Electrical – Direction of flow of current in secondary circuit in AC transformer/rectifier

Others have straightened you out on the "12V secondary" part. As far as substituting a DC adapter (wall wart or laptop PSU) for the transformer; this will generally be safe, but a couple of points to note will help avoid surprises :

1) If the project is driving motors, remember that motors are nasty loads that can generate high voltage spikes when running, and 5-10x their rated current for a moment when starting (or permanently when stalled). Transformers are pretty tough and take this abuse well; laptop PSUs may shut down and cheap wall warts can make smoke under the same stress. (If the project has current limiting and noise suppression around the motors, you should be OK.)

2) a "12V" secondary generates 12V RMS at the rated load current. That means the peak voltage is about 1.4* higher, plus 10% or so at no load. Subtract a couple of diode drops and ripple on the resulting DC and you'll get 15V or so there. If the project needs that extra voltage, a 12V adapter won't do the job.

Pl. connect a fast acting fuse (capacity depends on the flux level in transformer you desire to experiment with) )in line with battery and transforme.connect a fast acting FREE wheeling diode across the transformer primary with cathode to positive (reverse biased) and do the testing. V T Ingole