t V a, u a. u a, pd bx pd 



x 1 y 2x1 



a i y 



u + — uu + — vu +£-n ♦ -t t. +-r 



a pd bx pd 



% a 2 a 2 ^y / a °2 Vy V X /a 2 °1 



— — V - 



x u U a a y m 



y x 2 1 



P xx m \ xy 



34) 



V + — UV + — VV + -S— ti + -=• T, +^r 



1 U x a l U y a 2 % a 2 P y P 



-( S ) + - l S 



(35) 



-e — — U - -V- f ! V 



y U U a a 2 I x / a, 



x y 12 u x \ /^ 



The continuity equation becomes 



H + — (Ud) + — (Vd) = 

 t M a, W a 

 x 1 y 2 



(36) 



where the subscripts t , "'it an ^ a. indicate partial derivatives with 

 respect to time, a.. , and a„ , respectively, and the grid expansion coef- 

 ficients p and \i are defined by Equations 4 and 6. Note that in 

 obtaining Equations 34, 35, and 36, the assumptions made in paragraph 29 were 

 used. 



31. The nonlinear advective terms in the equations of motion often pose 

 stability problems. These terms are handled in the present model by using a 

 special scheme which will be described in the next paragraph. The eddy 

 viscosity terms can cause difficulties also during the numerical computation. 

 The finite difference schemes selected in the model and the formulation for 

 eddy viscosity adopted in the model minimize such difficulties and stability 

 problems provided that time and space steps and eddy viscosity coefficients 

 are properly selected for the phenomena being simulated. 



Computational techniques 



32. In order to solve the problem under consideration on a digital com- 

 puter, the differential equations (Equations 34-36) have to be expressed in a 

 finite difference form. In the present case, an alternating direction, 

 implicit, finite difference scheme is employed. In view of the presence of 



24 



