Fig. 1
Lid-driven in 2D analysis [2]
Fig. 2
Free-slip BC.
Fig. 3
Finite difference in explicit form [16].
Fig. 4
Graphic User Interface (GUI).
Fig. 5
Contour plot of u− and v−velocity distributions at (a) t = 1s (b) t = 2s (c) t = 4s and (d) t = 10s (Re=200 with no-slip BC).
Fig. 6
Comparison of contour plot of evolution (Re=200 with no-slip BC) at steady-state condition; (a) u−velocity distribution via FDM, (b) u−velocity distribution via FEM, (c) v−velocity distribution via FDM, (d) v−velocity distribution via FEM.
Fig. 7
Contour plot of streamline distribution at (a) t = 1 s, (b) t = 2 s, (c) t = 4 s, (d) t = 10 s and, (e) t = 169.70 s (Re=200 with no-slip BC).
Fig. 8
Contour plot of streamline distribution at steady-state condition (Re=200 with free-slip BC).
Fig. 9
Contour plot of u− and v−velocity distributions at steady-state condition (Re=200 with free-slip BC).
Fig. 10
Contour plot of streamline distribution at (a) t = 5.50 s, (b) t = 24.90 s, (c) t = 51.60 s, (d) t = 65.60 s, (e) t = 67.30 s, (f) t = 152.90 s and (g) t = 336.60 s (Re=325.8 with no-slip BC).
Fig. 11
Contour plots of u−and v−velocity distributions at (a) t = 5.50 s, (b) t = 24.90 s, (c) t = 51.60 s, (d) t = 65.60 s, (e) t = 67.30 s, (f) t = 152.90 s and (g) t = 336.60 s (Re=325.8 with no-slip BC).
Fig. 12
Maximum value of stream function along the iteration (a) Standard view (b) Zoomed view around coordinate (t(ψmax), ψmax).
Table 1
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 |
Fig. 13
Contour plot of streamline distribution at steady-state condition Re=325.8 with a) threshold free-slip BC and b) significant free-slip BC.
Fig. 14
Contour plot of u− and v−velocity distributions at steady-state condition (Re=325.8 with threshold free-slip BC).
Fig. 15
Contour plot of u− and v−velocity distributions at steady-state condition (Re=325.8 with significant free-slip BC).
Table 2
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 | – |