Mesh quality: skewness, aspect ratio, orthogonality

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.