Funduino motor shield is not supplying power to Arduino


I am doing an obstacle avoiding robot project. This is my very first project. I am using Arduino mega 2560, a Funduino L298P 2A Motor Shield Drive Two 7-12V DC Motor Driver and two DC motor that I got with my chassis. I have connected the DC motors with the two channels of Funduino motor shield and connected my motor shield on top of the Arduino.

Now my problem is that when I am powering my Arduino via the USB cable, both my motors are working.The motor shield is also blinking LEDs according to my program.

But whenever I am trying to drive the DC motors using external power source (12V AC to DC adapter) in the motor shield's power channel plug (I am not providing any power to the Arduino USB port or power jack in this case), nothing happens....!!! The motor shield doesn't get any power, no blinking lights, nothing.

Now I am not sure what is actually wrong with it. Do I need to power both the Arduino and motor shield separately or I need to use any other power source...??? or do I need to put any other connection between the Arduino and the motor shield (though I have not seen anyone doing this in any tutorial.) Sorry if I am sounding like a dumb. I am a complete newbie in this field.I have searched this forum and got some people having similar problems like me. But unlike mine, only their DC motors were not working but the motor shield was getting power. :( :(

Another problem is, if I am providing this 12V in Arduino's power jack ( in this case I am not providing any power source to the motor shield), the motor shield LEDs blinks quite fast, not clear enough. But it can not drive the motor. What can be the reason in this case..??

Could you please help me with your advice...??? Any help will be highly appreciable...!!! Thanks in advance. :) :)