Natural convection in 3D irregular domain

The classical De Vahl Davis natural convection test can be extended to 3D. In below figures steady state temperature contour and velocity quiver plots for Ra$=10^6$ case in 3D are presented. A more quantitative analysis is done by comparing characteristic values, i.e.\ peak positions and values of cross section velocities, with data available in literature~\cite{fusegi1991numerical}. We analyze six different cases, namely Ra$=10^6,10^7,10^8$ in 2D, and Ra$=10^4,10^5,10^6$ in 3D. The comparison in presented in below table.

All spatial operators are discretized using RBF-FD with $r^3$ PHS radial basis functions, augmented with monomials up to order $2$, with the closest $25$ nodes used as a stencil. For the time discretization time step $\Delta t=10^{-3}$ was used for all cases. Nodal distance $h=0.01$ is used for simulations in 2D and $h=0.25$ for simulations in 3D. Boundaries with Neumann boundary conditions are additionally treated with ghost nodes Ghost nodes (theory).

	Ra	$v_{max}$			$x$			$u_{max}$			$y$
		present	ref a	ref b	present	ref a	ref b	present	ref a	ref b	present	ref a	ref b
2D	$10^6$	0.2628	0.2604	0.2627	0.0378	0.0380	0.0390	0.0781	0.0765	0.0782	0.8476	0.8510	0.0390
	$10^7$	0.2633	0.2580	0.2579	0.0226	0.0230	0.0210	0.0588	0.0547	0.0561	0.8705	0.8880	0.0210
	$10^8$	0.2557	0.2587	0.2487	0.0149	0.0110	0.0090	0.0314	0.0379	0.0331	0.9189	0.9430	0.0090
3D	$10^4$	0.2495	0.2218	0.2252	0.8500	0.8873	0.8833	0.2435	0.1968	0.2013	0.1611	0.1799	0.1833
	$10^5$	0.2545	0.2442	0.2471	0.9402	0.9317	0.9353	0.1564	0.1426	0.1468	0.1447	0.1493	0.1453
	$10^6$	0.2564	0.2556	0.2588	0.9614	0.9653	0.9669	0.0841	0.0816	0.0841	0.1435	0.1403	0.1443

[ref a for 2D] [ref b for 2D] [ref a for 3D] [ref b for 3D]

@Article{couturier2000performance,
 author    = {Couturier, H. and Sadat, S.},
 title     = {Performance and accuracy of a meshless method for laminar natural convection},
 journal   = {Numerical Heat Transfer: Part B: Fundamentals},
 year      = {2000},
 volume    = {37},
 number    = {4},
 pages     = {455--467},
 doi       = {10.1080/10407790050051146},
 publisher = {Taylor \& Francis},

}