Hirt 



For both free-slip and no-slip walls, boundary conditions on the pressure 

 are derived directly from the difference equations. Since u|^j 2 is identically 

 zero, Eq. (12) relates cp^ + i to (p^^ and other known quantities in the wall vicinity. 

 Remembering that D|^; is zero, 



'!., 



^"i-.^^-.. (^" 



Similar conditions can easily be derived when the rigid wall coincides with a 

 different side of cell ( i , j ). 



This completes the necessary boundary conditions for rigid walls. Condi- 

 tions for prescribed input boundaries are straightforward and can be found in 

 detail in Ref. 1. For output boundaries, there are no unique prescriptions. The 

 investigator is free to choose conditions consistent with whatever flow he imag- 

 ines to exist outside the region being studied. One choice that has worked quite 

 well for many applications is described in Ref, 1. 



FREE -SURFACE BOUNDARY CONDITIONS 



With various combinations of rigid-wall, input, and output boundaries, it is 

 possible to simulate a great variety of confined flows. Many interesting incom- 

 pressible flows, however, involve free boundaries. Waves, jets, and splashing 

 drops are good examples. To treat these free surface flows we must have some 

 means of locating the free surface and of satisfying the surface boundary condi- 

 tions . 



Marker particles are used to solve the first of these problems. They rep- 

 resent selected points whose coordinates are calculated to move with the fluid. 

 An analogy may be drawn with the hydrogen bubbles often used in laboratory ex- 

 periments as a means of visualizing a flow. Any cell that contains a marker 

 particle is assumed to contain fluid. If such a cell is next to a cell containing 

 no marker particles, then it is designated as a surface cell, i.e., it contains the 

 free surface. 



Free-surface boundary conditions are applied at each surface cell. The 

 correct boundary conditions are the vanishing of the normal and tangential sur- 

 face stresses. If the curvature of the surface is small, the stress conditions in 

 two dimensions can be approximated by 



(p - 2v — - - (p„ (22) 



+ 



= , (23) 



where n refers to the outward normal direction to the surface and m to the tan- 

 gential direction. Allowance is made for an applied surface pressure cpg, which 

 can be useful on occasion. 



424 



