Lagrangian Solutions of Unsteady Free Surface Flows 



Making the assumption of straight sides the actual area of a 

 cell in the physical plane is 



A = i[ (X, - X3)(Y2 - Y4) - (X2 - X^){Y^ - Y3)] (15) 



Number suffices refer to the four vertices, numbered anticlockwisej 

 other node numbering conventions are shown in Fig, 1. If this area 

 is to remain unaltered after proceeding in time from station p to 

 p + 1 through Eq, (14) then 



Imag {(Zg - Z/(W, - w/*'^- (Z, - z/(W2 - W^)*"*'"^} 



"*"*+ 2(aP - A°) 



T 



= = R, (16) 



+ t{(U, - U^HY^ - V^) - (U2 - U^)(V, - V3)} +ii^__A.J 



where the terms on the L.H, S. , second line are numerical cor- 

 rections required to preserve continuity more exactly and prevent 

 accumulation of error over a large number of time steps. The nu- 

 merical value of the L,H.S. at some point in the iterative solution 

 is termed the continuity residual, Rg. 



Assuming linear variation in velocity along each side of the 

 cell, evaluating the circulation around 1234 and setting this equal to 

 the known. Initial circulation, Fc* yields (In the case of a homo- 

 geneous fluid): 



Real {(Z2 - Z4)(W, - W3) - (Z, - Z3)(W2 - W^)} - ZF^ = = Rj (17) 



Slight hesitation Is required here since, for validity, the Z ajid W 

 values In this equation should relate to the same station In time. 

 But by choosing to apply It at the midway stations and substituting 

 2P*i/2_ 2.^ + (T/2)zf*"^the T terms are found to cancel and (17) per- 

 sists when the vcilues referred to are Z and W Rj Is the circu- 

 lation residual. The modification of (17) In the case of a hetero- 

 geneous fluid Is delayed until section 4E. 



Combining (16) and (17) produces the cell equation: 

 (Zg - Z^)(W, - W3) - (Z, - Z3)(W2 - W^) Main Part 



+ IT ((U, - U3)(V2 - V,) - (U2- U,)(V, - V3)} .11^!^ Cotr'e'caonI 

 - ZTf, Permanent Cell Circulation Term 



123 



