ABSTRACT 



In this report, a variational principle for unsteady body 

 wave problems is treated both with and without a convolution 

 integral, and with both linear and nonlinear free surface con- 

 ditions. Functionals are obtained for the numerical computation 

 of unsteady flow fields near a body that moves on or beneath the 

 free surface. This formulation can be applied to ship hydro- 

 dynamic performance problems of water entry and body slamming, 

 as well as to arbitrary body motion. 



ADMINISTRATIVE INFORMATION 

 The work reported herein has been supported by the Numerical Naval Hydrodynamics 

 Program at the David W. Taylor Naval Ship Research and Development Center. This 

 program is jointly sponsored by the Office of Naval Research and DTNSRDC under Task 

 Area RR0140302, Work Unit 1542-018. 



INTRODUCTION 



In the early 1970' s the David W. Taylor Naval Ship Research and Development 

 Center (DTNSRDC) recognized the demand for advanced numerical methods to predict the 

 hydrodynamic performance characteristics of naval ships, particularly when classical 

 methods proved inadequate. 



Thus, in 1974 the Numerical Naval Ship Hydrodynamics Program was begun at 

 DTNSRDC. Under this program the author previously investigated the steady ship-wave 

 problem using a variational principle associated with a localized finite-element 



1 "k 



technique. This method is useful particularly to analyze the flow field near the 

 ship in detail; in the far field, the Michell approximation can be used. This report 

 extends the problem to the unsteady case. 



For both the steady and unsteady problems, the simple calculation is for a 

 linear free surface condition with exact body boundary conditions. An iterative 

 method is needed for a nonlinear free surface condition. However, in the unsteady 

 problem, the variational principle requires an integration with respect to time using 

 the initial conditions; in this instance, a convolution integral is useful. The 

 variational principle for the unsteady body wave problem with exact body boundary 

 conditions is treated both with and without convolution, and with both linear and 



*A complete listing of references is given on page 13. 



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