We propose a non-iterative monolithic projection-based method to examine the nonlinear dynamics of time-dependent chemotaxis-driven bioconvection problems. In the proposed method, all the terms are advanced using the Crank–Nicolson scheme in time along with the second-order central difference in space. Linearizations, approximate block lower–upper decompositions, and an approximate factorization technique are adopted to improve the computational efficiency while preserving the second-order temporal accuracy. We perform numerical simulations of quasi-homogeneous bioconvection, two-dimensional forced chemotaxis bioconvection, and two-dimensional chemotaxis-driven bioconvection to test the numerical performance of the proposed method. The results show that the proposed method provides predictions that are in good agreement with those in previous works. Moreover, it preserves the second-order accuracy in time, significantly reduces the time-step limitation, and improves the computational efficiency. Finally, the proposed method is employed to investigate the nonlinear dynamics of chemotaxis-driven bioconvection problems with varying characteristic bacterial concentration and chamber depth. Four regimes were classified based on the fluid and bacterial motions: stable shallow-chamber, unstable shallow-chamber, unstable deep-chamber, and chaotic deep-chamber flows. We show the formation and merging of falling plumes and their surrounding fluid motion under random initial conditions as well as their convergence toward stationary or chaotic bacterial plumes. To track the dynamical regimes over the entire considered domain, we designed normalized variance and kurtosis, which reflect formation and merging of plumes and intermittency in chaotic cases, respectively. A posterior classification, which provides a rough outline of the characteristic features of the different regimes, was also carried out.

我们提出了一种基于非迭代整体式投影的方法，以研究时间依赖性趋化性驱动的生物对流问题的非线性动力学。在所提出的方法中，所有术语都使用Crank-Nicolson方案及时进行了扩展，并附带了空间的二阶中心差。采用线性化，近似块上下分解以及近似因式分解技术来提高计算效率，同时保留二阶时间精度。我们进行了准均匀生物对流，二维强制趋化生物对流和二维趋化驱动生物对流的数值模拟，以测试该方法的数值性能。结果表明，所提出的方法提供的预测结果与以前的工作相吻合。此外，它保留了时间上的二阶精度，大大减少了时间步限制，并提高了计算效率。最后，该方法被用于研究具有变化特征细菌浓度和腔室深度的趋化性驱动的生物对流问题的非线性动力学。根据流体和细菌运动将四种状态分类为：稳定的浅室，不稳定的浅室，不稳定的深室和混乱的深室流动。我们显示了下降的羽流的形成和合并以及它们在随机初始条件下的周围流体运动，以及它们趋向于平稳或混乱的细菌羽流的收敛。为了跟踪整个考虑域内的动态机制，我们设计了归一化方差和峰度，它们分别反映了混乱情况下羽流的形成和合并以及间歇性。还进行了后验分类，粗略地概述了不同制度的特征。