Fig. 1
![Lid-driven in 2D analysis [2]](https://sciendo-parsed.s3.eu-central-1.amazonaws.com/6471f878215d2f6c89db71c3/j_ijmce-2025-0006_fig_001.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Content-Sha256=UNSIGNED-PAYLOAD&X-Amz-Credential=ASIA6AP2G7AKDKDSK32F%2F20260305%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20260305T212854Z&X-Amz-Expires=3600&X-Amz-Security-Token=IQoJb3JpZ2luX2VjEAgaDGV1LWNlbnRyYWwtMSJHMEUCIHZNSHV9qAn9G%2FYdEeJzoqWDC%2BhxfvvjikV%2FiftG%2B7cIAiEA9nIRXSbm3%2Ffz0F0s52IkQmcdCrKveqt2hoPZ3GYSCLUqxgUI0f%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARACGgw5NjMxMzQyODk5NDAiDHuiXmauKnNDATpnUyqaBXY4VMnAFI4WxpQ031%2FKf6xUqPd4t3OVQPTu4QCrLUUWsxKm%2BBtwr23mWf78kkIawLxP0foRJuaWKCJIcgYGGcWx6GNc30wOFycuXr35zTECwU%2FSXxUnON3EXnuTPR%2Ben7QqW94wP2YwXSb1F99N%2Bb9GYUhE8AB6bzZfjeGsqYUKIdB4VYcaeMbEcJSFS7X1IU3cTLktkuiWcj2Vh%2BfzdE%2BJxpTS0EssORNf5Pe%2FBjt2%2FEURqqdEeBx7xrE19etH%2BcYKJcK1Lvv47zpsusSTQW8wLWH2OeMDvrn%2FR6QLAHywnOthnZvcQh8QpQeI1YbZDB0dyEAkvBDbEz5J3uPP%2F0elei8ln5VHgS%2BzGl3QZV7d9VP%2BiWEO2vVluB6kFAP6jhSLcNh3NpngByusvjOGurMf7HGlm3QNCS9G5caG1dv1ZWoKWYIAi%2BXjj1Nw1loK78Vq8XNweAKnY0YRr%2F2XSuBytJsJXZv1VplWhFFaEvJCBlhsLyB6jiVKMf9jdq%2BkFj%2FNddJWE8MlKPBhd5Z9X9%2BN%2FFVUZNIRlqJvL5CR8Ydp4h3ew21NtbGb9Bpo9kLOm1AlmpxisFJmSr6189V%2Ftv2xfvSuU%2FvqJLVcgsJVNEljZULestsMXLafFGec8DhQ22%2BbqsIhjm3uarBfWOWG4ptYoCiHiWkdVegj9TxYdG3U8T7Kw9tJlggt717A9gn%2BHHM6VXKthht6CN2ULazdLLgDGfPUp%2BBX96SVjRMCTJd8rDpbAE5Chw8JZ8qyyPGVRjQzIjh9OI%2FY2nQ0%2F%2B8yKOKJO6QNKuamjJgoplwkUmWqfyI%2FnzgnFo6jNRx1YP5FH7npxwSj040k4yX9i6vzCTE7fbuJ7SXZLrBdrzmIbXCAbsNhKbSMSZySwjD8y6bNBjqxAfE8I%2BqXkOwbhxdftbkNjmM1Tk5trpu5EL%2BYDc8GS%2B1QJOlrbfdhh%2BK8wwzTUG4l9GCkoF9sUbIrnaBYYfqwJu7z4KKfmG%2FN8%2BUjF0w19AMco079L8%2BW3QengL8E4gBK9rE%2FfcTwWzOodVu799yYHHFreo0DgJrxGNmV0zZMtXOmNTH4gQA6UjijmDgq3sH0OzctFGkfhQ6WqmCpdqaurL1bjVFRo77yyqu3j5DV85tR1w%3D%3D&X-Amz-Signature=0814d2e370bf07fa7f0f8f88185fd422dfeeb0e5f3aaf320e5fdbfa6ffe249ac&X-Amz-SignedHeaders=host&x-amz-checksum-mode=ENABLED&x-id=GetObject)
Fig. 2
Fig. 3
![Finite difference in explicit form [16]](https://sciendo-parsed.s3.eu-central-1.amazonaws.com/6471f878215d2f6c89db71c3/j_ijmce-2025-0006_fig_003.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Content-Sha256=UNSIGNED-PAYLOAD&X-Amz-Credential=ASIA6AP2G7AKDKDSK32F%2F20260305%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20260305T212854Z&X-Amz-Expires=3600&X-Amz-Security-Token=IQoJb3JpZ2luX2VjEAgaDGV1LWNlbnRyYWwtMSJHMEUCIHZNSHV9qAn9G%2FYdEeJzoqWDC%2BhxfvvjikV%2FiftG%2B7cIAiEA9nIRXSbm3%2Ffz0F0s52IkQmcdCrKveqt2hoPZ3GYSCLUqxgUI0f%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARACGgw5NjMxMzQyODk5NDAiDHuiXmauKnNDATpnUyqaBXY4VMnAFI4WxpQ031%2FKf6xUqPd4t3OVQPTu4QCrLUUWsxKm%2BBtwr23mWf78kkIawLxP0foRJuaWKCJIcgYGGcWx6GNc30wOFycuXr35zTECwU%2FSXxUnON3EXnuTPR%2Ben7QqW94wP2YwXSb1F99N%2Bb9GYUhE8AB6bzZfjeGsqYUKIdB4VYcaeMbEcJSFS7X1IU3cTLktkuiWcj2Vh%2BfzdE%2BJxpTS0EssORNf5Pe%2FBjt2%2FEURqqdEeBx7xrE19etH%2BcYKJcK1Lvv47zpsusSTQW8wLWH2OeMDvrn%2FR6QLAHywnOthnZvcQh8QpQeI1YbZDB0dyEAkvBDbEz5J3uPP%2F0elei8ln5VHgS%2BzGl