Imagine a bent tube in a 'u' shape with a fan in the middle. Both openings of the tube point toward the front of the plane.
That's basically what a propeller engine with a thrust reverser is. And I think you can visualize how it will do nothing if you turn on the fan. The force from the air exiting and entering the tube will cancel out.
Consider it as a mass flow problem. You're ingesting a certain mass of air m at a velocity v and releasing it in the opposite direction. Assuming no losses in the U tube, the only thing that changes is the direction of v. What matters is the total momentum, mv.
If the tube were straight, mv would point out the back and the airplane would go forward. With a U shaped tube, you have instead -mv which is the same momentum, but a different direction, and the airplane moves backwards.
Now that I think of it, it might be easier to visualize as a conservation of energy problem.
Do you think so? The outward stream would probably be more directed than the inward stream, which comes in more diffuse. (Though I don't know if that matters.)
A difference in kinetic energy -- E(outflow)-E(inflow) -- of air would result in net force pointing forward (a decelerating force, if we consider a plane). Basically, we'd need the air to get accelerated forward more than rearward, producing net force.
Kinetic energy is E = (mV^2)/2
Obviously the mass of inflow is equal to the mas of the outflow. We can achieve a difference in speeds if the effective cross-section of the (forward-pointing) outlet was smaller than cross-section of the inlet.
That's basically what a propeller engine with a thrust reverser is. And I think you can visualize how it will do nothing if you turn on the fan. The force from the air exiting and entering the tube will cancel out.