Le Mehaute 



promising method applicable to this problem seems to be the MAC 

 Method developed by Harlow and his group at the Los Alamos 

 Scientific Laboratory. Further study to improve both accuracy and 

 efficiency (with respect to computer time) has led to development 

 of other techniques, such as SUMMAC (Chan, ^ ai. [ 1969]). Despite 

 the degree of sophistication that has been achieved for treating free 

 surface flow problems by numerical techniques, problems still re- 

 main which require some approximation. These are related to the 

 amount of energy dissipated by viscous turbulence associated with 

 the plume and base-surge radiating from the explosion. Energy 

 dissipated by the radiating surge is similar to that in a tidal bore, 

 except for the difference in water depth. The choice of a suitable 

 viscosity coefficient that realistically accounts for turbulent dissi- 

 pation can only be made empirically, and will be related to the mesh 

 size of the numerical model. This choice is also subject to the con- 

 straint of numerical stability. 



III, WATER WAVE FORMULATION 



3, 1 General Analytical Generation Model 



Using the initial conditions obtained by the above methods, 

 one can determine the water waves generated by such disturbance 

 analytically. 



The problem of surface waves generated by an arbitrary -- 

 but localized -- disturbance of the free surface has been investigated 

 by Kajiura [ 1963] , who has derived very general solutions incor- 

 porating the effects of initial displacement, velocity, pressure, and 

 bottom motion. Kranzer and Keller [ 1959] present a simplified 

 approach through the assumption of radial symmetry. The two solu- 

 tions are equivalent under appropriate conditions, but, because the 

 former permits utilization of the previous methods in the form of a 

 time-dependent input condition, the approach of Kajiura [ 1963] will 

 be adopted here. 



The problem may be formulated as follows. In water of 

 constant depth D, the coordinate system is established with x and 

 y' in the horizontal plane of the undisturbed surface and z'' taken 

 vertically upward: t is the time, r|''(x"^,y ,t ) the surface eleva- 

 tion, V (x , y ,z ,t ) the particle velocity, and p (x ,y ,z ,t ) 

 the pressure. The motion is assumed irrotational , implying the 

 existence of a potential function ^(x*,y*,z*,t*). Dimensionless 

 quantities are introduced as follows: 



X = X 



* /t-v * 



/D y = y*/D z = z*/D 



t = t*Vg7D Ti = 7i*/D V = v^/Vg/D 



$ = ^*/(dV^) P = P*/pgD 



76 



I 



