at the edge of the boundary layer and 9 is the streamline component of 

 the momentum thickness made dimensionless by the half length L of the 

 ship. 



The final empirical formulas that are used to reduce the number of 

 unknown quantities to three, in order to match the number of differential 

 Equations (3a, 3b) and (7), are the formulas that relate the crossflow 

 momentum and displacement thicknesses 9^„, 9^-, » ^oo' ^'^^ ^o ^° *-^^ stream- 

 line momentum and displacement thicknesses 9 and & . These relationships 

 constitute the critical approximations of three-dimensional boundary layer 

 theory in the momentum- integral framework. It is easiest to simply make 

 crossflow and streamline flow boundary-layer profile assumptions and derive 

 the corresponding relationships that result from the definitions (5a-5f ) . 

 However it is not really necessary to proceed in this way. Basically, 

 definitions (5a-5f) simply say that if the streamwise velocity profile u is 



described parametrically by parameters [a.] and the crossflow pro- 



^ i=l,...,n 

 file V is described parametrically by the parameters [3.] then the 



^ j=l,...,m 



boundary layer thicknesses 9, „ and 6„, k and £ = 1,2 are each functions of 

 the parameters [a.], [3.] such as 9, „ (a, , . . . ,a ; 3^,..., 3 )• One could 

 determine these functions experimentally and thereby completely bypass any 

 velocity profile assumptions. 



In fact, such a scheme has already been used for the streamwise flow 

 by using the Head entrainment method. From two-dimensional and axisymmetric 

 flow theory, the entrainment method gives 



6^ = 6^(H,0^^) (15a) 



C^ = C^(H,9^^) (15b) 



directly without any profile assumptions. In the most highly developed 

 entrainment method of Green et al. , a third independent parameter is added 

 to 9tt and H, namely the entrainment coefficient C , so that 



10 



