--- title: "Mesh quality: skewness, aspect ratio, orthogonality" subject: "Applied Computational Fluid Dynamics" chapter: 4 key: "semester_7/applied_computational_fluid_dynamics:mesh-quality-metrics" source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/mesh-quality-metrics site: BCT Engineering --- # Mesh quality: skewness, aspect ratio, orthogonality Applied Computational Fluid Dynamics · Chapter 4: Mesh Generation and Solver Setup - Asked in: year not given, 6 marks - Video: [How to check the MESH Quality in Ansys Workbench | Mesh Metrics | Skewness | Orthogonal Quality](https://www.youtube.com/watch?v=X1km2gg_FTA) — MECH Tech Simulations - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/mesh-quality-metrics - Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-4 ## Question Explain the mesh quality measures skewness, aspect ratio and orthogonality. State the acceptable ranges and the effect of poor quality on the solution. ## Short answer Skewness measures cell shape deviation from ideal (0 perfect, 1 degenerate); keep maximum below 0.85. Aspect ratio is longest/shortest edge; ideal 1, high values acceptable only in aligned boundary layers, elsewhere keep below 5. Orthogonality (0 bad, 1 best) should be above 0.1. Poor quality causes large truncation error, slow convergence, and inaccurate forces. ## Answer Mesh quality strongly affects accuracy, stability and convergence. #### Skewness It measures how far a cell is from an ideal shape (equilateral triangle, rectangle). - **Equiangular skewness:** $$ S=\max\left[\frac{\theta_{max}-\theta_e}{180^\circ-\theta_e},\ \frac{\theta_e-\theta_{min}}{\theta_e}\right] $$ where $\theta_e=60^\circ$ for triangles/tetrahedra and $90^\circ$ for quads/hexahedra. - Range 0 (perfect) to 1 (degenerate). Typical guide: 0 to 0.25 excellent, 0.25 to 0.5 good, 0.5 to 0.8 acceptable, 0.8 to 0.95 poor, above 0.95 unacceptable. Aim for maximum below 0.85 (below 0.9 in 3D tetra). #### Aspect ratio Ratio of the longest to the shortest edge (or dimension) of a cell: $$ AR=\frac{\Delta_{max}}{\Delta_{min}} $$ - Ideal value is 1. High values (10 to 1000) are acceptable only inside boundary layers where gradients are in one direction and the flow is aligned with the long side. - Elsewhere keep $AR<5$; avoid sudden changes in cell size (size change ratio below about 1.2 to 1.3). #### Orthogonality Measures the angle between the face normal vector and the vector joining the centres of the two adjacent cells (and between the face normal and the vector to the face centre). - Orthogonal quality $=\min\left(\cos\theta\right)$ where $\theta$ is that angle; range 0 (bad) to 1 (best). Acceptable above about 0.1 (best above 0.2), and non-orthogonality (the angle itself) should be below about 70 to 75°. - A perfect Cartesian cell has orthogonality 1. #### Effects of poor quality - Large truncation error and numerical diffusion, as the central gradient assumption fails. - Slow convergence, divergence or negative cell volumes. - Wrong pressure gradients and forces, even when residuals look small. - Need for extra non-orthogonal correction loops. Quality should be checked before solving, and bad cells fixed by re-meshing, smoothing or local refinement. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For equiangular skewness, what do values 0 and 1 mean? A. 0 is degenerate, 1 is perfect B. 0 is perfect (ideal shape), 1 is degenerate C. 0 is the limit for acceptable cells, 1 is excellent D. Both are acceptable Answer: B. Skewness ranges from 0 (ideal equilateral triangle or rectangle) to 1 (degenerate cell). (Key point: Skewness meaning) 2. Where is a high aspect ratio (10 to 1000) acceptable? A. Anywhere in the core flow B. Near sudden size changes C. In regions of strong swirl D. Inside boundary layers where gradients are one-directional and the long side is aligned with the flow Answer: D. Stretched cells are fine when gradients act mainly normal to the wall; elsewhere AR should stay below about 5. (Key point: Aspect ratio use) 3. How is orthogonal quality read? A. 0 is best and 1 is worst B. Higher is better, with 1 best; a Cartesian cell has 1 and below about 0.1 is unacceptable C. It only matters for triangles D. It must stay below 0.1 Answer: B. Orthogonal quality is the minimum cos(theta) between face normal and centroid vectors, from 0 (bad) to 1 (best). (Key point: Orthogonal quality) 4. Which is an effect of poor mesh quality? A. Faster convergence due to larger cells B. Lower truncation error C. Slow convergence or divergence, and wrong forces even when residuals look small D. No need for non-orthogonal correction Answer: C. Poor cells increase truncation error and numerical diffusion and can wreck stability and pressure gradients. 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