Concrete is an wonderfully easy product to shape. For your scenario, all you really need is make a form for the concrete that matches the concavity of your drain, and to set it in place when you place the concrete.
It's not clear when you say curved if the drain itself curves along its length or if by curve you refer to the concave portion of the drain.
If your drain is basically straight with a concave portion, the problem is very easy - PVC drain pipe of the appropriate diameter, plugged tight with rags and capped off with duct tape will do the trick nicely.
If your drain itself curves, you'll have to get more creative in making a form - possibly shaping one from wood.
Regardless of the shape, in this scenario you will want to suspend the form from above, so build a series of wooden supports and put hooks in the top of the form in a clean straight line and tie each hook to a frame so that the form hangs ROUGLY halfway into the trough when the frame is set over it.
You must also make sure that the form is lowest wherever the water is to exit the trough and highest at the trough's ends, so that water will flow properly in the trough. A 1/8" drop per linear foot of trough should suffice.
What you want is a form that from its end looks like this picture, and which has the slightest tilt to it to make the water run downhill in the trough in the proper places.
(Ground - brown, frame - red, form - grey, black - line suspending form to frame)
Once your form is ready, move it out of the way, mix your concrete and fill the trough up until the concrete is high enough that the form will set down into it some. Make sure there are no air gaps in the concrete so far.
Put the form in place, carefully making sure it settles into the concrete to the proper depth (ie: the suspending wire is taut). Now fill the rest of the trough's sides with concrete until its just level with the soil.
Wait until the concrete begins to set. When you can push on the concrete with your finger and not easily make a dent, remove the form, and use a wet sponge to clean up the edges where you removed it.
Viola - a concrete drain with concave drain channel!