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

Figures & Tables

Symbol	Meanings	Symbol	Meanings

ρ	density [kg/m³]	ψ	stream function [m²/s]
U	applied velocity [m/s]	ψ_max	maximum stream function [m²/s]
L	length of cavity [m]	U_wall	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]	u_s	slip velocity [m/s]
g	gravity acceleration [m/s²]	l_s	slip length [m]
u	applied velocity [m/s]	g_x	gravity acceleration in x–direction [m/s²]
θ	angle [deg]	g_y	gravity acceleration in y–direction [m/s²]
t	time [s]	Δt	step time [s]
∇	Gradient operator	h	step size [m]
ζ_top	vorticity at the top [m/s²]	s f_j,i,n	stream function at previous time [m²/s]
ζ_left	vorticity at the top [m/s²]	s f_j,i,n+1	stream function at previous time [m²/s]
V	velocity vector [m/s]	ζ_right	vorticity at the right [m/s²]
ζ	vorticity [m/s²]	ζ_bottom	vorticity at the bottom [m/s²]
e	steady-state criterion

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	–