Bvennen and Whitney 



invariably seemed to employ the Eulerian description of the motion 

 though the Lagrangian concept has been used for some time in the 

 much simpler one -dimensional case (e.g. , Heitner [ 1969] , Erode 

 [ 1969] ) and to make small time expansions (Pohle [ 1952]). Perhaps 

 the best known of these Eulerian methods is the Marker- and-Cell 

 technique (MAC) begun by Fromm and Harlow [ 1963] and further 

 refined by Welch, et al. [ 1966] , Hirt [ 1968] , Amsden and Harlow 

 [ 1970] , Chan, Street and Strelkoff [ 1969] and others. The most 

 difficult problem arises in attempting to reconcile the initially un- 

 known shape and position of a free surface with a finite difference 

 schenne and the necessity of determining derivatives at that surface. 

 In the same way, few solutions exist with curved or irregular solid 

 boundaries. In steady flows, mapping techniques have been em- 

 ployed to transform the free surface to a known position (e.g. , 

 Brennen [ 1969]). It would therefore seem useful to examine the use 

 of parametric planes for unsteady flows. The Lagrangian description 

 in its most general form (Lamb [ 1932] ) involves such a plane and by 

 suitable choice of the reference coordinates , the free surface can 

 be reduced to a known and fixed straight line. However a discussion 

 of other parametric planes and mapping techniques Is Included In 

 Section 3. 



The major part of this paper is devoted to the development 

 of a numerical method for the solution of the Lagrangian equations 

 of motion In which full use is naade of the flexibility allowed In the 

 choice of reference coordinates. For the moment, we have restricted 

 ourselves to cases of Invlscid flow. Very recently, Hirt, Cook and 

 Butler [ 1970] published details of a method which employs a 

 Lagrangian tagging space but is otherwise similar to the MAC tech- 

 nique. This Is further discussed in Section 4B. 



II. LAGRANGIAN EQUATIONS OF MOTION 



The general invlscid dynamical equations of motion In 

 Lagrangian form are (Lamb [ 1932]): 



(X,,.F) 



Xo) 



Xb 

 X„ 



+ (Y^t-G) 



Yb 

 Yc 



+ (Z,,-H) 



tt 



Zb +^ Pb 

 Zcl iPc 



= 



(1) 



where X,Y,Z are the Cartesian coordinates of a fluid particle at 

 time t, F, G, H are the conmponents of extraneous force acting upon 

 It, P Is the pressure, p the density and a, b, c are any three 

 quantities which serve to Identify the particle and which vary con- 

 tinuously from one particle to the next. For ease of reference 

 (X,Y,Z) are termed Eulerian coordinates , (a,b,c) Lagrangian co- 

 ordinates. Suffices a,b,c,t denote differentiation. 



118 



