View source for 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. 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. Domain is discretized with our [[Positioning of computational nodes|fill algoritm]] with
Nodal distance $h=0.025$. Boundaries with Neumann
boundary conditions are additionally treated with ghost nodes [[Ghost nodes (theory)]].

[[File:DVD_3D.png|400px]]
[[File:DVD_3D_irreg.png|400px]]

{| class="wikitable"
! 
! style="text-align: center;" | 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] Couturier, H. & Sadat, S. Performance and accuracy of a meshless method for laminar natural convection Numerical Heat Transfer: Part B: Fundamentals, Taylor & Francis, 2000 , 37 , 455-467

[ref b for 2D] Kosec, G. & Šarler, B. Solution of thermo-fluid problems by collocation with local pressure correction International Journal of Numerical Methods for Heat & Fluid Flow, Emerald Group Publishing Limited, 2008 , 18 , 868-882

[ref a for 3D] Wang, P.; Zhang, Y. & Guo, Z. Numerical study of three-dimensional natural convection in a cubical cavity at high Rayleigh numbers Int. J. Heat Mass Transfer, Elsevier, 2017 , 113 , 217-228

[ref b for 3D] Fusegi, T.; Hyun, J. M.; Kuwahara, K. & Farouk, B. A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure  Int. J. Heat Mass Transfer, Elsevier, 1991 , 34 , 1543-1557