Configuration

The lid-driven cavity flow is a well known benchmark largely studied in the litterature. We will compare our work to the ones of Bruneau [1] and Botella [2], for \( Re=1000 \). The domain is \( [0, 1] \times [0, 1] \), wall are set to all the boundaries except the top one, on which \( (u,v)=(-1,0) \). A set of vortices can be identified, from the main one, to secondary and eventually tertiary ones on bottom left and right corners.

Runtime parameters

Regular mesh size varies from \( 128^2 \) to \( 1024^2 \). The second order centered advection scheme is used.

Comments

Tertiary vortices can be identified thanks to a Chebyshev mesh that is refined in the corners.

Results

Figures 1 and 2 show respectively streamlines that put in evidence the main and secondary vortices, and vorticy isolines.

Figure 1: Streamlines of the velocity field

Figure 2: Vorticity isolines (values of isolines as in [1])

We compare in next Table the mean kinetic energy to value obtained in [1]:

Mesh	Kinetic Energy
128x128	4.341489E-02
256x256	4.423212E-02
512x512	4.444592E-02
1024x1024	4.450005E-02
Order	2.34
Extrapolated	4.4510889E-2
Bruneau \( 1024^2 \) [1]	4.4503E-02

Values of the vorticity at the center of the main vortex (and its position) are given in next Table and compared to the one od [1] and [2].

Mesh	Vorticity	x	y
256x256	2.06216	0.469112	0.565375
512x512	2.06634	0.469475	0.565544
1024x1024	2.06741	0.469205	0.565249
Order	1.97	N/A	N/A
Extrapolated	2.06778	N/A	N/A
Bruneau \( 1024^2 \) [1]	2.0674	0.46875	0.56543
Botella \( N=160 \) [2]	2.067753	0.4692	0.5652

Values of the vorticity at the center of the second left vortex (and its position) are given in next Table and compared to the one od [1] and [2].

Mesh	Vorticity	x	y
256x256	-1.08285	0.136055	0.111901
512x512	-1.11163	0.136327	0.112096
1024x1024	-1.11335	0.136005	0.111861
Order	4.06	N/A	N/A
Extrapolated	-1.11346	N/A	N/A
Bruneau \( 1024^2 \) [1]	-1.1120	0.13672	0.11230
Botella \( N=160 \) [2]	-1.109789	0.13602	0.1118

References

[1] Charles-Henri Bruneau, Mazen Saad, The 2D lid-driven cavity problem revisited, Computers & Fluids 35 (2006) 326–348.

[2] Botella O, Peyret R. Benchmark spectral results on the lid-driven cavity flow. Comput Fluids 1998;27(4).