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Variant slip effects, aspect ratios and steady-state criteria in unsteady cavities with direct PDE simulation of eddies

Hon Fei Wong
Hon Fei Wong
, 
Muhammad Sohail
Muhammad Sohail
 e 
Noor Fadiya Mohd Noor
Noor Fadiya Mohd Noor
   | 02 giu 2024
International Journal of Mathematics and Computer in Engineering's Cover Image
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Article Category: Original Study
Pubblicato online: 02 giu 2024
Pagine: -
Ricevuto: 12 lug 2023
Accettato: 21 gen 2024
DOI: https://doi.org/10.2478/ijmce-2025-0006
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© 2025 Hon Fei Wong et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Lid-driven in 2D analysis [2]
Lid-driven in 2D analysis [2]

Fig. 2

Free-slip BC
Free-slip BC

Fig. 3

Finite difference in explicit form [16]
Finite difference in explicit form [16]

Fig. 4

Graphic User Interface (GUI).
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).
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.
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).
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).
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).
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).
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).
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).
Maximum value of stream function along the iteration (a) Standard view (b) Zoomed view around coordinate (t(ψmax), ψmax).

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.
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).
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).
Contour plot of u− and v-velocity distributions at steady-state condition (Re=325.8 with significant free-slip BC).

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 –
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