Numerical investigations on the finite time singularity in two-dimensional Boussinesq equations

Abstract

To investigate the finite time singularity in three-dimensional (3D) Euler flows, the simplified model of 3D axisymmetric incompressible fluids (i.e., two-dimensional Boussinesq approximation equations) is studied numerically. The system describes a cap-like hot zone of fluid rising from the bottom, while the edges of the cap lag behind, forming eye-like vortices. The hot liquid is driven by the buoyancy and meanwhile attracted by the vortices, which leads to the singularity-forming mechanism in our simulation. In the previous 2D Boussinesq simulations, the symmetricial initial data is used. However, it is observed that the adoption of symmetry leads to coordinate singularity. Moreover, as demonstrated in this work that the locations of peak values for the vorticity and the temperature gradient becomes far apart as $t$ approaches the predicted blow-up time. This suggests that the symmetry assumption may be unreasonable for searching solution blow-ups. One of the main contributions of this work is to propose an appropriate asymmetric initial condition, which avoids coordinate singularity and also makes the blow-up to occur much earlier than that given by the previously simulations. The shorter simulation time suppresses the development of the round-off error. On the numerical side, the pseudo-spectral method with filtering technique is adopted. The resolutions adopted in this study vary from $1024^2$, $2048^2$, $4096^2$ to $6144^2$. With our proposed asymmetric initial condition, it is shown that the $4096^2$ and $6144^2$ runs yield convergent results when $t$ is fairly close to the predicted blow-up time. Moreover, as expected the locations of peak values for the vorticity and the temperature gradient are very close to each other as $t$ approaches the predicted blow-up time.