Skip to main content
Have a personal or library account? Click to login
Variant slip effects, aspect ratios and steady-state criteria in unsteady cavities with direct PDE simulation of eddies Cover

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

International Journal of Mathematics and Computer in Engineering
Volume 3 (2025): Issue 1 (June 2025)
By: Hon Fei Wong,  Muhammad Sohail and  Noor Fadiya Mohd Noor  
Open Access
|Jun 2024
Figures & tables

Figures & Tables

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).

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).

Nomenclatures_

SymbolMeaningsSymbolMeanings
ρdensity [kg/m3]ψstream function [m2/s]
Uapplied velocity [m/s]ψmaxmaximum stream function [m2/s]
Llength of cavity [m]Uwallapplied velocity at top wall [m/s]
μdynamic viscosity [Pa.s]nnon-negative integer, number of nodes
ReReynolds numbericolumn
Aaspect ratiojrow
Hheight of cavity [m]Δxstep size [m]
uvelocity in x–direction [m/s]Δystep size [m]
vvelocity in y–direction [m/s]usslip velocity [m/s]
ggravity acceleration [m/s2]lsslip length [m]
uapplied velocity [m/s]gxgravity acceleration in x–direction [m/s2]
θangle [deg]gygravity acceleration in y–direction [m/s2]
ttime [s]Δtstep time [s]
∇Gradient operatorhstep size [m]
ζtopvorticity at the top [m/s2]s fj,i,nstream function at previous time [m2/s]
ζleftvorticity at the top [m/s2]s fj,i,n+1stream function at previous time [m2/s]
Vvelocity vector [m/s]ζrightvorticity at the right [m/s2]
ζvorticity [m/s2]ζbottomvorticity at the bottom [m/s2]
esteady-state criterion

Value and location of ψmax at different steady-state criteria (AR= 1_629)_

Noκ1κ2κ3κ4Remark
1.e ≤ 10−25.5 s0.0089x=0.775, y=1.466Primary main cell is formed
2.e ≤ 10−324.9 s0.0186x=0.675, y=1.262Primary main cell is growing
3.e ≤ 10−451.6 s0.0209x=0.625, y=1.140Secondary eddies are formed
4.e ≤ 10−565.6 s0.0210x=0.600, y=1.100Secondary eddies are growing
5.e ≤ 10−667.2 s0.0210x=0.600, y=1.100–
6.e ≤ 10−767.3 s0.0210x=0.600, y=1.100–
7.e ≤ 10−8152.9 s0.0207x=0.600, y=1.100Secondary main cell is formed and fixed in size
8.e ≤ 10−10336.6 s0.0207x=0.600, y=1.100–
DOI: https://doi.org/10.2478/ijmce-2025-0006 | Journal eISSN: 2956-7068
Journal RSS Feed
Language: English
Page range: 57 - 74
Submitted on: Jul 12, 2023
Accepted on: Jan 21, 2024
Published on: Jun 2, 2024
Published by: Harran University
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:
Numerical finite difference,
vorticity-stream function,
slip effects,
eddies simulation,
Matlab®
Related subjects:
Computer sciences,
Computer sciences, other,
Engineering,
Introductions and overviews,
Engineering, other,
Mathematics,
General mathematics,
Physics,
Physics, other

© 2024 Hon Fei Wong, Muhammad Sohail, Noor Fadiya Mohd Noor, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 3 (2025): Issue 1 (June 2025)
Previous article
Next article