FLOW AROUND BENDS 119 



at an angle 6 < do are given by 



V ^ 

 D = -^ sin^ (^0 ± B/2) [1005] 



g 



the positive sign giving depths along the outside wall and the negative 

 sign depths along the inside wall. The depth at the maximum height 

 of the first disturbance is obtained by substituting the value of ^o given 

 by equation (1004) for 6 in equation (1005). 



Knapp and Ippen also describe measures which they found to be 

 effective in preventing the formation of excessively high standing waves 

 in bends of channels having a rectangular cross section.^ For the 

 correction of undesirable flow conditions in existing channels, diagonal 

 sills may be installed at the beginning and end of the curve. For new 

 designs good flow conditions may be insured by the use of circular 

 transition curves, or by banking. 



The purpose of the diagonal sills, and also of the circular transition 

 curves, is to introduce a counter-disturbance of just the right magnitude, 

 phase, and shape to neutralize the undesirable oscillations which would 

 form at the changes of curvature. It was found that this could be 

 accomplished by building simple rectangular sills diagonally across the 

 bottom of the channel. The details of the design had to be determined 

 experimentally. 



Circular transition curves of comparatively simple design were also 

 found to function satisfactorily. The transition curve should have 

 twice the radius of the central curve. It should curve in the same direc- 

 tion and be a half wave length long. The central angle of the transition 

 curve is given with sufficient accuracy by the formula 



W 

 St = tan-i r-—— [1006] 



Rt tan /3o 



in which Rt{ = 2Rc) is the radius of the transition curve. If the radius 

 and length of the transition curves have these values the counter- 

 disturbance set up will be of the correct magnitude and phase relation 

 to minimize the height of the standing waves in the curve. A similar 

 circular transition at the lower end of the curve is necessary to prevent 

 disturbances in the downstream tangent. 



The method of banking permits equilibrium conditions to be set up 

 without the introduction of counter-disturbances. It was the most 

 effective method tested. For equilibrium, the cross-slope should be 



^ " Experimental Investigation of Flow in Curved Channels — Abstract of Re- 

 sults and Recommendations " (2 volumes), A. T. Ippen and R. T. Knapp. July 6, 

 1938, U. S. Engineer Office, Los Angeles. 



