A General Method for Calculating 

 Three-Dimensional Laminar and 

 Turbulent Boundary Layers on Ship Hulls 



Tuncer Cebeci , K. C. Chang, and Kalle Kaups 

 Douglas Aircraft Company, 

 Long Beach, California 



ABSTRACT 



A general method for representing the flow proper- 

 ties in the three-dimensional boundary layers around 

 ship hulls of arbitrary shape is described. It makes 

 use of an efficient two-point finite-difference 

 scheme to solve the boundary-layer equations and in- 

 cludes an algebraic eddy- viscosity representation 

 of the Reynolds-stress tensor. The numerical method 

 contains novel and desirable features and allows the 

 calculation of flows in which the circumferential 

 velocity component contains regions of flow reversal 

 across the boundary layer. The inviscid pressure 

 distribution is determined with the Douglas -Neumann 

 method which, if necessary, can conveniently allow 

 for the boundary-layer displacement surface . To 

 allow its application to ships, and particularly to 

 those with doiible-elliptic and flat-bottomed hulls, 

 a nonorthogonal coordinate system has been developed 

 and is shown to be economical, precise, and compara- 

 tively easy to use. Present calculations relate to 

 zero Froude number but they can readily be extended 

 to include the effects of a water wave and the local 

 regions of flow separation which may stem from bul- 

 bous-bow geometries . 



1 . INTRODUCTION 



A general method for determining the local flow 

 properties and the overall drag on ship hulls is 

 very desirable and particularly so with the present 

 need to conserve energy resources. It is difficult 

 to achieve for a number of reasons including the 

 turbulent nature of the three-dimensional boundary 

 layer, the complexity and wide range of geometrical 

 configurations employed, the possibility of local 

 regions of separated flow, and the existence of the 

 free surface. In addition, and although these dif- 

 ficulties may be overcome in total or in part, the 

 resulting calculation method must have the essential 

 features of generality, efficiency and accuracy. 



The purpose of this paper is to describe a general 

 method which is capable of representing the flow 

 properties in the boundary layer around ship hulls 

 of arbitrary shape. It is based on the general 

 method of Cebeci, Kaups, and Ramsey (1977) , developed 

 for calculating three-dimensional, compressible lami- 

 nar and turbulent boundary layers on arbitrary wings 

 and previously proved to satisfy the requirements 

 of numerical economy and precision. To allow its 

 application to ships in general, and to double- 

 elliptic and flat-bottomed hulls in particular, 

 an appropriate coordinate system has been developed. 

 Previously described coordinate systems, for example 

 a streamline system such as that of Lin and Hall 

 (1966) or the orthogonal arrangement of Miloh and 

 Patel (1972) are limited in their applicability and 

 the present nonorthogonal arrangement is similar 

 to that of Cebeci, Kaups, and Ramsey (1977). 



The numerical procedure for solving the three- 

 dimensional boundary-layer equations makes use of 

 Keller's two-point finite-difference method (1970) 

 and Cebeci and Stewartson's procedure (1977) in 

 computing flows in which the transverse velocity 

 component contains regions of reverse flow. This 

 is in contrast to previous investigations, for 

 example those of Lin and Hall (1966) and Gadd (1970) , 

 which are limited either to zero crossflow or to a 

 unidirectional and small crossflow. It is also in 

 contrast to the previous methods of Chang and Patel 

 (1975) and Cebeci and Chang (1977) which did not 

 have a good and reliable procedure for computing 

 the flow in which the transverse velocity component 

 contained flow reversal. 



In representing turbulent flow by time-averaged 

 equations , a turbulence model is required and an 

 algebraic eddy-viscosity formulation, similar to 

 that of Cebeci, Kaups, and Ramsey (1977), is used. 

 This is in contrast to the two-equation approach 

 which Rastagi and Rodi (1978) have applied to three- 

 dimensional boundary layers and which, in principle, 

 should be better cible to represent flows which are 

 far from equilibrium. The previous comparisons pre- 



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