Ignoring all important friction, you are left with very simple equation

Velocity = sqrt(2*g*H)
g=9.81 m2/sec
H=Head Loss, m = (slope in fraction) * (Distance travelled)

Now, you notice it for youself that velocity goes on increasing with distance.

But no such system of "channel" has actually been experienced where you can ignore friction. I am just being curious, what do you want to do with this correct but useless equation?

Thanks for the response - greatly appreciated!

I have a whole set of streams with gradients, but no info for the "roughness" or friction for the streams. I was hoping to be able to derive at least something for stream velocity, based purely on the gradient, but as you show, velocity increases with distance making this useless!

Do you by any chance know if anyone has derived an average friction coefficient for hydrologic flow? This would at least give me a starting point for my application, pending being able to get friction data.

Thanks again.

In a channel, the equation we generally use is Manning's. I do not want to eloborate Manning's equation here because of its fame and availability in most of the hydraulic text books. When you go through it, you will see a friction coefficient n which depends on the material the channel is composed of.
Approximate values of 'n' are

concrete =0.014-0.018
steel = 0.01
Earthen channel (smooth) =0.03
Channels with vegitation=0.05

You can refer to Ven Te Chow's book "Open Channel Hydraulics" for details.