it works, using a cooking analogy:

• The Old Way: You try to mix a sauce by stirring it once with a giant spoon. If the pot is weirdly shaped, you miss spots or splash sauce everywhere.
• The New Way (Recursive Filtering): Instead of one big stir, the new filter does a series of small, careful steps.
  1. Face-Averaging: Imagine looking at the walls of your kitchen cells and averaging the temperature based on the walls, not the center of the room. This avoids the confusion of the "stretched floor."
  2. Recursive Steps: It doesn't just do this once. It does it, then checks the result, then does it again, and again (3 or 4 times). Each time, it gently refines the picture, smoothing out the noise without losing the important details.

3. The "Tuning" Process

You can't just guess how strong these steps should be. If you stir too hard, you ruin the sauce; too soft, and it's still lumpy.

The authors used a computer optimizer (like a super-smart robot chef) to find the perfect "recipe." They told the robot:

• "Don't change the flavor of the big notes (low frequencies)."
• "Completely silence the high-pitched squeaks (high frequencies) so they don't cause noise."
• "Make sure the filter works the same way whether the net is stretched or square."
• "Don't let the simulation crash (stability)."

The robot ran thousands of simulations to find the perfect set of numbers (coefficients) that made the filter work perfectly on any shape of net.

4. The Results: A Clearer Symphony

When they tested this new filter on two famous "turbulence test cases" (a flow between two walls and a swirling vortex):

• The Old Filters: The simulations either crashed (Laplace) or gave blurry, inaccurate results (Simple).
• The New Filter: It kept the simulation stable even on the most stretched, difficult grids. It predicted the speed of the wind and the stress on the walls much more accurately.

The Bottom Line:
Think of this new filter as a universal translator for turbulence. No matter how weird or stretched the computer grid is, this filter translates the chaotic motion of the fluid into a clear, stable, and accurate picture. It stops the computer from crashing and helps engineers design better airplanes, cars, and wind turbines by giving them a more reliable view of how air and water move.

Technical Summary

1. Problem Statement

Large-Eddy Simulation (LES) relies on spatial filtering to separate resolved scales from sub-grid scales (SGS). In Approximate Deconvolution (AD) methods, an explicit filter is crucial for scale separation and the reconstruction of sub-filter scales (SFS). However, existing explicit filters designed for general unstructured grids (common in complex engineering applications) suffer from significant limitations:

• Grid Sensitivity: Conventional filters (e.g., Laplace differential filters and simple face-averaging filters) exhibit strong dependence on the cell aspect ratio. On highly stretched grids (typical of boundary layers), their spectral properties degrade.
• Instability and Divergence: Filters like the Laplace filter violate stability and positivity criteria on grids with high aspect ratios, leading to solution divergence and unphysical oscillations near walls.
• Anisotropy: Existing filters often apply filtering anisotropically, failing to provide consistent attenuation across all spatial directions (streamwise, spanwise, and wall-normal).
• Commutation Errors: On non-uniform grids, standard filters introduce significant commutation errors (non-commutativity between filtering and differentiation operators), leading to dispersion errors.

2. Methodology

2.1. Mathematical Framework

The study utilizes an AD-LES approach coupled with the Mixed Alternative Linear Dynamic Model with Equilibrium assumption (ALDME). The governing equations involve:

• Soft Deconvolution: Reconstructing SFS stresses using an approximate inverse of the explicit filter (Van Cittert iterative algorithm).
• Hard Deconvolution (Regularization): Modeling the remaining SGS stress using a dynamic eddy-viscosity model.

2.2. Analysis of Existing Filters

The authors performed a rigorous a priori analysis of two common filters on various grid configurations (uniform, non-uniform, and unstructured):

• Laplace Filter (Differential): Found to be unstable on grids with aspect ratios $R > (c/2)^{3/4}$. It violates positivity and stability criteria on stretched grids, causing divergence.
• Simple Filter (Face-Averaging): While stable and positive, it suffers from high commutation errors and anisotropic attenuation that worsens with grid stretching.

