The periodic unsteady mixed convective flow and heat transfer in a square enclosure driven by a concentric rotating circular cylinder is investigated numerically. The top and bottom walls of the enclosure are adiabatic and the side walls keep lower constant temperature, while the temperature of the inner circular cylinder fluctuates periodically with time at higher averaged value. The mixed convection is driven by the temperature difference and the velocity of the inner cylinder. The two-dimensional mixed convection is simulated with high accuracy temporal spectral method and local radial basis functions method. The Richardson number is studied in the range 1 <= Ri <= 20, the temperature pulsating period ranges from 1 to 10,000 and the temperature pulsating amplitudes are delta = 0.5,1.0 and 1.5. The numerical results show that the fluid flow and heat transfer are strongly dependent on the pulsating temperature of inner cylinder. Comparing with the steady state mixed convection, the heat transfer is enhanced generally for the time-periodic unsteady mixed convection. Moreover, the phenomenon of backward heat transfer is discussed quantitatively. Also, the influence of pulsating temperature on the unsteady fluid flow and heat transfer are discussed and analysed. (C) 2017 Published by Elsevier Ltd.