Just my casual non-scientific observations:
In unrestricted free-flow, the fan's power draw (Watts) is roughly proportional to the voltage squared down to the stall voltage.
Fan power draw is related to the back-pressure. Fans will draw more power in free-air, than when acting against a restriction.
Panasonic gives PQ curves for their fans at different voltages, which can be seen here:
http://www.panasonic.com/industrial/...pdf/fba12g.pdf
Pretty sure that Panasonic actually measured such. From this we can make some further observations:
Fan peak pressure is roughly proportional to the voltage squared.
Fan pressure - while peaking at proportional to the voltage squared at "dead head", is not proportional to the voltage squared once air-flow occurs. Pressure seems to be proportional to the voltage squared divided by the ratio of the new and old air-flow rates against a resistance.
Fan peak flow is roughly directly proportional to voltage
In fact axial fans can be observed to act very similarly to (DC) water pumps. Both of these observations from Panasonic's axial fan PQ curves also seem to match my own observations of DC water-cooling pumps, however the peak flow of water pumps does not seem to be as linear with voltage as axial fans, but I suspect this is related to fitting resistance effects.
None of the above contradicts the Vent-Axia fan laws. A doubling of voltage will result in a quadrupling of motor power, but
will not result in a doubling of air-flow against resistance. To do so requires roughly sqrt(8) or ~2.8x the original voltage. The actual proportional values of this will depend on whether various sections of the "system" are in laminar or turbulent flow.