Leveraging unstructured grids for direct numerical simulations of wall turbulence

This paper introduces the {\eta}-grid, an unstructured grid-generation framework for direct numerical simulations of wall turbulence that scales grid sizes with the local Kolmogorov scale, achieving accuracy comparable to conventional Cartesian grids while significantly reducing computational cost, particularly at high Reynolds numbers and over complex riblet geometries.

The Gist

Imagine trying to simulate how air flows over a car or water moves past a ship's hull. To do this accurately on a computer, scientists use a technique called Direct Numerical Simulation (DNS). Think of DNS as creating a giant, 3D digital microscope that breaks the fluid (air or water) into millions of tiny, invisible cubes (a grid). The computer then calculates how every single cube moves and interacts with its neighbors.

The problem is that fluid flow near a surface (like the side of a ship) is incredibly chaotic and detailed. To get a clear picture, you need a massive number of these tiny cubes right next to the surface. However, as you move further away from the surface, the chaos smooths out, and you don't need such tiny cubes.

The Old Way: The "Rigid Brick Wall"

Traditionally, scientists used a Cartesian grid. Imagine building a wall out of identical, rigid bricks.

• The Problem: To see the tiny details near the surface, you have to make the bricks at the bottom very small. But because these bricks are rigid and connected in a straight line, you are forced to use those same tiny bricks all the way up to the top of your wall, even where the details aren't important.
• The Result: You end up with a wall made of billions of tiny bricks, most of which are unnecessary. This makes the computer simulation incredibly slow and expensive, like trying to count every grain of sand on a beach just to measure the tide.

The New Solution: The "Smart, Stretchy Net"

This paper introduces a new method called the $\eta$-grid (eta-grid). Instead of rigid bricks, imagine a smart, stretchy fishing net.

• How it works: The authors designed a system where the size of the net's holes changes automatically based on how much detail is needed.
  • Near the surface (The "Inner Layer"): The net has very small, tight holes to catch the tiny, chaotic swirls of the fluid.
  • Further away (The "Outer Layer"): As the fluid gets calmer, the net automatically stretches, making the holes much larger.
• The Secret Ingredient: The size of these holes is based on something called the Kolmogorov scale (denoted as $\eta$). Think of this as the "smallest possible swirl" that can exist in the fluid at any given height. The new grid simply says: "Make the hole size just big enough to catch the smallest swirl at this specific height, and no bigger."

Why This is a Big Deal

The authors tested this "smart net" on two different types of computer codes (one like a spectral element method, the other like a finite volume method) and compared it to the old "rigid brick" method.

1. Accuracy: The results were nearly identical. The "smart net" captured the physics just as well as the "rigid bricks," with less than a 1% difference in key measurements like friction and speed.
2. Massive Savings: This is where the magic happens.
   • For smooth surfaces (like a flat wall), the new grid reduced the number of required "bricks" (grid points) by about 90% at high speeds.
   • For rough surfaces (like a wall with tiny grooves called "riblets" designed to reduce drag), the savings were even more dramatic—up to 97% fewer grid points.

The Analogy of the "Grooved Wall"

To understand the riblet part, imagine a wall covered in tiny, parallel grooves (like the texture of a golf ball or a shark's skin).

• The Old Way: To simulate this, the rigid brick method had to keep the bricks tiny everywhere because the grooves forced the grid to be fine all the way up. It was like trying to count every single thread in a sweater, even the parts far away from the fabric.
• The New Way: The "smart net" knows that once you are a few inches above those tiny grooves, the flow becomes smooth again. It stretches the holes immediately above the grooves, ignoring the tiny details that no longer matter.

The Bottom Line

The authors have created a framework that acts like a smart zoom lens for fluid simulations. It focuses its computational power exactly where it's needed (near the wall) and relaxes where it doesn't.

• For Smooth Walls: It scales the effort much more slowly as the simulation gets bigger.
• For Rough Walls: It scales even better, making simulations of complex, drag-reducing surfaces feasible on computers that previously couldn't handle them.

In short, they found a way to do the same high-quality work with a fraction of the computing power, turning a task that might take a supercomputer a month into one that could be done in a few days.

Technical Summary

Technical Summary: Leveraging Unstructured Grids for Direct Numerical Simulations of Wall Turbulence

Problem Statement
Direct Numerical Simulation (DNS) of wall turbulence is computationally expensive, primarily due to the requirement of resolving the Kolmogorov length scale ($\eta$) across the entire domain. Conventional Cartesian grid solvers, widely used for smooth and non-smooth surfaces (e.g., via Immersed Boundary Methods), typically enforce a fixed or slowly varying streamwise ($\Delta x^+$) and spanwise ($\Delta z^+$) grid resolution throughout the domain. This approach leads to significant over-resolution in the outer regions where the Kolmogorov scale grows, creating a computational bottleneck. This is particularly acute for simulations involving complex surfaces like riblets, where the grid must be extremely fine near the surface to resolve micro-grooves, yet this fine resolution is unnecessarily maintained throughout the entire boundary layer thickness in Cartesian frameworks.

