Lagrangian Solutions of Unsteady Free Surface Flows 



F(x'' + TU'^*'^,Y^ + Tv''*''^,t)=0 (19) 



Dynamic free surface conditions are simply constructed from 

 Eqs. (1). If, for example, the only extraneous force is that due to 

 gravity, g, in the negative Y direction, the condition on a free 

 surface such as AB, Fig. 1, is 



X,,X, . (Y, . g)Y, = 1 ^ f;"^"'^/^") (20) 



p da\ ^^2 + Ya)^ ^ 



where T is the surface tension if this is required. 



Unlike the field Eqs, (8) or (18) these boundary conditions 

 may not be homogeneous in all the variables , In a given problem 

 only the boundary conditions are altered by different choices of 

 typic al len gth, h (perhaps an initial water depth), and typical time, 

 say V h/g in the above example. Then, using the same letters for 

 the dlmenslonless variables . g and T/p In Eq. (20) would be re- 

 placed by 1 and S = T/pgh . The numerical form of that condition 

 used at a free surface node such as (Fig. i) Is: 



p P+l/2 p-\/2, P p + i/2 p-i/2 



(X, - X/(U; ^ - U; 1 + (Y, - Y3)'(Vo - Vo ' + T) 



= tS(P,'' - Pj) (21) 



where P Is assessed at each node as 



[ (X, - X3)(Y, + Y3 - 2YJ - (Y, - Y^){X^ + X^ - 2X,)] 



[(X, - X3) +(Y, -Y/r^ 



and the accelerations have been replaced by the expressions given 

 In Section 4A. Again, Eq. (21) relates Uq to Y^**^^ since all 



other quantities are known. 



If the liquid starts from rest at t = (as In the examples of 

 Section 6) then difficulties at the singular point t = can be avoided 

 by choosing to apply the condition at t = t/4 rather thaii t = 0, 

 Using Ztt = 2Z[^/t and Z = Z° + (t/4)z}/^ at that station the special 

 boundary condition becomes 



ui^{(X, - X3)° +1 (U, - Tj/^} +1 (V, - V3)'^^vf + Tg/2) = (22) 



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