Sec. 61.16 



PREDICTED BEHAVIOR IN CONFINED WATERS 



411 



When the substitution of Ru for h is made, the 

 potential-flow ratio Vh /Vi for confined waters 

 becomes a function of y/ Ax/Rh for the restricted 

 channels. A procedure corresponding exactly to 

 the Schlichting method described in Sec. 61.5 can 

 then be employed. This means that Fig. 61. G 

 serves for making the numerical calculations as 

 before, provided the user remembers that the 

 upper scale is a square-draft to hydraulic-radius 

 ratio. 



From the theoretical curves of Fig. 61.E and 

 the experimental curves of Fig. 61. G the speed 

 and the resistance of a ship in a restricted channel 

 can then be computed when its deep-water speed 

 and resistance are known. The procedure to be 

 followed is the same as for computing shallow- 

 water resistance; several examples follow. 



Exam-pie 61.VII. Take the case of the 370-ft shallow- 

 water ship of Example 61.1 preceding, moving in the 

 channel depicted at 1 in Fig. 61.N. The essential model 

 and ship data are given on SNAME RD sheet 9, covering 

 TMB model 3818. What would be the actual ship speed 

 at a resistance equal to that for 8 kt in deep water? The 

 ship has a maximum section area Ax of 1,111.3 ft^ and the 

 water is at sea level, with a temperature of 80 deg F. The 

 value of the square draft y/Ax is 33.34 ft. The value of 

 g is 32.174 ft per sec^. 



The section area of the water around the ship, using 

 the values in diagram 1 of the figure, is 



= [(250)(35)] + 



+ 



(35)^ 

 2 



35^ 



[(.o.e)(f)] 



1,111.3 



= 9,311.7 ft' 



The wetted perimeter, including that of the ship, is 

 P = 250 + 35 cosec (45 deg) 



+ 35 cosec (30 deg) + 



= 467.5 ft. 



The hydraulic radius is then 9,311.7/467.5 = 19.92 ft, 

 only a little more than half the channel depth. The 

 Va^/Rh ratio is 33.34/19.92 or 1.67. 



The equivalent "rectangular" depth Aeq of the channel 

 is the section area without the ship, divided by the surface 

 width, or (9,311.7 + l,111.3)/(250 + 35 -|- 60.6) = 

 10,423/345.6 = 30.16 ft. 



The value of Vao is 8 kt or 13.51 ft per sec. Then 



13.51 



•v/32. 174(30. 16) 



0.43. 



From the theoretical curve of Fig. 61.E, the correspond- 

 ing value of the intermediate speed ratio Vj/Va, is 1.00 

 and Vi = Va, . Hence all the speed reduction is due to 



potential flow. Since the points corresponding to Ai and 

 Bi of Fig. 61. B coincide, the friction resistance does not 

 come into the picture, nor is it necessary to construct any 

 deep-water and restricted-channel resistance curves. 



Entering the extrapolated broken-line portion of the 

 curve of Fig. 61. G for a square-draft to hydraulic-radius 

 ratio of 1.67, the value of the potential-flow ratio Vk/Vi 

 is 0.783. Since F/ = Va> in this case, Vh = 0.783Fco = 

 0.783(13.51) = 10.6 ft per sec or 6.28 kt. This is the 

 required speed. 



It is pointed out in Eq. (61.iii) of Sec. 61.14 

 that when the channel width b becomes large in 

 proportion to the channel depth h, as when a 

 shallow river Avidens into a shallow estuary, the 

 term 2h in the expression for the hydrauhc 

 radius drops out, leaving simply the quotient 

 hh/b, whereupon the hydraulic radius Ru becomes 

 equal to the depth h. 



The question now arises, what constitutes 

 unrestricted shallow water? This is difficult to 

 answer explicitly because it depends upon the 

 maximum-section area of the ship being con- 

 sidered with the water and upon the square-draft 

 to hydraulic-radius ratio. Put i n a nother way, 

 the effect of using the ratio '\/ Ax/Rh instead 

 of the ratio vA^/Zi, where In, is the restricted- 

 water depth, depends to some extent upon the 

 position of the ratio point along the graph of 

 Fig. 61. G. At small values of the ratio vAxA, 

 toward the left end of the diagram, the potential- 

 flow speed ratio F^/F/ changes very little with 

 change in water depth. In any case, one or two 

 calculations involving the hydraulic radius, along 

 the lines of Example 61. VII, should clear up the 

 matter readily. When the channel width becomes 

 from 100 to 200 times the depth, the table in 

 Sec. 61.14 indicates that the restricted channel 

 has become the practical equivalent of open, 

 unlimited shallow water. 



61.16 Lack of Reliable Data on Power and 

 Propulsion-Device Performance. No satisfactory 

 method has yet been developed for estimating 

 the increase in shaft or propeller power, the 

 change in rate of propulsion-device rotation, or 

 the variations in other propulsion factors due to 

 shallow and restricted waters. E. A. Wright 

 touches briefly on these matters [SNAME, 1946, 

 Fig. 10, p. 381]. The present unsatisfactory 

 situation is due partly to the limitations imposed 

 by various kinds of propelling machinery on the 

 combinations of rotational speeds, torques, and 

 powers developed by them. The usual ship- 

 performance data are rarely of much help because, 

 for example, the throttle setting may be held 



