Lagrangian Solutions of Unsteady Free Surface Flows 



If Xq, Yq, Zq is the position of a particle at some reference 

 time t(j (when the density is Pg) then the equation of continuity is 

 simply 



3(X.Y,Z) ^ 3(X „.Y,.Z,) ^ 

 a(a,b,c) "^^ a(a,b,c) 



Frequently it Is convenient to define a, b, c as identical to Xq, Yq, Z^, 

 thus reducing the R.H.S. of (2) to PqJ however it will be seen in the 

 following sections that flexibility in the definition of a , b , c is of 

 considerable value when designing numerical methods of solution. 



If the extraneous forces, F, G, H, have a potential !i2 and 

 p, If not uniform, Is a function only of P then, eliminating fi + P/p 

 from (1): 



W^^b^c - UcXb + V^Y^ - V^Y, + W^^Z^ - W^Z^) = ^ = 

 ^(U^X, - U,X, + V,Y, - V,Y, + W^Z^ - U,Z^) = -^ = (3) 



^ (U,Xb - U^X, + V^Yb - V^Y, + W^Z^, . Wj,Z^) = -^ = 



where, for convenience, the velocities X^, Y^, Z^ are denoted by 

 U, V, W. The quantities F, , Fg, Fj are related to the Eulerlan 

 vortlclty components, ^| , ^.g, ^3 by 



r, = ;,(YbZ, - Y^Z,) + ^(Z^X^ - Z^X,) + ^(X,Y^ - XJ^) 



r^ = C,(YcZ, - Y,Z,) + ^^t^cXa - Z,X,) + ^(X.Y, - X^Y^) (4) 



F^ = ;,(YoZb - Y,Z,) + ^(Z^X, - Z^X,) + ^(X,Y, - X^Y^) 



(Thus, of course, vortlclty changes with time are due solely to 

 changes In the coefficients of the L.H.S. of (4) which. In turn, 

 represents stretching and twisting of the vortex line.) Given the 

 vortlclty distribution t,(X,Y,Z) at some Initial time, tg, F(a,b,c) 

 (which Is Independent of time) nnay be obtained through Eqs . (4) and 

 used In the final form of the dynamical equations of motion, namely 

 Eqs, (3) Integrated with respect to time. 



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