CALCULATIONS ILLUSTRATING THE EFFECT OF BOUNDARY LAYER ON WAVE RESISTANCE 
new streamline then leaves the form tangentially, and 
gradually becomes parallel to the central line at some 
small distance to the rear of the stern. It may be noted 
that this departure of the virtual streamline from the 
form does not necessarily mean separation of flow in the 
usual sense; the latter phenomenon might be represented 
on this scheme by the new line leaving at an angle to the 
form. We shall take two examples and in both we shall 
suppose the new curve to reach the central line at its 
rear end; we are then dealing effectively with closed 
forms and this simplifies the work, though it could be 
extended to include a permanent wake to the rear. 
First, considering a 16 ft. model, we take the point of 
departure to be 1 ft. before the stern and suppose the 
new line to close in at a point 2 ft. behind the stern. If 
this new curve is given by 
n= H+agé+al+aeé (17) 
the conditions are 
15 dyn 7 a 
eet oa Wane aac Saline g 
4 A (18) 
7 = 0; qe7% for Sen 
These determine the coefficients in (17) and we get 
1075 35 41 32 
2 3 
Ds ays wa ae cys CY) 
This curve is shown starting from the point B on the 
section in Fig. 1. It passes the stern at a transverse dis- 
tance 55/72 from it, and represents little more than 
smoothing out the stern angle of the model. 
We now have instead of (14) 
1 
I+iJ=—2]| €etvidé 
5 
amit 
+|EBerBejencae . ao 
By 
From (13) this gives the result 
R 16p[2 2128 1 32768 1 32 
Poaay Romande BIS 9 34) 
32 64 2 
— 3s Od t age (p,) + yf Pd 
62 64 32 
+ zyaPs (Pd) — 303 Po (Pa) + 93 Ps (Ps) 
320 2048 
sf ee PE (pee | 21 
3% 6 (Ps) 9" 7 (Ps) (21) 
where py = 15 yof/8; P2 = 9 yol4; P3= 3 yol/8 
This is graphed in curve B of Fig. 1. It shows how 
this small modification practically eliminates the humps 
532 
and hollows at very low speeds and reduces them con- 
siderably up to about f = 0-24. 
To make a rather larger change, we suppose next that 
the point of departure is 2 ft. before the stern, the line 
closing in as before at 2ft. behind the stern. The 
coefficients in (17) are now determined from 
3. 
URI Gar Ae = = 7 
4 (22) 
Rayan len (ye Ea 
7=9; Bs for € = 4 ; 
These give the curve 
D Samal Ss ; 
Diet 16 Sr Do (23) 
The curve is shown starting from C on the section in 
Fig. 1. It passes the stern of the model at a transverse 
distance $5 from it. In this case, we have 
1 
1+is=—2| eertag 
-3 
+ [(G-s¢-se) ema . (24) 
and hence we obtain 
R 16 p[2 731 , 288 1 
3 + 
Pe wy 15iy2 1) 354 
19 6 9 
~— P;(p,) — aP, — pp 
one 5 (Pi) ¥ s(t 7, 4 (P2) 
7 
12 
Yo ‘4 (P1) 
a 
27 
3 18 
+ 5 Pee) — Pr 0 
6 63 
P; (2) + 3 Pola) + gia Ps (Ps) 
(25) 
where py = 7 yol4; P2 = 9 yol4; Ps = vol2 
This is graphed in curve C of Fig. 1. Here the dif- 
ference from curve A for the original model is very 
marked, and the modification is probably more than is 
needed so far as low and medium speeds are concerned. 
It remains to be seen what difference is made at high 
speeds, but a model of infinite draught is not suitable 
owing to the exaggerated values obtained at high speeds. 
Parabolic Model of Finite Draught 
We turn now to a model of the same form, with 
vertical sides, and of draught d. We shall take the 
draught d to be one-twentieth of the length 2 I, because 
this ratio was used in some previous work!? and the 
results given there can be used to check the present work. 
For the model itself, we have from (7) and (8) 