3QZV7d9VP%2BiWEO2vVluB6kFAP6jhSLcNh3NpngByusvjOGurMf7HGlm3QNCS9G5caG1dv1ZWoKWYIAi%2BXjj1Nw1loK78Vq8XNweAKnY0YRr%2F2XSuBytJsJXZv1VplWhFFaEvJCBlhsLyB6jiVKMf9jdq%2BkFj%2FNddJWE8MlKPBhd5Z9X9%2BN%2FFVUZNIRlqJvL5CR8Ydp4h3ew21NtbGb9Bpo9kLOm1AlmpxisFJmSr6189V%2Ftv2xfvSuU%2FvqJLVcgsJVNEljZULestsMXLafFGec8DhQ22%2BbqsIhjm3uarBfWOWG4ptYoCiHiWkdVegj9TxYdG3U8T7Kw9tJlggt717A9gn%2BHHM6VXKthht6CN2ULazdLLgDGfPUp%2BBX96SVjRMCTJd8rDpbAE5Chw8JZ8qyyPGVRjQzIjh9OI%2FY2nQ0%2F%2B8yKOKJO6QNKuamjJgoplwkUmWqfyI%2FnzgnFo6jNRx1YP5FH7npxwSj040k4yX9i6vzCTE7fbuJ7SXZLrBdrzmIbXCAbsNhKbSMSZySwjD8y6bNBjqxAfE8I%2BqXkOwbhxdftbkNjmM1Tk5trpu5EL%2BYDc8GS%2B1QJOlrbfdhh%2BK8wwzTUG4l9GCkoF9sUbIrnaBYYfqwJu7z4KKfmG%2FN8%2BUjF0w19AMco079L8%2BW3QengL8E4gBK9rE%2FfcTwWzOodVu799yYHHFreo0DgJrxGNmV0zZMtXOmNTH4gQA6UjijmDgq3sH0OzctFGkfhQ6WqmCpdqaurL1bjVFRo77yyqu3j5DV85tR1w%3D%3D&X-Amz-Signature=3c10e268366ad3d749f6e148fe599ae63bcb44e5851dd3a988272fa02ea79165&X-Amz-SignedHeaders=host&x-amz-checksum-mode=ENABLED&x-id=GetObject)
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Nomenclatures_
| Symbol | Meanings | Symbol | Meanings |
|---|---|---|---|
| ρ | density [kg/m3] | ψ | stream function [m2/s] |
| U | applied velocity [m/s] | ψmax | maximum stream function [m2/s] |
| L | length of cavity [m] | Uwall | applied velocity at top wall [m/s] |
| μ | dynamic viscosity [Pa.s] | n | non-negative integer, number of nodes |
| Re | Reynolds number | i | column |
| A | aspect ratio | j | row |
| H | height of cavity [m] | Δx | step size [m] |
| u | velocity in x–direction [m/s] | Δy | step size [m] |
| v | velocity in y–direction [m/s] | us | slip velocity [m/s] |
| g | gravity acceleration [m/s2] | ls | slip length [m] |
| u | applied velocity [m/s] | gx | gravity acceleration in x–direction [m/s2] |
| θ | angle [deg] | gy | gravity acceleration in y–direction [m/s2] |
| t | time [s] | Δt | step time [s] |
| ∇ | Gradient operator | h | step size [m] |
| ζtop | vorticity at the top [m/s2] | s fj,i,n | stream function at previous time [m2/s] |
| ζleft | vorticity at the top [m/s2] | s fj,i,n+1 | stream function at previous time [m2/s] |
| V | velocity vector [m/s] | ζright | vorticity at the right [m/s2] |
| ζ | vorticity [m/s2] | ζbottom | vorticity at the bottom [m/s2] |
| e | steady-state criterion | ||
Value and location of ψmax at different steady-state criteria (AR= 1_629)_
| No | κ1 | κ2 | κ3 | κ4 | Remark |
|---|---|---|---|---|---|
| 1. | e ≤ 10−2 | 5.5 s | 0.0089 | x=0.775, y=1.466 | Primary main cell is formed |
| 2. | e ≤ 10−3 | 24.9 s | 0.0186 | x=0.675, y=1.262 | Primary main cell is growing |
| 3. | e ≤ 10−4 | 51.6 s | 0.0209 | x=0.625, y=1.140 | Secondary eddies are formed |
| 4. | e ≤ 10−5 | 65.6 s | 0.0210 | x=0.600, y=1.100 | Secondary eddies are growing |
| 5. | e ≤ 10−6 | 67.2 s | 0.0210 | x=0.600, y=1.100 | – |
| 6. | e ≤ 10−7 | 67.3 s | 0.0210 | x=0.600, y=1.100 | – |
| 7. | e ≤ 10−8 | 152.9 s | 0.0207 | x=0.600, y=1.100 | Secondary main cell is formed and fixed in size |
| 8. | e ≤ 10−10 | 336.6 s | 0.0207 | x=0.600, y=1.100 | – |