We conducted a numerical study on the onset of electro-convection as well as the complex flow phenomena of dielectric liquid subjected to unipolar autonomous charge injection in the annular gap between two concentric circular cylindrical electrodes. The Nernst-Planck equations governing the charge density transport, the Poisson equation for the electric potential and the Navier-Stokes equations for the fluid flow are solved numerically using the finite volume method. The developed code is validated by comparing the critical stability parameter values for the onset of electro-convection with those obtained from the linear stability analysis. We identify in a parameter space the stable hydrostatic state and the electro-convection state. The electro-convection is again divided into three regimes: stationary, oscillatory and chaotic. For inner cylinder radius $r_i{geq}1.0$ we observed an increase in the number of charged plumes and vortex pairs with stability parameter T before the electro-convection becomes chaotic. For outer injection, although the onset of electro-convection starts at T higher than the inner injection, the onset of chaotic motion occurs at lower T.