Computation of the Motion of Long Water Waves 



j ' JMAX- 



j = 3 



J = 2 



FREE 



U, i- 



SURFACE 



^H. 



"■•i: 



8X 



8Y 



i° 2 



i= 3 



v.. I 



Vii»i 



I ly;-: p. 



♦fj 



Vi».i-i 



i ° IMAX 



Fig. 7. Cell setup and position of variables 



Poisson's equation, whose source term is a function of the velocities. 

 This equation was derived subject to the requirement that the resulting 

 finite- difference momentum equations should produce a new velocity- 

 field that satisfies the continuity equation (conservation of mass). The 

 finite-difference equations of motion are then used to compute the new 

 velocities throughout the mesh. Finally, the marker particles are 

 moved to their new positions, their velocities being interpolated from 

 the nearby cells. The new flow configuration now serves as the initial 

 condition for the next time step and the foregoing procedure is repeated 

 as many times as necessary for the investigation. With proper choice 

 of 6x, 6y and 6t , the SUMMAC algorithm is capable of yielding so- 

 lutions that are computationally stable and also reasonably faithful in 

 simulating the physical phenomena. 



Dimensionless variables are used throughout (cf. , Sec. 3.1). 

 The governing equations for an incompressible, inviscld fluid are 



9u 

 8t 



8u 



^ s— 

 ox 



8u 



ay 



ap 



8x 



= - «t-^g 



(33) 



171 



