street^ Chan and Fromm 



caused by the explicit nature of the r\^^ equation. Second, a series 

 of simulations of specific hydraulic models should be made to deter- 

 mine the grid size A required to resolve the smallest significant 

 feature of a problem and to determine the sensitivity of the simula- 

 tion to discontinuous bottom topography. 



IV. SUMMAC 



Chan and Street [ 1970a] proposed the SUMMAC computing 

 technique as a tool for analyzing two-dimensional finite- amplitude 

 water waves under transient conditions. The method is, as noted 

 above, a modified version of MAC which was developed by Welch, 

 et al. [ 1966] . The essence of the initial modifications consisted of 

 a rigorous application of the pressure boundary condition at the free 

 surface and extrapolation of velocity components from the fluid 

 interior so that inaccuracy in shifting the surface boundary is kept 

 at a minimum. 



The objective of this section is to provide a summary of the 

 SUMMAC method, of its application to water wave problems and of 

 a number of new improvements added to SUMMAC since Chan and 

 Street [ i970a] was written. 



4.1. Summary of the Method 



The fluid is regarded as incompressible and the effect of 

 viscosity on the macroscopic behavior of flow is considered to be 

 negligible. The entire flow field is covered with a rectangular mesh 

 of cells, each of dimensions 6x and 6y. The center of each cell 

 is numbered by the indices i and j, with i counting the columns 

 in the x-direction and j counting the rows in the y-direction of a 

 fixed Cartesian coordinate system (Fig. 7). The field-variable 

 values describing the flow are directly associated with these cells 

 [ Welch, et al. 1966] . The fluid velocity components u and v and 

 the pressure p are the dependent variables while the independent 

 variables are x, y and the time variable t. 



In addition to the cell system which represents the flow field 

 by a finite number of data points , there is a line of marker particles 

 whose sole purpose is to indicate where the free surface is located. 

 These hypothetical particles may or may not represent the actual 

 fluid particles at the free surface, depending on whether one chooses 

 the Lagrangian or the Eulerian point of view to calculate the motion of 

 free surface. 



The marker-and-cell system provides an instantaneous repre- 

 sentation of the flow field for any particular time. When an initial 

 set of conditions is given, the entire fluid configuration can be ad- 

 vanced through a small but finite increment of time 6t, First, the 

 pressure for each cell is obtained by solving a finite- difference 



170 



