Get a 12 volt power supply. If you want to know 'why'... read on.
A typical class 2 power supply, which is what a wall-wart type supply is... is unregulated. This means that the voltage coming out of it is not going to remain constant. It will change with line voltage (wall socket voltage) and the load applied to the power supply. These power supplies are designed with this in mind. If you buy a 12 volt power supply and read the voltage coming out of it with a meter, with (in this case) no fan attached, don't be surprised to see 15 or more volts. The power supply will experience what is called a 'voltage drop' when you apply a load to it. Any device attached to a power supply is a 'load'. The greater the resistance of the device, the more it loads down the power supply and thus the lower the actual voltage is going to be. All class 2 power supplies will read a higher voltage when unloaded, than is written on the housing. When an appropriate load is applied to the power supply, such as hooking up a fan to it... then the voltage will drop.
In the case of your 15 volt class 2 supply... the 'actual' voltage of that supply is going to more like 18 volts. When you apply a load to it, that voltage will drop to near 15 volts. Depending on the resistance of the load itself. If the load is very small... then the voltage drop will be less and you may still have more than 15 volts running through the device. Your fan may not like that for long. Computer fans are not like bathroom fans. They are DC, electronically controlled, three phase motors. The electronics in those fans may or may not handle the added voltage. Depends on the quality of the parts in the controller circuit. Better to use a class 2 supply that is meant for a 12 volt device.