Although the instability of an initially motionless fluid subjected to a Rayleigh-type acceleration is well known, this type of excitation applied to an initially moving fluid has not been studied previously. We have therefore performed numerical analyses for two-dimensional unsteady thermal convection in circular cylinders under variable low-g conditions associated with space flight. When an acceleration vector is applied parallel to the thermal gradient for a fluid at rest, no convection results for the stable direction, and an instability leads to Rayleigh convection for the opposite direction. However, when the acceleration has a component which is orthogonal to the gradient, convection always results at any Rayleigh number; this is the usual situation during space flight. We therefore study the resultant effect on convection when both types of acceleration are applied, concurrently or sequentially, and when the resultant vector varies in direction with time. Our results indicate that for space flight conditions, the Rayleigh accelerations impose significant, but not dominating, alterations in the established convection even when the Rayleigh number is less than critical.