Lagrangian Solutions of Unsteady Free Surface Flows 



VII. CONCLUDING REMARKS 



Rather severe examples were taken In order to test the 

 limiting characteristics of the method developed. Provided the 

 various interval limitations were adhered to only two problems arose 

 which could prematurely conclude a computation. First, excessive 

 elongation of the cells in regions of the nnost violent motion could 

 cause the mesh points to be excessively widely spaced; rezoning 

 could, however, make it possible to continue. Secondly, it would 

 appear that a more detailed knowledge and treatment of some singu- 

 larities is required. Work on this, and especially on the shoreline 

 singularity of example three, is in progress at the moment. 



Other types of examples which have been only briefly investi- 

 gated thus far are: the matching with a semi- infinite region in which 

 some analytic solution is used; the inclusion of surface tension; the 

 extension of the method to three dimensions; examples in which the 

 fluid is inhomogeneous. It is hoped to present such results in the 

 near future. 



The authors are deeply appreciative of the kind and considerate 

 help given by Professor T. Y. Wu. 



This work was partially sponsored by the National Science 

 Foundation under grant GK 23 70 eind by the Office of Naval Research 

 under contract N000l4-67-A-0094-00i2. 



REFERENCES 



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Biesel, F, , "Stujiy of wave propagation in water of gradually varying 

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 NBS Circular 521, 195 2. 



Brennen, C, , "A numerical solution of axisymmietric cavity flows ," 

 J. Fluid Mech. , ll_, 4, 1969. 



Erode, H, L. , "Gas dynamic motion with radiation: a general numeri- 

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Carrier, G. F. aiid Greenspan, H. P., "Water waves of finite 



amplitude on a sloping beach," J. Fluid Mech. , 4, 1958. 



143 



