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=ASIA6AP2G7AKORPE3AVD%2F20260305%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20260305T213001Z&X-Amz-Expires=3600&X-Amz-Security-Token=IQoJb3JpZ2luX2VjEAkaDGV1LWNlbnRyYWwtMSJHMEUCIQDqDrGeiuQguEXgAW%2BvilrOwBZmWllRdXEGUmiOpZxKsgIgVcEMvWfXkipZ4y4bMybhUX%2FLtc8%2FDl96dn0dDxDSdJEqxgUI0v%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARACGgw5NjMxMzQyODk5NDAiDGxt%2BpohN4u4b5v7pCqaBU30dT77HEicPiJP6Q6btmLF53Ywx5l4mGo3KnFQn%2FN949XfMOtmcDAuz3TVwExhi8gmufmI6oowZ5W%2FbslLiytKNQHYMYwtz4sdHJKABujwUjyj7RpKK2ons8h89S46WiZwJHS71Ko7UsO%2FALtivtDe26bMf1PobrBTqDdjMpeunx7SjltLTc9RZ6o1X5enNXyvkNF0rl2I94U%2B2H6NL975OPkZE%2FIvRZY33wGFFjnUB%2FXjMO%2FnPQWYRL7MHOWma%2FoooTMHV0qOFcGIb34DN1thGFS25to%2FIPnGwzTwj5EPyNflNlO%2FfqOXbooZTNxn%2B0SSKPrSn19YNJlFQNiOsJKJaavTvADJLqFNHTswdUb2ZN1MnIDhUzm%2BGSgf5YsunqNS19h8SFHlRaEw%2BdW%2BsNlVbAOJxMqwY3dTjnCCo79UKd0KYrOMDUANrZRd5vgsJkO6PVJjSaiX4ecrJbbXQ9%2F%2FonRj5gM9V3ailL7qyEnZ1gntkrqU1%2FDBxflsXCnylgOqs5u6eiSHm%2FAPWvNjKH7%2BnlPMyCaHHrMgsmqxYRmDbZXnOH7VWoJAs941vk59iq85gSln8EAOgOnl3hbHub2NvXwgODjgs0hpmR9x3UmPgMa%2FrfRRvQbdteYwSW8JHyb87Hh765NyAnzJPvyJlU69RhO1M3E%2F5Y8Ylkf4SwyTabTcai%2FSKBcyMjlGB0mQkgOueW7xUuwwMimuQi8MGk7tKcwo954MKqYr5bMxDoLrD%2B2Pu2zGRYMTCHwz2VAIefzxQIGOdKevX5bE3hl7O7cL7DEC%2BASqJM7FBVFDxHxZdxot4nblPJe6tc%2BW9KbsNCzhTQAB4Rl3pkmLAh55VhgEB%2B7gNyerMwy4%2BzuuE544P6PhWpRA9TVPMjCg8KbNBjqxAU8ldAKRmm%2B9xrvhVhVw6OUXcvF%2BYcFtBMsS%2BpWbzl74FpM5p0tpCSDHnO2RkjGjs6FivS4QtzjuiiAZ33rkxw1ltZrNY8xrJEfp7P2ivcuIYgfChXpqHe2ZkvVc1yuRB43J5FikZeAKQ00Ft24XUFg1fBaW6K%2FJz7W6MM4jyDUMw%2Ba7ILFmp5F3GimbysYEDn%2BGWAuBLB6qwW5PADHVO8g0c4n2aWECxdilPMdKanT5yA%3D%3D&X-Amz-Signature=6af101c0b1279e1259d86c302e5f948fef8dcee450d6e1bbc624942167a0e368&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=ASIA6AP2G7AKORPE3AVD%2F20260305%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20260305T213001Z&X-Amz-Expires=3600&X-Amz-Security-Token=IQoJb3JpZ2luX2VjEAkaDGV1LWNlbnRyYWwtMSJHMEUCIQDqDrGeiuQguEXgAW%2BvilrOwBZmWllRdXEGUmiOpZxKsgIgVcEMvWfXkipZ4y4bMybhUX%2FLtc8%2FDl96dn0dDxDSdJEqxgUI0v%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARACGgw5NjMxMzQyODk5NDAiDGxt%2BpohN4u4b5v7pCqaBU30dT77HEicPiJP6Q6btmLF53Ywx5l4mGo3KnFQn%2FN949XfMOtmcDAuz3TVwExhi8gmufmI6oowZ5W%2FbslLiytKNQHYMYwtz4sdHJKABujwUjyj7RpKK2ons8h89S46WiZwJHS71Ko7UsO%2FALtivtDe26bMf1PobrBTqDdjMpeunx7SjltLTc9RZ6o1X5enNXyvkNF0rl2I94U%2B2H6NL975OPkZE%2FIvRZY33wGFFjnUB%2FXjMO%2FnPQWYRL7MHOWma%2FoooTMHV0qOFcGIb34DN1thGFS25to%2FIPnGwzTwj5EPyNflNlO%2FfqOXbooZTNxn%2B0SSKPrSn19YNJlFQNiOsJKJaavTvADJLqFNHTswdUb2ZN1MnIDhUzm%2BGSgf5YsunqNS19h8SFHlRaEw%2BdW%2BsNlVbAOJxMqwY3dTjnCCo79UKd0KYrOMDUANrZRd5vgsJkO6PVJjSaiX4ecrJbbXQ9%2F%2FonRj5gM9V3ailL7qyEnZ1gntkrqU1%2FDBxflsXCnylgOqs5u6eiSHm%2FAPWvNjKH7%2BnlPMyCaHHrMgsmqxYRmDbZXnOH7VWoJAs941vk59iq85gSln8EAOgOnl3hbHub2NvXwgODjgs0hpmR9x3UmPgMa%2FrfRRvQbdteYwSW8JHyb87Hh765NyAnzJPvyJlU69RhO1M3E%2F5Y8Ylkf4SwyTabTcai%2FSKBcyMjlGB0mQkgOueW7xUuwwMimuQi8MGk7tKcwo954MKqYr5bMxDoLrD%2B2Pu2zGRYMTCHwz2VAIefzxQIGOdKevX5bE3hl7O7cL7DEC%2BASqJM7FBVFDxHxZdxot4nblPJe6tc%2BW9KbsNCzhTQAB4Rl3pkmLAh55VhgEB%2B7gNyerMwy4%2BzuuE544P6PhWpRA9TVPMjCg8KbNBjqxAU8ldAKRmm%2B9xrvhVhVw6OUXcvF%2BYcFtBMsS%2BpWbzl74FpM5p0tpCSDHnO2RkjGjs6FivS4QtzjuiiAZ33rkxw1ltZrNY8xrJEfp7P2ivcuIYgfChXpqHe2ZkvVc1yuRB43J5FikZeAKQ00Ft24XUFg1fBaW6K%2FJz7W6MM4jyDUMw%2Ba7ILFmp5F3GimbysYEDn%2BGWAuBLB6qwW5PADHVO8g0c4n2aWECxdilPMdKanT5yA%3D%3D&X-Amz-Signature=372095e58273bec3510947f4193e599aff1abc43b9af3ac6ae926a41e39d0bc6&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 | – |