Dynamically mesh adaptation and Boussinesq Convection

Evolution of the temperature distribution

Vorticity concentration and the moving mesh

Vorticity and the moving mesh

How can we compute accurately the collapse to very small length scales and the rapid loss of regularity of a time-evolving solution? A solution-adaptive mesh is indispensable for this task. We have developed (with T. Y. Hou) a painless and efficient dynamically adaptive mesh to compute accurately time-dependent multi-dimensional solutions that develop singular or nearly singular behavior in regions of complex geometry.

The design of our dynamically adaptive mesh was motivated by the fascinating and still open problem about whether a finite-time singularity can form out of smooth initial data in inviscid and incompressible 3-D Euler flows. This is not just a mathematical question. The finding and understanding of finite-time singularities may be crucial to explain small scale structures in viscous turbulent flows. Using our dynamically adaptive mesh we investigated the production and concentration of vorticity in 2-D Boussinesq convection of a strongly layered flow. The governing equations of Boussinesq convection are analogous to those of 3-D axi-symmetric Euler flow with swirl. Does the vorticity blow up in finite time?