410 



HYDRODYNAMICS IN SHIP DESIGN 



Sec. 61.15 



two sides. With the same water area as the duct, 

 its Rh'is, larger. For an open channel with a ship, 

 as at 4, Rh is the ratio of (1) the cross-section area 

 of the water in the channel to (2) the wetted 

 perimeter of the solid boundaries of both channel 

 and ship. In shallow water of infinite width, 

 when 6 — * CO, the depth h, the ship girth G, 

 and the ship section area ^4^ become negligible 

 with respect to the water-area and width factors. 

 The hydraulic radius Rh then becomes equal to 

 the depth h, as at 3 in the figure. For a rectangular 

 canal this is expressed in symbols as: 



Rh = 



bh 



b + 2h + G 



b is large 



bh 



compared to = -r- = h 



other factors ,„, ..., 



(61.111) 



Assuming a waterway with horizontal bottom 

 and vertical sides, not occupied by a ship, the 

 hydraulic radius is related to the water depth h 

 in the following manner: 



serves as such, apparently because of the very 

 large bed clearance under the largest (and 

 widest) flat-bottomed model that is towed in it. 

 Since more of the water goes under the bottom 

 of a model with a large B/H ratio than with a 

 small one, the effect of small bed clearance 

 becomes greater as the B/H ratio increases. 

 Unfortunately, not enough is known of these and 

 other effects to take account of them quantita- 

 tively at the present time. 



61.15 Estimating the Effect of Lateral Restric- 

 tions in Shallow Water in the Subcritical Range. 

 Since the speed of a wave of translation in a 

 restricted channel depends only on the depth of 

 the channel, it appears plausible to assume that 

 O. Schlichting's theoretical assumption concern- 

 ing the equality of pressure resistance due to 

 wavemaking at the speed F„ and V, remains 

 valid for these restricted channels. The second 

 assumption of SchUchting concerning the speed 

 correction due to the potential flow around the 



Width 6 in terms of h 



50 



100 



200 



Rh in terms of h 



0.33 



0.60 



0.833 



0.909 



0.962 



0.990 



For open channels of non-rectangular and 

 irregular sections, with ships in them, the hy- 

 draulic radii are determined by exactly the same 

 procedure, governed by the same rule. Example 

 61. VII, in the next section, illustrates the method. 



Studies made in connection with the prepara- 

 tion of Fig. 61. L, combined with analyses under- 

 taken (in 1956) subsequent to those reported in 

 the remaining sections of this chapter, indicate 

 rather definitely that shallow-water effects cannot 

 be correlated on the basi s of the single "trans- 

 verse" parameter 'VAx/h, nor can confined- 

 water effects be correlated solely on a basis of 

 ■V Ax/Rh- This applies particularly to effects 

 associated with the potential-flow ratio of Sec. 

 61.5, between the shallow-water speed Vh and 

 the Schlichting intermediate speed Vi . Analyses 

 of the blocking effect of model basins upon the 

 ship models towed in them indicate that the 

 interference effects are negligible even when, 

 because of the limited width of the basin, the 

 hydraulic radius of its section is not much more 

 than half of its actual depth. By this criterion, the 

 basin is by no means the equivalent of unlimited 

 deep water of the same depth. Nevertheless, it 



ship hull requires modification to take account of 

 the width of the channel. The relationship devel- 

 oped by L. Landweber [TMB Rep. 460, May 

 1939, p. 10] involves, instead of the depth of 

 water h as before, the hydraulic radius. The 

 necessity for taking full account of the lateral 

 restrictions is emphasized by the following 

 comments, quoted from a discussion by F. 

 Rayner on page 114 of a paper by A. F. Yarrow 

 fINA, 1903]: 



". . . one of the greatest difficulties in towing on inland 

 waters is the friction between the boats and the sides and 

 bottom of the water way. I have myself seen, on some of 

 the narrow canals, steam barges almost stationary when 

 going through what are called, in canal language, "bridge 

 holes," where you get the minimum width, and conse- 

 quently enormous friction; as soon as the boat gets away 

 from the bridge, she shoots ahead." 



The ratio -vAx/h, relating the square draft 

 to the water depth, then becomes \/Ax/Rh , 

 relating the square draft to the hydrauUc radius. 

 The fact that Landweber used, in the reference 

 cited, a value twice as large as that defined here 

 was compensated for by his use of a factor 2 in 

 the ratio of square draft to hydraulic radius. 



