Sec. 45.7 



FRICTION-RESISTANCE CALCULATIONS 



99 



(c) U. S. S. Hamilton (DD 141), destroyer. The TMB 



data are unpublished except for one model-ship 

 velocity profile comparison by E. A. Wright 

 [SNAME, 1946, Fig. 24, p. 393]. 



(d) TMB model 3898, representing a proposed twin-skeg 



Manhattan, described by the present author in 

 SNAME, 1947, pp. 112-125. Unpublished data are 

 on file at the David Taylor Model Basin. 



(e) Baker, G. S., NECI, 1929-1930, Vol. XLVI, pp. 



83-106 and Pis. Ill, IV, and V; also pp. 141-146 

 (includes tests on models). On page 86 of this 

 reference. Table I gives other sources of test data 

 on models. 



(f) Baker, G. S., NECI, 1934-1935, Vol. LI, pp. 303-320; 



also SBSR, 28 Mar 1935, Fig. 5, p. 353. 



(g) Calvert, G. A., INA, 1893, Vol. XXXIV, pp. 61-77. 



This reference describes one of the first, if not the 

 first measurement of the velocity profile in the 

 boundary layer of a friction plank, 28 ft long. In 

 Fig. 9 of PI. Ill Calvert gives three velocity profiles, 

 for speeds of 2, 3, and 4 kt, for a transverse distance 

 of 0.44 ft from the plank surface. 



45.7 The Development of Formulas for Calcu- 

 lating Ship Friction Resistance. Friction resist- 

 ance was recognized as a sort of separate entity 

 in the ship-resistance picture as far back as the 

 1790's. Attempts were made then, by Mark 

 Beaufoy and others, to determine its magnitude 

 [INA, 1925, pp. 109, 115]. It remained, however, 

 for the eminent William Froude to conduct the 

 first systematic friction experiments on flat sur- 

 faces and to establish the first systematic basis 

 for the calculation of friction drag. Following 

 extensive towing experiments with thin planks 

 having various coatings, he developed the 

 expression 



R^ = fSV (45.i) 



where / was his own friction-drag coefficient and 

 the exponent n approximated 1.83 for smooth, 

 varnished surfaces. He found a length effect in 

 addition, but this was not reduced to mathematical 

 terms. R. E. Froude, the son of William Froude, 

 after a re-analysis of the observed data, later 

 changed the speed exponent n to 1.825. 



Following the elder Froude's work the principal 

 landmarks in the evolution of a suitable formula 

 for calculating ship friction resistance were: 



(1) Osborne Reynolds' development in the early 

 1880's of the_ dimensionless relationship UL/v, 

 now named the Reynolds number and expressed 

 SLsR„ 



(2) The development of the boundary-layer 

 theory by Ludwig Prandtl in the early 1900's. 

 The highlights and details of the last three- 

 quarters of a century of progress on this project 



have been told many times so that they are not 

 summarized or repeated here. A historical sum- 

 mary was given some time ago by K. S. M. 

 Davidson [PNA, 1939, Vol. II, pp. 76-83] and 

 others more recently by F. H. Todd [SBMEB, 

 Jan 1947, pp. 3-7; SNAME, 1951, pp. 315-317]. 

 A list of the principal references is given in Sec. 

 45.26 for the benefit of the interested reader. 



It has been recognized, practically from the 

 beginning of this development, that a friction 

 formulation which correctly expresses the drag 

 of a thin, flat plank or friction plane by no means 

 applies directly to the prediction of ship friction 

 resistance. There are a number of reasons for this: 



(a) The possibility of laminar flow on the plank 

 or plane and on a towing model whose friction 

 drag is calculated from such data, as compared 

 to the fully turbulent flow over practically the 

 entire wetted area of the ship 



(b) The effects of transverse curvature on the 

 submerged edge or edges of the plank or friction 

 plane as well as of both transverse and longitudinal 

 curvature on a towing model and on the ship 



(c) The effects of various pressure gradients, 

 especially the longitudinal gradients, on the 

 curved surfaces of model and ship 



(d) The variation between the calculated at-rest 

 wetted surface of a model or ship and the actual 

 wetted surface when it is moving and making 

 waves, as well as changes in flow caused by 

 orbital wave motion and the like 



(e) The smoothness of the plank or friction-plane 

 surface as compared to the relatively rougher ship 

 surface, especially in view of the need for greater 

 absolute smoothness on the ship to insure hydro- 

 dynamic smoothness in the larger scale 



(f) Other effects of differences in absolute size 

 or scale. 



William Froude, B. J. Tideman, and others, 

 working in the 1860's, the 1870's, and later, 

 recognized that no ship is ever as smooth as a 

 friction plane towed in a model basin. They 

 provided in their friction coefficients a series of 

 positive allowances for what they considered to 

 be unavoidable roughnesses in the ships of their 

 day. They did not know, in those years, that the 

 ships had actually to be smoother, in an absolute 

 sense, to afford the same degree of hydrodynamic 

 smoothness as obtained on their models. However, 

 the modern friction formulations are all developed 

 for and apply strictly to fiat, smooth plates, 



