A chemotaxis-diffusion-convection coupling system for describing a form of buoyant convection in which the fluid develops convection cells and plume patterns will be investigated numerically in this study. Based on the two-dimensional convective chemotaxis-fluid model proposed in the literature, we developed an upwind finite element method to investigate the pattern formation and the hydrodynamical stability of the system. The numerical simulations illustrate different predicted physical regimes in the system. In the convective regime, the predicted plumes resemble B\'enard instabilities. Our numerical results show how structured layers of bacteria are formed before bacterium rich plumes fall in the fluid. The plumes have a well defined spectrum of wavelengths and have an exponential growth rate, yet their position can only be predicted in very simple examples. In the chemotactic and diffusive regimes, the effects of chemotaxis are investigated. Our results indicate that the chemotaxis can stabilize the overall system. A time scale analysis has been performed to demonstrate that the critical taxis Rayleigh number for which instabilities set in depends on the chemotaxis head and sensitivity. In addition, the comparison of the differential systems of chemotaxis-diffusion-convection, double diffusive convection, and Rayleigh-B\'enard convection establishes a set of evidences that even if the physical mechanisms are different at the same time the dimensionless systems are strongly related to each other.