Sec. 50A 



CALCULATION OF VVAVEMAKINC RESISTANCE 



213 



equivalent to that encountered when viscous flow 

 is present, as on an actual ship. In other words, 

 the friction resistance and the resistance due to 

 wavemaking do not influence each other, at 

 least as a first approximation, and may therefore 

 be calculated separately. 



(k) The ship form possesses easy waterlines and 

 buttocks (with small slopes), so that separation 

 does not occur in any region around its boundary 

 (1) The length-beam ratio is large and the beam- 

 draft ratio moderately small 

 (m) Change of trim and sinkage due to the ship 

 motion do not affect the pressure resistance. This 

 is a simplifying but not a necessary assumption. 



The wave height, the particle velocity in the 

 wave, and the slopes of the ship waterhnes and 

 buttocks are limited in magnitude only by the 

 necessity of keeping the general expressions for 

 velocity potential and wave resistance in linear 

 form [Lunde, J. K., SNAME, 1951, p. 83]. 

 Actually, the ship forms to which the analytic 

 procedure can be and has been applied are by no 

 means excessively thin and fine, as witness the 

 forms depicted in SNAME, 1951, Fig. 6 on page 

 18, Fig. 8 on page 20, Fig. 9 on page 21, Fig. 7 on 

 page 72, and Fig. 3.Q in Sec. 3.13 of Volume I of 

 this book. 



Change of trim, involving slope drag in addition 

 to hydrodynamic resistance, is not a sizable factor 

 except in vessels intended to run at high speed- 

 length quotients. The pressure resistance due to 

 wavemaking, in the usual model-testing and ship- 

 powering technique, is considered to be inde- 

 pendent of the friction resistance, although there 

 are known to be interactions between the two, 

 discussed in (d) of Sec. 12.1. 



This leaves but two limitations of consequence. 

 The fact that the run of the ship must be suffi- 

 ciently tapering, both horizontally and vertically, 

 to avoid all separation means that the analytic 

 method as now developed is limited definitely to 

 pressure resistance due to wavemaking. Pressure 

 resistances due to separation and other factors 

 have to be calculated separately. It is effective, 

 therefore, in determining only a part of what is 

 generally classed as residuary resistance. 



The major limitation of the analytic procedure 

 is that it fails to take account of the effective 

 change in form of the run, especially near and at 

 the stern, caused by the displacement thickness 

 of the boundary layer and by whatever separation 

 zones exist there. While the displacement thickness 



\* Lenc^th of Model, 16.0 ft : 



I Beam, 1.5 ft 



I iBoundoru-Layer Displacement Thickness 



1 16 is Greotly Exoqaeroted Here 



-4 



Condition 'A, with Mode.1 T 



Symmetrical Fore and Aft Mid length 



1 Direction of Motion 



1-^3 ft-^ 

 I — |lftl^ 



Modification B Shown on this Side 



^ ' ^ - ■ • ^ ^ ^ ^ ^' 'i -I I • • — 

 /. ti J Modification C Shown on This Side Z. 



NPL Model l790B,with Wall Sides and Porobolic WL 



Graph A is for a-dimi Model 

 in Condition A, Diaqrom 1 

 Graph B for Conditit 



0.40 



0.35 



0, 



OZS 



O.ZO 



0.15 



°°° 0.16 0.18 0.20 02Z 0.2A O.Zb 0.28 



Froude Number y-^oL 



Fig. 50. B Calculated Resistance Data for 



Slender 2-Diml Ship Forms with and without 



Allowances fob Boundary-Layer Thickness 



and Separation 



is small in proportion to the ship's transverse 

 dimensions, indicated at 1 in Fig. 50. B, this is 

 true only from the stem to that region along the 

 run where the boundary layer thickens perceptibly 

 as the ship surface recedes from the water flowing 

 past it. When it recedes at too rapid a rate separa- 

 tion occurs in a form such as to fill out or fair out 

 the blunt endings. 



The separation-zone effect can be allowed for 

 after a fashion by a method developed by T. H. 

 Havelock and illustrated schematically at 2 in 

 Fig. 50. B, adapted from Fig. 1 on page 262 of 

 INA, 1948. Here the ship form is enlarged by the 

 addition of an "appendage" at the stern which is 

 estimated to be of the proper size and shape to 

 produce a potential flow (and a surface-wave 

 system) about the ideahzed ship that is found 

 around the actual ship. Unfortunately, this 

 estimate involves a knowledge of and an abihty 

 to predict separation which is greater than that 



