Experiments at Stanford University have shown that it is possible to generate high electron number density DC and pulsed diffuse discharges in atmospheric pressure air. We are interested in scaling up these discharges to produce larger volumes of plasma. Thus, the physical mechanisms that determine the limiting size of the discharges must be understood. In this paper, we focus on the radial spreading mechanism and carry out computational experiments with a three-temperature, finite-rate computational fluid dynamics code. We find that axisymmetric simulations of the jet are unstable, with the jet shear layer rolling up into vortical structures. The jet non-uniformities pinch the discharge which causes a breakdown into what appears to be an arc. This is contrary to the experimental observations, and therefore we postulate that turbulent mixing may increase the stability of the jet through enhanced mixing. We use an approximate formulation that produces a steady jet flow, resulting in stable diffuse discharges with reasonable experimental agreement. We then vary the flow parameters and find that laminar diffusive processes are too weak to produce much change in the discharge radius.