Two and three-dimensional transient models of a thermal plasma torch are presented to analyse the effect of operating conditions on arc root fluctuations for various types of anode configuations. The computational fluid dynamics code, OpenFOAM(c), is utilised in a modified form to model the arc and plasma flow inside the argon plasma torch. Modeling a thermal plasma requires a combination of mutually related fluid dynamics and electromagnetic phenomena which dictate the size and stability of the arc column. While the 2D axisymmetric model is effective in depicting the average energy and average velocity profiles of plasma gas, it fails to describe the fluctuations in arc behaviour and its mode of attachment on the anode. In a 3D model, the arc movement is captured in both axial and azimuthal directions wherein the arc-root attachment on anode surface seems to fluctuates in a periodic manner and the voltage fluctuations mirrors the arc-root attachment behaviour within the system. We have here investigated the effect of operating conditions on the frequency of this arc reattachment, finding that the input current and gas flow rate have a significant effect on arc behaviour. We have also reported the effect of arc voltage perturbations on the exit plume properties (velocity and average temperature) and finally the effect of anode nozzle geometry on the fluctuations and plume behavior is presented.