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 fill algoritm with Nodal distance $h=0.025$. Boundaries with Neumann boundary conditions are additionally treated with 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] 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