2.3. Proposed Novel Filter

A new recursive explicit filter is introduced, combining:

1. A New Face-Averaging Technique ($F_M$): Unlike standard area-weighted averaging, this technique is designed to be less sensitive to grid topology.
2. Recursive Filtering: The filter is applied iteratively ($N_R$ times) with relaxation coefficients ($b_n$) to tune the filter width and spectral properties.
   • Formulation: $\bar{\phi}_i^n = (1 - b_n)\bar{\phi}_i^{n-1} + b_n F_M(\bar{\phi}_i^{n-1})$

2.4. Multi-Objective Optimization

The filter parameters ($N_R$ and $b_n$) were determined via a constrained multi-objective optimization using a Genetic Algorithm (GA). The optimization aimed to satisfy:

• High-wavenumber attenuation: Vanishing transfer function at the Nyquist frequency.
• Commutativity: Minimizing the first moments of the filter in physical space.
• Dispersion Error: Minimizing the imaginary part of the transfer function.
• Stability & Positivity: Ensuring the transfer function magnitude $\le 1$ and real part $> 0$.
• Filter Width: Precise control of the cutoff frequency (targeting $\Delta = 2h$).

The optimization was performed on a representative non-uniform Cartesian grid (growth rate 1.1, aspect ratio 50) to ensure robustness for boundary layer flows.

3. Key Contributions

1. Novel Filter Design: Development of a recursive, face-averaging based explicit filter that is independent of grid aspect ratio, addressing the primary failure mode of existing filters on stretched grids.
2. Optimization Framework: A systematic methodology to tune filter coefficients for unstructured grids, ensuring desirable spectral properties (isotropy, stability, low dispersion) simultaneously.
3. Comprehensive A Priori Analysis: Detailed spectral analysis demonstrating that the new filter maintains isotropic behavior and stability even on grids with aspect ratios up to 50, whereas conventional filters fail.
4. Open-Source Tools: Provision of MATLAB code and user guides (Supplementary Materials) for calculating filter coefficients on arbitrary grids and performing the optimization, facilitating adoption by the community.

4. Results

4.1. A Priori Analysis (Spectral Properties)

• Isotropy: The new filter exhibits nearly identical transfer functions in all spatial directions, regardless of grid stretching.
• Stability: Unlike the Laplace filter, the new filter remains stable and positive on grids with aspect ratios up to 50.
• Attenuation: It achieves sufficient attenuation near the Nyquist wavenumber in all directions, a property lacking in the "simple" filter on stretched grids.
• Commutation: The first moments (commutation error) are significantly reduced compared to the simple filter.

4.2. A Posteriori Analysis (Turbulent Channel Flow)

Simulations were conducted at friction Reynolds numbers $Re_\tau = 395$ and $590$ on highly stretched unstructured grids.

• Stability: Simulations using the Laplace filter diverged immediately. The simple filter remained stable but produced inaccurate results.
• Mean Velocity Profile: The new filter significantly reduced the log-layer mismatch in the mean velocity profile compared to the simple filter, bringing results closer to Direct Numerical Simulation (DNS) data.
• Reynolds Stresses: Improved prediction of streamwise and shear Reynolds stresses, particularly at higher $Re_\tau$.
• Energy Budget: The new filter increased the contribution of deconvolved SFS stresses and reduced the directly resolved kinetic energy (by ~30%), indicating better scale separation. The modeled SGS energy remained below 10%, satisfying Pope's criterion for well-resolved LES.
• LES Quality: The ratio of eddy viscosity to molecular viscosity remained well below the critical value of 20.

4.3. A Posteriori Analysis (Taylor-Green Vortex)

The filter was tested on 3D unstructured prism grids for the decaying Taylor-Green Vortex benchmark.

• The new filter maintained closer agreement with DNS data for kinetic energy decay, especially in the later stages of the simulation where small-scale structures dominate and grid stretching effects are most pronounced.

5. Significance and Conclusion

This study addresses a critical bottleneck in applying high-fidelity AD-LES to complex geometries using unstructured grids. By demonstrating that grid aspect ratio sensitivity is the primary cause of failure for existing filters, the authors propose a solution that decouples filter performance from grid topology.

The new recursive filter enables stable, high-fidelity LES on highly stretched boundary-layer grids without the need for ad-hoc boundary treatments or excessive computational cost. The results confirm that proper explicit filter design is the key to unlocking the full potential of AD-LES methods, yielding superior accuracy in predicting turbulent statistics compared to conventional approaches. The work provides a robust, optimized, and reproducible framework for future LES applications on general unstructured meshes.