CHAPTER II 



BERNOULLI'S THEOREM APPLIED TO A FRICTIONLESS 

 RECTANGULAR OPEN CHANNEL 



Consider first, that water flowing along a horizontal rectangular 

 channel with a high velocity is to be slowed up by means of a smooth 

 gradual enlargement in the cross section. Assume that the flow is 

 steady and that the effect of friction may be ignored. Bernoulli's 

 theorem, then, states that if water moves in such a stream from one 

 point to another at the same level, without loss of energy by friction or 

 impact, the sum of the static head and the velocity head at the first 

 point is equal to the sum of the same quantities at the second point. 



i,^ 



Hjfi 



Fig. 201. Diagram Illustrating Flow in an Expanding Tube. 



Consteint width, level bottom, no friction or impact. The ordinates of the curve EFGH represent 

 the static pressure on the bottom AM; the height of the curve EFGH above the straight line EH 

 represents the static pressure on the top EH ; and the distance of the curve EFGH below the line OL 

 represents the head corresponding to the velocity within the tube. The critical conditions of flow exist 

 at the section NP. 



Let Fig. 201 represent such a condition. Suppose the original velocity 

 is Vi and depth Di. If the channel is open at the top, the static head 

 at the point K is Di and the velocity head is V\ llg. 



Suppose the channel to be of uniform width and the bottom level. 

 In order to secure the gradual smooth enlargement of cross section, 

 suppose a rigid top to be added to the channel as shown from K to B., 

 rising along a straight Ime. We know by experience that if the angle 

 is not made too great, the water will cling to the inclined plane and as the 

 cross section increases the velocity will be smoothly and gradually 

 reduced, as exemplified in the expanding tube of the Venturi meter. 



16 



