You are feeling the sensation of blood circulating through an artery near your fallopian tube. I recommend that you have an arterial Doppler ultrasound to check the flow of your arteries in the area of your left fallopian tube. You might have a blockage in an artery which is causing a bruit. A "bruit" is an unusual sound that blood makes when it rushes past an obstruction in an artery when the sound is auscultated with the bell portion of a stethoscope.
If you have any questions please ask me.
Without a doppler I cannot tell you which specific artery could be blocked nor the amount of the blockage; however, from your description it sounds like your left ovarian artery might have some plaque causing it to become partially obstructed so that you feel a fluttery sensation as blood tries to flow past the obstruction. http://upload.wikimedia.org/wikipedia/commons/d/d4/Gray589.png
This is a common problem among obese women and women who have peripheral artery disease. I advise you to be examined by a vascular specialist who should perform a doppler ultrasound of your left lower quadrant.