Methodology
The authors propose a new unstructured grid-generation framework termed the $\eta$-grid. The core principle is to set the wall-normal ($\Delta y^+$) and spanwise ($\Delta z^+$) grid sizes proportional to the local Kolmogorov scale ($\eta^+$), allowing grid coarsening in the outer flow regions where turbulence scales are larger.

1. Grid Formulation:

   • Inner Layer: A thin layer ($\sim 50$ viscous units) adjacent to the wall uses viscous-scaled grid sizes similar to conventional DNS ($0.3 \lesssim \Delta y^+ \lesssim 4$, $\Delta z^+ \simeq 5$). For non-smooth surfaces (e.g., riblets), the grid resolves the smallest surface wavelength ($\ell^+$) with $\ell^+/30 \lesssim \Delta y^+, \Delta z^+ \lesssim 4$.
   • Outer Region: Above the inner layer, grid sizes increase proportionally to the local Kolmogorov scale ($\Delta y^+ \simeq \Delta z^+ \simeq 2\eta^+$).
   • Scaling Laws: The authors refine semi-empirical fits for $\eta^+$ in both the logarithmic and wake regions for turbulent channel flow and zero-pressure-gradient (ZPG) turbulent boundary layers (TBL). They propose piecewise power-law fits for $\eta^+$ that transition at $y^+ \approx \delta^+/2$.
   • Riblet Adaptation: For riblet geometries, the framework introduces a "riblet sublayer" ($\delta_r^+$) filled with square cells to resolve secondary flows and Kelvin-Helmholtz rollers, transitioning smoothly into the $\eta$-based scaling above.

2. Implementation:

   • The framework utilizes the open-source package Gmsh to generate multi-block unstructured grids.
   • The grids are tested using two distinct solvers: SOD2D (Spectral Element Method) and OpenFOAM (Finite Volume Method).
   • Simulations cover turbulent channel flow and ZPG TBLs over smooth walls and various riblet geometries (triangular grooves) up to a friction Reynolds number $\delta_0^+ = 1000$.

Key Contributions

• Framework Development: A robust, multi-block unstructured grid generation strategy that dynamically adapts grid spacing to the local Kolmogorov scale, applicable to both smooth and complex rough surfaces.
• Refined Scaling Laws: Updated semi-empirical fits for the Kolmogorov scale in the wake region of TBLs and channel flows, improving the accuracy of grid sizing in the outer layer.
• Solver Agnosticism: Demonstration that the $\eta$-grid framework yields consistent, high-fidelity results across different numerical discretization schemes (SEM and FVM).
• Riblet-Specific Formulation: A tailored grid strategy for riblets that resolves the sublayer physics (secondary flows, KH rollers) while allowing rapid coarsening in the outer flow, overcoming the limitations of Cartesian/IBM approaches.

Results

• Accuracy: The $\eta$-grid produces results with less than 1% difference compared to conventional Cartesian grids and reference DNS/experimental data. Metrics include skin-friction coefficients ($C_f$), mean velocity profiles ($U^+$), turbulent stresses, and spectrograms.
• Grid Efficiency:
  • Smooth Walls: The number of grid points ($N_\eta$) scales as $\propto \delta_0^{+2.5}$, compared to $\propto \delta_0^{+3.0}$ for Cartesian grids with hyperbolic tangent stretching. At $\delta_0^+ = 6000$, the $\eta$-grid requires only $\sim 10\%$ of the grid points of a standard Cartesian grid ($N_\eta/N_{Tanh} \simeq 0.1$).
  • Riblets: The efficiency gain is even more pronounced. For drag-reducing triangular riblets, $N_\eta$ scales as $\propto \delta_0^{+2.0}$, while Cartesian grids scale as $\propto \delta_0^{+3.0}$. At $\delta_0^+ = 6000$, the $\eta$-grid requires only $\sim 3\%$ of the grid points of a Cartesian grid ($N_\eta/N_{Tanh} \simeq 0.03$).
• Riblet Physics: The framework successfully captures complex flow mechanisms over riblets, including the virtual origin, secondary flows, and Kelvin-Helmholtz rollers, with grid convergence achieved using moderate grid parameters ($C_y=2.0, C_z=2.5, y_{in}^+=50$).
• TBL Validation: DNS of ZPG TBLs over riblets showed good agreement with experimental data (Baron & Quadrio; Choi & Orchard) regarding drag reduction percentages and velocity profiles, despite inherent experimental-DNS discrepancies in wake regions.

Significance
The paper claims that the $\eta$-grid framework offers an "enormous grid saving," making high-Reynolds-number DNS of wall turbulence over complex surfaces computationally feasible. By aligning grid resolution with the local physical scales (Kolmogorov scale) rather than maintaining fixed resolutions, the method drastically reduces the computational cost (grid points and time-step constraints) without sacrificing accuracy. The authors emphasize that this approach enables the affordability of conducting DNS of TBLs over complex surfaces at Reynolds numbers that match experimental conditions, a task previously hindered by the prohibitive cost of Cartesian-based methods. The work validates the efficacy of unstructured grids for high-fidelity turbulence simulations, moving beyond the constraints of structured Cartesian meshes.