Sec. 45.1 



FRICTION-RESISTANCE CALCULATIONS 87 



Turbulent Flow a-l-:— Transition —>4-<: Laminar Flow 



r~~ — ^ -. I Logarithmic 



FOR TURBULENT FLOW 



V/?T?/y>, 



Tr- 



mA 



X to Leading Edqft 



■\i I rorabolio Approximation 



I |_ U^- — 1^ Leodii 



Parobolol I ^■«:- ?~T6 



I I ^~~^-^_ U ? I 



Equation of Profile, | Equation of Profile, 



U-Cyf or U = U„f^)^| U-Clogy or U-U„-|^ 



Nominal Thickness of La\/er ot Points B and C, 



6 - 0.38 ^{^^'^ - 0.38 X R^"''^ 



Displacement Thickness 6*= 0.146, obt. 



5ublQx;er Thickness S^_- k-^ , where ll.6-i k*: IS.6 



Momentum Thickness 6- 0.097 6 



Shape Factor H = -|- - 1.3 to 1.5 



Enerqiy Thickness 0*- 0.175 5, abt. 



Shear Stress at Plote TQ-CLp-q -0.059q-Rx°"^ 



Sheor Velocity ^t'{-^)°''' 



Shearinq-Stress Coefficient Ct - . ."^ , , ; - -?- 

 ^ ^^ 0.5/0 U^ 1 



Local Frictional-Drag Coefficient at Points B ond C, 



Clf = 0.059 Rx""'* 



Mean Drog Coefficient for x-rDistance 



to B or C, Cf - 0.074 R^ °"^ 



FOR LAMINAR FLOW 

 Equation of Profile, U„-U-a(6-y)^or U=U^--^(6-y)2 

 Nominol Thickness of Lo^er ot ony Point A, 

 6 = 5.(-^)''== 5.Ri°-= 

 Displocement Thickness 6*= l.73x R^'^ = 0.55S,abt. 

 Sublayer Thickness 6l- (Entire Loijer is Laminar) 



Momentum Thickness 9= 0.133(5, abt. 



6* 

 Shape Factor H = -q-" 2. 6, obt. 



Enerqy Thickness ©*= (Not Applicable in This Case) 



Shear Stress at Plate To=CLF-q- 0.66q-R^°-^ 



Shear Velocity \Jt-(^)°-^ 



Shearing-Stress Coefficient (^r~ oJoU ^ " ~3~ 



Local Frictionol-Droq Coefficient at Any Point A, 

 Clf" 0.66 R.£°-5 



Mean Drag Coefficient for x-Distance to A, 

 Cf- 1.33 Rx'" 



Fig. 45.A Summary op Viscous-Flow Formulas for Laminar and Turbulent Flow 



the entire length L of a body or ship, to use the 

 symbol R„ . When this is done it almost invariably 

 signifies that the space dimension in the Reynolds 

 number is that entire length. However, in many 

 studies of viscous flow and boundary-layer 

 development it becomes necessary to consider 

 the situation at or ahead of a certain point along 

 the body or ship which has an x-distance less 

 than the length L. In these cases the Reynolds 

 number for the situation under study is the 

 x-Reynolds number, expressed as R^ , with the 

 x-distance from the leading edge or stem stated 

 in each case. For example, in the formulas of 

 Figs. 5.R and 45.A, the symbol R^ is used in 

 connection with values averaged for the whole 

 length, while R^ is used for local values, or for 

 situations where the x-distance is itself a factor 

 in the formula. 



It seems almost certain that there are factors 

 in viscous flow not taken into account by the 

 Reynolds number, but until more is known of 

 them, they can not well be considered in any 

 quantitative treatment of friction resistance. 



Friction-resistance calculations are necessary 

 for the Froude method of predicting ship resist- 

 ance, in which the residuary resistance derived 

 from tests of a model of the same proportions and 

 shape is added to the friction resistance deduced 

 from resistance tests of flat, smooth surfaces in 

 the form of thin planks or friction planes. 



Because of the limitations of model-testing 

 equipment it is still necessary to extrapolate the 

 flat, smooth-plate data well beyond the experi- 

 mental range, especially for large, fast vessels. It 

 has not been possible to check this extrapolation 

 in the latter range by full-scale ship measurements 



