CHAPTER X 

 BENDS, TRANSITIONS, AND OBSTRUCTIONS 



The theories of the backwater curves and of the hydraulic jump, as 

 elaborated in the preceding chapters, will suffice for most practical 

 problems in determining the longitudinal water-surface profiles of canals 

 and rivers. Cases will occasionally arise, however, in which water- 

 surface curves are needed that cannot be computed under the assump- 

 tions of either of these theories. Though not even the simplest 

 hydraulics of all the possible problems of this type has been worked out 

 on a mathematical basis, certain of them have been studied enough to 

 yield results of practical value. It is the purpose of the present chapter 

 to discuss the more important of these in some detail. A good knowl- 

 edge of the preceding chapters, and especially of the first four chapters, 

 is a necessary prerequisite to a complete understanding of the material 

 presented in this chapter. 



Flow around bends: velocity less than the critical. One of the 

 troublesome but unavoidable characteristics of open channels is the 

 presence of curves in alignment. When water flows in a straight uniform 

 channel the transverse profile is probably a horizontal line. Various 

 observers have claimed that the water in the middle of the stream is 

 higher than at the banks. If waves are present, they are much larger 

 in the middle than near the banks, and their crests are therefore corre- 

 spondingly higher; but it has not yet been satisfactorily proved that 

 the average surface elevation is any greater at the center. 



On a bend or curve the transverse surface profile cannot be level. 

 Water, like all matter, when in motion moves in accordance with 

 Newton's first law of motion, that is, it moves in a straight line unless 

 deflected by the action of some unbalanced force. If water moves in a 

 curve, there must be an unbalanced force acting against the water and 

 directed towards the center of curvature. Let Fig. 1001 represent the 

 transverse cross section of a stream at a bend. Consider that portion 

 of the stream whose cross section is represented by the area ABFE. 

 As this portion flows around the bend it is deflected toward the center 

 of curvature on the inside of the bend. This deflection is caused by the 

 excess of pressure on the face BF over the pressure on the face AE. 

 The excess of pressure can exist only when the water surface at B is 



110 



