Sec. 61.12 



PREDICTED BEHAVIOR IN CONFINED WATERS 



407 



/. = ^^^= 191.8 ft. 

 (0.645) g 



The limiting depth required is therefore approximately 

 192 ft. 



To determine the limiting depth at which the 

 shallow-water effects become negligible, for all 

 practical purposes, it may be assumed that this 

 depth corresponds to a condition where the shal- 

 low-water resistance does not exceed 1.005 of 

 the deep-water resistance. As the resistance may 

 again be assumed, for a first approximation, to 

 vary between the square and the cube of the 

 ship's speed, an increase in resistance of 0.005 

 corresponds to an increase in speed of the order 

 of about 0.002. Similarly, limiting the resistance 

 in shallow water to that encountered in deep 

 water involves a speed reduction to the order of 

 0.998 y„ in the shallow water whose depth is to 

 be determined. 



If the product of the VJVi and the 7//7„ 

 ratios is to exceed 0.998, the values of both ratios 

 must be close to 1.000. Assuming that the value 

 of the VJV I speed ratio must exceed 0.998, 

 examination of the corresponding curves of Fig. 

 61. K shows that the value of W Ax/h must be 

 less than about 0.218, regardless of the size of 

 the ship. Furthermore, the percentage of critical 

 velocity must be less than 0.567, regardless of the 

 speed. This is equivalent to shrinking the diagram 

 of Fig. 61. K to the point where the vertical limits 

 on both end scales are 0.998 and 1.000. Two 

 values of the limiting depth h are first derived 

 from the two relationships given. If they are 

 not nearly the same the depth h is determined 

 by trial and error as before. 



Example 61.V. The data for this example are taken 

 from SNAME RD sheet 39; TMB model 3796. A liner of 

 62,660 tons displacement, 962 ft long by 117.8 ft beam 

 by 34.39 ft draft, having a Cx of 0.981, is expected to 

 run at a speed of 32 kt. What is the limiting depth of 

 unrestricted shallow water in which the speed with a 

 given deep-water resistance does not drop below 0.998 Fa> ? 

 No account is taken of other ship-performance factors 

 such as possible vibration. 



The maximum-section area Ax of the ship is 117.8 

 (34.39)0.981 = 3,974 ft^. The square draft VaI is 63.03 

 ft. The speed of 32 kt is equivalent to 54.05 ft per sec. 

 The value of g is taken as 32.174 ft per sec^. 



By the potential-flow criterion alone (point F on Fig. 

 61. K), the ratio y/Ax/h is 0.218 and the limiting depth 

 is 63.03/0.218 = 289.1 ft. 



By the critical-speed criterion alone (point B on Fig. 

 61. K), the value of V^iy/gh is 0.567 and the limiting 

 depth is 



h = 



2,921.4 



(0.567)'^ (0.3215)(32.174) 



= 282.4 ft. 



These two values of h are close enough so that the 

 larger of the two may be assumed as the limiting one. 

 In this case it might be well to set the minimum depth as 

 290 ft. 



Example 6 l.V I. At the opposite extreme, assume a 

 motorboat having a maximum section area Ax of 6.25 ft", 

 running at 9.5 kt, or 16.05 ft per sec. What is the limiting 

 depth of unrestricted shallow water in which the speed 

 with a given deep-water resistance does not drop below 

 0.998Fco ? The value of g is taken as 32.174 ft per sec". 



As a starter, consider that the overall velocity ratio 

 Fft/Fco = 0.998 is made up of the two ratios Vh/Vi = 

 0.9997 and Vj/V^ = 0.9982. These values are admittedly 

 arbitrary but with a little experience they can be estimated 

 rather closely. Then from Fig. 61. K, at a Vh/Vr ratio of 

 0.9997, the square-draft to water-depth ratio \/ Axlh is 

 0.115. Hence, transposing. 



K 



VAx V6.25 



0.115 



0.115 



= 21.7 ft. 



For a Vi/Vcc, ratio of 0.9982, the critical-speed ratio 

 Vaily/ gh from Fig. 61. K is 0.563. Again transposing. 



hn = 



Yl 



(16.05)' 



fir(0.563)' (32.174)(0.563)' 



= 25.3 ft. 



The potential-flow effect is so small here that the square- 

 draft to water-depth ratio is somewhat indeterminate. 

 Nevertheless, it is apparent that, because of the greater 

 depth required to produce the assumed wave-speed ratio, 

 the latter factor is also the determining one in this case. 

 The minimum depth is therefore of the order of 25 or 26 ft. 



61.12 D. W. Taylor's Criterion for the Limiting 

 Depth of Water for Ship Trials. A simple formula 

 is given by D. W. Taylor for the minimum depth 

 of water In, involving "no increase of resistance" 

 [S and P, 1943, p. 79]. This is the dimensional 

 expression: Minimum depth = 10 (draft //) 

 (F/a/L), where the depth h, the draft U, and 

 the length L are in ft, and the speed V is in 

 kt. Taylor gives the following limitations for this 

 formula: 



"1. To vessels not of abnormal form or proportions up 

 to a block coefficient Cg of 0.65 



2. For speeds for which V /\/l is not greater than 0.9 



3. The formula may be of use beyond the limits indicated 

 above, but in such cases (it) needs to be applied with 

 caution and discretion." 



Despite these limits, expressly stated, this 

 formula has been used rather widely for estimating 

 minimum depths of water in which to conduct ship 

 trials. 



Taylor's formula as it stands is not consistent 

 dimensionally, for the reasons given in Appendix 



