Mesh independence is the foundation of credible CFD results. A simulation that changes significantly with mesh refinement cannot be trusted for engineering decisions. Yet mesh independence studies are often rushed or skipped entirely, leading to unreliable predictions and costly design mistakes.
The Richardson Extrapolation Method
The Richardson Extrapolation method provides a rigorous framework for quantifying mesh-related uncertainty.
The Process
By running simulations on systematically refined grids (typically with refinement ratios of 1.5-2.0), we can estimate:
- •The grid-converged solution
- •The discretization error
"This uncertainty should always accompany CFD predictions."
Mathematical Framework
For a quantity φ computed on grids with spacing h₁ < h₂ < h₃:
φ_exact ≈ φ₁ + (φ₁ - φ₂) / (r^p - 1)
where:
r = h₂/h₁ (refinement ratio)
p = ln[(φ₃ - φ₂)/(φ₂ - φ₁)] / ln(r) (order of convergence)
Different Quantities, Different Convergence
Critical insight: Different quantities converge at different rates.
| Quantity Type | Convergence Rate | Example |
|---|---|---|
| Integral | Fast | Drag coefficient |
| Surface | Medium | Pressure distribution |
| Local | Slow | Peak shear stress |
Critical design parameters should be identified early, and mesh refinement focused on achieving acceptable convergence for these specific outputs.
y+ Requirements: A Common Pitfall
y+ requirements for turbulence models often dictate near-wall mesh resolution:
Wall-function approach: 30 < y+ < 300
Low-Reynolds-number model: y+ < 1
The Problem
Inconsistent y+ treatment across the domain can lead to:
- •⚠️ Solution artifacts
- •⚠️ Slow convergence
- •⚠️ Incorrect wall shear prediction
Best Practice
Maintain consistent y+ across all walls, and document the values in your simulation report.
Adaptive Mesh Refinement
Automated mesh adaptation offers an alternative to manual mesh refinement studies.
By refining based on solution gradients or error indicators, adaptive meshing can achieve target accuracy with fewer cells than uniform refinement.
Adaptation Criteria
| Criterion | Best For |
|---|---|
| Gradient-based | General flows |
| Adjoint-based | Specific outputs |
| Feature-based | Known phenomena (shocks, wakes) |
Caution: Care must be taken to ensure the adaptation criteria capture the physics of interest. Adapting on the wrong feature can miss critical flow structures.
Documentation: As Important as the Study
Documentation of mesh independence studies is as important as the studies themselves.
What to Document
For engineering projects, we maintain detailed records of:
Mesh Study Report
├── Grid parameters (cell count, quality metrics)
├── y+ distribution maps
├── Convergence plots for key quantities
├── Richardson extrapolation analysis
├── Uncertainty estimates
└── Recommendations for production mesh
This documentation supports:
- •Design reviews
- •Regulatory submissions
- •Future reference and reproducibility
Quick Reference: Mesh Study Checklist
- • Define quantities of interest before meshing
- • Create at least 3 systematically refined meshes
- • Use refinement ratio of 1.5-2.0
- • Check y+ consistency across walls
- • Plot convergence for all critical quantities
- • Perform Richardson extrapolation
- • Document uncertainty estimates
- • Archive meshes and results
Key Takeaways
- •Richardson Extrapolation quantifies discretization error rigorously
- •Different output quantities converge at different rates
- •y+ consistency is critical for turbulence model accuracy
- •Adaptive meshing can reduce computational cost while maintaining accuracy
- •Documentation of mesh studies is essential for engineering credibility
Need help with mesh independence studies for your CFD project? Contact our experts for guidance.
