--- title: "Skewness and aspect ratio of cells" subject: "Applied Computational Fluid Dynamics" chapter: 4 key: "semester_7/applied_computational_fluid_dynamics:skewness-aspect-ratio-numerical" source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/skewness-aspect-ratio-numerical site: BCT Engineering --- # Skewness and aspect ratio of cells Applied Computational Fluid Dynamics · Chapter 4: Mesh Generation and Solver Setup - Asked in: year not given, 6 marks - Video: [What is Skewness and Aspect ratio in Ansys Meshing? In Hindi](https://www.youtube.com/watch?v=BqoCOu5pxUw) — Shubham Lad - 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/skewness-aspect-ratio-numerical - Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-4 ## Question A triangular cell has vertices at (0, 0), (2, 0) and (0.5, 1.2) (units in mm). A quadrilateral cell has vertices at (0, 0), (4, 0), (4.6, 0.5) and (0.6, 0.5) (mm). For each cell find the interior angles, the equiangular skewness and the edge-based aspect ratio, and comment on the quality. ## Short answer Triangle: angles 67.4°, 38.7°, 74.0°; equiangular skewness 0.36; aspect ratio 1.54 (good). Quadrilateral: angles 39.8° and 140.2°; skewness 0.56; aspect ratio 5.1 (fair, skewed and elongated). ## Answer Equiangular skewness: $S=\max\left[\dfrac{\theta_{max}-\theta_e}{180-\theta_e},\ \dfrac{\theta_e-\theta_{min}}{\theta_e}\right]$, with $\theta_e=60^\circ$ (triangle) and $90^\circ$ (quad). Aspect ratio = longest edge / shortest edge. #### Triangle Edge lengths: - A(0,0)-B(2,0): $2.000$ mm - B(2,0)-C(0.5,1.2): $\sqrt{1.5^2+1.2^2}=1.921$ mm - C-A: $\sqrt{0.5^2+1.2^2}=1.300$ mm Angles from the dot product: - At A: vectors AB=(2,0), AC=(0.5,1.2): $\cos\theta=\dfrac{1.0}{2\times1.3}=0.3846$, $\theta_A=67.4^\circ$ - At B: BA=(-2,0), BC=(-1.5,1.2): $\cos\theta=\dfrac{3.0}{2\times1.921}=0.7809$, $\theta_B=38.7^\circ$ - At C: $180-67.4-38.7=74.0^\circ$ $$ S=\max\left[\frac{74.0-60}{120},\ \frac{60-38.7}{60}\right]=\max[0.116,\ 0.356]=0.356 $$ $$ AR=\frac{2.000}{1.300}=1.54 $$ Skewness 0.36 falls in the "good" range (0.25 to 0.5), and AR is small, so the triangle is acceptable. #### Quadrilateral (a leaning parallelogram) Edges: bottom $4.0$ mm, right side $\sqrt{0.6^2+0.5^2}=0.781$ mm, top $4.0$ mm, left side $0.781$ mm. Angles: at (0,0) between (4,0) and (0.6,0.5): $\cos\theta=\dfrac{2.4}{4\times0.781}=0.768$, $\theta=39.8^\circ$. The adjacent angle is $180-39.8=140.2^\circ$. $$ S=\max\left[\frac{140.2-90}{90},\ \frac{90-39.8}{90}\right]=0.558 $$ $$ AR=\frac{4.0}{0.781}=5.12 $$ #### Comment | Cell | Skewness | AR | Quality | |---|---|---|---| | Triangle | 0.36 | 1.54 | Good | | Quad | 0.56 | 5.12 | Acceptable only; skewed (39.8° angle) and elongated | The quadrilateral is both strongly skewed and stretched. A stretched cell is fine in a boundary layer if aligned with the wall, but the sheared shape (not rectangular) reduces orthogonality and accuracy; it should be improved by reducing the shear (making the angles closer to 90°). **Answer:** Triangle $S=0.36$, $AR=1.54$ (good); quad $S=0.56$, $AR=5.1$ (fair). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For the triangle (0,0), (2,0), (0.5,1.2), which set of interior angles is correct? A. A = 67.4, B = 38.7, C = 74.0 degrees B. A = 38.7, B = 67.4, C = 74.0 degrees C. A = 60, B = 60, C = 60 degrees D. A = 74.0, B = 67.4, C = 38.7 degrees Answer: A. Dot products give cos A = 1.0/(2 x 1.3) = 0.3846 (67.4 deg) and cos B = 0.7809 (38.7 deg), so C = 74.0 deg. (Key point: Triangle angles) 2. What is the triangle's equiangular skewness? A. 0.116 B. 0.36 C. 0.56 D. 1.54 Answer: B. S = max[(74-60)/120, (60-38.7)/60] = max[0.116, 0.356] = 0.36, in the good range. (Key point: Triangle skewness) 3. For the leaning parallelogram, why is the skewness 0.56? A. Its angles are 39.8 and 140.2 degrees, so S = max[(140.2-90)/90, (90-39.8)/90] = 0.56 B. Its angles are 45 and 135 degrees C. Its aspect ratio is 5.12 D. Its angles are 90 degrees Answer: A. cos theta = 2.4/(4 x 0.781) = 0.768 gives 39.8 deg and 140.2 deg, so the skewness is 50.2/90 = 0.558. (Key point: Quad angles and skewness) 4. The quad has edges 4.0 mm and 0.781 mm. What is its AR and conclusion? A. AR = 0.195; excellent B. AR = 5.12; acceptable only, since it is both skewed and stretched C. AR = 1.54; good D. AR = 5.12; excellent because stretched cells are always fine Answer: B. AR = longest/shortest = 4.0/0.781 = 5.12. With S = 0.56 and sheared angles, the quad is only fair; reducing shear toward 90 degrees would help. 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