UNSTEADY, FREE SURFACE FLOWS; 



SOLUTIONS EMPLOYING THE LAGRANGIAN 



DESCRIPTION OF THE MOTION 



Christopher Brennen, Arthur K. Whitney 



California Institute of Technology 

 Pasadena^ California 



ABSTRACT 



Numerical techniques for the solution of unsteady free 

 surface flows are briefly reviewed and consideration 

 is given to the feasibility of methods involving param- 

 etric planes where the position and shape of the free 

 surface are known in advance. A method for inviscid 

 flows which uses the Lagrangian description of the 

 motion is developed. This exploits the flexibility in 

 the choice of Lagrangian reference coordinates and is 

 readily adapted to include terms due to inhomogeneity 

 of the fluid. Numerical results are compared in two 

 cases of irrotational flow of a homogeneous fluid for 

 which Lagrangian linearized solutions can be con- 

 structed. Some examples of wave run-up on a beach 

 and a shelf are then computed. 



I. INTRODUCTION 



There are many instances of unsteady flows in which ajnalytic 

 solutions, even approximate ones, are not available. This is par- 

 ticularly true of free surface flows when, for example, non-linear 

 waves or even slightly complicated boundaries are involved. Though 

 analytical methods are progressing, especially through the use of 

 variationail principles (Whitham [1965]) and, in some cases, the 

 non-linear shallow water wave equations yield important results 

 (Carrier and Greenspan [ 1958]) there is still a need for numerical 

 methods. Indeed, nunnerical "experiments" can be used to comple- 

 ment actual experiments. 



Until very recently numerical solutions in two dimensions 



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