Efficient monolithic projection-based method for chemotaxis-driven bioconvection problems

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.