Modeling density-driven flow in porous media is challenging due to the nonlinear coupling between flow and transport equations, the large domains of interest and the wide range of time and space scales involved. Solving this type of problem numerically using a fixed mesh can be prohibitively expensive. Here, we apply a dynamic mesh optimization (DMO) technique along with a control-volume-finite element method to simulate density-driven flows. DMO allows the mesh resolution and geometry to vary during a simulation to minimize an error metric for one or more solution fields of interest, refining where needed and coarsening elsewhere. We apply DMO to the Elder problem for several Rayleigh numbers. It is demonstrated that DMO accurately reproduces the unique two-dimensional (2D) solutions for low Rayleigh number cases at significantly lower computational cost compared to an equivalent fixed mesh, with speedup of order ×16. For unstable, high Rayleigh number 2D cases, multiple steady-state fingering solutions exist and are all captured by our approach with high accuracy and significantly reduced computational cost, with speedup of order ×6. Velocity-dependent dispersion is shown to have a small impact on the 2D numerical solutions. The lower computational cost of simulations using DMO allows extension of the high Rayleigh number case to a three dimensional (3D) configuration. We demonstrate new 3D fingering patterns that have not been observed previously. Early time, transient 3D patterns represent combinations of the previously observed, steady-state 2D solutions, but all evolve to a single, steady-state finger in the late time limit.

中文翻译：

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密度驱动流动的有效数值模拟：应用于 2 维和 3 维老问题

由于流动和传输方程之间的非线性耦合、感兴趣的大域以及所涉及的时间和空间尺度范围广泛，对多孔介质中的密度驱动流动进行建模具有挑战性。使用固定网格以数值方式解决此类问题可能非常昂贵。在这里，我们应用动态网格优化 (DMO) 技术以及控制体积有限元方法来模拟密度驱动的流动。DMO 允许在模拟过程中改变网格分辨率和几何形状，以最小化一个或多个感兴趣的解决方案领域的误差度量，在需要的地方细化并在其他地方粗化。我们将 DMO 应用于几个 Rayleigh 数的 Elder 问题。结果表明，与等效的固定网格相比，DMO 准确地再现了低瑞利数情况下的独特二维 (2D) 解决方案，计算成本显着降低，速度提高了 ×16。对于不稳定的高瑞利数 2D 情况，存在多个稳态指法解决方案，并且都被我们的方法以高精度捕获，并且显着降低了计算成本，速度提高了 ×6。速度相关色散对二维数值解的影响很小。使用 DMO 进行模拟的较低计算成本允许将高瑞利数情况扩展到三维 (3D) 配置。我们展示了以前未观察到的新 3D 指法模式。早期，瞬态 3D 模式代表先前观察到的组合，