A numerical study of the complete system of reaction-diffusion-convection equations that govern the dynamics of Belousov-Zhabotinsky (BZ) reactive fluid has been conducted in order to understand the interaction of chemical waves with fluid convection induced by chemical reactions. In the regime studied, it was found that although the heat of reaction alone could drive convection, it was too weak to destabilize the wave. When the influence of the isothermal chemical reactions was combined with that of the heat of reaction, however, convective instability of the wave structure could result for both up-going and down-going waves. The regime in the parameter space was defined within which instability of the up-going wave and stable propagation of the down-going wave is expected, as observed in a magnetic resonance imaging (MRI) experiment. The velocity of an undistorted wave front in the presence of convection was found to be independent of the lateral scale of the system and the direction of wave propagation, as was also observed in the MRI experiment.