Lagrangian Solutions of Unsteady Free Surface Flows 



(U- X^)Y, - (V- Y,)X„ = (11) 



(Ut + F)Xo + (Vt + G)Ya + (U - Xt)Uo - (V - Y^)V^ = 0. (12) 



Now a useful choice concerning the (a,b) plane would be to 

 require the mapping from (X,Y) to be conformal. Then, of course, 

 (10) simply reduces to the Cauchy-Riemann conditions Ug = - Vi,, 

 Ujj = Vq so that W = U - iV is an analytic function of c = a + ib or 

 of Z. 



In this way, John [ 1953] has constructed some special, exact 

 analytic solutions . The kinematic condition, (11), has the particular 

 solution W(a,t) = Z^(a,tl on. the free surface, which implies W(c,t) = 

 Zj("C",t) by analytic continuation. If, in addition, 



Z^^ +(F +iG) = iZ^K(c,t) (13) 



where K is real on the free surface, then the dynamic condition 

 thereon is also satisfied. John discusses several examples for various 

 choices of the function K, 



The potential of such methods may not have been fully realized 

 either analytically or numerically. In the latter case, however, the 

 conformality of the (X, Y) to (a,b) mapping is not necessarily a 

 great advantage, whereas a fixed and known free surface position 

 most certainly is. 



The digression ends here and the following sections develop 

 a Lagrajiglan numerical method from the equations of Section 2. 



IV. A NUMERICAL METHOD EMPLOYING LAGRANGIAN 

 COORDINATES 



A method for the numerical solution of Incompressible, 

 planar flows Is now described. It attempts to take full advantage of 

 the flexibility In the choice of Lagranglan coordinates. 



A. Time Variant Part 



The method uses an Implicit scheme with central differencing 

 over time, t. Thus Z'^(a,b) Is determined at a series of stations 

 In time, distinguished by the Integer, p. Knowledge of velocity Vcdues , 

 2,p^ , at a midway station p + 2 enables Z (a,b) to be found from 

 2^ through the numerlceil approximation 



Z***' = Z" + TZJ**'^ (error order t^Z^^^) (14) 



121 



