FLOW AROUND BENDS HI 



slightly higher than at A . The amount of excess can be calculated by 

 the formula for centrifugal force when the velocity of the moving water 

 and the curvature of its path are known. For an actual stream such 

 calculations are subject to some degree of uncertainty because the 

 velocity varies throughout the cross section, and because the radius of 

 curvature cannot be precisely determined. Subject to these uncer- 

 tainties, however, the calculated difference in surface elevation between 

 the inner and outer banks at a bend agrees with the observed difference 

 within the limit of possible error in the observation. The surface at 

 the outer bank, for flow at velocities well below the critical, is rarely 

 higher than the surface at the inner bank by more than a few inches. 



^ A B j/Water Surface^ 



InnerW^ — ■ y—L^L^-- _Jy Outer 



Rank "^Horizontals C D Jfr Bank 



Fig. 1001. Channel Cross Section Illustrating Conditions in Bend of River. 



Formulas for the difference in water-surtace elevation between the 

 inner and outer banks of a stream flowing around a curve at less than 

 the critical velocity can be derived as follows: Consider the forces 

 acting on an element of the stream represented by ABEF, Fig. 1001. 

 Assume that all parts of this element have the same velocity and are 

 moving in circular paths around the same center of curvature. 



Let b = breadth of the stream, 



r = radius of curvature of flow at the element ABFE, 

 fi = radius of curvature of inner bank, 

 r2 = radius of curvature of outer bank, 

 dr = distance AB, 

 y = depth of element AE, 

 dy = height of B above A , 

 V = velocity of water, 

 w = weight of unit volume of water, 

 R = radius of curvature at center of stream. 



The centrifugal force acting on the element ABFE is equal to the 

 excess pressure on the face BF over the pressure on the face AE, due 

 to the height of the surface at B above the surface at A . Considering 

 the element to have a length of unity up and down stream, its volume 

 is ydr, its weight is wydr, and its mass is {w/g)ydr. The excess pressure 

 on the face BF is wydy. From mechanics it is known that centrifugal 



