112 



IIM^RODN \ WIIC.S I\ Sllll' IMSICX 



^^rr.n.n 



ratio of 0.07. a li II ratio of t.JO, ami a (',. of 

 0.705: 



Ai?, = 0.0i p^^-^^-^»^)V 2.5] 



•(0.8.-, + 0.70o)[l.3 -^J 



= 0.0320 



For a paddle steamer like the lAtcij Ashton, 

 especially of shallow draft, the augment of velocity 

 in the watcrlinc region arising from the induced 

 velocity generated ahead of and abaft the paddles 

 acts to increase the friction drag. This amount is 

 difficult to determine in figures but theoretically 

 it always involves an increase in resistance. 



Despite the hydrodynamically correct basis for 

 increases in the friction drag due to convex 

 curvature in a ship, both transverse and longi- 

 tudinal, many cases occur in practice which cast 

 doubt upon the validity of those additions to the 

 friction drag for a smooth, flat plate in turbulent 

 flow. Indeed, there are cases where the entire 

 -(AC^) allowance, for all types of roughness as 

 well as for both types of curvature, is practically 

 zero. There are those who say, and with some 

 reason, that results of this kind lead them to 

 question the smooth, flat-plate, turbulent-flow 

 friction formulation it.sclf, which gives higher C^ 

 values than it should in certain ranges of R„ . 



In any event, the analyses made to date of 

 existing propeller-tiirust data on ships are in- 

 sufficient to indicate whether the curvature 

 allowances, as estimated by the procedures 

 described, are reasonable or not. All that can be 

 said with certainty is that A,CV and A^Cy are 

 probably no larger than indicated, and that they 

 are both [)ositivc. 



45.15 Criterion for a Hydrodynamically Smooth 

 Surface. A.s a crilcrion for liydrudynaiiiic smooth- 

 ness in turbulent n(jw S. CJoldstein has suggested 

 the exprcs-sion |R and M 1703, Jul 1930, p. 113] 



k,Mr 



< 5 



(4.'i.vi) 



where fc^, is the average height of the hills or 

 roughnesses above tlio limen, (/, is the shear 

 velocity V T„/p, and v is the kinematic viscosity. 

 Ah pointed out in Sec. l.'j.lO, the left-hand expres- 

 «ian of the ini'<|uality ha.s the form of a Heynoltis 

 number, so that it can be rrlatcd to a simple 

 number. 



Of intcre.st in this connection i.s a .set of crilcri.a 



given by .1. S. Hay in Portoti Technical Paper 428 

 (unclassified) of 24 June 1954, issued bj'' the 

 Chemical Defense Experimental Ivstablishment of 

 the Ministry of Supply in Great Britain (copy in 

 TMB librarj')- I'l the .symbols of the present 

 volume, and where the factor A"o is definetl solely 

 as a "roughness parameter," these are: 



(1) Aerodynamically rough flow. 



A-„ > 2..'5 



U, 



(2) Tran.'^itiiina! flow. 



2.0 > -^ > 0.13 



(3) Aeroilynamically smooth flow, 



ko < 0.13 



U, 



To apply Goldstein's criterion to the practical 

 case, it is first neces.sary to know the local inten- 

 sitj' of shear tq . For turbulent flow, from Fig. 45. A, 



To _ , _ 0.0.')9 



— ^ I.F — nOi 



whence 



0.059 



ii4 



{R.y 



(5.iii) 



As an example, assume for a destroyer a point 

 200 ft abaft the stem and a speed of 30 kt, 

 ecjuivalent to 50. G7 fps. The kinematic viscosity 

 V for salt water is 1.2817(10"^) ft" per sec and p/2 

 is 0.995 slugs per ft^. The value of U^ is 2,507 fps'. 

 From Table 45. b li, is about 790 million, whence 

 IC = 60.2. Then u = 0.059(0.995) (2,507)/G0.2 

 = 2.504 lb per ft". The shear velocity U, = 

 {to/pT' = [(2.504)/1.9905]"' = 1.1215 ft per sec. 

 From Ef|. (45. vi), assuming that [(A-^.f^,)/*'] is 

 as large as 5, 



,^, = (5,)/f;, = (5)1.28n(10-') ^ 3 ., j(jQ..j ^^^ 



whence A\, = 0.08()(10"') ui. 



This means that the maximum permissible 

 rougluie.ss heights, for a hydrodynamically smooth 

 surface, would i)rol)ably be of the order of one- 

 thousandth of an inch. It amounts practically to 

 a laboratory smoothness, exceedingly expensive 

 and laborious if not practically impo.ssible to 

 achieve on a i.'irge ve.s.sel, even for a .sjiecial trial. 



