Surface dielectric barrier discharges have been proposed as means of airflow actuation. A simplified air plasma model fully coupled with gas dynamics is presented and solved numerically using asynchronous mesh adaptation and time integration. Two modes of actuation depending on the driving voltage waveform are presented and analyzed. The first one uses high-voltage sine waveform in the kilohertz frequency range to transfer momentum from ions to gas molecules. The second one uses high-voltage nanosecond pulses to transfer energy to the neutral gas on a short time scale thus generating shockwaves.